Full Text
Band Spectra and Their Significance for Chemistry.^1
R. Mecke, Bonn.
V. Determination of Heats of Dissociation.
To understand the course of a chemical reaction, knowledge of its thermal effect at absolute zero is required.
The thermal effect can, of course, be determined calorimetrically, i.e., by a purely experimental route; but it is also possible—and for chemical kinetics this is the most fruitful method—to calculate the thermal effect from the dissociation energies of the components of the reaction, since these dissociation energies are known with sufficient accuracy.
In a considerable, and moreover ever increasing, number of cases the dissociation energies can be calculated with great accuracy from band spectra, so that here too band spectroscopy proves to be a fruitful auxiliary method of chemistry.
By dissociation energy in what follows we shall always understand the amount of energy required to decompose a molecule into its separate constituent parts, i.e., in the case of simple molecules—into gaseous atoms (example: $\mathrm{H_2}=\mathrm{H}+\mathrm{H}$ or $\mathrm{H_2O}=\mathrm{H}+\mathrm{H}+\mathrm{O}$), and in the case of complex atoms—into atoms and atomic groups (example: $\mathrm{CO_2}=\mathrm{CO}+\mathrm{O}$). In special cases the work of separating an atom from the remainder of the molecule may also be of interest, i.e., the so-called energy of valence bonds, for example $\mathrm{C-H}$, $\mathrm{C-C}$, $\mathrm{C=C}$, $\mathrm{O::C}$.
^1 Conclusion; see issue 5, p. 680.
We shall denote all dissociation energies by the letter \(D\), in distinction to the heat effect of a reaction \(Q\), which—as we shall now see—is additively composed of dissociation energies. If, for example, we recall the combustion of hydrogen and oxygen to water vapor, then here the initial products are always oxygen molecules and hydrogen molecules, and the final products are molecules of water vapor. In this reaction, as is known, the heat effect \(Q\) is equal to \(58.3\) kg cal per molecule of water vapor. This heat effect is calculated from the dissociation energies of hydrogen \((D_{\mathrm H}=100\ \text{kg cal})\), oxygen \((D_{\mathrm O}=162\ \text{kg cal})\), and water vapor \((D_{\mathrm{H_2O}}=239\ \text{kg cal})\), by means of the following three equalities:
- Dissociation of hydrogen: \( \mathrm H_2 + D_{\mathrm H} = 2\mathrm H.\)
- Dissociation of oxygen: \( \frac{1}{2}\mathrm O_2 + \frac{1}{2}D_{\mathrm O} = \mathrm O.\)
- Dissociation of water vapor: \( \mathrm H_2O + D_{\mathrm{H_2O}} = 2\mathrm H + \mathrm O.\)
Formation of \(\mathrm H_2O\) from \(\mathrm H_2\) and \(\mathrm O_2\): \( \mathrm H_2 + \frac{1}{2}\mathrm O_2 = \mathrm H_2O + Q.\)
Thus the heat effect is
\[ Q = D_{\mathrm{H_2O}} - D_{\mathrm H_2} - \frac{1}{2}D_{\mathrm O} \]
\[ 58 = 239 - 100 - 81\ \text{kg cal}. \]
As a second example let us consider the successive addition of an atom: the combustion of carbon (diamond) to carbon monoxide and then to carbon dioxide. Here, in addition to the dissociation energies of oxygen \((162\ \text{kg cal})\) and carbon monoxide \((249\ \text{kg/cal})\) (for carbon dioxide, \(148\ \text{kg cal}\)), one must also take into account the thermochemically determined heat of sublimation \(S\) \((141\ \text{kg cal})\) of carbon, i.e., the quantity of energy required to convert solid carbon into monatomic gaseous carbon.
