Bulletin of the Tomsk Technological Institute

of Emperor Nicholas II.

Vol. 15. 1909. No. 3.

I.

V. L. Malyshev.

TEMPERATURE MEASUREMENT.

1–80.

PREFACE.

Despite the ever-increasing importance, in science and in technology, of the accurate measurement of temperatures, up to the present time there has not been, not only in Russian but also in other languages, a single manual on this subject in the strict sense of the word. There are either works that examine the question from a purely theoretical point of view, or practical manuals that consider the matter from some special point of view—for example, the measurement of temperatures in the metallurgical industry, or as applied to the investigation of steam installations, and so forth. In view of the immense importance of the question of measuring temperatures, attention is devoted to it and space is allotted to it in the most varied works and manuals; but each of the authors only touches upon it, and moreover from his own point of view, not only failing to exhaust it, but not even giving, in general, a complete picture of the ways in which this question is being solved in the present state of science and technology.

In contrast to K. Boltz, who twenty years ago published the first systematic work on the measurement of high temperatures and complained of the insufficiency of material on this question, at the present time one must note the great abundance of literary data, which, however, are scattered chiefly through numerous scientific and technical journals. As is always the case when extensive material accumulates, contradictions are inevitably encountered, often undermining confidence in work that is otherwise valuable.

In the present work the author has endeavored to present the accumulated material in systematized form, freeing it from contradictory and already obsolete data. The task set by the author was to give, in a generally comprehensible but at the same time scientific exposition, a survey of all methods and instruments for measuring various temperatures, to indicate briefly their theory, to give a description of their construction and particular features, of the manner of handling them, and to examine

their advantages and disadvantages, in order thereby to provide all the data necessary for choosing the instrument most suitable in each given case, and, finally, to indicate methods and devices for the graduation of new instruments and the calibration of working instruments.

Special attention has been devoted to the latter section, both because of its importance and because of the difficult-to-explain paucity of corresponding information in the literature.

Alongside a critical presentation of others’ data and views on the question under consideration, the author has considered it his duty to share his own personal observations and methods, developed in his practice both in using various instruments and, especially, in calibrating them.

Taking into account the ever-increasing importance of temperature measurement and the growing requirements regarding the accuracy of these measurements, the author hopes that the present work will prove a useful aid both in scientific investigations—chiefly in engineering laboratories, which have at last begun to gain full recognition here in Russia as well—and for persons concerned with the measurement of temperatures in various fields of practical activity.

In view of the breadth of the subject, on the one hand, and the author’s intention to provide not a monograph on the given question but rather a guide for practical use, on the other, the author fears that some sections may perhaps have been treated in insufficient detail, and will be very grateful for all indications of any shortcomings and omissions that may be noticed.

As for the order of exposition, all the material concerning the theory and construction of the instruments under consideration is divided into the following principal sections, according to the various phenomena on which the instruments are based:

I, expansion of bodies:
    1—gaseous,
    2—liquid,
    3—solid,

II, heat transfer:
    4—increase in the temperature of water,
    5—melting of bodies,
    6—rate of heat transfer,

III, optical phenomena:
    7—change of color,
    8—change of light intensity,

IV, electrical phenomena:
    9—excitation of electromotive force,
    10—change of electrical conductivity.

SECTION I.

Theory and Construction of Various Instruments.

1. Temperature scales.—The concept of temperature, or intensity of heat, is given to us directly by touch: one body seems hot to us, another warm, a third cold, and so on.

The unit of temperature—the degree—is a wholly conventional quantity. Temperature itself, unlike other quantities, is not measurable in the exact sense of the word. For a quantity to be directly measurable it must obey two physical laws: equivalence and additivity. Temperature obeys only the first of these; therefore the temperatures of two bodies cannot be compared directly, as we compare, for example, the lengths or weights of two bodies, saying that one of them is so many times greater or heavier than the other; instead one must resort to the aid of a third object, serving as an indicator, in other words, a measuring instrument.

As an indicator of temperature one may take the most varied properties of bodies, such as: change of volume, color, electrical conductivity, electromotive force, and others. In order to make use of these properties, one must choose a strictly definite unit of temperature, which at the present time has been established by using two conventional fundamental points: the temperature of melting ice, taken as \(0^\circ\) (Celsius), and the temperature of boiling water at a barometric pressure of 760 mm of mercury, taken as \(100^\circ\) C. By this definition, what is established is not the “degree,” but a quantity 100 times greater than it. The question arises: how is it to be divided into 100 parts? Depending on what property and what body are taken as the indicator, the law of variation will be different, and consequently the scales established according to different properties, while coinciding at \(0^\circ\) and \(100^\circ\), will diverge at intermediate divisions.

For a long time, as the principal instrument—the normal thermometer—a mercury thermometer was used, consisting of a glass vessel with mercury, which, when the temperature rises, ascends through a thin, capillary glass tube having along its entire length a possible [[unclear: continuation cut off]]

one may obtain the same transverse section. In this case 1 degree is found by dividing the tube between two principal points into 100 parts equal in length.

However, already from the beginning of the last century attention was drawn to the fact that gases expand especially uniformly, approximately by \(\frac{1}{273}\) part of their volume (at \(0^\circ\) C.) per \(1^\circ\) C., and thermometers based on the expansion of gases began to be constructed (Pouillet, Becquerel, Regnault, and others).

Of all gases, hydrogen has the most constant coefficient of expansion. A comparison of the scale of a mercury thermometer and a hydrogen thermometer shows that mercury expands according to a different law than a gas; i.e., the coefficient of expansion of mercury, as of all solid and liquid bodies, is not constant, but depends on temperature. In Fig. 1

is shown the difference between the scales of mercury and hydrogen thermometers: the first indicates more than the second, depending on the kind of glass; the deviations, reaching their greatest value between \(0^\circ\) and \(100^\circ\) at about \(45^\circ\), amount to \(+0.11\) to \(0.15^\circ\). In the same Fig. 1 a comparison is given of the readings obtained when the instrument is filled with other gases—nitrogen and carbon dioxide; the difference for nitrogen is practically negligibly small, whereas carbon dioxide proves unsuitable for this purpose. At present the gaseous hydrogen scale is adopted as the basis of temperature measurement.

A perfectly exact, i.e. uniform, scale, however, can be found only by using the second law of thermodynamics—the Carnot cycle. According to Carnot’s law, the efficiency of a completely reversible cycle does not depend on the nature of the working body, in which lies the great advantage of this method. Thomson, who first introduced it, proceeded in the following way: take some gas and change its state, forcing the gas to perform a Carnot cycle, wherein, during the change along the isotherm \(T_1 = const.\), heat \(Q_1\) was imparted to the gas, and along the isotherm \(T_2\) heat \(Q_2\) was taken from it; for such a change of state, the efficiency of the cycle is

\[ \eta=\frac{Q_1-Q_2}{Q_1}. \]

On the basis of the properties of gases,

\[ \frac{Q_1-Q_2}{Q_1}=\frac{T_1-T_2}{T_1}, \]

whence

\[ \frac{T_1}{T_2}=\frac{Q_1}{Q_2}. \]

Repeating the cycle between the isotherms \(T_1\) and \(T_3\), we obtain in the same way

\[ \eta'=\frac{Q_1-Q_3}{Q_1}=\frac{T_1-T_3}{T_1}, \]

whence

\[ \frac{T_1}{T_3}=\frac{Q_1}{Q_3}. \tag{2} \]

Comparing eqs. (1) and (2), we obtain in general

\[ \frac{T_1}{T_n}=\frac{Q_1}{Q_n}. \tag{3} \]

Determining the quantities of heat \(Q_1\) and \(Q_n\) and knowing the temperatures \(t_1\) and \(t_n\) of the isotherms in \(^{0}\)C, it is easy to find that

\[ T=t+273^0. \tag{4} \]

This scale, by the method of its derivation, is called thermodynamic; its \(0^0\) corresponds to \(-273^0\) C. The real value of this zero is absolute

the absence of any heat content, any molecular motion, i.e. absolute rest, owing to which this scale is often called absolute, and its \(0^\circ\) the absolute zero.

If one takes a perfect gas, strictly obeying the Boyle–Mariotte and Gay-Lussac laws, and makes it perform a Carnot cycle, taking as the scale the gas scale of this gas, then one obtains the same relation (3); consequently, the gas scale of a perfect gas coincides with the absolute scale.

Hydrogen, in its properties, is so close to a perfect gas that the gas scale obtained with it differs from the absolute scale by only hundredths of a degree. In view of this, and also because the method of measurement by means of the Carnot cycle is very complicated, in practice hydrogen-filled gas thermometers are used as the fundamental, standard thermometers.

In conclusion it should also be mentioned that, besides the so-called centigrade Celsius scale, now adopted in all scientific investigations, and also in everyday practice in most countries, in Russia and in some Balkan states the 80-degree Réaumur scale is still used; on it \(0^\circ\) coincides with \(0^\circ\) C., and the boiling point of water is denoted by \(80^\circ\), i.e. it differs from the Celsius scale only by larger divisions. In Great Britain, its colonies, and in the U.S. United States, the Fahrenheit scale is still often used. On this scale \(0^\circ\) is obtained artificially by means of a cooling mixture of definite composition, while the temperature of melting ice is denoted by \(+32^\circ\), and that of boiling water by \(+212^\circ\) F.

Denoting by \(t\) the number of degrees on the Celsius scale, by \(t_{\mathrm{p}}\) that on the Réaumur scale, and by \(t_{\phi}\) that on the Fahrenheit scale, we obtain, for converting the readings of one scale into those of another, the following expressions:

\[ t = \frac{5}{4} t_{\mathrm{p}} = \frac{5}{9}\left(t_{\phi} - 32^\circ\right), \tag{5} \]

\[ t_{\mathrm{p}} = \frac{4}{5} t = \frac{4}{9}\left(t_{\phi} - 32^\circ\right), \tag{6} \]

\[ t_{\phi} = \frac{9}{5} t + 32^\circ = \frac{9}{4} t_{\mathrm{p}} + 32^\circ. \tag{7} \]

In what follows we shall everywhere speak only of the centigrade scale.

Gas thermometers.

2. General remarks.—The use of a gas for measuring temperatures is based on the characteristic equation of perfect gases

\[ PV = GRT = GR\left(t + \frac{1}{\alpha}\right), \tag{8} \]

where \(P\) is the pressure in kg/m\(^2\), \(V\) the volume of the gas in m\(^3\), \(G\) the weight of the gas taken in kg, \(R\) its constant, varying depending on the kind of gas and known from chemistry for all gases, \(T\) the absolute temperature, and \(\alpha\) the coefficient of expansion of the gas, taken for perfect gases as \(\alpha = \frac{1}{273}\), and for other gases having a somewhat larger value.

At first sight it would seem sufficient to measure \(P\) and \(G\) of the working gas and, knowing the volume \(V\) occupied by it, the constant \(R\), and the coefficient \(\alpha\), to calculate \(t\). However, the measurement of \(G\) is associated with very great difficulties; likewise, it is difficult to find with sufficient accuracy the volume \(V\), which inevitably changes during the measurement of temperature. Therefore, for any reasonably reliable reports, the temperature \(t\) is found not from a single observation, but from two, arbitrarily changing one or several of the quantities entering into equation (8), and taking the ratio of two characteristic equations written for different conditions.

Since in equation (8) one may vary either 2 quantities, or at once 3, or 4, or even all 5, theoretically there may be 26 different methods of measuring temperatures by means of gases. Since changing \(R\) is equivalent to changing the working gas, which is connected with a considerable complication of the whole operation, not at all repaid by anything, in practice observations are always made with one and the same gas. In that case there remain only 4 variable quantities, still allowing 11 different combinations, i.e. 11 methods. Of these 11 possible methods we shall indicate only the principal ones, which have been made the basis of one or another constructed instrument.

Method 1: first we shall indicate the expression for finding \(t\) in the most general case, when all 4 quantities vary. The course of observations should be as follows: one must first measure the pressure \(P_0\) at some

at a second known temperature \(t_0\) (melting ice or boiling water); the gas in the amount \(G_0\) kg then occupies the volume \(V_0\), i.e., we have

\[ P_0 V_0 = G_0 R \left(t_0 + \frac{1}{\alpha}\right). \]

Then the pressure \(P\) is measured at the desired temperature \(t\), leaving in the apparatus only \(G\) kg of gas, which occupies the known volume \(V\); then we have

\[ P V = G R \left(t + \frac{1}{\alpha}\right); \]

dividing this equation by the preceding one, we obtain

\[ \frac{P V}{P_0 V_0} = \frac{G\left(t+\frac{1}{\alpha}\right)} {G_0\left(t_0+\frac{1}{\alpha}\right)}, \]

whence

\[ t= \frac{P}{P_0}\frac{V}{V_0}\frac{G_0}{G} \left(t_0+\frac{1}{\alpha}\right) -\frac{1}{\alpha}. \tag{9} \]

In practice this method cannot be recommended, since it requires too many measurements. Equation (9) has been derived only as a general expression, from which the remaining methods will be obtained as special cases.

Method 2: work with one and the same amount of gas, \(G=\mathrm{const.}\); the volume is also kept constant, \(V=\mathrm{const.}\); the change in temperature \(t\) is judged from the change in \(P\). For this, one first measures the pressure \(P_0\) at some known temperature \(t_0\), as indicated above, and then determines the pressure \(P\) at the desired temperature \(t\), forcing the gas again to assume the initial volume \(V\). Since \(G=G_0\) and \(V=V_0\), from Eq. (9) we obtain directly

\[ t=\frac{P}{P_0}\left(t_0+\frac{1}{\alpha}\right)-\frac{1}{\alpha}. \tag{10} \]

This method is very convenient to carry out in practice, and the precise instruments described below are based on it. The disadvantage of this method is that, when high temperatures are measured, excessively large pressures are obtained.

Method 3: instead of the pressure \(P\), the volume \(V\) is varied; proceeding otherwise as in the preceding case, we obtain in the same way from equation (9)

\[ t=\frac{V}{V_0}\left(t_0+\frac{1}{\alpha}\right)-\frac{1}{\alpha}. \tag{11} \]

To carry out this method accurately is practically difficult; namely, when \(V\) is changed, part of the gas will be in a space with a lower temperature than the measured \(t\); a correction will have to be introduced, which is equivalent to changing the working mass of the gas \(G\).

Method 4: only the volume \(V\) is kept constant; the remaining quantities—\(P, T\), and \(G\)—are varied. This method is an improvement of method 2 in the measurement of high temperatures. The measurements are made in the same way as in method 2, but in order to reduce \(P\) part of the gas is released as the temperature rises. We therefore have, similarly to the preceding case, from equation (9)

\[ t=\frac{P}{P_0}\frac{G_0}{G}\left(t_0+\frac{1}{\alpha}\right)-\frac{1}{\alpha}. \tag{12} \]

Method 5: in particular, it is possible, while releasing gas, to keep the pressure \(P_0\) constant; then expression (12) is simplified and takes the form

\[ t=\frac{G_0}{G}\left(t_0+\frac{1}{\alpha}\right)-\frac{1}{\alpha}. \tag{13} \]

Method 6: instead of taking readings at two different temperatures, one may take them at one and the same temperature sought, varying only \(G\), whereby \(P\) will of course change; the volume \(V=\mathrm{const.}\); for this case we have

\[ P_1V=G_1R\left(t+\frac{1}{\alpha}\right) \quad\text{and}\quad P_2V=G_2R\left(t+\frac{1}{\alpha}\right), \]

or, subtracting one from the other, we obtain

\[ (P_1-P_2)V=(G_1-G_2)R\left(t+\frac{1}{\alpha}\right), \]

whence

\[ t=\frac{P_1-P_2}{G_1-G_2}\cdot\frac{V}{R}-\frac{1}{\alpha}. \tag{14} \]

This method is theoretically equivalent to direct use of equation (8), with the change that instead of the pressure \(P\) and the weight of the gas \(G\), the differences of two pressures and two weights enter. But since \(G_1 - G_2\) is precisely equal to the quantity of gas released, which can be determined far more accurately than the total quantity \(G_1\) contained in the apparatus, this method, in respect of accuracy, has an undoubted advantage.

To increase the accuracy of the measurements, one must have as large as possible a difference in the quantities of gas \(G_1 - G_2\), which gives a correspondingly large difference of pressures \(P_1 - P_2\). Usually \(P_1\) may be taken equal to atmospheric pressure, and \(P_2\) to the rarefaction obtained by means of a water-jet pump, i.e. approximately 15–20 mm of mercury.

This method was first applied by É. Becquerel, who called his apparatus a thermo-volumenometer. Owing to the use of small pressures, this method is especially valuable for measuring high temperatures. Its other merit, which will become clear below, consists in the fact that the unavoidable corrections for the influence of various circumstances on the temperature found by means of it have the smallest magnitude; consequently, other conditions being equal, its accuracy is higher than that of the other methods.

3. Sources of error.—In the practical realization of the gas thermometer, a whole series of circumstances is encountered that affect the accuracy of its readings; the most important of these circumstances are as follows:

1. The coefficient of expansion \(\alpha\) is in fact not the same for different gases, as was indicated in the diagram in Fig. 1, and, moreover, although only weakly, depends on temperature and pressure; namely, \(\alpha\) decreases as the temperature rises and, conversely, increases as the pressure increases. Of all gases, hydrogen has the most constant coefficient; for it \(1/\alpha\) varies only from 272.7 to 273.2; with sufficient accuracy one may take, for all temperatures and pressures,

\[ \frac{1}{\alpha}=273. \]

For nitrogen, for temperatures from 0 to \(100^\circ\) C., one may also take

\[ \frac{1}{\alpha}=273. \]

For air between the same temperatures, at a pressure of 760 mm of mercury,

\[ \frac{1}{\alpha}=273, \]

falling at a pressure of 3600 mm to 270.

Other gases have an even more variable coefficient \(\alpha\), and therefore they should not be used for measuring temperatures.

1. At high temperatures the walls of the vessel containing the gas become permeable to gases. Thus, for example, incandescent platinum becomes highly permeable to \(H_2\); when a platinum bulb filled with some gas other than \(H_2\) is heated in an open flame, which almost always contains \(H_2\), the latter penetrates into the bulb and, by increasing the pressure inside the bulb, causes an error in the sense of exaggerating the temperature readings; conversely, from a platinum bulb filled with \(H_2\), the latter will flow out. For the other gases platinum is apparently impermeable; therefore, for high temperatures it is better to replace hydrogen by nitrogen.

