I. A. Sokolov.

Transition Points of Elementary Substances on a Logarithmic Temperature Scale

As is known, Kelvin, in his well-known construction of the thermodynamic absolute temperature scale, proceeded from the relation

\[ \frac{Q_1}{T_1}=\frac{Q_2}{T_2}; \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (1) \]

which follows from Carnot’s theorem:

\[ \frac{Q_1-Q_2}{Q_1}=\frac{T_1-T_2}{T_1} \qquad \ldots \ldots \ldots \ldots \ldots \ldots (2) \]

In such a definition of temperature, the connection between the heat content and the absolute temperature of a substance is emphasized, first of all that of an ideal gas.

Since, on the other hand, on the basis of the well-known relation

\[ K=\frac{3}{2}RT, \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots (3) \]

where \(K\) is the kinetic energy of a gram-molecule of gas, and \(R\) is the universal gas constant, the temperature of a gas is determined by the kinetic energy of the translational motion of its particles, it becomes natural to estimate an increase in the temperature of a gas by the ratio of its kinetic energies, and not by the usual difference of temperatures.

If the gas previously had temperature \(T_1\), and now has temperature \(T_2\), with \(T_2 > T_1\), then it is natural to indicate how many times the gas has increased its temperature, proceeding from the relation

\[ \frac{T_2}{T_1}=\frac{K_2}{K_1} \qquad \ldots \ldots \ldots \ldots \ldots \ldots (4) \]

which obviously follows from (3).

In the graphical representation of thermal processes it is possible to use a logarithmic temperature scale; moreover, from (4) we have

\[ \log T_2-\log T_1=\log \frac{K_2}{K_1}, \]

i.e., the logarithm of the number indicating how many times the energy of the gas has increased is equal to the increment of the logarithm of the absolute temperature.

Let us construct such a logarithmic temperature scale, using ordinary decimal logarithms. One degree Kelvin will be

then represented by zero and will be the initial point of our scale (Fig. 1); \(10^\circ K\) will be represented by a segment equal to unity, which will give us the scale for further construction; a temperature of \(100^\circ K\) will be represented by point 2, \(1000^\circ K\) by point 3, etc.

It is curious how, in such a representation, the “unattainability” of absolute zero is vividly expressed, or, more precisely, the practical difficulties of attaining low temperatures, known to us from historical data on this question. In order to reach absolute zero one must descend downward from the zero of the logarithmic scale by an infinitely great distance. To descend to \(0.1^\circ\) means to go down from this zero by the same segment as from \(1000^\circ\) to \(100^\circ\). In other words, the energy of an ideal gas, in passing from \(0.1^\circ\) abs to \(1^\circ\) abs, increases by the same factor as in passing from \(100^\circ\) abs to \(1000^\circ\) abs.

How narrow, on this scale, is the interval separating the fixed points of the Celsius thermometer; how attainment of high temperatures is unfolded! The temperature of the sun (about \(10000^\circ\)) is as much higher than the melting temperature of copper (about \(1000^\circ\)) as this latter is higher than the temperature of liquid air, so common in scientific usage (about \(100^\circ\) abs).

It seemed to me not uninteresting to compare on this scale the principal temperature data known for the chemical elements (melting point, boiling point at normal pressure, and critical temperature), connecting them with the ordinal number determining the position of the element in the Mendeleev system, as a result of which the appended table was obtained. (For the table see pp. 86–88).

To make the survey of the table clear and instructive, I represented the tabular material in the form of the appended diagram.

For this purpose I placed the axis of ordinal numbers perpendicular to the logarithmic temperature scale and plotted in the vertical direction the values of the logarithms of the temperatures of melting, boiling at normal pressure, and critical temperature, since these data are available in the scientific literature¹.