In this case:
I. 1. Sublimation of carbon: \(\mathrm C_{\text{solid}} + S_{\mathrm C} = \mathrm C_{\text{gas}}.\)
-
Dissociation of oxygen: \(\frac{1}{2}\mathrm O_2 + \frac{1}{2}D_{\mathrm O} = \mathrm O.\)
-
Dissociation of carbon monoxide: \(\mathrm{CO} + D_{\mathrm{CO}} = \mathrm C_{\text{gas}} + \mathrm O.\)
Formation of \(\mathrm{CO}\) from \(\mathrm C\) and \(\mathrm O\): \(\mathrm C_{\text{solid}} + \frac{1}{2}\mathrm O_2 = \mathrm{CO} + Q_1.\)
II. 1. Dissociation of oxygen: \(\frac{1}{2} O_2 + \frac{1}{2} D_O = O\).
- Dissociation of carbon dioxide: \(CO_2 + D_{CO_2} = CO + O\).
Formation of \(CO_2\) from \(CO\) and \(O\): \(CO + \frac{1}{2} O_2 = CO_2 + Q_2\).
Hence
\[ \begin{aligned} \mathrm{I}\quad & Q_1 = D_{CO} - D_O - S_C \\ & 27 = 249 - 81 - 141, \end{aligned} \]
\[ \begin{aligned} \mathrm{II}\quad & Q_2 = D_{CO_2} - D_O \\ & 67 = 148 - 81. \end{aligned} \]
In other words, in order to separate the first atom of oxygen from a molecule of carbon dioxide, we need 148 kg cal per mole; to separate the second atom of oxygen—already 249 kg cal.
Both of these simple examples show that the thermal effect of chemical reactions, generally speaking, is equal to the sum of the dissociation energies (plus the heats of evaporation and sublimation in the case of liquid and solid bodies) of the reaction components entering into the right-hand side of our reaction equation minus the sum of the dissociation energies of the left-hand side. Thus the following important law is obtained:
\[ Q = \Sigma D\ \text{(right-hand side)} - \Sigma D\ \text{(left-hand side)}, \tag{24} \]
which connects the thermal effects, accessible to experimental determination, with the theoretically important dissociation energies. Depending on which of the sums of energies (of the right or left side) has the greater value, we encounter a positive or negative thermal effect, i.e. exothermic or endothermic reactions. However, one further important point must be taken into account. We have called the dissociation energy that energy which was necessary in order to decompose a molecule into normal unexcited atoms. However, we shall see below that in the primary process of decomposition a molecule by no means always gives normal atoms. If, for example, we wish to decompose a molecule of iodine photochemically by absorption of light—which is entirely possible—then for this a dissociation energy \(Z\) of 57 kg cal per mole is required, but in this primary process
there arise not two ordinary iodine atoms, but only one, while the second is in the more energy-rich, i.e. excited, \(^{2}P_{1}\) state of its electron shell. This excitation energy of the atom \(A_{n}\)—in round numbers \(22\) kg cal—is released after the decay of the molecule, and therefore it must be subtracted from the decomposition energy in order to obtain the true dissociation energy \(D\), which is equal to \(35\) kg cal. Thus the relation always holds
\[ D_{m}=U_{m}-A_{n}. \tag{25} \]
In the case of stable molecules in the normal state (i.e. at not too high temperatures) the excitation energies of the resulting atoms will always be less than the energies expended in decomposition, so that the dissociation energies always turn out to be positive; we may regard them as the “absolute” heats of the reaction of molecular decomposition. But in that case equation (24) represents the most general expression for the important law of additivity of the thermal effects of chemical reactions.
The dissociation energy is not always amenable to chemical determination, since for this relatively high temperatures are constantly required, as well as exact measurements of the equilibrium constants and of their temperature dependence. In this way, for example, the dissociation energies of the halides \(Cl_{2}\), \(Br_{2}\), and \(I_{2}\) were determined, and the dissociation energy of hydrogen was approximately estimated. On the other hand, thanks to the existence of the law expressed by equality (24), it is sufficient to know the dissociation energies of a few molecules in order, from the known thermal effects of chemical reactions, to calculate the dissociation energies of a whole series of compounds. Thus, for the majority of organic substances it is sufficient to know the dissociation energies of hydrogen (\(D=100.5\) kg cal), oxygen (\(D=162\) kg cal), nitrogen (\(270\) kg cal), and the heat of sublimation of carbon (\(S=140\) kg cal), and in addition the heats of combustion, which are easily determined calorimetrically. But precisely the elements just enumerated form very stable molecules, whose decomposition into atoms is not readily effected thermally. It is precisely here that spectroscopic methods are most convenient.