Another material often used for gas thermometers—porcelain—proves, at high temperatures, to be permeable to water. According to Crafts’s experiments, into a pyrometer with a capacity of about \(65\ \mathrm{cm.^3}\), at a temperature of \(1350^\circ\), up to \(2\ \mathrm{mg}\) of water penetrated, which gives readings exaggerated by \(70^\circ\). Porcelain itself is a porous material; its impermeability is achieved only thanks to a layer of glaze; unfortunately, this layer softens soon after \(1000^\circ\), and therefore at higher temperatures even a small excess of pressure is enough to lift away from the porcelain and rupture the glaze layer. Only by coating the gas vessel with glaze both outside and inside can the probability of gas penetration into the thickness of the wall, and the associated danger of damaging the glaze layer, be considerably reduced.

Glass, although it does not let gases pass through, does dissolve them in itself, especially at higher temperatures; furthermore, it is usable only up to \(500\text{–}600^\circ\mathrm{C}\). Recently it has become possible to make glass from pure quartz (\(SiO_2\)); this material, which melts at \(t > 2000^\circ\mathrm{C}\), is impermeable up to \(1200^\circ\) neither to hydrogen nor to other gases; at temperatures above \(1300^\circ\) it begins to soften appreciably. Owing to its very small coefficient of expansion, quartz glass is entirely unaffected by abrupt changes of temperature, so that at present it is the best material for gas thermometers up to \(1000^\circ\).

1. When the temperature changes, the capacity of the gas vessel also changes, which cannot at all be neglected even in the roughest calculations. In the following Table 1 are given the mean cubic coefficients of expansion \(\beta\) for the more important materials.

For porcelain, \(\beta\) is given for hard porcelain, i.e. fired at a temperature of about \(1400^\circ\). For glass, \(\beta\) is very non-uniform; in general, the more refractory the glass, the smaller \(\beta\) is.

Table 1.

material	temperature limits	β
platinum	0—100°	0.000021
”	0—1000°	0.000027
Bayeux porcelain	0—1000°	0.000013—17
” Meissen	0—100°	0.0000081
” ”	0—1400°	0.0000094
Jena normal glass	0—100°	0.000024
” ” 59^III	0—100°	0.000017
” quartz	0—1000°	0.0000016

4, for the accuracy of measurements it is very important that the working gas be absolutely dry; the presence of even a small quantity of moisture distorts the readings. At the same time it is very difficult to get rid of moisture completely; in particular, it is difficult to dry the vessel itself; even with repeated pumping of the gas out of the apparatus, with the rarefaction brought to a considerable value, and subsequent filling with well-dried gas, a certain quantity of moisture will remain on the walls of the apparatus, and it will indicate an exaggerated temperature. It is especially difficult to free oneself from moisture in a porcelain pyrometer; it must necessarily be glazed on the inside as well.

The only way to avoid moisture in the vessel is to fill it with gas while hot, at a temperature of up to 400 degrees, and, of course, also with repeated removal of the gas.

5, finally, the magnitude of the harmful space, i.e. the volume of the tube connecting the gas vessel with the mercury manometer, has an even greater influence on the accuracy of the readings. It is rather difficult to take exact account of the influence of the harmful space, since, on the one hand, it is difficult to measure this volume, and on the other, it is even more difficult to establish what part of it should be considered filled with gas having the ambient temperature, and what part with gas having the measured temperature. In reality there is no sharp boundary between these temperatures; the temperature of the gas in the harmful space is po-

gradually pass from the highest temperature to the temperature of the surrounding medium.

To increase the accuracy of the instrument, the relative harmful space must be made as small as possible; in view of this, the volume of the vessel with the gas must not be too small, in no case less than \(100\ \mathrm{cm.^3}\), preferably up to 500 or even \(1000\ \mathrm{cm.^3}\); the tube to the manometer must be as short as possible, capillary, with a cross-section of not more than \(0.5\ \mathrm{mm.^2}\) in the case of platinum or quartz, and up to \(1.0\ \mathrm{mm.^2}\) if it is of porcelain. In good instruments the relative harmful space amounts to from 0.1 and not more than \(0.5\%\) of the working volume.

4. Calculation of corrections.—Only in the case where measurements are carried out by method 1, i.e. when each time all quantities are measured exactly, is it possible, at least theoretically, to obtain correct results at once; in all other cases it is necessary to introduce certain corrections, since the quantities which, in the different methods, we regard as constant in reality change all the time under the influence of various conditions.

Let us take the general equation (9), replacing in it the temperatures \(t\) by absolute temperatures \(T\), i.e. let us write

\[ T = T_0 \frac{P V G_0}{P_0 V_0 G}. \]

Let us replace the ratio of pressures \(\frac{P}{P_0}\) by \(\frac{H\delta}{H_0\delta_0}\), where \(H\) and \(H_0\) are the heights of the mercury column in the manometer, and \(\delta\) and \(\delta_0\) its densities; then we obtain

\[ T = T_0 \frac{H\delta V G_0}{H_0\delta_0 V_0 G}. \]

If by \(T'\) we denote the temperature obtained from the calculations before the introduction of any corrections, and by \(dT = T - T'\) the error in the determination of this temperature, then, taking logarithmic differentials and noting that the initial temperature \(T_0\) is a quantity exactly known and, consequently, constant, we have

\[ \frac{dT'}{T'} = \frac{dH}{H_0} + \frac{d\delta}{\delta_0} + \frac{dV}{V_0} - \frac{dG}{G_0}. \tag{15} \]

Let us consider these corrections as applied to the various methods of measurement indicated above.

For method 2: the heights \(H\) and \(H_0\) are measured separately; therefore \(dH = 0\), and

\[ dT = T\left(\frac{d\delta}{\delta_0}+\frac{dV}{V_0}-\frac{dG}{G_0}\right). \tag{16} \]

The density of mercury changes as a result of the change in the temperature of the surrounding air from \(t_1\) in °C at the first reading to \(t_2\) at the second reading;

\[ \frac{d\delta}{\delta_0}=\frac{\delta-\delta_0}{\delta}; \]

but, as is known,

\[ \delta=\delta_0\left[1-\beta'(t_2-t_1)\right], \]

where the coefficient of expansion of mercury is \(\beta' = 0.000182\); whence

\[ \frac{d\delta}{\delta_0}=-0.000182\,(t_2-t_1). \tag{17} \]

The volume of the vessel with gas, \(V\), changes as a result of the expansion of the walls when heated from \(T_0\) to \(T\),

\[ \frac{dV}{V_0}=\frac{V-V_0}{V_0}, \]

\[ V=V_0\left[1+\beta(T-T_0)\right], \]

where \(\beta\) is the coefficient of cubical expansion of the material of the vessel. Noting that if \(T_0\) is the melting temperature of ice, then \(T - T_0 = t\), i.e. the sought temperature in °C, we obtain

\[ \frac{dV}{V_0}=\beta t. \tag{18} \]

The change in volume due to the change in pressure may be entirely neglected. Thus, for example, the platinum vessel of Chappuis’ apparatus, when the pressure is increased by 1000 mm of mercury, increases in volume by only \(0.002\%\).

The change in the quantity of gas \(G\) takes place in the sense that, owing to the presence of a harmful space, the quantity of gas contained in it will change with the change in pressure and in the ambient temperature.

If, of the total quantity of gas \(G\) kg, in the harmful space there are \(g_0\) kg at the first reading and \(g\) kg at the second, then the working quantities of gas will be, for the first reading,

\[ G_0 = G - g_0, \]

and for the second,

\[ G' = G - g, \]

whence the correction for the change in the quantity of gas is

\[ \frac{dG}{G}=\frac{G'-G_0}{G_0}=\frac{g_0-g}{G_0}. \tag{19} \]

We find the quantities \(g_0\) and \(g\) from the equations

\[ P_0 V_{\mathrm{в}} = g_0 R T_1 \quad \text{and} \quad P V_{\mathrm{в}} = g R T_2, \]

where \(V_{\mathrm{в}}\) is the volume of the harmful space, and \(T_1\) and \(T_2\) are the temperatures of the surrounding air at the moments of observation in \({}^\circ\) abs.; next, replacing \(G_0\) from the equation

\[ P_0 V_0 = G_0 R T_0, \]

we then obtain from equation (19)

\[ \frac{dG}{G_0} = \left( \frac{P_0 V_{\mathrm{в}}}{R T_1} - \frac{P V_{\mathrm{в}}}{R T_2} \right) : \frac{P_0 V_0}{R T_0}. \]

Replacing \(V_{\mathrm{в}}\) by \(\nu V\), where \(\nu\) is the relative harmful space, and observing that with sufficient accuracy one may in the present case take \(V_0 = V\), we obtain

\[ \frac{dG}{G_0} = \nu \left( \frac{P_0}{T_1} - \frac{P}{T_2} \right) \cdot \frac{T_0}{P_0}; \]

observing also that, with sufficient accuracy,

\[ \frac{P}{P_0}=\frac{T}{T_0}, \]

we finally obtain

\[ \frac{dG}{G_0} = \nu \left( \frac{T_0}{T_1} - \frac{T}{T_2} \right). \tag{20} \]

Thus the total correction is

\[ dT=T'\left[0.000182\,(t_1-t_2)+\beta t+\nu\left(\frac{T'}{T_2}-\frac{T_0}{T_1}\right)\right]. \tag{21} \]

To determine the magnitude of this correction, let us give the following example: suppose there is a platinum pyrometer; \(\nu=0.005\); \(T_0=273^\circ\); \(t_1=17^\circ\), and \(t_2=27^\circ\) C. Table 2 gives all the corrections for three measured temperatures \(t\) in \({}^\circ\)C.

Table 2.

\(t\)	\(T'\dfrac{d\delta}{\delta_0}\)	\(T'\dfrac{dV}{V_0}\)	\(T'\dfrac{dG}{G_0}\)	\(dT\)	\(d\theta\)
\(500^\circ\)	\(-1.4\)	\(+10.4\)	\(+6.3\)	\(+15.3^\circ\)	\(\pm0.63\)
\(1000^\circ\)	\(-2.3\)	\(+34.4\)	\(+21.0\)	\(+53.1\)	\(\pm2.10\)
\(1500^\circ\)	\(-3.2\)	\(+71.8\)	\(+44.0\)	\(+112.6\)	\(\pm4.40\)

From Table 2 it is evident that the correction for method 2 under consideration reaches a magnitude which in no case may be neglected. To this it must be added that the correction for the influence of the dead space, unfortunately, is far from being as definite as one might think from equation (20). As was already indicated above, the temperature of the gas in the connecting tube over some part of its length is not equal to the temperatures \(t_1\), respectively \(t_2\), but higher than these. This circumstance can be taken into account only by taking for the dead space \(V_{\mathrm{B}}' < V_{\mathrm{B}}\), estimating by eye what part of the tube should be assigned to the volume of the working vessel. Depending on the length of the tube, the uncertainty in determining the point of division may give an error \(d\theta\), which can be estimated from an equation similar to equation (20), replacing only \(\nu=\dfrac{V_{\mathrm{B}}}{V_0}\) by

\[ \frac{dV_{\mathrm{B}}}{V_0}=\frac{V_{\mathrm{B}}-V_{\mathrm{B}}'}{V_0}, \]

i.e. we obtain

\[ d\theta=T'\frac{dV_{\mathrm{B}}}{V_0}\left(\frac{T_0}{T_1}-\frac{T'}{T_2}\right). \tag{22} \]

To determine the possible inaccuracy of the whole measurement owing to the presence of the dead space, the quantity \(d\theta\) is indicated in Table 2.

for the following data: the volume of the gas vessel \(V = 100\ \mathrm{cm.^3}\); \(V_b = vV = 0.5\ \mathrm{cm.^3}\); the cross-section of the capillary tube \(0.5\ \mathrm{mm.^2}\), its length \(1000\ \mathrm{mm}\).

In calculating the correction \(T' \dfrac{dG}{G_0}\) for \(V_b\), the full length of the tube was taken; if it is assumed that \(100\ \mathrm{mm}\) of its length from the gas vessel should be assigned to the vessel, and not to the harmful space, then the values \(d\theta\) indicated in the table are obtained. The uncertainty for \(100\ \mathrm{mm}\) of length can by no means be considered exaggerated.

The quantity \(d\theta\) gives an indication of the degree of accuracy of the measurements with this instrument.

Correction for method 3: in this case

\[ \frac{dT'}{T'} = \frac{dH}{H_0} + \frac{d\delta}{\delta_0} + \frac{dV}{V_0} - \frac{dG}{G_0}; \tag{23} \]

here

\[ \frac{dH}{H_0} = \frac{H - H_0}{H_0}, \tag{24} \]

where \(H\) and \(H_0\) are the heights of the mercury column at two successive readings, though differing little from one another, since practically it is impossible to achieve \(P = const.\).

\[ \frac{d\delta}{\delta_0} \quad\text{and}\quad \frac{dV}{V_0} \]

are calculated by the equations (17) and (18) indicated above.

We find the last term by denoting the volume of the unheated gas at the first reading by \(V'\), at the second by \(V''\), and the weights of gas contained in them, respectively, by \(g_0\) and \(g\); then, as before, we have

\[ \frac{dG}{G_0} = \frac{g_0 - g}{G_0}, \]

but

\[ PV' = g_0RT_1, \]

\[ PV'' = gRT_2, \]

\[ PV_0 = G_0RT_0, \]

substituting from this \(g_0\), \(g\), and \(G_0\) into the preceding equation, we obtain

\[ \frac{dG}{G_0} = \left( \frac{V'}{T_1} - \frac{V''}{T_2} \right) \frac{T_0}{V_0}. \tag{25} \]

As can be seen, the total correction \(dT\) for this method is numerically greater than for the preceding one.

Correction for method 4: since here the quantity of gas varies and, consequently, must be measured accurately, equation (15) is transformed into

\[ \frac{dT}{T'}=\frac{d\delta}{\delta_0}+\frac{dV}{V_0}. \tag{26} \]

For the quantities of gas we have, using the notation indicated above, the relations

\[ G_0=G-g_0=\frac{H_0\delta_0 V_0}{R T_0}; \tag{27} \]

\[ G'=G-g=G_0-(g-g_0), \tag{28} \]

\[ g_0=\frac{H_0\delta_0 V_{\mathrm{b}1}}{R T}, \tag{29} \]

\[ g_0=\frac{H\delta V_{\mathrm{b}2}}{R T_2}, \tag{30} \]

where \(V_{\mathrm{b}1}\) and \(V_{\mathrm{b}2}\) are the volumes of the harmful space in the first and second readings.

The magnitude of the possible error due to uncertainty in the length of the heated part of the connecting tube is

\[ \frac{d\theta}{T'}=\frac{dG_0}{G_0}-\frac{dG'}{G'}=\frac{dG_0}{G_0}\left(1-\frac{T'}{T_0}\right). \tag{31} \]

Table 3 gives all the corrections for the same instrument that was indicated above in the example for method 2.

Table 3.

\(t\)	\(T'\dfrac{d\delta}{\delta_0}\)	\(T'\dfrac{dV}{V_0}\)	\(dT\)	\(d\theta\)
\(500^\circ\)	\(-1.4^\circ\)	\(+10.4^\circ\)	\(+9.0^\circ\)	\(\pm 1.4^\circ\)
\(1000\)	\(-2.3\)	\(+34.4\)	\(32.1\)	\(4.7\)
\(1500\)	\(-3.2\)	\(+71.8\)	\(68.6\)	\(9.7\)

Table 3 shows that the corrections for this method are smaller than for the preceding ones, but the possible error \(d\theta\) is approximately twice as large.

Correction for method 5: as in the preceding case,

\[ \frac{dT}{T'}=\frac{dH}{H_0}+\frac{d\delta}{\delta_0}+\frac{dV}{V_0}, \tag{32} \]

where the last two terms are calculated from equations (17) and (18), while the quantity

\[ \frac{dH}{H_0}=\frac{H-H_0}{H_0}, \]

which depends on the inaccurate observance of the condition \(P=\mathrm{const.}\), depends exclusively on the skill of the person taking the readings; on average one may take \(H-H_0=20\) mm. Table 4 gives the corrections for the same instrument as above, with \(H=1000\) mm assumed.

Table 4.

\(t\)	\(T'\dfrac{dH}{H_0}\)	\(T'\dfrac{d\delta}{\delta_0}\)	\(T'\dfrac{dV}{V_0}\)	\(dT\)	\(d\theta\)
\(500^\circ\)	\(+15.5^\circ\)	\(-1.4^\circ\)	\(+10.4^\circ\)	\(+24.5^\circ\)	\(\pm 1.4^\circ\)
\(1000\)	\(+25.5\)	\(-2.3\)	\(+34.4\)	\(57.6\)	\(4.7\)
\(1500\)	\(+35.5\)	\(-3.2\)	\(+71.8\)	\(103.1\)	\(9.7\)

Table 4 shows a large increase of \(dT\) in comparison with method 4, in exchange for which method 5 gives nothing.

Correction for method 6: although the readings are made at one and the same temperature, the external temperature may change, which causes a change in the density of the mercury \(\delta\); further, the volume \(V_0\) of the gas vessel is measured at a low temperature; consequently, as above, its expansion must be allowed for, and the influence of the harmful space must also be taken into account. Thus we obtain

\[ \frac{dT}{T'}=\frac{d\delta}{\delta_0}+\frac{dV}{V_0}-\frac{dG}{G_0}. \tag{33} \]

The first two terms are calculated from equations (17) and (18); the last we obtain by denoting by \(G'_1\) and \(G'_2\) the actual working quantities of gas, in the form

\[ \frac{dG}{G}=\frac{(G'_1-G'_2)-(G_1-G_2)}{G_1-G_2}, \]

where

\[ G_1=G'_1+g_1 \quad \text{and} \quad G_2=G'_2+g_2, \]

where \(g_1\) and \(g_2\) are the amounts of gas in the harmful space at the first and second readings; whence

\[ dG=-(g_1-g_2), \]

or

\[ -\frac{dG}{G}=\frac{g_1-g_2}{G_1-G_2}; \]

further, from equation (14) we have

\[ G_1-G_2=\frac{(P_1-P_2)V}{RT}; \]

in the same way we obtain for the harmful space

\[ g_1-g_2=\frac{(P_1-P_2)V_{\mathrm{в}}}{RT}, \]

whence finally we obtain

\[ -\frac{dG}{G}=\frac{V_{\mathrm{в}}}{V}=\nu . \tag{34} \]

Thus the total correction is

\[ dT=T'[0{,}000182(t_1-t_2)+\beta't+\nu]. \tag{35} \]

The possible error due to ignorance of the boundary between the hot and the cold space, using the notation indicated above, is expressed as

\[ \frac{d\theta}{T'}=\frac{dV_{\mathrm{в}}}{V}. \tag{36} \]

Table 5 shows all the corresponding corrections for the same numerical data as in the preceding cases.

Table 5.