EXPLANATION OF THE TABLE

In the first column of the table is given the ordinal number of the element.
In the second " " " the chemical symbol of the element.
In the third " " " the absolute melting temperature \(T_f\).
In the fourth " " is given \(\log T_f\).
In the fifth " " are given the differences of the values \(\log T_f\) for two neighboring elements—denoted \(\Delta T_f\).
In the sixth " " is indicated the absolute boiling temperature at normal pressure \(T_d\).
In the seventh " " is given \(\log T_d\).
In the eighth " " are given the differences of the values \(\log T_d\) for two neighboring elements—\(\Delta T_d\).
In the ninth " " is indicated the absolute critical temperature of the element \(T_k\).
In the tenth " " \(\log T_k\).
In the eleventh " " \(\Delta T_k\).
In the twelfth " " the differences: \(\log T_d - \log T_f\).
In the thirteenth " " the differences: \(\log T_k - \log T_f\).
In the fourteenth " " the differences: \(\log T_k - \log T_d\).

When the plotted points are connected (in the case of insufficiently reliable data, a question mark is placed at the corresponding points of the diagram) by rectilinear segments, as is customary in analogous representations of the properties of elements, very instructive “curves” are obtained, which we shall now survey.

1. The curve of melting points reveals a periodicity that has long been noticed. Each period of Mendeleev’s table¹, beginning with the 2nd, is characterized by such a course of the “curve.” The initial element of the period (an alkali metal) has a low melting temperature².

Further the “curve” rises to the element of group IV: in the second period to \(C\) (\(T_f\) for graphite \(=3773.2\); \(\log T_f=3.5766\)), in the third to \(Si\) (\(T_f=1693.2\); \(\log T_f=3.2287\)); in the 4th to \(Ti\) (\(T_f=2073\); \(\log T_f=3.3166\)).

In the second period the curve, after the maximum reached, falls rapidly—for nitrogen \(T_f\) is only 63.3, \(\log T_f=1.8014\)—the fall amounts to almost two units of our logarithmic scale. Further the fall continues to \(Ne\), \(T_f=24.53\), \(\log T_f=1.3897\), whence there is a steep rise to the first element of the 3rd period—\(Na\).

In the third period, after the maximum \(\log T_f\) reached at \(Li\), a descent begins toward \(P\)³ and \(S\), and further toward \(A\) (\(T_f=84.0\); \(\log T_f=1.9243\)⁴).

These values (except \(Ne\)) fit well on the ascending straight line drawn from \(A\) to \(Rn\).

In the fourth—large—period, after the indicated maximum at $Ti$, the melting curve remains up to the first member of the triad $(Fe, Ni, Co)$, i.e. $Fe$, at approximately the same height (with a slight depression at $Mn$), after which it begins to fall rapidly, descending at $Ga$ to the value: $T_f = 302.9$, $\log T_f = 2.4813$. After this there is again a rise toward $Ge$, and from there again a descent to the element of the zero group—krypton $(Kr)$.

In the fifth period the course of the melting curve is very similar to its course in the fourth period: a steep rise from $Rb$ to $Sr$, then a smooth rise to $Mo$ (and not to the element of the fourth group—as in the 4th period), a descent in the triad $Ru, Rh, Pd$ to $Jn$ (the analogue of $Ga$, where in the preceding period a minimum was likewise observed); then a rise through $Sn$ to $Sb, Te$ and finally to $J$.

The sixth period, as is known, is “loaded” with the elements of the rare earths. However, at its beginning (as also at the beginning of the 4th and 5th periods), there is the usual rise from $Cs$ to $Ba$.

If earlier—more precisely in the 2nd, 3rd, and 4th periods—the rise of the melting curve led us to a maximum in the 4th group, then here this rise apparently extends over the whole region of the rare earths; the element of the fourth group $Hf$ has a higher $T_f$ than $Jb$,—the maximum is observed at $W$, and both the preceding $Ta$ and the following $Re$ have, on the logarithmic scale, a temperature close to the melting point of $W$:

The fall of the curve continues through the triad $Os, Jr, Pt$ to $Au$. Next there follows a sharp fall to $Hg$—an element of group 2, and not to $Tl$ (an element of the 3rd group, whose analogues—$Ga$ in the 4th period and $Jn$ in the fifth—gave a minimum of the melting temperatures).

$Tl$ and $Pb$ have almost the same melting temperature; $Bi$—the analogue of $Sb$—has a lower melting temperature, not a higher one, than $Pb$.