However, before I pass to consideration of the latter, it is necessary first to dwell briefly on the conversion factors from spectroscopic units of energy to the thermal units more customary for chemists. According to the well-known equation
\[ \text{Energy} = h\nu = eV \tag{26} \]
we can measure energies also by specifying the corresponding frequency or, when energy is supplied by electron impacts, by specifying the accelerating voltage of the electron. In these cases the energies of individual atoms or molecules are measured, in contrast to thermal quantities, which, as a rule, refer to one mole, i.e. \(6.06 \cdot 10^{23}\) molecules. Using the known values of the quantities \(h\) and \(e\), we obtain the following conversion factors:
1 volt is equivalent to \(8100\ \text{cm}^{-1}\), or equivalent to
\(23.07\ \text{kg cal}/\text{mole}\).
All three units will be used by us in what follows.
The decomposition of a molecule into its constituent parts can be carried out spectroscopically in the following three ways: 1) one may break the molecular bond by bombarding it with electrons and determine the voltage required for this by the known methods of electron impacts; 2) one may impart to the molecule, by excitation (thermal or electrical), such a rotation that the centrifugal force exceeds the bond and the molecule flies apart; 3) one may impart to the vibration of the nuclei such an energy that the chemical bond will be broken and the molecule will again dissociate.
The practical applicability of the first method of decomposition is still hindered by the difficulties encountered in interpreting the results obtained: with the present state of our knowledge of band spectra from spark currents and of the voltages determining critical potentials, we cannot with sufficient certainty distinguish the excitation potential from the potential of ionization or dissociation. Only in some cases \((\mathrm{H}_2, \mathrm{N}_2)\) has it proved ...
possible to draw conclusions about dissociation energies by indirect methods, and moreover with a rather large error. Thus, for the present this method can be used only to confirm results obtained by other methods.
As for the second method, several examples may be cited from which it follows that even at a relatively small rotational energy a molecule may become unstable. Spectroscopically this is manifested in the fact that within a band, the individual lines of which represent different rotational velocities of the molecule, beginning with some definite line, the intensity of the series suddenly falls, the lines themselves become very diffuse and, in the end, disappear altogether. Thus, for example, in the HgH molecule the series breaks off at \(m = 31\), which corresponds, in round numbers, to a rotational energy of \(5000\ \mathrm{cm}^{-1}\); in the case of AlH the series breaks off at \(m = 23\) (energy \(3400\ \mathrm{cm}^{-1}\)); and in the apparently very unstable molecule CaH the series breaks off already at \(m = 11\) (energy \(540\ \mathrm{cm}^{-1}\)).
As Ludloff has shown1, the condition for this instability of rotation is the vanishing of the second derivative, in finite differences, of the rotational energy, i.e.,
\[ \frac{\Delta^2 F(m)}{\Delta m^2}=0. \tag{27} \]
However, it is hardly permissible, without any reservations, to identify this limiting rotational energy with the dissociation energy, since at present we still can say nothing about the mechanism of such a separation and must even assume that the atoms of the molecule, owing to the centrifugal force, fly apart with an excess of kinetic energy. If this is so, then the true values of the dissociation energy must be smaller than those found in this way. The only case in which this has so far been verifiable is the HgH molecule. In this case it turns out that the dissociation energy determined by the third
to the method, with sufficient accuracy, is \(2990\ \mathrm{cm}^{-1} = 0.37\,V\), i.e. in fact considerably less than the maximum rotational energy \((5000\ \mathrm{cm}^{-1})\). For the two other molecules mentioned above, unfortunately, results of determining the dissociation energy from the vibrations of the nuclei are still lacking.