\(t\)	\(T'\dfrac{d\delta}{\delta_0}\)	\(T'\dfrac{dV}{V_0}\)	\(-T'\dfrac{dG}{G_0}\)	\(dT\)	\(d\theta\)
\(500^\circ\)	\(-1{,}4^\circ\)	\(+10{,}4^\circ\)	\(+3{,}9^\circ\)	\(+12{,}9^\circ\)	\(+0{,}77^\circ\)
\(1000\)	\(-2{,}3\)	\(+34{,}4\)	\(+6{,}4\)	\(38{,}5\)	\(1{,}27\)
\(1500\)	\(-3{,}2\)	\(+71{,}8\)	\(+8{,}9\)	\(77{,}5\)	\(1{,}77\)

From Table 5 it is evident that, with this method of correction, not only is the correction for the expansion of the vessel, assumed the same for all methods of correction, comparatively very small in magnitude, but also absolutely very small; the possible error \(d\theta\) is practically negligible.

Thus it is indeed clear that this method surpasses all the others in the accuracy of its indications; the possible error, with a good instrument and careful conduct of the observations, does not exceed \(0.15\%\) of the temperature obtained.

5. Construction of gas instruments.—We have already seen what difficulties are involved in the manufacture and use of gas thermometers. To this must also be added that, owing to the large dimensions of the measuring part of the instrument, the gas vessel, it is very difficult to attain uniformity of its temperature; moreover, because of these same large dimensions, such instruments can be used to measure the temperature of by no means all bodies, but only of gaseous and liquid ones, and, in addition, of those occupying a considerable volume—for example, the temperature of the interior space in furnaces, or of molten metal in a crucible of sufficient size. It will therefore become clear why gas instruments are used chiefly for establishing the fundamental temperature scale, then for calibrating other instruments more suitable for practical use, and only in rare cases—and then with a corresponding change of construction—for direct measurement of temperatures. In view of this, we shall begin the description with the normal instrument, i.e. the instrument by means of which the presently accepted temperature scale is established.

In order to have a strictly defined temperature scale, it is necessary to establish once and for all which instrument is to be considered normal, i.e. to choose the working gas, the initial pressure, and the method of measurement.

By the resolution of the International Commission of Weights and Measures, now universally accepted, an instrument with hydrogen, operating at constant volume and quantity of gas, with an initial pressure at \(0^\circ\) C of 1000 mm of mercury, was chosen as the normal thermometer.

In Figs. 2–5 such a normal thermometer is shown, constructed by Chappuis and located—

Fig. 2 and 3.

…in the north; it is, therefore, an apparatus operating by method 2. It consists of a cylindrical vessel \(a\) of iridium platinum, with a capacity of \(1038.98\ \mathrm{cm}^3\) at \(0^\circ\) C.; this vessel is connected to a mercury manometer by a platinum capillary \(b\) 1 meter long; by filling it with mercury its capacity was found to be \(0.567\ \mathrm{cm}^3\); the total harmful space is equal to \(1.072\ \mathrm{cm}^3\); \(c\) is a metal trough with double walls, filled with water, into which the thermometers being verified are immersed. For constancy of temperature, the trough \(c\) is also immersed in water; \(f\) is a pulley which sets in motion the stirrers \(e_1\) and \(e_2\). The trough \(c\) is closed on top by a glass cover, not shown in the drawing, through which the readings of the mercury thermometers are taken.

For measuring temperatures below \(0^\circ\) C., vessel \(c\) is filled with alcohol, and vessel \(d\) with a cooling mixture.

For the determination of \(100^\circ\), vessel \(c\) is replaced by a horizontal steam bath, likewise made of two concentric shells with a double cover into which vertical tubes are set for the mercury thermometers being verified.

The measuring apparatus—the manometer—is shown in Drawing 4: the gas pressure is transmitted by capillary \(b\) to the mercury in the glass vessel \(g\), which communicates at the same time with the glass vessels \(h\) and \(i\), open at the top; \(h\), together with the dark cup, serves for the barometer \(k\). Vessel \(i\), connected with \(g\) and \(h\) by a flexible iron tube \(o\), can be moved along the rail-stand \(p\)—roughly, directly by hand, and precisely by means of the micrometer screw \(l\). Drawing 5 shows, in section and on a larger scale, the middle part of tube \(g\); here the platinum point is visible, up to contact with which the mercury in \(g\) is raised by moving \(i\); the distance from the platinum point to the beginning of the capillary is only \(0.6\) mm. The barometric tube \(k\) can be moved 70 mm upward and downward along the stand \(p\). The stopcocks \(m\) and \(n\) serve—the first to disconnect the bottom of \(g\) from \(h\), and the second, a three-way cock, to disconnect \(i\) from \(h\) and to release

Drawing 4.

Drawing 5.

mercury. The readings are made with the aid of a cathetometer; the height \(H\) gives the total pressure of the gas in the thermometer.

One of the advantages of this instrument is that, in order to determine the pressure \(H\), only two readings need be taken, whereas in other instruments four are required—two on the manometer and two on the barometer.

This instrument is very accurate and does not require any special aging. Repeated observations have established that heating it to \(100^\circ\) does not change the point \(0^\circ\). We shall apply this instrument to the determination of temperatures from \(-25^\circ\) to \(+100^\circ\) C.

For high temperatures, such an accurate gas instrument, which could be regarded as a normal one, has still not yet been constructed. Comparing everything that was said above about the various practical difficulties in constructing such an instrument, one can outline the conditions that will have to be observed: for temperatures up to \(1000^\circ\) C the vessel must be of quartz glass, and up to \(1300^\circ\)—of porcelain, with a layer of glaze inside and outside; the working gas must be nitrogen; the measurements must be carried out by method 6—the thermovolumenometer—or, in general, by a method that does not presuppose constancy of the mass of the gas.

One of the chief difficulties in using a gas instrument at high temperatures consists in the difficulty of attaining uniform heating of the instrument. In this respect, the apparatus constructed by Barus for aging thermoelements is noteworthy, Fig. 6:

Fig. 6.

the porcelain gas vessel \(a\), with a capacity somewhat greater than \(500\ \mathrm{cm}^3\), is surrounded by a cast-iron muffle \(b\), composed of two halves, which, by means of the wheels \(c c\), rests on the rollers \(d, d, d, d\). With the aid of the pulley \(e\), the muffle is set into rotation, whereby an absolutely uniform heating is achieved of the vessel \(a\) and of the thermoelement \(t\) inserted into the recess made in it. The muffle

is heated by gas burners. The entire apparatus is enclosed in a shell \(f\) of refractory material. Barus conducted all measurements by method 5—with constant gas pressure.

Let us now proceed to the description of instruments that can be used in practice for the direct measurement of temperatures.

Figure 7 shows Joly’s apparatus, often called an air thermometer, since the hydrogen in it is replaced by air when especially high accuracy is not required. The measurements are carried out by method 2, i.e. with a constant quantity and volume of gas. The establishment of the constant initial volume is achieved by the mutual displacement of the tubes with mercury \(b\) and \(c\) along the stand \(d\), with divisions in mm; \(b\) and \(c\) communicate with each other by means of a rubber, braided tube \(e\). The level of the mercury in \(b\) is always brought into contact with the glass point in the upper part of \(b\).

Fig. 7.

Instead of making two pressure readings and calculating \(t\) by equation (10), in practice it is preferred to calibrate the apparatus in advance so as to obtain the reading directly from a single pressure reading. This calibration is carried out in the following way: the gas vessel \(a\) is placed in melting ice and, having set the level of mercury in \(b\) accordingly, the difference of levels \(h_0\) is measured; the pressure \(p_0\) in mm of mercury will obviously be

\[ p_0 = h_0 + B, \]

where \(B\) is the height of the barometer at that time. Then \(a\) is placed in the vapor of boiling water at a pressure of 760 mm of mercury; moreover, if the barometric pressure \(B < 760\) mm, the vapor pressure is artificially increased by intensifying the boiling; the level difference \(h_{100}\) is again measured, and the pressure is

\[ p_{100} = h_{100} + 760. \]

For each \(1^\circ\) C, the gas pressure therefore increases by

\[ \frac{P_{100} - P_0}{100} \]

mm of mercury. If at some sought temper-

temperature \(t_x\) the difference of levels will be \(h_x\); then, neglecting the change in barometric pressure, we have

\[ t_x=(h_x-h_0):\frac{P_{100}-P_0}{100}=\frac{h_x-h_0}{P_{100}-P_0}\,100 . \tag{37} \]

With this method of measurement there is no need to introduce a correction for the expansion of the vessel \(a\), assuming that its coefficient of expansion changes little with temperature. For a more accurate determination of \(t\) it is only necessary to compute, by the correction indicated above, the harmful volume, and to take into account the change in the barometer.

For temperatures up to \(500^\circ\) C, the bulb \(a\) is made of borosilicate glass; up to \(1000—1100^\circ\), of quartz glass; still higher—up to \(1300—1400^\circ\)—of porcelain.

Quite convenient for practice, though considerably less accurate, is the apparatus of the Richard firm, which in essence also operates according to method 2. The apparatus, Fig. 8, consists of a steel vessel filled with nitrogen and connected by means of a capillary with a Bourdon metal manometer, whose pointer indicates the increase in nitrogen pressure when the vessel is heated. The apparatus is provided with a self-recording device. The divisions in \(^{\circ}\)C are established empirically, by comparison with another instrument; corrections for changes in the outside temperature and barometric pressure are neglected. The limits of application of the apparatus are \(100—700^\circ\) C.

Fig. 8.

Of the instruments operating according to method 3, i.e. with a constant quantity of gas but with variable volume, we shall describe only the Ziegert-Dürr pyrometer, Fig. 9. The porcelain vessel \(a\) communicates

by a copper capillary \(b\) with a copper bell \(c\); the bell \(c\) is suspended from the arm of a short lever, which can rock on a steel knife-edge; to the other arm of the lever is attached a counterweight \(d\), almost balancing \(c\); the latter is immersed with its lower edge in paraffin oil, which separates the space under the bell from the outside air. When the air in \(a\) expands as a result of heating, part of it passes into \(c\), which as a result rises and turns the pointer, which directly indicates the temperature sought. The instrument can be used for temperatures from \(0^\circ\) to \(500^\circ\) C. The graduation is carried out empirically.

Drawing 9.

In order to avoid calculating corrections for the change of the ambient temperature and barometric pressure as compared with those prevailing during the graduation, the instrument is supplied with a special indicator, which consists of a U-shaped glass tube, one end of which is sealed and blown into a small bulb, while the other is open; paraffin oil is also poured into the indicator in such an amount that, when the bulb is immersed in water of a definite temperature, equal to the temperature at which the instrument was graduated, the level of the liquid in the open branch coincides with a mark placed on it. Before using the pyrometer, the indicator is inserted into the internal cavity of the instrument, which by means of the cover \(e\) is hermetically separated from the outside atmosphere; then the pointer is set to the temperature of the surrounding medium by exhausting or blowing air into the bell \(c\) with the aid of a rubber tube through the cock \(f\); then, in the same way, by means of the cock \(g\), the level of the liquid in the indicator is brought to its mark. The instrument is very convenient, but apparently is no longer manufactured by the firm at present, since the porcelain vessels broke too easily. This circumstance can be remedied by making the vessels, for example, of nickel, but then the temperature limit is restricted to approximately \(1000^\circ\) C.

One may also mention instruments working with a variable mass of gas: when the latter is heated, it is allowed to escape freely

from the instrument, so that its pressure remains equal to atmospheric; when the temperature has become established, the remaining quantity of gas is measured, either by pumping it out with a mercury pump (Deville and Troost) or by displacing it with carbon dioxide, which is then absorbed by alkali (V. Meyer and Crafts). The required temperature is calculated from equation (13), or in general from equation (12). In practice these instruments do not have to be used.

However, there is an instrument, also working with a variable quantity of gas, which, owing to its construction, is applicable in practice: this is Wiborg’s pyrometer. Its idea is as follows: to a definite volume \(V\) of air, which has assumed the temperature \(t\) to be measured but has atmospheric pressure \(B\), there is added a second volume of air \(V'\) of the same pressure but of another temperature \(t'\); after this all the air is compressed to the former volume \(V\) and allowed to assume the temperature \(t\) to be measured, which is found from the increase of pressure by the amount \(b\).

If all the air had remained under the pressure \(B\), it would occupy the volume \(V + V'[1+\alpha(t-t')]\). Since gas volumes are inversely proportional to the pressures, we have the relation

\[ \frac{V + V'[1+\alpha(t-t')]}{V}=\frac{B+b}{B}, \]

whence the required temperature is

\[ t=t' + \frac{Vb - V'B}{\alpha V'B}. \tag{38} \]

In Figs. 10 and 11, p. 28, Wiborg’s instrument is shown, adapted for technical measurements, e.g., for determining the temperature of heated air during blast: the gas vessel \(V\), of capacity about \(12\ \mathrm{cm.^3}\), forms one end of the porcelain tube \(A\), whose outside diameter is about \(20\ \mathrm{mm.}\), and inside diameter \(0.5\ \mathrm{mm.}\), so that it may be regarded as a capillary; the tube \(A\) is fastened, by means of a special cement (lead oxide finely ground with glycerin, rapidly hardening and withstanding a temperature up to \(250^\circ\ \mathrm{C}\)), into a metal end-piece \(H\), by which it can be screwed into the nozzle \(H'\), either vertically, as shown in Fig. 10, or horizontally, after changing places with the plug \(U\). In the same cement there is set into \(H'\) a glass capillary, which from the point \(m\), over a length of \(10\ \mathrm{mm.}\), widens into a tube about \(2\ \mathrm{mm.}\) in diameter, and then passes into a pear-shaped enlargement, which contains the additional volume \(V'\), and whose capacity from the lower edge \(m\)

to \(m\) is about \(0.1\,V\). To the right of \(m'\) upward goes the manometric tube \(B'\), about 440 mm high; at the lower branch pipe \(R\) the glass tubes communicate with the rubber bag \(K\), in which there is mercury; by turning the screw \(S\) in one direction or the other, one may either press on the bag with the cover \(N\) and force the mercury to rise through the tubes, or lower it.

Observations with the aid of this instrument are carried out as follows: the end \(V\), protected from mechanical damage by the perforated casing \(X\), is inserted into the space under investigation; the air in \(V\) is heated to the desired temperature \(t\), and its excess escapes through \(B\) and \(B'\). After this, the mercury is raised to \(m'\), when \(B\) is separated from \(B'\), and then further to the mark \(m\), driving the volume \(V'\) with temperature \(t'\), read from the mercury thermometer \(T\), into \(V\). As this air is heated, its pressure will increase, and it will be necessary to raise the screw \(S\) until the mercury finally comes to rest at the mark \(m\); the height of the mercury in the tube \(B'\) from \(m'\) to the end of the column will give the desired pressure \(b\) in equation (38), from which \(t\) is found.

Figs. 10 and 11.

In practice, instead of calculating \(t\) from equation (38), it is preferable to obtain the reading directly from a scale. However, depending on the different barometric pressure and the temperature \(t'\), the quantity of air

in the added volume \(V'\) will be different, and, together with that, the height \(b\), for the same difference \(t - t'\), will be different, i.e. the length of the scale must change, increasing as \(B\) increases and \(t'\) decreases, and conversely; moreover, the very beginning of the scale is accordingly shifted. This latter circumstance makes it easy to find the required scale by turning the wooden cylinder \(O\), on which are arranged the scales for different quantities of air, i.e. \(B\) and \(t'\): namely, if the mercury is raised to the mark \(m\), on the condition that the air in \(V'\) and \(V\) has the same temperature, then the scale whose beginning coincides with the level of the mercury in \(B'\) is the one to be used in the subsequent reading.

This method, however, is inapplicable if the instrument has already been installed and the temperatures in \(V\) and \(V'\) are different. For this case there is an auxiliary tube \(Q\), expanded at the end into a small bulb \(Q'\); the mercury level in \(B'\), corresponding to 0 of the scale, is found if the mercury in \(Q\) is raised to the mark \(x\). This additional device is based on the same idea as the whole instrument, namely that a definite quantity of air, in the present case from \(m''\) to \(x\), is driven into a known volume, and then the resulting pressure is measured.

When measuring especially high temperatures, the porcelain tube must be wrapped with asbestos, so as not to subject it to excessively abrupt fluctuations of temperature, which can easily cause it to crack.

As can be seen, handling the instrument is very convenient and simple, and its readings are sufficiently accurate, since the scales are obtained not empirically but by calculation.

For less exact observations, Wiborgh reconstructed his instrument, replacing the mercury manometer with a metal one. Although the instrument

Figs. 12 and 13.

turned out to be less accurate, it was also less fragile and still more convenient to use. This pyrometer is shown in Figs. 12 and 13:

a porcelain vessel \(V\) containing air, with a capacity of about \(100\ \mathrm{cm}^3\), is connected by a capillary channel with a cavity \(V'\) formed by two springy disks; when the rod \(s\) is pressed against these disks, they assume a perfectly flat shape and come tightly into contact with one another, and when the pressure ceases they again assume their original form. The disks have small openings through which the vessel \(V\), on one side, communicates with the atmosphere, and on the other, by means of a flexible lead capillary \(m\), with the cavity of a manometric spring, which actuates the main pointer \(Z\). To measure with this instrument the temperature \(t\) of some space, the vessel \(V\) is placed in it, and when it has assumed this temperature, by means of the external bow \(G L G\) one presses \(s\) against \(d\); at this, first the spaces \(V\) and \(V'\) are disconnected from the atmosphere, and then the disks of \(V'\) are pressed against each other; the air contained between them is driven into \(V\) and expands owing to the rise in temperature; the resulting increase of pressure in \(V\) is transmitted to the manometric spring, which turns the pointer \(Z\) by a corresponding deflection.

It remains to discuss the corrections for changes in the temperature \(t_0\) of the surrounding air and in the barometric pressure \(B\), as compared with those that prevailed during the graduation of the instrument. These corrections amount to ensuring that, independently of temperature and pressure, one and the same quantity of air by weight is enclosed in the vessel \(V\) at the moment action begins. Since, as the temperature \(t_0\) increases, the specific weight decreases, the volume \(V'\) must correspondingly be increased; when the pressure \(B\) increases, on the contrary, it must be decreased. This change in the volume \(V'\) is achieved by the corresponding turning of the ring \(E\), which, by means of the finger \(n\), turns the lever \(o\) with the ring \(g\); \(g\) is connected on one side obliquely and, resting with its flat side against the fixed part \(f\), when turned presses back to a greater or lesser extent the cup \(d\), and with it the left disk of \(V'\).