It is hardly credible that the rise of the melting curve toward $Po$ given in the tables of the TE is reliable. We would be inclined to expect a fall of our curve through the halogen $Wg$ to the noble gas $Rn$.

The course of the curve in the seventh period from the alkali metal $Am$ should begin with a rise toward the alkaline-earth $Ra$; $Ac, Th, Pa$, and $U$ ought to have nearly identical and, moreover, high melting temperatures. Such values are reliably known only for $Th$ and $U$.

1. Let us pass to a survey of the “curve” of boiling of the simple substances. We note at once, first, the still greater approximate character of the tabulated data for high temperatures, and, secondly—what is perhaps still more important—the uncertainty of the very content of the concept of boiling in such cases. If one adheres to the classical definition of the boiling temperature as that point at which the elasticity of the saturated vapor of a liquid becomes equal to normal pressure, then in our cases the complete conventionality of this definition is revealed, both because hardly any investigator has troubled to observe pressure at sufficiently high temperatures, reaching 3000° and more degrees of the absolute scale, and because boiling no longer marks any particularly significant moment in the history of a substance. In this respect, the concepts of the critical temperature of a substance, marking an unconditional tran—

Transition points of elementary substances on a logarithmic temperature scale

Dedicated to the centenary of Prof. Svante A. [[unclear: surname]]

Figure

Visible labels and legend

Curves:

• Melting curve
• Boiling curve
• Critical-temperature curve
• Atomic-weights curve
• Curve \((\mathrm{cm}\cdot \mathrm{T})\) [[unclear: abbreviated label]]

Element sequence along the horizontal axis:

H, He, Li, Be, B, C, N, O, F, Ne, Na, Mg, Al, Si, P, S, Cl, A, K, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ga, Ge, As, Se, Br, Kr, Rb, Sr, Y, Zr, Nb, Mo, Ma, Ru, Rh, Pd, Ag, Cd, In, Sn, Sb, Te, J, Cs, Ba, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, Ta, W, Re, Os, Ir, Pt, Au, Hg, Tl, Pb, Bi, Po, An, [[unclear: element symbols]], Ra, Ac, Th, Pa, U

Group labels visible above the plot:

I, II, Θ, Б, И, O, A, 6, Ы

Numerical scale labels visible:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Horizontal axis numbering visible:

5, 10, 20, 30, 40, 50, 60, 70, 80, 90

transition of a substance into the gaseous state—the temperatures of “absolute boiling”—have immeasurably great physical significance. Unfortunately, the critical temperatures could have been determined by different investigators only for a small number of simple bodies.

Be that as it may, a survey of the “boiling” curve is of considerable interest, especially in comparison with the melting curve constructed by us. In the diagram illustrating our considerations, there is of course again revealed the familiar periodicity of this property for simple substances, as well as an almost complete parallelism in the course of the two curves throughout the entire periodic system.

As in the preceding case, let us begin the survey with the second period. The initial element \(Li\) has a rather high boiling temperature: \(T_d\) 1473° K, \(\log T_d = 3.1683\). Further these temperatures rise to \(C\) \((T_d = 4473?; \log T_d = 3.6506)\), after which there is a steep descent to \(N\) \((T_d = 77.4, \log T_d = 1.8887)\)—again a slight rise to \(O\) \((T_d = 90.2, \log T_d = 1.9552)\), then a descent—at first not very steep—to \(F\) \((T = 86.2; \log T_d = 1.9355)\) and a steeper one to \(Ne\) \((T_d = 27.3, \log T_d = 1.4362)\), whence a rise to the alkali metal \(Na\), which begins the third period.

In the third period—the principal rise of the melting curve is through \(Mg\) and \(Al\) to \(Si\) \((T_d = 2873\ (?); \log T_d = 3.4584)\); thence—a fall to \(P\) \((T_d = 553.2; \log T_d = 2.7429)\), again a slight rise to \(S\) \((T_d = 717.8; \log T_d = 2.8560)\) and again—a fall through \(Cl\) to \(A\) \((T_d = 90.2; \log T_d = 1.9552)\) and a rise to the alkali metal \(K\), opening the 4th period¹.