The phenomenon mentioned above (the blurring of lines and the weakening of the intensity of series) must be closely connected with a completely analogous phenomenon observed by Henri¹ in a large number of spectra, namely—in most spectra of polyatomic molecules. This phenomenon, which Henri called “predissociation” of the molecule, consists in the following. If one investigates the structure of a series of bands corresponding to one and the same electronic transition, then, as the vibration increases, it turns out that, beginning from a certain boundary, the bands become blurred and continuous, the fine rotational structure disappears, and in the end only the nuclear vibrations alone prove to be quantized. In known cases, as, for example, in \(S_2\), \(NO_2\), \(SO_2\), the transition from a spectrum with quantized rotation to a spectrum with continuous bands occurs quite abruptly, so that the boundary can be determined with an accuracy of a few tenths of an Angstrom. The number of successive vibrational states with quantized rotation which the molecule possesses before predissociation sets in varies greatly depending on the substance. Investigation of the properties of molecules in this state of predissociation has further shown that the molecules become more active with respect to chemical reactions, and that fluorescence upon excitation by the corresponding rays becomes very weak; that, finally, the frequency of the vibrations of the atoms becomes smaller, and consequently the internuclear distance larger, and that predissociation, as the temperature rises, appears already at small vibrational quantum numbers. All this speaks in favor of the fact that here a transition to a more unstable molecule is observed. However, what exactly
¹ V. Henri, Structure des molécules. Paris, 1925; Z. Physik. 49, 774, 1928; cf. also K. F. Bonhoeffer and L. Farkas, Z. Phys. Chem. 134, 337, 1927.
what lies behind this phenomenon is not yet entirely clear. I shall give a table compiled on the basis of Henri’s data.
TABLE 10.
Predissociation.
| Molecule | Frequency of vibrations \(a\) | Vibrational levels stable \(n\) | Vibrational levels unstable \(n'\) | Predissociation limit |
|---|---|---|---|---|
| Sulfur \(S_2\) | 924 | 14 | 4 | 2794 Å |
| Sulfur dioxide \(SO_2\) | 220 | 22 | 0 | 2800 |
| Formaldehyde \(H_2CO\) | 1190 | 8 | 6 | 2700 |
| Phosgene \(Cl_2CO\) | 457 | 9 | 7 | 2700 |
| Methylamine \(CH_3NH_2\) | 280 | 3 | 14 | 2450 |
| Benzene \(C_6H_6\) | 920 | 10 | 1 | 2200 |
| \(p\)-Xylene \(C_6C_6H_4\) | 1100 | 4 | 3 | 4420 |
| Pyridine \(NC_5H_5\) | 480 | 4 | 2 | 2750 |
The possibility of the third and most interesting case of decomposition of molecules—the instability of nuclear vibrations—was first indicated and investigated in the most detailed manner by Franck. For this purpose Franck considered a number of typical absorption spectra (\(Cl_2, Br_2, I_2\)), in which the bands of the quantum series (i.e. the bands \(n''=\mathrm{const}\), \(n'=0, 1, 2, 3,\ldots\)) merge at a certain point, which, just as at the boundary of a line spectrum, is adjoined by a region of continuous absorption. In the case of an atom this phenomenon, as is known, is explained by its ionization; in the case of a molecule Franck sees in this a sign of the breakup of the molecule into the atoms composing it, with both atoms of the molecule here having to fly apart with an excess of kinetic energy and thereby give rise to continuous absorption. For a better understanding of this phenomenon let us return to Fig. 1 (see issue 5, p. 686), which shows the dependence of the potential energy of binding of a molecule on the distance between the atoms. We saw that the potential energy, as the distance between the atoms decreases to a minimum—the position of equilibrium—
decreased, and then rapidly increased. The ordinate of this minimum of the potential is nothing other than our dissociation energy, i.e. the potential energy that must be expended in order completely to separate the two atoms from one another.