On the fixed inner dial there is a scale of barometric pressure, and on the outer dial, which turns together with the ring \(E\), a scale of temperatures. In order that, at a given temperature \(t_0\) and pressure \(B\), the required quantity of air be present in \(V'\), \(E\) must be turned so that the division marked \(t_0\) coincides with the division marked \(B\). Then the pointer \(Z\) will directly indicate the desired temperature, and no calculations of corrections need be made.

The temperature $t_0$ is read from the thermometer $L$ located on the dial plate, and the pressure $B$ from the aneroid barometer $Q$ located there as well. The divisions of the main scale are, of course, made empirically, as are those of the auxiliary scales. The instrument is suitable for measurements up to $1400^\circ$ C. The accuracy of its readings is hardly better than $\pm 2\%$, but in use it is very convenient.

In conclusion, we shall also describe Joly’s pyrometer, improved by Steinbart, which is based on an entirely different principle than all the methods and instruments discussed above; namely, it is based on the laws of the passage of gas through small openings. The operation of this instrument is readily understood from the diagram in Fig. 14: in the space $e_2$, rarefaction is produced by means of the ejector $d$, under the action of which the air located in $e_1$ flows into $e_2$ through a very small opening $b$; the loss of air in $e_1$ is replenished by its entry from the external atmosphere through the second small opening $a$; owing to the large resistances in passing through the openings $a$ and $b$, with continuous operation of the ejector in $e_1$ there will always be maintained a certain rarefaction, measured by the vacuum gauge $f$, and in $e_2$ an even greater rarefaction, measured by the vacuum gauge $g$.

Fig. 14.

We shall call the difference between the readings of $f$ and $g$ the relative rarefaction. In steady motion the rarefaction in $e_2$ will be constant; but if less air begins to enter through the opening $a$ than exits through $b$, then the rarefaction in $e_1$ will begin to increase, and the relative rarefaction to fall; as a result of this the velocity in $b$ will begin to decrease, and soon a new steady motion will set in, but with a different quantity of flowing air and with a different relative rarefaction. The indicated change in the quantity of flowing air occurs chiefly at the expense of a change in its specific gravity, and the latter decreases with increasing temperature, and conversely. If the air entering through $a$ is heated, then the rarefaction in $e_1$ will increase, and the relative rarefaction will fall. Thus the magnitude of the relative rarefaction, measured by the vacuum gauges $f$ and $g$, is a function of temperature, and, for a definite constant rarefaction in $e_2$ and a definite size of the openings $a$ and $b$, from it one can judge the temperature of the incoming air.

Although the decrease in the density of the air in \(e_1\), owing to the change in the velocities \(a\) and \(b\), does not proceed proportionally to the rise in temperature, but rather somewhat more slowly, so that with increasing temperature the accuracy of the readings and the sensitivity of the instrument fall somewhat, nevertheless they remain quite sufficient for practical purposes. Thus, for example, in the instrument described below, with a rarefaction in \(e_2\) (below atmospheric pressure) of 1220 mm of water column, a change in temperature by \(10^\circ\) C corresponds to a change in the relative rarefaction: between 650 and \(800^\circ\) C—about 6 mm of water column; between 800 and \(1100^\circ\)—about 4.5 mm; and, finally, between 1100 and \(1650^\circ\)—still about 2 mm.

Fig. 15.

In Fig. 15 the entire instrument is represented schematically: under the action of a steam ejector \(d\), which sucks air out of the space \(e_2\), air flows in there from the external atmosphere, passing through the filter \(i\) with cotton wool, then through the tube \(k\) with the platinum tip \(o\), whence, through the small opening \(a\) in the same platinum tip \(e\), along the tube \(ll\) and the coil \(m\), it passes through the second small opening \(b\) and the tube \(n\). The tips \(oe_1\) are inserted into the space whose temperature is to be determined.

In order that, from the vacuum gauge \(g\), filled with colored water, or from the self-recording spring vacuum gauge \(G\), one may at once obtain the reading in \({}^\circ\) C without any corrections, it is necessary, first, that the rarefaction in \(e_2\) be maintained exactly at a known height, and second, that the air, when passing through the opening \(b\), always have a constant temperature. The first condition is satisfied by the fact that the vessel \(H\) with the cavity \(e_2\) is filled with water, into which there is immersed a tube \(s\), sealed into the cover and open at both ends; the ejec-

pump \(d\) draws out somewhat more air than can enter through the instrument; but however vigorously it may work, the rarefaction cannot rise above a column of water equal to the distance from the lower end of tube \(s\) to the water level in \(H\), since the deficiency of air is constantly replenished from the atmosphere through tube \(s\). In existing instruments the rarefaction is maintained at \(1220\) mm of water column. Owing to evaporation of the water, the level in \(H\) falls with time, which is revealed by a lowering of the level in the vacuum gauge \(f\) below pointer \(r\); to make up the loss it is enough to open cock 3, and water from vessel \(t\) will rush into \(H\) owing to the rarefaction in \(e_2\).

The second condition—the constancy of the temperature before \(b\)—is satisfied by the fact that the air enters \(b\) through the coil \(m\), which is continuously heated by steam coming from the ejector; since the pressure in \(h\) is atmospheric, the temperature is maintained at about \(100^\circ\) C.

Finally, in order that the frame into which the platinum filaments are fitted should not suffer from the high temperature, it is surrounded by a water jacket, through which passes a continuous flow of water entering through branch pipe \(x\) and leaving through \(y\).

In Fig. 16 this part of the instrument is shown in its constructive form; the same parts as in Fig. 15 are denoted by identical letters. Openings \(a\) and \(b\) are made 0.4 mm in diameter.

Fig. 16.

The length of tube \(l\), from opening \(a\) to \(b\), has no influence on the accuracy of the readings; it is necessary only that all joints be perfectly tight. Tube \(ll\) may be taken up to 150 meters long, with a diameter of \(3/8''\); thus this pyrometer is also suitable for readings at a distance.

The temperature limits for which it can serve are from \(100^\circ\) to \(1650^\circ\) C. (in Fig. 15 the Fahrenheit scale is shown); moreover, according to the observations of Prof. Denton, at the highest temperature the accuracy of the readings is not less than \(\pm 1.5\%\); as the temperature decreases, the accuracy increases. The graduation of the scale is performed, of course, experimentally, by comparison with some other instrument. As to the sensitivity of the instrument, it has already been discussed above.

In conclusion, one may also mention the measurement of temperatures by means of an especially sensitive liquid manometer. The arrangement of such an installation, as applied to a smokestack, is shown in

Fig. 17: \(a\) is a vertical tube, closed at both ends, made of copper, porcelain, or platinum, depending on the magnitude of the temperature being measured; to the ends in \(a\) are soldered two horizontal branch tubes \(b\) and \(c\) of the same material; by means of these branch tubes and rubber or copper thin tubes, the internal cavity \(a\) communicates with the two tubes of Krell’s micromanometer, which indicates the pressure corresponding to the height \(H\) of the column of air; the higher the temperature of the air in \(a\), the lighter it is, and the greater the height of the liquid column in tube \(f\), which communicates with branch tube \(c\). This is not the place to describe the very ingenious construction of this micromanometer; we shall confine ourselves only to indicating the scale, graduated directly in °C, for an instrument in which the measuring tube has an inclination of 1:200, and the height \(H = 1000\) mm, Fig. 18.

Fig. 17.

In order that the pressure of the air in \(a\) should always be equal to atmospheric pressure, a pressure regulator \(e\), consisting of two glass bulbs open at the top and communicating at the bottom, forming a water seal, is attached to the connecting tube by means of the tee \(d\); when the temperature rises and the air in \(a\) expands as a result, its excess passes out through \(e\); when the temperature falls, air, conversely, is drawn into \(a\) through \(e\).

The micromanometer may be set up at a distance of several meters from tube \(a\), which is an additional advantage of this instrument, very convenient, though only for certain cases, such as: for measuring tempera-

tures in vertical channels, flues, hot-air pipe stands, etc.

The sensitivity and accuracy of the readings may be judged from the scale shown in Fig. 18 at natural size.

Instruments with Liquids

6. Subdivision.—Instruments in which the working body is a liquid may be subdivided into two groups: the first includes instruments by means of which temperature is measured by the visible increase in the volume of the liquid; the second, instruments that indicate temperature by the increase, associated with the increase of volume in a closed space, in the pressure of the liquid. Since at high temperatures the liquid in these instruments passes into the vaporous state, these instruments are, as it were, a transitional stage to gas pyrometers, which work with a constant quantity and volume.

We shall first discuss instruments of the first group, then those of the second.

7. Glass thermometers.—As the working body one may use mercury, alcohol, toluene, petroleum ether, pentane, etc.

A thermometer with alcohol, usually colored blue for clear reading, can be used from \(-100^\circ\) to \(+75^\circ\) C. The drawback of the alcohol thermometer is that alcohol wets glass, and therefore, when the temperature is falling, until all the alcohol has run down, the instrument gives incorrect, diminished readings. However, when measuring temperatures below \(-10^\circ\), the alcohol thermometer should be preferred to the mercury one, since mercury becomes viscous at these temperatures. In addition, the alcohol thermometer is indispensable for measurements in an alternating-current field, since the mercury one in this case gives incorrect readings under the influence of eddy currents.

Instead of alcohol, thermometers are often filled with toluene, \(C_6H_5CH_3\); the measuring range of such an instrument is \(-100^\circ\) to \(+50^\circ\) C. For especially low temperatures, down to \(-200^\circ\) C., thermometers are filled with petroleum ether or pentane, \(C_5H_{12}\).

The most widespread instrument for measuring temperature is the glass mercury thermometer, consisting of a lower vessel, a bulb, and a capillary tube provided with a scale, which is mounted either on the tube itself or on a separate one arranged ...

alongside it. For measuring the temperature of liquids, the tube together with the scale is enclosed in a glass tube— a casing. For better visibility of the mercury column, the capillary tube is usually strongly flattened.

In choosing the type of glass, two principal requirements must be kept in mind: the coefficient of expansion of the glass should be as uniform as possible and small, and the thermal after-effect of the glass should, as far as possible, be small. By thermal after-effect is meant the change in the density of the glass, persisting for some time, under the influence of a change in temperature, and the associated change in the capacity of the vessel containing the mercury, which entails a displacement of the fixed points \((0^\circ\) and \(100^\circ)\), and consequently of the entire scale of the instrument. This phenomenon is explained by molecular displacements occurring in the glass over a long period after the glass has been subjected to the action of another temperature. Thus, after the manufacture of the instrument, the capacity of the vessel decreases over a long interval of time, sometimes several years; this produces a continuous increase in all the readings of the instrument. Conversely, any heating causes a temporary lowering of the readings. Careful experiments at the Imperial German Physico-Technical Institute have established that the lowering of the readings disappears the more rapidly, the higher the temperature which the thermometer had before being immersed in melting ice; in view of this, when determining the fixed points, one must first determine \(100^\circ\) and then transfer the thermometer as quickly as possible into melting ice.

The magnitude of the after-effect, as well as of the temporary lowerings, is especially great in glasses containing lead, and also in glasses containing potassium and sodium simultaneously, which therefore are unsuitable for thermometers; thus the best are refractory glasses.

The change of the fixed points is reduced to a considerable extent if the finished thermometer is subjected to very prolonged annealing, over several weeks, in vapors of sulfur, mercury, or cadmium. French thermometers treated in this way are provided with the inscription “recuit.”

Thus, far from all kinds of glass satisfy the stated requirements.

Good thermometers are made in Germany from Jena glass, the so-called normal thermometer glass, or from glass of grade \(16^{\mathrm{III}}\), or from an even better grade—borosilicate \(59^{\mathrm{III}}\); for very high temperatures, above \(500^\circ\) and up to \(575^\circ\), the more refractory Jena combustion glass is used.

glass (Verbrennungsglas). In recent times mercury thermometers have even been made up to \(+750^\circ\) C., using quartz glass for this purpose; however, these thermometers cannot become at all widespread because of their extreme cost. Good French thermometers are made of a special, very refractory glass, verre dur.

As regards the expansion of mercury, then, as has already been mentioned, it too does not expand quite uniformly, although more uniformly than other liquids; specifically, its coefficient of expansion increases with temperature. The German Imperial Physico-Technical Institute has found that, for an entirely cylindrical tube of Jena glass 59″, in which the interval \(0^\circ\)—\(100^\circ\), taking into account the expansion of both mercury and glass, is equal, for example, to 100 mm, the readings \(200^\circ\), \(300^\circ\), \(400^\circ\), and \(500^\circ\) C. already correspond not to 200, 300, 400, and 500 mm, but to 200.4, 304.1, 412.3, and 527.8 mm above \(0^\circ\). This increase in the coefficient of expansion of mercury is, of course, taken into account in the manufacture of thermometers.

A special advantage of mercury over other liquids consists in the fact that it is easy to obtain it absolutely pure from all impurities. Thanks to this, complete comparability is achieved among the readings of thermometers of different origins.

Another advantage is the wide temperature limits: an ordinary mercury thermometer with an evacuated capillary tube can be used from \(-35^\circ\) to \(+330^\circ\), even \(340^\circ\) C.; at \(357^\circ\) mercury boils. To raise the upper limit, the boiling point of mercury is raised by filling the space above the mercury with some gas indifferent with respect to it, nitrogen or carbon dioxide, under a pressure of 10–25 atm. In this way it becomes possible to measure temperatures up to \(550^\circ\), even \(575^\circ\), and, with quartz glass, up to \(750^\circ\) C. However, the presence of gas above the mercury affects the accuracy of the readings at low temperatures, so such instruments should be used only for temperatures above \(300^\circ\) C.

Finally, it is necessary to mention the relation between the capacity of the vessel and the dimensions of the capillary tube. To increase the sensitivity of a thermometer in the sense of indicating small changes in temperature, the vessel must be of large capacity and the tube of small cross-section; however, with an excessively narrow tube the resistance to the motion of the mercury increases greatly. On the other hand, in order that the thermometer readings lag as little as possible in time behind changes in temperature, the capacity of the vessel must be as small as possible. Likewise, a large vessel with

accordingly, a large quantity of mercury takes much heat away from the body whose temperature is to be determined, which may entail a noticeable lowering of this temperature. Depending on the ratio of the capacity of the vessel to the cross-section of the tube, one obtains the length of \(1^\circ\), which usually ranges from \(0.7\) mm when divided into whole degrees, to \(6\)–\(7\) mm when divided into \(0.1^\circ\); in thermometers for calorimetric purposes, with divisions of \(0.01^\circ\), the length of \(1^\circ\) reaches \(70\) mm.

In good instruments the capacity of the vessels is made \(50\)–\(60\) times greater than the capacity of the tube from \(0^\circ\) to \(100^\circ\).

8. Corrections to the readings of a mercury thermometer.—In scientific investigations, when it is required to determine the temperature with an accuracy of \(0.1^\circ\), or even more accurately, the readings even of the best normal thermometer, supplied with a certificate from the German Imperial Physico-Technical Institute or another official institution, require a whole series of corrections.

1. The effect of the thermal after-effect is taken into account in the most reliable way by calibrating, as often as possible, the fundamental points \((0^\circ\) and \(100^\circ)\) and thereby finding the correction \(\lambda_t\), which must be added to the reading of the instrument already corrected according to the certificate or the table compiled by comparison of the given instrument with the normal one. The method of determining the fundamental points is indicated below, in the section on calibration of instruments.

Strictly speaking, a change in the position of the fundamental points occurs with every change in the temperature being measured, i.e. each temperature corresponds to its own special position of the fundamental points. Thus the correction for the depression of the zero must be introduced for every temperature reading; only by introducing this correction do results consistent with one another and with the gas scale result when measuring temperatures with thermometers differing in the composition of the glass and in the magnitude of the depression. To obtain these corrections, the most reliable procedure is to compare the given instrument at different temperatures with a gas thermometer and to compile a corresponding table of corrections. Guillaume compiled such a table for a French thermometer made of verre dur; the depression at \(100^\circ\) reaches almost \(0.1^\circ\). German investigators give empirical formulas, for example, of the form

\[ \lambda_t = at + bt^2; \tag{39} \]

for glass \(16^{\mathrm{III}}\) Böttcher gives \(a = 0.00071,\ b = 0.00008\); this gives, at \(100^\circ\), the value \(\lambda_t = 0.871^\circ\). For glass \(59^{\mathrm{III}}\), \(a\) and \(b\) are somewhat smaller.

If possible, this correction is best determined directly by observation.

2, a correction for the non-uniform expansion of mercury, i.e., for the deviation of the reading from the gas scale, if, with a strictly cylindrical tube, the length of each degree is constant. As already indicated earlier, for thermometers above \(200^\circ\) the expansion of mercury is taken into account in the very making of the scale. Thermometers up to \(100^\circ\) may be corrected by using special tables or a graphical representation, Fig. 1.

If the above-mentioned table of corrections for the change in the fundamental points has been compiled by comparison with a gas thermometer, then this correction has entered into it by itself.

3, a correction for the effect of pressure. A vessel with mercury is subjected to pressure, external—atmospheric—and internal—from the column of mercury. The effects of these pressures are mutually opposite, but both tend to change the capacity of the vessel and thereby to shift the fundamental points, and consequently the whole scale.

If the thermometer was graduated at a barometer height of 760 mm, then at another height \(B\) one must introduce the correction

\[ \mu' = -\beta'(B - 760), \tag{40} \]

where \(\beta'\) is a coefficient determined experimentally. Numerically \(\beta'\) varies from 0.0001 to 0.0002; for thermometers made of verre dur \(\beta' = 0.000121\); for glass 59III \(\beta' = 0.000152\). Thus the entire correction \(\mu'\) does not exceed \(0.01^\circ\), and it may almost always be neglected.

The pressure of the mercury column, distending the vessel, lowers the readings. The correction may be calculated in the form

\[ \mu'' = \beta'' l \frac{\delta}{\delta_0}\cos\alpha, \tag{41} \]

where \(l\) is the length of the mercury column in mm, \(\delta_0\) is the density of mercury at \(0^\circ\), and \(\delta\) at the measured temperature \(t\), \(\alpha\) is the angle of the thermometer axis with the plumb line; the coefficient \(\beta''\) is somewhat larger than \(\beta'\), since to the expansion of the vessel is added the compression of the mercury column itself. Approximately one may take

\[ \beta'' = \beta' + 0.000015. \]

The direct determination of \(\beta''\) is made by observing the thermometer readings at \(100^\circ\) in the vertical and horizontal positions.

For a long thermometer the correction \(\beta''\) may reach as much as \(0.1^\circ\).

The total correction is \(\mu = \mu' + \mu''\); it increases if traces of gas, compressed when the temperature rises, are present above the mercury. Of course, this correction may be entirely neglected in instruments for high temperatures, filled with gas under high pressure.