The parallelism with the melting curve is complete, partly to the point of coincidence.

In the fourth period the boiling curve runs very similarly to the melting curve. Almost everywhere the differences \(\log T_d - \log T_f\) fluctuate within comparatively small limits (apart from the initial \(K\)): from \(Br\) \((\log T_d - \log T_f = 0.0961)\) to \(Se\) (the same quantity \(= 0.2897\)), with a mean value of the difference 0.2114, i.e. of the ratio

\[ \frac{T_d}{T_f} = 1.628. \]

Sharply falling out of the general picture are \(Ga\), for which \(\log T_d - \log T_f = 0.7912,\ \dfrac{T_d}{T_f} = 6.183\), and \(Ge\) \((\log T_d - \log T_f = 0.5087;\ \dfrac{T_d}{T_f} = 3.226)\).

In the diagram the behavior of these elements recalls the zigzag formed by the boiling curve from \(P\) to \(S\).

The course of the melting curve in the 5th period is similar to its course in the 4th period. Some uncertainty is introduced into this assertion by the absence of a number of data for \(T_d\) \((Nb, Ma, Rh,\) etc.). The differences \(\log T_d - \log T_f\) also fluctuate within small limits (from \(I\)—0.0731 to \(Te\)—0.3605), on the average—

than with the exception of \(In\) (the analogue of \(Ga\)) with a difference of 0.6048 and \(Sn\) (the analogue of \(Ge\)) with a difference of 0.7011, with the corresponding ratios

\[ \frac{T_d}{T_f}\;(\text{for } In)=4.025 \]

\[ \text{»}\;(\text{for } Sn)=5.024. \]

In the sixth period, if its “rare earths” are left aside, the course is very similar to the course in the 4th and 5th periods. The place of the downward zigzag formed in the 4th period by \(Ga\), and in the 5th by \(In\), is here occupied by the fall of \(T_d\) for \(Hg\), which is not a chemical analogue of \(Ga\), but by virtue of its low melting temperature corresponds to it. In this same place there is the greatest divergence of the curves (i.e. the greatest values of the difference \(\log T_d - \log T_f\)) for \(Tl\) (the analogue of \(Ga\) and \(In\)), \(Pb\) (the analogue of \(Sn\)) and \(Bi\) (the analogue of \(Sb\)).

The seventh period again begins with a rise of the curve, and, if its development to the end is assumed, it would probably repeat for us the essential features of the fifth period.

1. As has already been indicated, the construction of a “curve” of critical temperatures on the same logarithmic scale would be of great interest.

But critical data are known\(^1\) only for seventeen elementary substances, which deprives us of the possibility of constructing the curve. However, these points have been plotted on our diagram.

In the second, third, and fourth periods it is possible to establish the emerging parallelism of the course of the critical data with the course of the boiling temperatures.

In the case when the critical temperature is known, it is possible to speak of the logarithmic difference of the critical and boiling temperatures, i.e. of the quantity:

\[ \log T_k - \log T_f, \]

and, correspondingly, of the logarithmic difference of the critical and boiling temperatures, i.e. of the quantity

\[ \log T_k - \log T_d. \]

In the first case, from 12 data of our table (with the exception of \(He\)) one obtains on average:

\[ \log T_k - \log T_d = 0.2464, \]

or

\[ \frac{T_k}{T_d}=1.764. \]

If, however, one excludes from consideration the anomalously large difference for \(Hg\) (\(\log T_k - \log T_d = 0.4846\)), then

\[ \frac{T_k}{T_d}=1.678. \]

This does not fully agree with the known rule, derived from the theory of corresponding states:

\[ \frac{T_k}{T_d}\ \text{approximately}=1.5, \]

but in any case it gives a close value.

\(^1\) Tekhn. Ents. Sprav., vol. 1, p. 125.