Upon excitation of the molecule the atoms perform oscillations about the equilibrium position, and with an amplitude that is the greater the more vibrational energy has been supplied, i.e. the greater the corresponding quantum number of the band. Thus, from the fact that the potential curve approaches the \(r\)-axis asymptotically, it follows directly that an arbitrarily large amount of vibrational energy cannot be imparted to the molecule and that, consequently, the sequence of bands, the length of which represents nothing other than the vibrational energy expressed in units of frequency, must asymptotically approach a certain limit.
Fig. 15.
The place of convergence of the bands, which in very small series can easily be found by extrapolation, is detected at the onset of molecular dissociation, and its distance from the first band of the series, expressed in units of energy, gives the energy required for this dissociation (Fig. 15). It should be noted that the continuous spectrum adjoining the place of convergence of the bands is, as a rule, observed only in absorption, for it corresponds to the excess energy that the molecule retains after dissociation. Thus, for example, if the dissociation is caused photochemically, then the atoms fly apart with the corresponding excess of potential energy. The reverse process of emission would mean that two atoms which already possessed precisely this energy in advance
as excess energy, are encountered in forming a molecule—a process relatively rare and capable of occurring only at high temperatures.
In this elegant way the dissociation energy can easily be determined spectroscopically. In doing so, however, one must take into account a whole series of further points of view. First of all, we can combine bands into systems of bands in two ways, depending on whether we choose bands corresponding to one and the same initial state (longitudinal series), or to one and the same final state (transverse series). In the case of completely excited molecules, i.e. those in which all three kinds of energy are excited—electronic energy, vibrational energy, and rotational energy—and in what follows we shall confine ourselves only to such molecules, we can always establish from the spectrum two series of decay. In other words, each state of excitation of the electronic system of our molecule possesses a spectroscopically determinable decay energy. But I have already pointed out that in the decay of a molecule one usually obtains not normal, but excited atoms, and that the relation between the dissociation energy, the decay energy of the molecule, and the excitation energy of the atoms is expressed by the equality
Fig. 16.
\[ D_m = Z_m - A_a . \tag{25} \]
From this follow certain important consequences, which we shall clarify with the aid of the energy diagram (Fig. 16). Here \(y_m\) denotes the excitation energy of the electronic system of the na-
...of a molecule, which is equivalent to the frequency of the zero point of the band system \(n', n''=0\). \(Z'\) and \(Z''\) are the lengths of the band sequences in the initial and final states, i.e., the corresponding dissociation energies of the molecule. Further, it is easy to see that \(\nu_k\) is the frequency of the convergence point in the excited state. To reach this point from the lowest energy level, we can proceed in two ways. First, we can initially excite the electronic system of the molecule and then decompose this excited molecule into atoms; then \(\nu_k=\nu_m+Z''\). Second, we can first decompose the unexcited molecule into two atoms (for simplicity we shall assume that one atom is always in the normal state, while the other possesses the excitation energy \(A_a'\)) and, after this decomposition, transfer one of the resulting atoms from the state \(A_a'\) to the state \(A_a''\), obtained upon the decay of the excited molecule. For this last operation an expenditure of energy \(\nu^a=A_a'-A_a''\) is required, and we shall obtain the same state as in the first case. Thus, in general, the equality holds
\[ \nu_k=\nu_m+Z''=\nu_a+Z'. \tag{28} \]
It follows from this, above all, that the frequency distance of the two convergence points, i.e. \(\nu_k-Z'\), is equal to the difference of the energies of two atomic states\(^1\) and at the same time to the frequency of a certain atomic line. If, in a particular case, the two convergence points coincide, then the products of decay in the initial and final states of the molecule are one and the same.
Thus, if the lengths of the band sequences can be determined (for example, by extrapolating a serial formula) and if the excitation states of the products of decay are known, then equation (25) immediately gives the dissociation energy of the corresponding state of the molecule. However, the dissociation energy of the lowest energy level of the molecule is chiefly of interest, i.e. of the normal—
\(^1\) Namely, of the states into which the molecule decomposes in the normal and in the excited state.