4, the correction for the emergent column of mercury. The normal calibration of thermometers is carried out in such a way that all the mercury is subjected to the action of the corresponding temperature. Therefore, if during operation a part of the capillary tube containing mercury protrudes outward, then this part of the mercury column will have a different temperature and, consequently, its length will be different than if it were wholly immersed in the given medium; as a result the thermometer reading will differ from the true temperature, and it is necessary to introduce a certain correction \(\nu\), called the “column correction,”

\[ \nu = n(t' - t_c)\alpha, \tag{42} \]

where \(t'\) is the temperature indicated by the thermometer, \(t_c\) is the temperature of the column, \(n\) is the number of degrees by which the column protrudes, and \(\alpha\) is the apparent coefficient of expansion of mercury, taking into account the expansion of the glass tube. Depending on the composition of the glass, this quantity ranges from \(\alpha = 0.00027\) for ordinary Jena glass to \(\alpha = 0.000155\) for glass 59^III.

\(t_c\) is measured with an auxiliary thermometer, which is suspended next to the main one so that its bulb lies at the mid-height of the protruding column. If the length of this column is not great, then, owing to the good heat conduction of the mercury, its mean temperature will be higher than the external temperature \(t_c\); as a result some authors consider it more correct to reduce \(\alpha\), namely to take \(\alpha = 0.000135\). There are also other expressions for \(\nu\) than equation (41), but since they are of more complicated form and therefore less convenient for practical use, and since they also do not give a fully reliable value, we do not cite them here.

For especially precise measurements the correction \(\nu\) is determined directly with a Mahlke thermometer, in which the vessel with mercury is made in the form of a thin long tube. These thermometers should be available as a set of different lengths from 50 to 200 mm, and one should choose one such that the length of the tube-vessel is equal to the height of the protruding column; then the Mahlke instrument, suspended alongside, will directly give the required correction \(\nu\) in \({}^{\circ}\mathrm{C}\).

The correction \(\nu\) is the most essential of all, and often it must not be neglected even in experiments that are not especially precise. The magnitude of the error eliminated by introducing this correction may be judged from the following examples: the temperature of flue gases is measured, \(t' = 378^\circ\); a thermometer made of Jena glass, \(\alpha = 0.0002\), was inserted into the flue to a division of \(50^\circ\); the surrounding temperature is \(t_c = +29^\circ\); then

\[ \nu=(378-50)(378-29)\cdot 0.0002 = 23^\circ. \]

The indication \(378^\circ\) or \(401^\circ\)—is a large difference, and the example chosen is by no means an extreme one.

In similar cases the error may reach \(50^\circ\)--\(70^\circ\), i.e. the thermometer may read \(550^\circ\), while the actual temperature of the bulb will be \(600^\circ\)--\(620^\circ\), when the glass is already becoming viscous, i.e. the instrument may easily be damaged if this correction is forgotten.

However, even at less high temperatures the correction for the column must not be neglected; for example, let the temperature of the water entering the jacket of a gas engine be \(t'_1 = 17.1^\circ\), and that of the water leaving it \(t'_2 = 59.6^\circ\); in the room \(t_c = 25^\circ\). Then, if both thermometers project by \(-5^\circ\), the corrections will be: \(\nu_1 = (17.1 + 5.0)(17.1 - 25.0)\cdot 0.0002 = -0.03^\circ\), and \(\nu_2 = (59.6 + 5.0)(59.6 - 25.0)\cdot 0.0002 = -0.45^\circ\). The temperature difference by which the heat carried away by the water is computed is, apparently, \(59.6 - 17.1 = 42.5^\circ\), whereas the actual one is greater by \(0.03 + 0.45 = 0.48^\circ\); the error is equal to

\[ \frac{0.48}{42.5}\cdot 100 = 1.1\%. \]

In those cases when one does not wish to apply a correction for the column, seeing by eye that it is not especially large, at the very least one must not give the results of experiments with many decimal places.

However, in such a case, when it is necessary to measure not an absolute temperature but only the difference of two temperatures, and moreover ones not very greatly different from one another, the error can almost be destroyed if the thermometers are inserted so that the mercury columns in both protrude by the same number of degrees.

It might seem simpler, already during calibration or during the aging of the thermometer, to have the mercury column protrude by as much as it will protrude during use. But since the temperature of the column, approximately equal to the temperature of the surrounding air, may also vary over wide limits, such a method is not very reliable. Only thermometers for temperatures above \(200^\circ\) are sometimes graduated “with a protruding column,” since fluctuations

air temperatures are negligible in comparison with such high measured temperatures. In such thermometers it is necessary to observe the condition that the entire column be outside; this is not always convenient, and therefore sometimes part of the tube from the bulb is made without divisions; this part must be entirely immersed in the medium whose temperature is being measured.

9. Designs of mercury thermometers.—The principal parts of every glass thermometer were indicated above.

Like gas thermometers, mercury thermometers may be divided into two groups: standard instruments, which are used only for the verification of others, and working thermometers, used for the direct measurement of temperatures. Besides special care in manufacture, standard thermometers must also satisfy certain special design requirements.

The German Imperial Institute worked out the following conditions for standard thermometers: principal standard pyrometers are mercury thermometers with the points \(0^\circ\) and \(100^\circ\) on the scale, which can be calibrated by themselves, without the aid of other instruments: the divisions must be of strictly equal length; errors in the divisions, in the caliber, are allowed to be no more than \(0.05\) of the smallest division; the width of the stroke not more than \(0.1\) of a division; the capillary tube must end in an enlargement from which the air has been completely pumped out. The graduation of these thermometers is carried out at least every \(10^\circ\). The greatest permissible errors in the scale are:

1. for the fundamental points, \(0.10^\circ\);
2. for errors in the caliber, the difference of the greatest deviations, \(0.25^\circ\);
3. the depression of the point \(0^\circ\) after half an hour of heating at \(100^\circ\), after 5 minutes’ stay of the thermometer in melting ice, not more than \(0.10^\circ\) C.

Standard thermometers are sometimes graduated in such a way that, when applying the divisions, the errors in the tube and the corrections to the gas scale are taken into account; such instruments are also called standard thermometers of the first rank, but no longer principal ones.

Standard thermometers of the second rank are instruments whose corrections cannot be determined by themselves, but only by comparison with a principal standard thermometer; moreover, somewhat larger errors in the divisions are permitted in them. Strictly speaking, these are no longer standard instruments, but only good working instruments.

In order that working thermometers should not become too long, but at the same time should have both fundamental points for precise verification,

the scale is often made shortened; for this purpose, after \(0^\circ\), the expansion of the part of the capillary tube corresponding to the expansion of the mercury up to the temperature from which the divisions of the scale are to begin is substituted.

Conversely, for technical purposes it is often desirable that the beginning of the scale be separated from the bulb with mercury by a considerable distance, for example, for measuring temperatures in ducts and chimneys, when the thermometer must be pushed through an opening in a wall of considerable thickness. This part of the capillary tube, from the bulb to the beginning, is made up to 50 cm long, and sometimes up to 100 and even up to 200 cm.

In terms of construction, thermometers may be further divided into two groups: thermometers with a scale and rod thermometers. In the former, behind the thin capillary tube there is a scale, usually of milk glass; the capillary tube together with the scale is enclosed in a second glass tube—a casing.

Rod thermometers consist of only a single capillary, but very thick-walled, tube, of outside diameter 4–10 mm; the divisions are applied directly to the tube in the form of incised strokes filled with black paint. The advantages of rod thermometers are great strength, smaller outside diameter, and the impossibility of accidental displacement of the scale; the disadvantage is that the paint on the divisions is easily worn off and is rather difficult to renew.

Thermometers used in engineering are often provided with a metal casing to protect them from mechanical damage. The lower part of the casing, surrounding the bulb itself, is made either with slots or closed and filled, for heat transfer, with oil, mercury, or fine metal filings for temperatures above \(300^\circ\). A thermometer of the first kind, for measuring the temperature of gases in chimneys, is shown in Fig. 19; the length of the inserted part \(l\) may be from 150 to 1500 mm, the angle of inclination—from \(0^\circ\) to \(90^\circ\). A thermometer of the second kind, for screwing into a superheated-steam pipeline, is shown in Fig. 20; length \(l \geq 80\) mm.

Fig. 19.

Fig. 20.

Finally, it is necessary also to mention thermometers which oper—

with a variable quantity of mercury. These are instruments with very small divisions of \(0.01^\circ\), or even smaller, which, in order to avoid too long a scale, are made for small ranges of temperature, only \(6–8^\circ\) (the length of the thermometer is 350 to 700 mm). In order to use a thermometer constructed, for example, for the range of temperatures from \(+5\) to \(+11^\circ\), for other, higher ranges, for example from 50 to \(56^\circ\), part of the mercury is driven over into the upper enlargement. Owing to this the column of mercury is, as it were, shortened, and when its end stands opposite the division \(5^\circ\), this will already correspond, for example, to \(50^\circ\). When a small quantity of mercury is driven over, the difference of readings will, as before, indicate with sufficient accuracy the difference of temperatures in \({}^\circ\mathrm{C}\); but in order to know the absolute number of degrees for such readings, the instrument must be compared with a normal thermometer. When large quantities are driven over, the value of the divisions must be determined anew.

Fig. 21.

In Fig. 21 a Beckmann thermometer is shown, in which the separation of part of the mercury column is made especially easy. Sometimes a Beckmann thermometer is made with a scale divided not into fractions of a degree, but into millimeters, or simply into a known number, for example 1000, of equal parts, which is more convenient when reading differences of temperatures.

10. Use of thermometers.—In measuring the temperatures of liquids, vapors, or gases, it is desirable that the bulb with the mercury should be in direct contact with the body whose temperature is being measured. In those cases where this body flows through pipes and, moreover, under pressure, such insertion presents certain difficulties: if a hole is drilled in the pipe and a thermometer is inserted into it, the pressure in the pipe will tend to throw it out; besides this, it is necessary to fill with some packing the gap between the edge of the opening and the thermometer, so that the liquid does not flow out. With pipes of large diameter a special stuffing box can be made; with a small pipe one can make do with a rubber stopper, through which the thermometer is pushed after being lightly greased with oil; the oil dissolves the rubber somewhat, and after a few days the stopper adheres to the glass so firmly that it cannot be removed except by cutting it. Such a thermometer, with a slightly conical stopper, is inserted as tightly as possible into the opening, which it is useful to cut slightly with a tap; by tying the stopper to the pipe with wire, one can in this way insert the thermometer even up to a fairly large ...

pressure in the pipe. If the thermometer has to be inserted into this opening and removed quite often, it is better to arrange a clamp of the kind used for water-gauge glasses, Fig. 22: a piece of copper tube \(a\), with a thread on the thickened outer end, is soldered or screwed tightly into the pipe; the tightening nut \(b\) presses on the rubber ring \(c\) and thereby provides the required tightness; the thermometer is held by friction; in addition, rubber may be glued to it, as above, after lubricating it with oil. When the thermometer is removed, an iron or cardboard disk is placed under the nut \(b\).

Fig. 22.

At very high pressures one must abandon direct insertion and use a fitting of the kind shown in Fig. 23. The fitting is made of copper tubing and filled with oil; for better heat transfer the oil is sometimes replaced by mercury; in that case the fitting must, of course, be iron; at higher temperatures oil is poured on top of the mercury in order to prevent its evaporation. The fitting is inserted obliquely into the vertical pipe at an angle of \(30\)–\(45^\circ\). The drawback of this method of insertion is that part of the heat passes through the fitting directly to the pipe and is radiated from the latter, so that at high temperatures the bulb of the thermometer has a temperature somewhat lower than that of the medium moving through the pipe. This error can be reduced, even almost eliminated, by surrounding the pipe around the fitting with a heat-impermeable layer. Another drawback of such a fitting is the lag of the thermometer readings behind changes in the temperature being measured, owing to the large heat capacity of the whole arrangement.

Fig. 23.

Applying the thermometer to the pipe and wrapping them with copper wire for better heat transfer cannot show the temperature inside the pipe, however well the thermometer is insulated from heat loss. By this method one can record only an increase or decrease in the temperature of the pipe walls, and not of the liquid flowing through it.

When measuring the temperature in long or wide channels, it is sometimes impossible to reach the place under investigation even with a long thermometer; then one may resort to an auxiliary metallic body, Fig. 24: this is a cylinder, best made of iron or cast iron, as sufficiently heat-capacious, yet at the same time not too heat-

of a conductive material, i.e., one that does not cool very quickly, about 40 mm in diameter, with a recess \(a\) drilled in it, into which mercury is poured, and an eye \(b\), into which the end of an iron rod is threaded; on this rod the cylinder is inserted into the space under investigation. After 10–15 minutes, when it has acquired the desired temperature, it is quickly removed and immersed in \(a\) mercury thermometer; the greatest rise of the mercury in it will indicate the desired temperature. Of course, this method gives a somewhat reduced temperature, but with skillful use the error is small.

Drawing 24.

If the temperature being measured is above \(350^\circ\) C., then the mercury in \(a\) will boil; instead of it, lead or zinc must then be used.

To measure the temperature of a solid body—for example, a bearing shell or the armature of an electric motor after operation—it is best to drill a hole for this purpose in the given part, fill it with mercury (or with oil, if the material of the body amalgamates with mercury), and insert a thermometer. Less reliable is to wrap the bulb of the thermometer with tin foil for better heat transfer and, pressing it as tightly as possible against the body being measured, cover it well from above with cotton wool.

An ingenious method for determining high temperatures with the aid of a glass mercury thermometer was proposed by Carpio: one must observe, by a stopwatch, the time during which the column of mercury rises by a definite number of degrees. The rate of rise of the mercury in each thermometer depends on the temperature level; this dependence can be found either by simultaneously measuring the temperature with some pyrometer, or by carrying out a series of such observations at different temperatures (within the limits of the scale of the given thermometer) and then continuing the curve of rates thus obtained.

In conclusion, it is necessary to mention also the rupture of the mercury column. The capillary tube usually ends at the top in an enlargement, into which the mercury enters if the thermometer is heated above its limiting temperature; without this enlargement it would burst. During work it is necessary to ensure that no mercury remains in this enlargement, since otherwise the thermometer readings will be lower than the actual temperature. If the mercury is not entirely pure, then part of the column may break off as a result of a shock, for example during shipment, even without heating. To join the column, one may lightly tap the lower end of the thermometer against a table (wood)

or, holding it in the right hand, strike it strongly and sharply against the left, or make use of centrifugal force: holding the thermometer by its upper end, swing the hand rapidly.

In instruments up to \(250^\circ\) C., in which there is an airless space above the mercury, a broken column can be joined by turning the thermometer bulb upward and tapping it lightly, so that all the mercury flows into the enlargement at the end of the tube. If the thermometer is then placed bulb downward, the mercury will run down and the break will disappear. Sometimes this operation has to be repeated.

In instruments filled at the top with gas under pressure, a break in the column occurs more rarely, almost exclusively as a result of heating above the limiting temperature or of excessively rapid cooling. In these instruments one must, by carefully heating the bulb with the mercury in a smoky flame, raise the level of the mercury to the upper enlargement and then allow it to cool slowly. The operation must be carried out very cautiously, so that the mercury by no means fills the entire upper enlargement, otherwise the instrument will burst.

If it is possible from the size of the separated part of the column, it is better to proceed in the opposite way: cool the thermometer enough that all the mercury collects in the bulb, of course with the aid of tapping or shaking.

In thermometers for very high temperatures, above \(500^\circ\), the upper enlargement with gas has a drawn-out tip \(a\), Fig. 25, which, after being filled with gas, is sealed with shellac and only then sealed off. So that, during heating, when the shellac can melt, it cannot enter the capillary tube, the enlargement at its end has a partition with an opening \(c\) in the form of a small branch tube. At an excessively high temperature the mercury, having filled the cavity \(b\), will pour out through the opening \(c\); it is then no longer possible to introduce it back into the capillary tube, and the instrument is ruined.

Fig. 25.

11. Maximum and minimum thermometers.—Although these instruments are used chiefly in meteorology, they may also be useful in technology. Usually these are glass thermometers, filled with liquid (mercury and alcohol), with fairly narrow temperature limits—from \(-40\) to \(+50^\circ\) C.

Maximum thermometers are usually filled with mercury. In Rutherford’s thermometer, with a horizontal capillary tube, in front of (above) the mercury there is a steel cylinder, which is not wetted by mercury and, when the column shortens as the temperature falls, remains

in the position occupied by it; conversely, the cylinder is drawn away by a magnet. The drawback of this instrument is that the cylinder sometimes sticks to the mercury, and then the instrument begins to act like an ordinary thermometer.

In other maximum thermometers the mercury column is simply broken; a small piece of it is separated from the rest by an air bubble. When the temperature falls, this detached piece continues to indicate the highest temperature reached before that time by the instrument. After the reading, the detached piece is brought closer to the rest by light tapping—in wall thermometers with a thick capillary—or by shaking—in medical thermometers. The drawback of these instruments is that the air bubble sometimes slips upward, and the column joins together.

In Negretti and Zambra instruments the capillary in the lower part is constricted by a narrowing; as a result, when the temperature falls the entire column breaks away from the main mass of mercury and remains in the capillary. Since, when the temperature falls, the column itself also shortens, a correction analogous to equation (42) must be added to the temperature read,

\[ \Delta t=(t_1-t_2)n.\alpha, \tag{43} \]

where \(t_1\) is the reading of the instrument in \({}^{\circ}\mathrm{C}\), \(t_2\) is the temperature in the room at the moment of observation, \(n\) is the total length of the separated column in \({}^{\circ}\mathrm{C}\), and \(\alpha\) is the same coefficient of expansion as in equation (42).

Minimum thermometers are filled with colored alcohol or toluene; at the end of the alcohol column in the capillary there lies a thin glass rod with thickenings at its ends; the thermometer is hung in a horizontal position. When the temperature rises, the alcohol passes by the glass rod, and when it falls, as a result of wetting it with alcohol, carries it along. The rod is moved back by light tapping.

Fig. 26.

Usually maximum and minimum thermometers (for observing the temperature in a room or for meteorological work) are mounted on a common board.

For the same purposes Six’s instrument has also been constructed, Fig. 26: bulb \(A\) is filled with alcohol, the expansion of which serves as the temperature measurer; in bulb \(B\) there is a small amount of ether, the elasticity of whose vapors constantly presses on the alcohol a column of mercury \(c\), which separates the alcohol from the ether. The purpose of this ...

of the column, to move the steel cylinders \(x\) and \(y\) located above it; when the temperature rises it stops, allowing the expanding alcohol to pass by and showing the lowest temperature, cylinder \(y\); when it falls, \(x\) stops. After the reading, \(x\) and \(y\) are again brought up to the mercury by means of a small magnet.