Let us calculate in the same way, for the 11 data points (again excluding \(He\) and \(Hg\)), the quantity \(\log T_k - \log T_f\); we obtain

\[ \log T_k - \log T_f = 0.3249. \]

Whence

\[ \frac{T_k}{T_f} = 2.113, \]

which gives us the possibility of stating a rule similar to the one given above, not encountered by us in the literature:

The absolute melting temperature is, approximately, half the critical temperature.

1. Above we have already had occasion to note that often the differences of the logarithmic boiling and melting temperatures \((\log T_d - \log T_k)\) turn out to be close to \(0.2\) ¹).

In reality these differences have very diverse values, and it is more correct to compare them by groups of the periodic system ²).

In the zero group

\[ \log T_d - \log T_f \]

has values from \(0.02\) \((Rn)\) to \(0.09\) \((Xe)\).

In the first group all alkali metals have an anomalously large value

\[ \log \frac{T_d}{T_f}, \]

namely from \(0.489\) for \(K\) to \(0.506\) for \(Li\) (and all, moreover, very close); \(Ag, Cu, Au\ldots\) from \(0.255\) \((Ag)\) to \(0.322\) \((Au)\), \(H\ldots\ 0.16\).

In the second group all elements (except \(Be\) and \(Hg\)) have a value

\[ \log \frac{T_d}{T_f} \]

from \(0.10\) \((Ba)\) to \(0.24\) \((Cd)\). \(Be\) has a very small value, \(0.04\), while \(Hg\) has an anomalously large one, as was already indicated above, namely \(0.43\).

The elements of the third group should be divided into two subgroups: the elements of the first half of the periods \((Y, Sc, La\ldots\) and apparently the rare earths) have a value

\[ \log \frac{T_d}{T_f} \]

from \(0.18\) \((Y)\) to \(0.28\) \((La)\), while the elements of the second half of the period \((Tl, In, Ga)\) all possess an anomalously large value of this ratio, from \(0.56\) \((Tl)\) to \(0.79\) \((Ga)\). \(Al\) occupies an intermediate position; for it

\[ \frac{T_d}{T_f} = 0.345. \]

\(B\) has a low value \(= 0.04\).

From this follows the generalization: low-melting metals require, in order to reach the boiling state, a greater increase than high-melting metals—\(A\).

(This conclusion could also have been drawn earlier: low-melting

¹) If, however, one bases oneself only on the approximate relations established here:

\[ \frac{T_k}{T_d} = 1.7 \]

\[ \frac{T_k}{T_f} = 2, \]

then

\[ \log T_d - \log T_f = \log \frac{2}{1.7} = 0.0705, \]

which fits well precisely the cases of those bodies whose critical temperatures are known.

²) On the diagram the values of the differences \(\log T_d - \log T_f\) are plotted at the bottom of the scale.

alkali metals also have a large interval between the melting and boiling points).

In the fourth group the low-melting elements $Pb$ and $Sn$ (cf. conclusion—A) have a large value of $\log \dfrac{T_d}{T_f}$, namely $Pb$... 0.5, $Sn$... 0.7. The refractory elements $Th$, $Ti$, $Si$, $Zr$, $Hf$—all have values from 0.19 ($Th$) to 0.29 ($Hf$); $C$ occupies, in this respect, an abnormally low place (0.073), while the comparatively refractory $Ge$—an abnormally high one (0.56).

In the fifth group the low-melting $Bi$ has a high value, 0.5; $N$ a very low one... 0.088, and all the others... from 0.22 ($V$) to 0.26 ($Sb$).

In the sixth group all the elements of the first half of the periods ($O$, $S$, $Se$, $Te$) (low-melting) have values from 0.19 to 0.36; while those of the second half (refractory) ($Cr$, $Mo$) have values from 0.12 to 0.14.

In the seventh group all elements have values from 0.07 ($J$) to 0.23 ($F$). In the eighth—the values are scattered (from 0.04... ($Ru$) to 0.35 ($Pt$).

In all, 48 of the elements given in the table have a value of $\log \dfrac{T_d}{T_f}$ up to 0.3, and 18 above this.

I have made these comparisons in order to show that within them one can discern a whole series of, perhaps, very important regularities.