… state. The determination of this energy is aided by equation (28), from which the relation follows directly
\[ D_n = D_m + A_m . \tag{29} \]
Consequently, to the dissociation energy determined from equation (25) we must add—in the majority of cases a known quantity—the excitation energy of the corresponding state of the molecule, in order at once to obtain the dissociation energy of the ground state. The circumstances are even simpler when we can decompose the molecule directly by absorption of light, i.e. photochemically: in this case—as has already been mentioned—in the absorption spectrum there is directly accessible to observation the point of convergence of the bands of the final state \(\nu_k\), situated at the beginning of continuous absorption. Then, from equations (28) and (25), there is the simple relation:
\[ D_n = \nu_k - A_k'' . \tag{30} \]
There is still another path which, under certain circumstances, leads us to the goal: we first decompose the molecule into neutral normal atoms, for which we need the dissociation energy \(D_n\); then we ionize one of the atoms, which requires the expenditure of the ionization work \(J_a\). On the other hand, we can first ionize the molecule (work \(J_m'\)), and then decompose this ionized molecule into a neutral and an ionized atom—both in the ground state (work \(D_i\)). The final state in both cases is one and the same, and therefore the equality is always valid:
\[ D_n + J_a = D_i + J_m . \tag{31} \]
We thus see that, by means of the indicated equalities, the results obtained from the line spectra of the atom (\(J_a, A_a\)) turn out to be closely connected with the known limiting values of the corresponding band spectra.
We shall now explain these equalities first with the example of the hydrogen molecule. Hydrogen possesses in the ultraviolet region (the Schumann region) at present
time by a well-investigated absorption spectrum, which consists of two systems of bands with a common initial state. The zero position \(n', n''=0\) of one system lies at \(99\,040\ \mathrm{cm}^{-1}\), the zero position of the other system at \(90\,080\ \mathrm{cm}^{-1}\); the convergence point for the normal state of the molecule, in the present case conveniently determined by extrapolation, lies below these zero positions, in round numbers at \(Z'=35\,250\ \mathrm{cm}^{-1}\) \((=4.34\ \mathrm{V})\). Extrapolation in both excited states, owing to the small length of the observed sequence of bands, gives rather inaccurate results; however, it turns out that the two convergence points coincide within the accuracy of the measurements. The mean value obtained is \(\nu_k=118\,000\) \((=14.6\ \mathrm{V})\). The distance between the convergence points of the initial and final states is therefore in both cases, in round numbers, \(118\,000-35\,250=82\,750\ \mathrm{cm}^{-1}\) \((=10.2\ \mathrm{V})\). But this is precisely the excitation energy of the first line of the Lyman series, which lies at \(\lambda 1215\) \((82\,250\ \mathrm{cm}^{-1}=10.15\ \mathrm{V})\). We may thus conclude from this that the initial state of the molecule (the ground state) in any case gives two normal atoms, whereas the excited molecule gives one normal and one excited atom, whose electron is in the two-quantum orbit \(2^2P_1\).