12. Pyrometers with liquids.—Instruments for measuring higher temperatures, approximately above \(300^\circ\)—\(400^\circ\) C., are usually called pyrometers. This name is somewhat misleading, since, with the introduction of mercury thermometers measuring up to \(550^\circ\)—\(700^\circ\) C., a strict distinction between a thermometer and a pyrometer cannot be drawn.

A mercury pyrometer, or thalpotasimeter, is based on the fact that, when a liquid boils, there exists a quite definite relation between the pressure and the temperature of its vapors. The iron vessel in which the mercury is located is connected by an iron capillary tube to a manometer, which is empirically graduated in \({}^\circ\)C. In this arrangement the connecting capillary tube and the spring tube of the manometer are filled with liquid mercury, which condenses there as in a colder space. The quantity of mercury must be such that at all temperatures there still remains liquid mercury in the vessel, and its vapors are therefore saturated.

The thalpotasimeter is theoretically inexact, since it depends on atmospheric pressure and, at different barometer heights, will give different readings: the measured temperature produces in the instrument a certain definite absolute pressure, whereas the manometer indicates not the absolute pressure, but only the excess over atmospheric pressure. In more exact experiments one must introduce a correction for the height of the barometer, determining it experimentally.

Such an instrument, manufactured by the firm Schäffer and Budenberg, is applicable for temperatures from \(300^\circ\) to \(750^\circ\) C. (the red heat of iron). A thalpotasimeter with ether instead of mercury is applicable for temperatures from \(50^\circ\) to \(180^\circ\) C.

If, in an instrument of the thalpotasimeter type, the entire space in the cold state is filled with mercury—the vessel, the connecting tube, and the spring of the manometer—then we obtain a mercury pyrometer based on the increase of pressure of liquid mercury owing to the impossibility of its expanding freely when the temperature rises. The scale of such an instrument is also divided empirically in \({}^\circ\)C. The connecting tube, having an external diameter of about 6 mm, can be made up to 50 m long, and the instrument can be used for readings at a distance. However, the temperature of the mercury in the connecting tube

affect the readings of the instrument; introducing a correction in this case is more difficult than with a glass thermometer; therefore efforts are made to keep the amount of mercury in the connecting tube and the manometer as small as possible; moreover, when there are fluctuations of temperature in the rooms through which the tube passes, the latter must be well insulated.

Fig. 27.

In Fig. 27 a similar instrument is shown for measuring the temperature of superheated steam, with the tube coiled for shipment. In those cases where the steel vessel containing mercury is exposed to the action of acids, it is covered with a lead sheath. Such instruments are suitable for temperatures from \(-20^\circ\) to \(+350^\circ\); if, however, the upper part of the manometric spring is filled with nitrogen under pressure, then from \(+50^\circ\) to \(+600^\circ\) C.

Instead of the ordinary Bourdon manometer, some factories give the spring a special form and enclose it in the cylindrical stem of the instrument. Such, for example, is the Zabel pyrometer, Figs. 28 and 29: when heated, the tube, bent in the form of a wave-like ribbon and entirely filled with mercury, under the pressure of the latter straightens somewhat; this motion is transmitted by means of a toothed segment to the pointer; owing to the peculiar form of the spring tube, the motion is strictly linear, and the divisions of the scale are therefore uniform. Such an instrument is suitable up to \(150^\circ\) C. For higher temperatures the spring is placed in the casing of the indicating device, and a special steel vessel is subjected to heating, connected with the spring by a thin steel tube. With nitrogen (15 atm.) above the mercury, such an instrument is built up to \(500^\circ\) C.

Figs. 28 and 29.

Steinle and Hartung pyrometers, Figs. 30 and 31, are based on the same principle and differ only in the construction of the elastic tube, which unwinds somewhat as the temperature rises.

The Dreyer, Rosenkranz and Droop works manufactures instruments with the same spiral tube for measurements from \(-20^\circ\) to \(+250^\circ\) C.; the length of the rod \(l\) is made from 70 to 3000 mm.

The firm Wegener and Mah uses a spring twisted from a steel flat strip, Fig. 32; the instrument has a scale from \(-20^\circ\) to \(+500^\circ\) C.; like all the preceding ones, with empirically marked divisions, it must from time to time be checked against an accurate, standard instrument and a table of corrections compiled. Some of

these instruments have a device that allows the pointer to be shifted somewhat in order to bring the readings into closer agreement with the true temperatures. For example, the instrument in Fig. 28 is adjusted by turning the small wheel \(a\); in the pyrometer in Fig. 32, for this purpose one must turn the button \(b\) in one direction or the other.

V. L. Maleev.

The instruments described are often furnished with a second—maximum—pointer, pushed along by the main one when the temperature rises and remaining in place when it falls.

Drawing 33.

Thermometers and pyrometers with pointers are often made self-recording, Drawing 33: a lever, actuated by the same mechanism as the pointer, rising or falling depending on the temperature, draws, with a pen attached to its end, a curve on a sheet of paper placed on a drum; the drum is rotated by a clockwork mechanism and makes 1 revolution in 12 or 24 hours. Such self-recording instruments are used for monitoring industrial installations; they are not suitable for precise experiments.

In general, the accuracy of instruments with a pointer is not great; the possible error, even with frequent checking, is \(\pm 10^\circ\); in the best case, with an especially carefully made instrument, \(\pm 5^\circ\), and with less proper handling it is much greater.

Expansion of Solids

13. Metallic and graphite pyrometers.—These instruments are based on the difference in the coefficients of expansion of different solids: for example, a plate composed of two strips of different metals soldered together bends when heated, the metal having the larger coefficient of expansion being on the convex side. By making a transmission of this motion to the pointer of a dial and applying graduations empirically, one can read the temperature to which such a plate has been heated. Such instruments are fairly widespread in meteorological observations, since they are very easy to make self-recording.

For higher temperatures, such metallic instruments have not been constructed.

Measurement of Temperature

Only in the very most recent times has a similar instrument appeared for measuring temperatures from \(500^\circ\) to \(1700^\circ\); this is Féry’s spiral pyrometer, Figs. 34 and 35. This instrument consists of two narrow strips of metal, only 2 mm wide, having different coefficients of expansion, soldered together and wound into a spiral in order to increase the deformations when the temperature changes; this spiral is shown enlarged (\(3/1\) natural size) in Fig. 35.

Fig. 34.

Fig. 35.

The rays from the incandescent body whose temperature is to be determined first fall upon the concave mirror \(C\), and from it, through a small opening in the mirror \(M\), onto the spiral \(S\); behind \(S\) there is a second concave mirror \(R\), which again reflects onto \(S\) the rays that passed the first time by it. The entire spiral \(S\) is blackened so that it may absorb the heat rays as well as possible. Under the influence of radiant heat the spiral \(S\), fixed at the center, unwinds, whereupon the aluminum pointer \(P\), attached to its free end, directly indicates the corresponding temperature on the dial \(D\).

In making observations, one must first set the instrument in focus, which is done with the aid of a toothed rack with pinion \(F\); the observer looks through the eyepiece \(E\) at the plane mirror \(M\), in which the body whose temperature is being determined is reflected; until the adjustment has been achieved, two halves of the image of the incandescent body, shifted relative to one another, are visible in \(M\); when these halves coincide, the instrument is correctly adjusted.

In making readings, the instrument is first set in focus, then the front opening of the instrument is closed and, by means of a special device, the pointer is set to zero; then the objective is opened and the motion of the pointer is watched, which at first deflects very rapidly and then more and more slowly. After reaching a certain division, which gives the desired temperature,

the pointer will gradually begin to move backward, owing to the fact that the whole instrument gradually heats up. For the next reading it is necessary again to set the pointer to zero. It takes about half a minute to make one reading.

This instrument is made with three different scales of divisions: from \(500^\circ\) to \(1100^\circ\), \(500^\circ\)—\(1400^\circ\), and \(500^\circ\)—\(1700^\circ\) C.

The accuracy of its readings is about \(\pm 1\%\). There are, however, cases in which it gives insufficiently reliable readings: for example, when cold air is blown into a furnace, or, for example, when it is directed at a shiny surface that radiates heat poorly, such as the surface of molten metal; in this case it indicates less than the actual value; for example, for the temperature of molten copper it indicates degrees by 100 below the true value. For such measurements it is necessary to prepare in advance a table of corrections, comparing its readings with those of some other accurate instrument.

The scale of the instrument described is, of course, obtained empirically, and therefore it must be checked from time to time. In general the instrument is very convenient and, if handled properly, sufficiently reliable.

In outward appearance and in the manner of handling, this instrument resembles the optical pyrometers described below.

Among instruments based on the expansion of solid bodies, mention should also be made of graphite pyrometers, which are subjected to the direct action of the temperature being investigated.

Of all solid bodies, graphite has one of the smallest coefficients of expansion; therefore it is convenient to take one plate of graphite and the other of metal (iron). If they are joined only at one (lower) end, then, upon heating, the difference in elongation of the free ends, transmitted to a pointer, will serve as a measure of the temperature attained.

Unfortunately, such pyrometers very soon become untrue, especially when heated above \(600^\circ\) C, since the plates acquire a residual elongation. Even with frequent checking, the accuracy of their readings is no more than \(\pm 20^\circ\) C, and with less careful handling the possible error is still greater.

Their range of application is from \(+350^\circ\) to \(+1000^\circ\) C; however, they may be subjected to temperatures above \(700^\circ\)—\(800^\circ\) only for a short time.

At one time graphite pyrometers, manufactured even now by very many factories alongside other instruments, gained in practice a rather wide, but wholly undeserved, distribution.

elimination. Their disadvantages are: inaccuracy, the need for frequent checking, very slow following of temperature fluctuations owing to the large mass of the measuring part, easy damage from impact, and, finally, bulky dimensions; at the same time they have no particular advantages, and therefore we shall not dwell on them in any greater detail.

Instruments Based on Heat Transfer

14. Calorimetric methods and instruments.—A convenient method of measuring temperature consists in placing a metal body of known mass in the space whose temperature is to be measured; after a certain time, when the body has assumed the desired temperature, it is removed and immersed as quickly as possible in a vessel containing an exactly known quantity of water. Knowing the heat capacity \(c\) of the body, its weight \(g\), the weight of the water in the vessel \(w\), including also the water equivalent of the vessel, and having the temperature of the water before immersion \(t_1\) and the final temperature of the body and water \(t_2\), for determining the temperature \(t\) of the body before immersion one has the relation

\[ cg(t-t_2)=w(t_2-t_1), \]

whence

\[ t=\frac{w}{cg}(t_2-t_1)+t_2 . \tag{44} \]

Fischer’s calorimeter, Figs. 36 and 37, p. 56, based on this idea, consists of a vessel \(A\) made of thin sheet copper, 50 mm in diameter and about 125 mm high; it is suspended in a wooden box \(m\); the space between \(A\) and \(m\) is filled with fibrous asbestos or glass wool \(B\). At the top, \(A\) is closed with a copper lid with two openings: the larger, about 30 mm in diameter, with a funnel \(v\) for the stirrer \(r\) and for lowering metal cylinders, the smaller—for the thermometer \(t\). The thermometer is taken with a very small mercury bulb and with a scale \(0^\circ\)–\(50^\circ\), subdivided into \(0.1^\circ\); a pocket \(s\), made of the same perforated copper sheet, protects the thermometer from blows by the stirrer; the latter consists of a copper disk soldered to a thick copper wire; at the top it has a handle, wooden or bone.

Nickel cylinders \(e\) (iron ones are sometimes used, but they require frequent cleaning and too quickly lose weight), weighing 20 g, about 12 mm in diameter and 20 mm high, are introduced into the space under investigation in an iron vessel \(a\), Figs. 38 and 39, connected with the handle \(f\) by a long rod \(b\), from 500 to 1000 mm long.

The weight of the water, together with the water equivalent of vessel \(A\), stirrer \(r\), and thermometer, should be about 250 g.

When the heated cylinder is lowered, it falls onto the stirrer \(r\), which is moved up and down; in this way the heat is communicated so rapidly and uniformly to the whole mass of water that the thermometer reaches the final temperature after 60–70 sec. Corrections for radiation and evaporation may be omitted, since the insulation of the apparatus is so good that, for example, when the temperature of the surrounding air is \(18^\circ\), the water in the apparatus, heated to \(25^\circ\), cools in an hour by only \(2\)–\(2.5^\circ\). Before the cylinder \(e\) is lowered, the water should have approximately the temperature of the outside air, or better still \(1\)–\(2^\circ\) lower.

Fig. 36 and 37.

Instead of copper, in good instruments the beaker and stirrer are often made of silver, which is less affected by water and air.

Fig. 38 and 39.

In the work, the sought temperature \(t\) is found from the equation (44) given above. If \(w\) is exactly 250 g, and the nickel cylinder \(g = 20\) g, then the following auxiliary Table 6 may be used.

Table 6.

\(t_2 - t_1\)	\(^{\circ}\mathrm{C}\).	4	5	6	7	8	9	10	11	12
\(t\)	”	391	486	581	673	763	852	939	1026	1110
for \(0.1\,(t_2 - t_1)\)	”	9	9	9	9	9	9	9	8

Example: before insertion of the cylinder the thermometer indicated \(16.1^\circ\); afterward it settled at \(24.5^\circ\); the rise of temperature is \(t_2-t_1=8.4^\circ\); we have

\[ \begin{aligned} 8^\circ &\quad \text{corresponds to} \quad \ldots\ldots\ldots\ldots 763^\circ,\\ 0.4 &\quad \text{”} \quad 9\times 0.4=\ldots\ldots\ldots 3.6,\\ &\quad \text{final temperature } t_2=\ldots\ldots 24.5,\\ &\quad \text{whence } t=\ldots\ldots 791.1^\circ \text{ C.} \end{aligned} \]

If a platinum cylinder is used, then temperatures up to \(1600^\circ\) can be determined. If \(w=250\), and \(g=20\) g, then in this case, to save time, one may use Table 7.

Table 7.

\(t_2-t_1\ ^\circ\text{C.}\)	1	2	3	4	5
\(t\) ”	369	696	995	1272	1530
for \(0.1(t_2-t_1)\) ”	33	30	28	26

For more accurate experiments one should take a vessel half as small, so that \(w\) is equal to 120 g; for this case Table 8 should be used.

Table 8

\(t_2-t_1\ ^\circ\text{C.}\)	2	3	4	5	6	7	8	9	10
\(t\) ”	368	519	670	820	963	1096	1230	1355	1473
for \(0.1(t_2-t_1)\) ”	17	15	15	14	14	13	13	12

In addition to being invariable, platinum cylinders have the advantage that, when they are immersed, the water does not boil owing to the very small heat capacity of platinum, and therefore no loss by evaporation occurs.

For better heat transfer the cylinders are often made not solid, but hollow and drilled through.

The calorimeters of Weinhold, Godlee, and others differ from the one described only in details of construction; their use is completely the same.

Ficks’s apparatus, Fig. 40, p. 58, differs from the preceding one by devices that allow the entire determination to be carried out very quickly and the report to be obtained at once without any calculations. Vessel \(d\), made of polished [[unclear: continuation cut off]] on both sides in order to reduce radiant emission, is lower—

the left tin is separated from the second similar vessel by an air layer. Everything is placed in a wooden box \(a\). The cylinders (of nickel or platinum) have two perpendicular drillings to increase the heat-transfer area. A very convenient device for carrying the cylinders and placing them in the space to be measured is shown in Fig. 41: to lower cylinder \(g\) into the calorimeter, it is enough to turn the vessel with the open end downward and, pulling the handle \(i\), draw out the latch.

Fig. 40.

Fig. 41.

In the experiment, water is poured into vessel \(d\), the quantity of which is once and for all determined in the case of a device with a mark; the temperature of the water must lie between \(16^\circ\) and \(17^\circ\) C. After stirring the water, the movable scale of the thermometer is set so that its \(0\) is at the height of the end of the mercury column. After introducing the heated cylinder into the calorimeter, the highest position of the mercury column will indicate on the scale directly the sought temperature \(t\) in \({}^\circ\) C.

It is convenient to work with this instrument, and much faster than with the preceding one, but the accuracy of its readings, owing to the empirical graduation of the scale and, moreover, under conditions that cannot always be observed in practice, is of course less.

Siemens’s pyrocalorimeter, Figs. 42 and 43, also has an empirically graduated scale, but gives results that are somewhat more exact, since the weight of the cylinders and of the water in it is greater; in addition, it permits corrections to be introduced for the temperature of the water and for changes in the weight of the cylinders. After pouring water into vessel \(a\) (568 cm.\(^3\)), the pointer of the movable scale \(c\) is set at the height of the mercury in thermometer \(b\). When, after the introduction of the heated cylinder, the temperature of the water reaches its greatest value, the required temperature \(t\) is found as the sum of the readings on the main scale of thermometer \(b\) and on the movable scale \(c\). Corrections for heat loss need not be made, in view of the good insulation of the instrument.

Cylinders of copper or iron can be used to measure temperatures up to \(1000^\circ\). A standard iron cylinder should weigh 112 g, a copper one 137 g, and a platinum one 402.6 g. With frequent use the first two lose weight, and then the readings taken on the movable scale must be multiplied by the ratio of the standard weight to the actual weight.

Example: a copper cylinder came to weigh 129 g; in the reading, \(36^\circ\) was read on the thermometer and \(870^\circ\) on the movable scale; the true temperature is

\[ t = 36 + 870 \frac{137}{129} = 960^\circ \text{C}. \]

Figs. 42 and 43.

Let us also mention the method of Sentinon, or of the Sèvres Manufactory: a double metallic tube (explorer) is inserted into the space under investigation; through the inner tube water flows in, and through the outer tube it flows out; the water is passed at a constant velocity sufficient to avoid the formation of steam. By measuring the temperatures of the inflowing and outgoing water, the temperature in the space under investigation is calculated from the given rate of flow. This method cannot give reliable results, since thermal conductivity.

the wall of the tube changes greatly, decreasing as a result of the soot and ash deposited on it.

15. Determination of the water equivalent.—In order for a water pyrometer (calorimeter) to give good results, all the data introduced into the calculations must be as accurate as possible.

The water equivalent of the apparatus, i.e. of the calorimetric vessel, stirrer, and thermometer, is determined for a Fischer calorimeter and similar ones in the following manner: the vessel and stirrer are made either of brass, whose heat capacity at room temperature is \(c_l = 0.093\), or of silver, whose heat capacity is \(c_c = 0.056\); if the vessel and stirrer together weigh \(m\) g, then their water equivalent is \(w_1 = m \cdot c\). The water equivalent \(w_2\) of the thermometer is best determined by direct experiment; for this the thermometer is heated by immersing it in a mercury bath heated to \(30^\circ\)—\(40^\circ\), and then it is quickly lowered into a small vessel containing an accurately weighed quantity \(p\) g of water. If the temperature of the thermometer before immersion, as indicated by the thermometer itself, is \(t_1\), the temperature of the water before immersion of the thermometer is \(t_2\), and after immersion is \(t_3\), then

\[ w_2 = p \frac{t_3 - t_2}{t_1 - t_3}, \tag{45} \]

and the total equivalent of the whole apparatus is \(w = w_1 + w_2\).