I. Melting temperatures have been established for almost all elements; for elements melting at high temperatures, the boiling temperatures are naturally unknown. It may be said with a considerable degree of probability that such an element $K$ has $\log \dfrac{T_d}{T_f}$ approximately 0.25; whence

\[ \log \frac{T_d}{T_f} = 0.25 \qquad \text{or} \qquad \frac{T_d}{T_f}\ \text{approx.} = 1.8 \qquad \text{or} \]

\[ T_d\ \text{approx.}\ 1.8\,T. \]

II. Critical temperatures may be estimated in a similar way,

\[ T_k\ \text{approx.}\ 2\,T_d, \]

and it is also evident from our diagram that the highest melting temperatures lie, on the logarithmic scale, a little above 3.5, i.e. 3200° $K$; consequently the highest critical temperatures must be considerably above 6400° $K$.

III. With what is the fusibility of elements connected? To clarify this I have given on my diagram a curve of “atomic radii,” understanding by this designation the quantity

\[ \rho = \sqrt[3]{\frac{A}{d}}, \]

where $A$ is the atomic weight, $d$ the density of the element—i.e. the cube root of the atomic volume at the melting temperature¹. (Of course, one could have given directly the curve of atomic volumes.) We see that (with the exception of the noble gases) the maxima of this curve, as a rule, correspond to low melting temperatures of the corresponding elements, and the minima, conversely, to high melting temperatures. Thus, elements with a large atomic volume—“loose” crystalline structures—melt easily. If we dwell on the alkaline-earth metals, which, as is known, possess a lattice in the form of a centered-

of the cube, then this is quite consistent with the fact that such a lattice does not satisfy the principle of “closest packing,” and, consequently, should be destroyed when its atoms attain a lower energy than in the case of close packings.

Thus, the other metals of the same 1st group: $Cu$, $Ag$, $Au$, having a face-centered structure, possess a higher melting point and an “abnormal” distance between the melting and boiling points (from 0.25 to 0.33 of our scale).

From this one might draw the conclusion that low-melting metals, owing to their looseness, are readily transformed into a liquid whose particles have a comparatively small kinetic energy. Therefore a relative increase of temperature, and consequently of the kinetic energy of the atoms, is required that is considerably greater than in the case of normally melting bodies in order for boiling to be possible. This also agrees with the presence, generally speaking, of low values of the latent heat of fusion of bodies with an abnormally large value of the interval $\log T_d - \log T_f$:

Whereas for $Ag$, $Au$, $Cu$ we have respectively

I propose to make the comparison of the regularities indicated by me with other physical properties the subject of a subsequent work.

I shall confine myself only to pointing out that many superconductors of the Kamerlingh-Onnes type ($Al$, $Ga$, $In$, $Sn$, $Hg$, $Tl$, $Pb$) also possess a low value of $\log \dfrac{T_d}{T_f}$. Superconductors of the Meissner type ($Ti$, $V$, $Ta$) do not possess this quality.

III.XII.1935

$^{1)}$ According to the data of the T. E. handbook, vol. II.

1. According to the data of V. Ya. Kurbatov, “Mendeleev’s Law,” L. 1925, p. 56. ↩↩↩↩

2. 	\(T_f\)	\(\lg T_f\)		\(T_f\)	\(\lg T_f\)		\(T_f\)	\(\lg T_f\)
   \(Li\) . .	459	2.6620	\(K\) . .	336	2.5256	\(Cs\) . .	299	2.4760
   \(Na\) . .	371	2.5690	\(Rb\) . .	312	2.4938

   These values fit fairly well on a descending straight line running from \(Li\) to \(Cs\). ↩

3. The melting point \(P\) is taken for yellow phosphorus. ↩

4. Let us compare the values of \(T_f\) and \(\log T_f\) for the elements of the zero group:

   	\(T_f\)	\(\log T_f\)		\(T_f\)	\(\log T_f\)		\(T_f\)	\(\log T_f\)
   \(Ne\) . .	24.3	1.3897	\(Kr\) . .	104.2	2.0179	\(Rn\) . .	201.2	2.3049
   \(A\) . .	84.0	1.9243	\(Xe\) . .	133.2	2.1245

   ↩