Further, from the theory of the hydrogen atom1 we know that this two-quantum state must be double, with an energy difference which, in the present case, is of course not appreciable \((\Delta \nu=0.366)\). We may therefore conclude that one system of bands upon dissociation gives the atom \(2_1\), and the other the atom \(2_2\). Both these systems of bands thus correspond to the close doublet of the Lyman line \(\lambda 1215\). From the length of the series of bands in the ground state one obtains a dissociation energy equal to \(4.34\ \mathrm{V}=100\ \mathrm{kg\ cal.}\); from the convergence point of the final states, determined by extrapolation \((118\,000=14.6\ \mathrm{V})\), after subtracting the excitation energy of the atoms \((10.15\ \mathrm{V})\), one obtains a dissociation energy of \(4.4\ \mathrm{V}\)—
\[ \nu=R\left(\frac{1}{1^2}-\frac{1}{2^2}\right). \]
with less accuracy. On the other hand, the hydrogen molecule can be decomposed directly photochemically, by absorption of light: at \(\lambda\ 849.4\) (\(14.53\ \mathrm{V}\)) a region of continuous absorption is sharply revealed, testifying to this decomposition. If, again, the excitation energy \(10.15\ \mathrm{V}\) [equation (30)] is subtracted, then, in good agreement with the preceding result, the value \(D_n = 4.38\ \mathrm{V} = 101\ \text{kg cal.}\) is obtained. Since, further, we know the ionization potentials of the atom (\(13.54\ \mathrm{V}\)) and, approximately, the ionization potential of the molecule (\(16.1 \pm 0.2\ \mathrm{V}\)), it follows from equation (31) that for the dissociation energy of the theoretically especially interesting ion \(\mathrm{H}_2^+\) the value is \(1.8\ \mathrm{V} = 42\ \text{kg cal.}\) The band spectrum of the ion \(\mathrm{H}_2^+\) has not yet been established; it is assumed, however, that parts of the known many-line spectrum of hydrogen are emitted by \(\mathrm{H}_2^+\).
In this most favorable case we thus have three mutually checking determinations of the dissociation energy, which make it possible to establish the latter with an accuracy, in round figures, of up to \(0.5\%\). For these results we are indebted chiefly to the work of Witmer1 and Diecke and Hopfield.2 Conditions as favorable as those for hydrogen are found in all those cases in which we can bring about the decomposition of the molecule photochemically, by absorption of light, i.e. practically for many absorption spectra. Therefore we shall first consider precisely these cases, which are limited to the elements of the first (\(\mathrm{Na}_2\), \(\mathrm{K}_2\), and \(\mathrm{NaK}\)), sixth (\(\mathrm{O}_2\), \(\mathrm{S}_2\), \(\mathrm{Se}_2\), \(\mathrm{Te}_2\)), and seventh groups (\(\mathrm{Cl}_2\), \(\mathrm{Br}_2\), \(\mathrm{J}_2\)) of the periodic system.
In particular, the group of the halides is distinguished by characteristic band spectra, in which the series of bands of the excited state are striking by their length: they can be traced almost directly up to the point of convergence, and the latter, moreover, can be determined from the beginning of continuous absorption. In chlorine the point of convergence lies at \(\lambda\ 4785\), in bromine—\(\lambda\ 5107\), in iodine—\(\lambda\ 4995\). The points of convergence of the principal co-
state, owing to the insufficient length of the observed sequence of bands, is difficult to establish; only in iodine, where with the aid of the resonance spectrum (fluorescence) approximately 27 members of a band series can be obtained, is an almost certain, though still rather considerable, extrapolation possible1 (from \(n'=27\) to \(n'=112\)). The convergence point should be sought approximately at \(13\,500\ \mathrm{cm}^{-1}\) \((=1.67\ \mathrm{V})\) below the zero point \(n,n''=0,0=15\,600\); consequently it is removed from the final state by approximately \(20\,050-13\,500=6550\ \mathrm{cm}^{-1}\) \((=0.8\pm0.2\ \mathrm{V})\). But this corresponds, within the limits of error, to the excitation energy of the iodine atom from the lower \({}^{2}P_{2}\) state to the next higher \({}^{2}P_{1}\) state (iodine has no other excitation energies of a similar order of magnitude). Thus we may assume that in iodine, and also in the other halogens, the molecule in the normal state dissociates into two normal atoms, while in the excited state it dissociates into a normal \({}^{2}P_{2}\) atom and an excited \({}^{2}P_{1}\) atom. The dissociation energies are thereby determined (cf. Table 11).