It is still more convenient to find \(w\) by the following experiment: into the vessel with the thermometer and stirrer inserted, pour a normal quantity of water heated to approximately \(30^\circ\)C; when the temperature \(t_1\) of the whole system has become established, quickly pour out this water, and in its place pour an accurately weighed quantity \(p\) of cold water at a temperature \(t_2\) of approximately \(15^\circ\)C, and observe the temperature \(t_3\) which is established this time. From the equality of the heats before and after mixing,

\[ w t_1 + p t_2 = (p + w)t_3, \]

we obtain

\[ w = p \frac{t_3 - t_2}{t_1 - t_3}, \tag{46} \]

an expression identical with equation (45).

16. Variability of heat capacities.—For the use of calorimeters and equation (44) it is also necessary to know the heat capacity \(c\) of the working cylinder. As is known, the heat capacity of metals is not constant, but varies,

increasing for most substances with increasing temperature. Unfortunately, up to now we do not have fully reliable values for heat capacities at various temperatures, despite the large number of experiments and the extensive literature on this question. The values found by different authors differ from one another by as much as 8% (platinum), which, when measuring high temperatures, e.g. \(1500^\circ\) C., is equivalent to a possible error of \(120^\circ\) C.!..

Nevertheless, let us present the most reliable data available for the metals used in calorimetry.

To determine the mean heat capacity of iron for temperatures from \(0^\circ\) to \(660^\circ\), Pionchon gives the following expression

\[ c_{\mathrm{ж}} = 0.1102 + 0.000025\,t + 0.0000000547\,t^2; \tag{47} \]

these data are confirmed fairly well by Oberhoffer’s recent experiments. It must be borne in mind here that the heat capacity of iron increases strongly with carbon content. Thus, according to Oberhoffer’s experiments, the mean heat capacities from \(0^\circ\) to \(650^\circ\) C. are: for pure iron \(c_{\mathrm{ж}} = 0.1432\), with a content of 1% carbon \(c = 0.1454\), and with 2% already \(c = 0.1476\).

The heat capacity of platinum is the most constant, which is especially valuable in the present case; Violle gives for the mean heat capacity of platinum for temperatures from \(0^\circ\) to \(1200^\circ\) C. the following expression

\[ c_{\mathrm{п}} = 0.0317 + 0.000006\,t; \tag{48} \]

In Table 9, p. 62, are given the mean heat capacities calculated from these expressions, as well as those established by direct experiments.

For greater reliability, instead of using Tables 5–9 one may determine the heat capacities experimentally, using the very calorimeter with which it is intended to work: the given cylinder is heated to some definite temperature \(t\), for example in steam at atmospheric pressure, and for higher temperatures in a furnace, gas or electric; here the actual temperature must be measured with a control instrument. The cylinder heated in this way is lowered just as quickly as in the actual work into the calorimeter, containing, including its equivalent, \(w\) kg of water, usually at \(15^\circ\) C. When, after thorough stirring, the temperature of the water ceases to rise, \(c\) is calculated from eq. (44), in which all the other quantities are now known,

\[ c = \frac{w(t_2 - t_1)}{g(t - t_2)}. \tag{49} \]

Table 9.

temperature range, °C	platinum, Violle	nickel	iron, Pionchon	copper	silver
0—100	0,0323	0,113	0,113	0,0933	0,056
0—200	329	115	117	944	—
0—300	335	120	123	955	—
0—400	341	127	129	966	—
0—500	347	128	137	977	—
0—600	353	129	145	—	—
0—700	359	130	157	—	—
0—800	365	131	169	—	—
0—900	371	133	175	—	—
0—1000	377	136	180	—	—
0—1100	383	—	—	—	—
0—1200	389	—	—	—	—
0—1300	(395)	—	—	—	—
0—1400	(401)	—	—	—	—

To simplify further calculations, one may determine not \(c\), which requires the determination of the weight of the body \(g\) on especially accurate balances, but the heat capacity of the given body \(C = cg\),

\[ C = \frac{w(t_2 - t_1)}{t - t_2}. \]

Having found \(c\) by the indicated method for 3 temperatures—the lowest, the middle, and the highest in the forthcoming work—one can, by interpolation, compile a table like Tables 5–8 indicated above. Such a direct determination of the heat capacity has at the same time the character of a calibration of the entire apparatus; moreover, all corrections for heat emission and cooling during the transfer of the heated body are introduced by themselves, i.e., as it were, a correction for the very handling of the apparatus by the given observer.

In conclusion it must be said that, despite everything, Fischer’s apparatus, along with gas thermometers, is one of the most reliab-

suitable for determining high temperatures. Its chief shortcomings are: the impossibility of making readings more often than every 10 or even 15 minutes, the dependence of the results on the dexterity and skill of the observer, and the comparatively limited field of its application; namely, with it one cannot measure the temperature of solid or liquid bodies, but only of spaces filled with a flame or with some other gas of high temperature.

17. Pyrometers based on melting points.—The idea of this method consists in exposing to the action of the heat source under investigation a series of bodies with different but previously known melting temperatures (melting points). The bodies are chosen so that their melting points differ from one another by a comparatively small number of degrees C., for example by \(20^\circ\)—\(30^\circ\). After these bodies have assumed the temperature under investigation, they are removed; some of the bodies will prove to have melted, the rest not. The temperature sought evidently lies between the melting temperature \(t_1\) of that one of the melted bodies whose melting point is higher than the others, and the temperature \(t_2\) of the most fusible body among those that have not melted. If one takes as the temperature sought

\[ t=\frac{t_1+t_2}{2}, \]

then the accuracy of the measurement is evidently equal to \(\pm \dfrac{t_2-t_1}{2}\).

As bodies with known melting points one may take, for example, metals and their alloys: cadmium \(315^\circ\), lead \(327^\circ\), zinc \(418^\circ\), silver \(955^\circ\), gold \(1064^\circ\), platinum \(1780^\circ\); alloys of silver with gold for temperatures intermediate between \(955^\circ\) and \(1064^\circ\), and of gold and platinum between \(1064^\circ\) and \(1780^\circ\). The addition of \(5\%\) platinum to gold raises the melting temperature by approximately \(30^\circ\)—\(35^\circ\); however, Zeger found that alloys of gold with a platinum content greater than \(15\%\) do not possess sufficient constancy of melting temperature, owing to the segregation that occurs. The alloys are taken in the form of small pieces of ribbon, weighing \(0.1\)—\(0.2\) g. Sets of a consecutive series of alloys are supplied ready-made by many firms. The principal disadvantages of this method are the small ranges of temperatures for which it is applicable, its low accuracy, and the expense when noble metals are used.

Zeger’s pyramids are far more widely used; they are made from a mixture of silica, alumina, lime, and potash; the more silica and alumina there are, the higher the melting point;

the lowering of the melting point is achieved, moreover, by adding to it a small amount of iron oxide. The dimensions of the little pyramids are: height 60 mm, side of the triangular base about 19 mm; for higher temperatures the pyramids are made of smaller dimensions—30 × 9 mm. On each pyramid, during its molding, a number is impressed, indicating its melting temperature. A pyramid is considered melted when it has softened so much that its top bends to the side and touches, or almost touches, the surface on which it stands. In Fig. 44 an example is shown of determining temperature by Seger cones: No. 6 has melted completely, No. 7 has just reached its melting temperature, \(1270^\circ\) C., No. 8 almost, while No. 9 is still quite untouched; the temperature was, consequently, about \(1270^\circ\). If in this case not a single pyramid had melted, it would have been necessary to set more fusible ones, Nos. 3–5.

Fig. 44.

These pyramids are at present manufactured in 59 numbers:

Nos.	For temperatures
022–010	from \(590^\circ\) to \(950^\circ\), in steps of \(30^\circ\),
09–01 and 1–20	from \(970^\circ\) to \(1530^\circ\), in steps of \(20^\circ\),
26–36	from \(1650^\circ\) to \(1850^\circ\), in steps of \(20^\circ\),
37–42	from \(1880^\circ\) to \(2030^\circ\), in steps of \(30^\circ\),

Nos. 21–25 (\(1550^\circ\)–\(1630^\circ\)) are no longer manufactured, since their melting points lay too close to one another.

In the initial determination of temperature with such pyramids, 10–12 pieces with consecutive melting points are placed, chosen above and below the presumed temperature of the furnace being investigated. So that the pyramids do not fall when they begin to bend, they are lightly attached with clay to a fireclay board and, moreover, somewhat inclined so that, when bending, the number will be on top.

When working with these pyramids, care must be taken that they are not subjected to the direct action of the flame; for this purpose they are surrounded with refractory bricks or placed on special stands made of refractory material with protective side and upper shields, Figs. 45–47.

Temperature Measurement

The accuracy of the indications of Seger cones may be judged from the detailed experiments carried out in 1907 by K. Lezer. Lezer observed not the temperatures at which these cones melt, but, in sequence, the melting of successive cone numbers. The results of his observations do not give absolute indications, but they do show very clearly the suitability of these cones for practical purposes, when it is important not so much to know the actual temperature in the kiln as to be able to ensure that a certain definite temperature, required in the given case, has been established.

Drawings 45–47.

Lezer proceeded in the following way: he placed in a special gas kiln a series of 14 consecutive numbers of cones, covering a temperature interval of 260°; he then gradually heated the kiln and, through special little windows, observed the successive changes that occurred in the cones as the temperature rose. The results of the observations he plotted as graphs, samples of which are given below. In these graphs, in the horizontal direction, along the abscissa axis, were laid off the minutes elapsed from the moment of the first change in the most fusible cone; in the vertical direction, at equal distances from one another, five lines, 1–5, were drawn, corresponding to the five stages of change in state that were recorded. Schematically these stages are shown in drawing 48.

Drawing 48.

Stage 1—the beginning of melting, the appearance of a slight bending of the tip; it is rather difficult to catch this moment, especially if, in doing so, the cone does not bend but only settles; therefore the mark for this stage on the graphs must be considered only approximate. Stage 2 may be considered the first fully characteristic one. Stages 3 and 4 are also easy to mark accurately, whereas the mark for stage 5, the end of melting, again

is associated with uncertainty. Thus, for the evaluation of the phenomena, only degrees 2–4 have real significance. For a complete characterization of the phenomena, the corresponding gas consumption per minute for heating the furnace should also be indicated on the graphs.

From the examples of graphs taken from Le Chatelier’s report, it is evident that the melting of the pyramids proceeds far from uniformly; for example, in Fig. 49, No. 15 outstripped the more fusible Nos. 17 and 16, No. 13 outstripped No. 14; in Fig. 50 Nos. 6 and 12 disturb the order, the latter falling into its proper place only at degree 5. The situation is somewhat better in Fig. 51; but, on the other hand, since

gas consumption 43 liters in 66 min. — 76 liters/min.

Fig. 50.

gas consumption 78 liters in 60 min.

Fig. 51.

90 liters — 95 liters — gas consumption 120 liters/min.

Fig. 52.

95 liters/min. — 105 — gas consumption 110 liters/min.

experiments, presented in Figs. 51 and 52, were carried out under almost identical conditions, then a comparison of these graphs shows how different the results obtained with individual specimens of one and the same No. of cones can be.

From Lezer’s experiments it is evident that the accuracy of the readings of the cones by no means corresponds to \(\pm 10^\circ\), as one might think from the tables supplied by their manufacturers. It would be more correct to regard the accuracy of the readings as only \(\pm 20^\circ\), and in some cases, for example Nos. 15, 13, 12, 6, into the manufacture of which, apparently, some error has crept in, even only \(\pm 40^\circ\). On the other hand, what is especially important for practice, individual specimens of one and the same No. in general give sufficiently close readings.

The cones described, as well as the stands for them, are manufactured by Seger’s laboratory in Berlin. Such cones, but of a somewhat different composition, are also manufactured by the Royal Porcelain Factory in Berlin.

The firm Foma Firt and Son in Riga manufactures similar pyrometers according to Brerley’s method under the name sentinel (watchman, guard). These are small cylinders 20 mm high and 12 mm in diameter, made from salts of various metals or mixtures of them; they are placed in small porcelain cups. The melting points of these salts are very sharp; an increase in temperature of \(1—2^\circ\) C is sufficient for the transition from the solid immediately into the liquid state to occur. If the temperature falls below the melting point, then these salts solidify, while the molten shiny surface becomes matte. The cylinders—sentinels can be made for any melting point; they are rather widespread in metallurgical production, and also in the hardening of steel.

To observe the moment of melting of a cylinder in a closed, for example, hardening furnace, the sentinel is inserted into the instrument shown in Fig. 53: by means of the screw spring \(m\), the small cylinder \(s\) is pressed against the bottom of a casing made from an iron gas pipe. When the required temperature, corresponding at the same time to the melting of \(s\), has been reached, \(s\) melts—

Figs. 53 and 54.

the rod resting upon it falls and, pressing on the projection \(n\), closes the circuit of the electric bell connected to the terminals \(i\) and \(k\).

In Fig. 54 another design is shown, simpler but less convenient, in which, at the moment when the cylinder melts, the current, on the contrary, is broken.

In conclusion we shall also mention a similar instrument that makes it possible to follow, from outside a furnace or firebox, the moment at which alloys melt—a kind of pyrometric sentinel, manufactured by the firm Ducomet in Paris. This instrument, Figs. 55 and 56, consists of a metal, iron or nickel tube \(A\) with a similar rod \(B\); the rod is constantly pulled upward by a spiral spring \(E\); the upper end of \(B\) is connected, by means of a toothed sector, with a pointer which stands at 0 when the rod \(B\) is in its lower position; onto the lower, projecting end of \(B\) is placed a set of cylinders \(M, N \ldots\) made of alloys with previously known, gradually increasing melting temperatures; between the cylinders—the pyrometers—non-melting iron disks are inserted; a check \(G\) holds the disks on the rod; the latter is brought into the lower position by means of a release screw with handle \(H\), which is then removed.

Figs. 55 and 56.

If a pyrometer thus “charged” is inserted through an opening in the masonry into the hot space under investigation, then, as the cylinders melt owing to the temperature prevailing there, the rod \(B\), under the action of the spring \(E\), will be drawn inward and will move the pointer \(C\). From the number of divisions passed by the pointer \(C\), one can judge how many cylinders have melted, and, knowing the sequence of their melting points, determine the temperature of the space under investigation.

Comparing all that has been said, one may arrive at the conclusion that pyrometers based on the melting of bodies make it possible only to determine the temperature with an accuracy of up to \(20^\circ\)—\(40^\circ\), and not to measure it. Nevertheless, in view of the extraordinary simplicity of handling them, they can be very valuable in certain practical cases, when it is not important to know the actual temperature, but it is important to maintain a certain definite temperature which has proved, by experience, to be the most advantageous in the given production process.

18. Thermophones.

The same single readings, and with the same degree of accuracy, are given by the so-called thermophone, or explosive pyrometer, proposed by Wiborgh and based on thermal conductivity. These are flat little cylinders about 20 mm in diameter and 10 mm high, stamped from clay of a definite, comparatively low thermal conductivity; inside them is a small quantity of a substance that explodes when the temperature rises to a certain value. In essence this method bears some resemblance to Carpio’s method, p. 46, for determining high temperatures by the time of the rise of mercury in a glass thermometer.

In using a thermophone, it is subjected to the action of the heat source being investigated, and the time that elapses before the explosion is observed with a stopwatch; from the tables supplied by the firm the corresponding temperature is found. The time before explosion depends on whether the thermophone is subjected only to the action of hot gases, or is heated by heat transfer from a heated solid body, which is taken into account in the tables. Thus, for example, in the first case 57″ corresponds to 800° C., while in the second only to 620° C.

Thermophones are applicable from 300° C. (3′24″.6) to 2400° P. (29.6″). Before use, the thermophone must necessarily have a temperature not below 18° and not above 22° C.

Usually the thermophone is thrown directly into the space being investigated; but in a case where this space cannot be littered with the flying fragments, it may be introduced into an iron or nickel tube closed at one end.

Figure 57 shows an arrangement for using a thermophone in determining the temperature of hot air supplied to a blast furnace: a is an iron tube about 23 mm in diameter, with wall thickness about 2 mm, inserted somewhat obliquely into the air duct; approximately in the middle of the tube a movable bottom is held by the handle b; when the thermophone has been dropped into tube a, it is closed with lid c, which is attached to a chain so that, during the explosion, it cannot fly off.

Fig. 57.

The accuracy of the readings of thermophones, their range of application, and, most importantly, their advantages and disadvantages are approximately the same as those of Seger cones; however, their use presupposes the availability of a stopwatch, whereas throwing them in is much simpler than setting up and removing the cones.

Instruments Based on Optical Phenomena

19. Optical pyrometers.—Since thermal and optical changes have the same nature, namely, they consist in a change in the number and amplitude of the vibrations of the molecules of a body, and since only the number of light waves per unit of time is greater than that of most heat waves, determining temperature by optical phenomena is one of the most natural and, perhaps, one of the most reliable methods. Since the change in the light emitted by a body, or, what is essentially the same thing, in its color, becomes noticeable only beginning at \(400^\circ\)—\(500^\circ\) C., this method is applicable only to higher temperatures, chiefly to various furnaces and fireboxes.

All methods and instruments for the optical determination of temperatures may be divided into two groups: those based on a change in color or spectrum, the qualitative method, and those based on a change in luminous intensity, the quantitative or photometric method.

To the first group belong the determination of temperature by eye, the instruments of Mesuré-Nouel and Tempel; to the second group, various pyrometers-photometers.

When heating some solid body, for example a piece of iron, it is easy to notice that its color changes as the heating proceeds: starting from dark, it gradually acquires a cherry-red, then red, yellow, and finally white color.

An approximate estimate of temperature by color may be made according to the following Table 10, established by Pouillet. According to other experiments, the temperatures for the corresponding colors are somewhat lower, by approximately \(100^\circ\)—\(150^\circ\). Although determining temperature by eye cannot give any reliable numerical values, we shall also give comparative Table 11 of temperature scales adopted in various countries; this is because in metallurgical practice this method is still not infrequently used, the temperature of a furnace or of incandescent metal being determined by their color.

Taking into account that different persons understand the same color name differently, and, chiefly, that the color sensitivity of observers’ eyes varies greatly, and finally that the color of an incandescent body appears to the eye to have different brightness depending on the lighting of the room in which the observer is located, the desire to replace such an estimate by instruments, in which the observer’s subjectivity would have less influence, becomes understandable.

Table 10.

Color	°C
beginning of incandescence	525
very dark red	600
dark red	700
beginning of cherry-red	800
cherry-red	900
bright cherry-red	1000
dark orange	1100
light orange (yellow)	1200
white	1300
bright white-hot	1400
dazzling white-hot	1500—1600

Table 11.

French scale	°C	German	°C	Gove (Amer.)	°C	Taylor–White	°C
incipient red	515					blood-red	532
very dark red	600	brownish-red	550	[[unclear: cell obscured]]	625	dark red	565
dark red	700	dark red	650	dark red	650	dark cherry-red	635
dark cherry-red	800					medium cherry-red	676
cherry-red	900	cherry-red	750	cherry-red	700	cherry-red	746
light cherry-red	1000	light cherry-red	800	light red	860	light cherry-red	843
orange	1100	orange	900	yellow	950	orange	899
yellow	1200	yellow	1000	yellow	1000	light orange	940
		light yellow	1100	light yellow	1050	fawn	996
				very light yellow	1100	light fawn	1080
white welding heat	1300	white	1200	white	1150	white	1204
brilliant white	1400	brilliant white	1300
dazzling white	1500	dazzling white	1500

An improvement of this determination by eye is provided by the pyrometric tube of Mesuré and Nouel, Figs. 58 and 59. Rays of light from an incandescent body, after passing through the frosted glass \(G\), by which uniform illumination of the field of view is obtained, fall upon the polarizer \(P\). The polarizer—a Nicol prism—consists of a crystal of calcite, cut perpendicular to its optical axis; both halves are then cemented together again with Canada balsam.

Figs. 58 and 59.

The ray incident on the prism consists of transverse vibrations of the ether. If the ray is simple, i.e. consists of only one color, then the prism will make it vibrate only in the plane of the principal section, i.e. in the plane passing through the crystallographic axis of the rhombohedron of spar perpendicular to one of its faces. Having passed through the polarizer, the ray reaches a second Nicol prism—the analyzer \(A\). If the analyzer is set so that its principal plane coincides with that of the polarizer, then the ray will pass through it without hindrance; but if the analyzer is turned through \(90^\circ\), then its plane of polarization will stand perpendicular to the plane of the polarizer, the ray will not be transmitted, and the field of view will be darkened.

A piece of quartz \(Q\), cut perpendicular to its axis and placed between the polarizer and the analyzer, will rotate the plane of polarization through a definite angle, and the field of view will become bright. If the analyzer is turned through this angle in the same direction, the field of view will again become dark.

In reality, through the glass \(G\) there enters not a simple ray, but a composite one, consisting of a collection of different colors; therefore the analyzer will extinguish not all the light, but only some colored ray; the field of view will be colored, and when the analyzer is turned a whole series of colors is obtained.

The quartz plate \(Q\) rotates the plane of polarization for different colored rays through different angles; therefore, depending on the composition of the light, the analyzer must be turned through one angle or another in order to obtain the same coloration of the field of view. Since the composition, i.e. the component colors of the ray, changes with temperature, approaching white as it increases, the angle of rotation of the analyzer when obtaining the same coloration of the field of view depends on the temperature. As the normal coloration in this instrument—

the borax takes on a yellow color, of a dirty lemon hue. For practical use of the instrument, a table is compiled indicating directly the temperature of the ray under investigation from the angle of rotation read on the circle of divisions \(L\). Thus, for example, an angle of rotation of \(6^\circ\) corresponds to \(900^\circ\) to \(1000^\circ\) C.; an angle of \(5^\circ\)—\(1200^\circ\) to \(1300^\circ\) C.; and \(3^\circ\)—\(1400^\circ\) to \(1500^\circ\) C.

The instrument also has an objective and a converging lens, necessary for determining temperatures below \(900^\circ\), when the intensity of the light is insufficiently great.

Since the colors of different bodies, solid and gaseous, at high temperatures, though close, are not identical, for greater accuracy it is useful always to direct the tube at a definite body, for example iron; and in that case, if it is not present in the furnace, deliberately place a piece of iron in it for this purpose.

As can be seen, the instrument cannot give exact readings, but for many practical cases, in the hands of an experienced observer who distinguishes color shades well, it fully serves its purpose. Moreover, it is very simple, inexpensive, its readings do not change with time, and they require no corrections of any kind.

Another, equally simple instrument, the Tempel spectral pyrometer, Fig. 60, is based on the observation that the length of the spectrum of a ray emitted by an incandescent body and refracted by a prism increases with temperature. The instrument consists of a prism enclosed in a casing, to which are attached a tube \(a\) with a scale, a tube \(b\) with cross-hairs, and an observing (magnifying) tube \(c\); the latter can be turned for observing any part of the spectrum. Opposite tube \(a\) is a lamp for illuminating the scale. The whole instrument is fixed on an adjustable stand.

Fig. 60.

To determine the temperature, tube \(b\) is directed at the incandescent body and one looks into \(c\) to see to which division of the scale the blue end of the spectrum reaches. The divisions of the scale are marked empirically and give directly the required temperature in °C.

Since the spectra of different bodies differ somewhat from one another, it is necessary either always to direct the tube at the body with which the scale was graduated—fireclay, with which most furnaces are lined inside—or to find a correction in °C, determining some average temperature of another body simultaneously with this instrument and with another, accurate one. The readings of the instrument with such a correction, when determining the temperature of the same body, will be correct at temperatures below the checked one with an accuracy of ±20°, and above the checked one with an accuracy of ±50°C. The limits of measurement with this instrument are 700°–1350°C, and, for a color-sensitive eye, up to 1600°C. The magnitude of the error when the correction is not observed may be judged from the fact that white bodies, for example, are known to give the same length of spectrum at a temperature 100° lower than the temperature of fireclay, whereas dark bodies, for example an open-hearth furnace, do so at a temperature almost 100° higher than that of fireclay. Thus this instrument, both with respect to simplicity of construction and handling, and also with respect to accuracy, is close to the preceding one.

20. Photometric pyrometers.—Photometric instruments are based on Wien’s law, which indicates the dependence between the intensity of light \(J\), the wavelength \(\lambda\), and the absolute temperature \(T\),

\[ J = a\lambda^{-5} e^{-\frac{b}{\lambda T}}. \tag{50} \]

where \(a\) and \(b\) are two constants determined from experiment, and \(e\) is the base of natural logarithms. This law is valid for waves of small length (red).

If the intensity of light of another source is \(J_1\), and its temperature is \(T_1\), then

\[ J_1 = a\lambda^{-5} e^{-\frac{b}{\lambda T_1}}, \]

whence

\[ \frac{J}{J_1} = e^{-\frac{b}{\lambda}\left(\frac{1}{T}-\frac{1}{T_1}\right)} = e^{\frac{b}{\lambda}\left(\frac{1}{T_1}-\frac{1}{T}\right)}, \tag{51} \]

the sought temperature \(T\) is found from the known \(b\), \(\lambda\), and \(T_1\), if \(\frac{J}{J_1}\) is found.

More accurate observations revealed deviations, especially with an increase of the product \(\lambda T\), of actual phenomena from equation (50). Planck proposed replacing it by another, more closely corresponding to observations, namely

\[ J = a\lambda^{-5}\left[e^{\frac{b}{\lambda T}} - 1\right]^{-1}, \tag{52} \]

whence the relation is obtained

\[ \frac{J}{J_1}= \frac{e^{\frac{b}{\lambda T_1}}-1} {e^{\frac{b}{\lambda T}}-1}. \tag{53} \]

The indicated dependence between \(J\), \(\lambda\), and \(T\) was confirmed by careful experiments carried out by Lummer and Pringsheim at the Imperial Physico-Technical Institute.

Equations (50) and (52) are valid only for rays emitted by absolutely black bodies. An absolutely black body is the name given to a body that absorbs all rays incident upon it. However, practically all solid and liquid bodies, especially at higher temperatures, may be regarded as absolutely black.

The light source used for comparison may also be a non-black body, but then one must determine in advance the temperature \(T_1\) that an absolutely black body would have at the same luminous intensity and the same wavelength.

To measure temperature by means of Wien’s law, it is necessary to isolate a simple ray with a definite wavelength \(\lambda\); for this purpose the rays are either passed through colored glass, or are decomposed into a spectrum by means of a prism and only rays of the same wavelength are selected by a narrow slit.

As for determining the luminous-intensity ratio \(\frac{J}{J_1}\), this is done in very different ways:

1. one may equalize the luminous intensity of the standard source and that of the source being compared with it, weakening the luminous intensity of one or the other of them, according to which has the higher temperature, by means of diaphragms or smoked glasses (Le Chatelier and Cornu);

2. one may compare the sources by changing the actual luminous intensity of the principal source (Holborn and Kurlbaum);

3. one may compare the sources by changing their apparent luminous intensity by means of a polarizer (Wanner).

The Le Chatelier and Cornu instrument, constructed according to scheme 1, Fig. 61, p. 76, consists of a viewing tube \(c-d\), to the side of which, perpendicularly, is attached a tube \(e\), at the end of which is fastened a lantern \(f\) with a lamp, whose light is compared with the light from the body under investigation. In tube \(e\) there is a plano-convex glass and a diaphragm through which the light from the lamp falls upon a small mirror \(i\), set at an angle of \(45^\circ\) at the principal focus of tube \(c-d\); the focus of the aforemen—

of this plano-convex glass coincides with the principal focus of the tube \(c\); \(g\) is the counterweight balancing the lamp.

Fig. 61.

As has already been said, for comparison one must take a simple ray; it is best of all to take a red one, since the intensity of red rays, owing to the special structure of our eye, changes very rapidly precisely in the ratio \(1:1000000\) between \(600^\circ\) and \(1800^\circ\) C.; in view of this, accurate results may be obtained even with a not especially precise measurement of the intensity of light. A red ray with wavelength \(\lambda = 659\) is obtained by placing, in front of the eyepiece, a glass \(h\) colored with copper oxide.

Sometimes the red glass is replaced by a green one, transmitting rays with \(\lambda = 546\), which, however, is applicable only for higher temperatures.

The intensity of the light of the incandescent body is compared with the indicated lamp, for which purpose the coincident images of the light of the lamp and of the body under investigation are brought to the same degree of illumination. Equality of illumination, i.e. the weakening of one or the other source of light, is achieved by changing the aperture of the diaphragm in front of the objective or in front of the lamp, depending on which of the light sources must be weakened, and by inserting dark, smoked glasses in front of the diaphragms.

If equality of illumination is attained with \(n_0\) smoked glasses, \(d_0\) the diameter of the aperture of the diaphragm and \(f_0\) the focal length in the tube \(c\), for the normal lamp, and correspondingly \(n_1\), \(d_1\), and \(f_1\) in the objective \(d\), for the light source under investigation, then the ratio we need of the light intensity \(J_1\) of the latter to the light intensity \(J_0\) of the normal lamp is expressed, if \(k\) is the absorption coefficient of the colored glass in the eyepiece, by the equation

\[ \frac{J_1}{J_0}=\frac{k^{n_1}}{k^{n_0}}\left(\frac{d_1}{d_0}\right)^2\left(\frac{f_1}{f_0}\right)^2 \]

or, more simply for computations,

\[ \frac{J_1}{J_0}=\left(\frac{1}{k}\right)^{n_0-n_1}\left(\frac{d_1}{d_0}\right)^2\left(\frac{f_0}{f_1}\right)^2 . \tag{54} \]

For the glasses used in the Le Chatelier instrument, for red \(k=\frac{1}{11}\), and for green \(k=\frac{1}{7}\).

The calculations according to eqs. (51), (53), or (54) may be made in advance and reduced to a table; then the use of the instrument is considerably simplified.

The instrument described is suitable for measuring temperatures approximately from \(700^\circ\) to \(2000^\circ\); for lower temperatures its sensitivity rapidly decreases.

In Fig. 62 a diagram is shown of the same instrument, improved by Prof. Féry. The principal difference is that the small mirror \(i\) is formed on a large glass by means of a narrow silvered strip, so that the reflection of the light of lamp \(f\) is obtained in the middle of the eyepiece in the form of the same narrow strip, cutting across the image of the body being investigated (in Fig. 63, a red-hot crucible). Then, instead of diaphragms in front of the objective, there are two absorbing prisms \(k\) and \(l\); by moving them perpendicular to the axis of the main tube, one can vary the thickness of the layer passed through by the rays of the body being investigated, and thereby vary their intensity; \(d\) is a dark butterfly glass for determining high temperatures. The desired temperature is found from a table as a function of the scale reading of the position of prisms \(k\) and \(l\). Fig. 64 shows the external appearance of this instrument.

Figs. 62 and 63.

Fig. 64.

The Holborn and Kurlbaum pyrometer, manufactured by Siemens & Halske and operating according to scheme 2, is in essence only a constructive improvement of the preceding one.

This instrument, Fig. 65, consists of a viewing tube with lenses \(L_1\) and \(L_2\) and a diaphragm \(D\) beside them. In the principal focus of the tube there is an electric incandescent lamp; the lamp is supplied with current (4 volts) from the storage battery \(E\); the current is regulated by the rheostat \(W\) so that the luminous intensity of the lamp becomes equal to the luminous intensity of the incandescent body under investigation; this moment is easily and sharply recognized from the fact that the filament of the lamp, formerly either lighter or darker than the background, suddenly becomes invisible; accordingly, the current indicated by the ammeter \(S\) gives the temperature of the filament, from which it is then not difficult either to calculate the temperature of the body or to take it from a calibration table prepared in advance. The calibration of the instrument is carried out at the Imperial Physico-Technical Institute with the aid of an actually black body.

Fig. 65.

To facilitate observation, glass \(G\), colored ruby-red with copper oxide, is inserted into the eyepiece.

The range of measurement with this instrument is from \(600^\circ\) to \(1900^\circ\) C.; in this case temperatures from \(600^\circ\) to \(800^\circ\) C. are measured without the red glass, and for measurement from \(1200^\circ\) to \(1900^\circ\) glass of double thickness (4 mm) is inserted. Finally, temperatures above \(1900^\circ\), i.e. above the temperature of the lamp filament, can be compared, without calculating their absolute value, by passing the light of the incandescent body, in order to weaken it, through a pair of prisms \(P\).

Handling the instrument is not difficult, and its indications are very precise.

We shall now describe Wanner’s pyrometer, which operates according to scheme 3, Figs. 66 and 67. Through slit \(a\) enter the rays of the body being investigated,

Figs. 66 and 67.

through slit \(b\), the rays from the light source adopted for comparison—a 6-volt incandescent lamp supplied with current from a small storage battery. Both beams of rays pass through a compound

lens \(L_1\), and then through the compound prism \(G\), which gives a direct image, the rays being refracted and giving two spectra. On further passage through a pair of prisms \(K\) of Iceland spar these two spectra are decomposed into two pairs of spectra, the spectra in each pair being polarized in mutually perpendicular directions. After passage through the double prism \(P\) the number of spectra is doubled, i.e. in all 8 spectra enter the eyepiece \(L_2\); the angle of the prism \(P\) is chosen so that, of these eight spectra, two—one horizontally polarized from the lamp and one vertically polarized from the source under investigation—are brought close together to facilitate their comparison. The red rays of a definite wavelength \(\lambda\) sent by them fill both halves of the field of view, which is observed through the Nicol prism \(N\). The slit \(S_2\) in front of this prism stops all the other rays of the spectra.

When two rays polarized in mutually perpendicular directions pass through the Nicol prism, for a certain position of the prism one of them is transmitted with full intensity and the other is extinguished; when the prism is rotated, the intensity of the first light will decrease and that of the second will increase, until at an angle of rotation of \(45^\circ\) the intensities of both rays become equal. If these two rays come from different sources and have different intensities, then after passage through the Nicol prism their intensities become equal not at \(45^\circ\), but at some other angle of rotation \(\varphi\); this angle determines the ratio of the intensities of the two rays, namely

\[ \frac{J}{J_1}=\operatorname{tg}^2\varphi . \tag{54} \]

The subsequent calculation may be carried out by eq. (51) or (53). In practice a table is used, in which, for successive angles \(\varphi\), the corresponding temperatures are shown directly. The angle of rotation \(\varphi\), at which both halves of the field of view appear illuminated quite equally, is read on the scale at the eyepiece of the instrument. The table for converting readings of Wanner’s pyrometer into degrees C. is most simply compiled by comparison with the readings of a thermoelectric pyrometer heated in an electric furnace with a platinum winding, the Wanner pyrometer being directed at the very junction of the thermoelement.

The intensity of the light of the osmium incandescent lamp changes with time and, in addition, depends on the voltage of the battery current; therefore its luminous intensity must be checked from time to time by comparing it in the same instrument with the luminous intensity of a standard lamp, in which

amyl-acetate ether is burning. If, in comparison with this lamp, unequal illumination of the two halves of the field of view is obtained, then the incandescent lamp is moved, by means of the setting screw, nearer to or farther from the objective until equal illumination of both halves of the field of view is attained. If, even when the lamp is brought as close as possible, the upper half of the field of view is illuminated more weakly than the lower, this indicates insufficient current; the storage battery must then be recharged, the voltage of which, when it is fully charged, should be 6.2 volts. If, with a sufficiently charged battery, the upper half is nevertheless illuminated more weakly than the lower, this indicates an abnormally large resistance in the circuit; all the connections must be examined and cleaned.

With an excessively strongly charged battery it may happen, on the contrary, that even when the lamp is removed as far as possible, the upper half of the field of view remains illuminated more strongly than the lower; then the battery must be somewhat discharged, by passing current from it through an ordinary incandescent lamp, until, upon a repeated check with the amyl-acetate lamp, the required result is obtained. Such a check should be made no less often than after 100, or at the very most 150, observations.

The range of application of this instrument is from \(900^\circ\) to \(2000^\circ\) C.; and if smoked glass is placed in front of the objective, then up to \(4000^\circ\) C. and even higher. The accuracy of its readings, with known skill, is \(0.5\)—\(1\%\); at temperatures above \(1400^\circ\) it decreases somewhat, falling by \(2000^\circ\) to \(3\%\).

Handling the instrument is very convenient; its verification is accurate and simple; it is based on a theoretically and practically proven law, so that the degree of accuracy of its readings is not open to doubt.

Fig. 68.

This explains the wide distribution which it has acquired in a comparatively short time. In the diagram, drawings 66 and 67 show the instrument at approximately \(1/3\) of its natural size.

In conclusion, let us also describe the newest construction of Wanner’s pyrometer, intended for measuring lower temperatures, from \(625^\circ\) to \(1000^\circ\) C. The special feature of this instrument, Fig. 68, consists, in comparison with the pre-