The circumstances are less favorable in group 6 of the periodic system. Oxygen is in the best situation, and is, incidentally, of the greatest interest. Here two absorption spectra are known: the atmospheric absorption bands of oxygen in the red region, and the absorption spectrum in the ultraviolet region. In the latter spectrum the band series of the excited state can be followed almost directly to the convergence point, so that the extrapolation is small (from \(n''=18\)—the last measured quantum at \(\lambda\ 1756\)—to \(n''=21\)); continuous absorption begins at \(\lambda\ 1751\) \((Z=7.05\pm0.03\ \mathrm{V}=162\ \text{kg cal})\). Similar values are given by the final state of the atmospheric bands (\(7.0\ \mathrm{V}\)) and the common ground state (\(6.7\ \mathrm{V}\)), although in both cases the extrapolation is considerable (from \(n''=4\) to \(n''=51\) and from \(n'=17\) to \(n'=69\)). The lowest term of the oxygen atom is the triplet term \({}^{3}P_{i}\), for which the level separations amount to only a few hundredths
volts (0.01 or 0.02 V), so that in the present case they cannot be detected. Thus we again have dissociation in the ground state of the molecule, probably into normal atoms and into atoms \(^{3}P_i\)—in excited states. There is no other possibility here, since the next higher term of the oxygen atom would require an excitation energy of \(9.1\ \mathrm{V}\).1 For the other elements—\(S_2\), \(Se_2\), and \(Te_2\)—the circumstances are analogous, though less favorable, since the convergence points for them cannot be determined with such accuracy. A further difficulty is added here, namely that for selenium and tellurium we do not yet know the lower excitation stages of the atoms, and therefore the latter have to be estimated with the aid of data for oxygen and sulfur. The resulting results are given in Table 11.
In the alkali metals, which in the vapor state are also sometimes encountered as diatomic molecules, the absorption spectra of \(Li_2\), \(Na_2\), \(K_2\), and even \(NaK\) have been established. For the energy
Thus,
\[ \text{photochemically:}\quad NO_2 + 77\ \mathrm{kg\ cal} = NO + O. \]
\[ \text{thermally:}\quad NO_2 + 13\ \mathrm{kg\ cal} = NO + \frac{1}{2}O_2. \]
From these equations, by subtraction, one immediately obtains for the dissociation energy of oxygen \(128\ \mathrm{kg\ cal}\), and hence the excitation energy of the \(^{1}D\)-term is equal to \(1.4\ \mathrm{V}\).
Note by E. Sh.
dissociation, however, reliable numerical data have been obtained only in the case of \(Na_2\). The diagram explains the circumstances that arise here, which are analogous to what is observed in the case of hydrogen. Here, too, there are two absorption spectra—the “green” and the “red.” The convergence point of the ground state lies \(1.0\,V\) above the zero level; the convergence points of both excitation stages again coincide within the limits of accuracy of the measurement, the series of bands of the green system having a length of \(4850\ \mathrm{cm}^{-1}\) \((0.6\,V)\), while the series of the red system has a length of \(101\,000\ \mathrm{cm}^{-1}\) \((1.25\,V)\) (Fig. 17).
Fig. 17.
The distance between the convergence points of the final and initial states corresponds, to within the accuracy, to the frequency of the \(D\)-lines of the sodium atom \(5890/96\). Hence, as before, we conclude that the ground state decomposes into two normal atoms, while the excited state decomposes into a normal atom and an atom in the state \({}^2P_i\). Probably one system again leads to an atom in the state \({}^2P_1\), the other—to the state \({}^2P_2\). The corresponding difference in energy upon extrapolation, however, cannot be established.
Consequently the dissociation energy of the \(Na_2\) molecule is equal to \(1.0\,V = 23\ \mathrm{kg\ cal.}\)
In the preceding examples it is especially noteworthy that, as might have been expected, only those absorption spectra are observed which are associated with the lowest terms of the corresponding atoms, namely with the terms \(1^2S\) and \(2^2P_i\) in hydrogen and sodium, with the terms \({}^2P_i\) in the halogens, and with the triplet \({}^3P_i\) in the oxygen atom.
In many cases, especially in emission spectra,
sequences of bands cannot be traced all the way to the place where they merge in the spectrum. In such a case one has to resort to a more or less considerable extrapolation of the serial formula—the \(an-bn^2\) extrapolation—which we have already used more than once in the examples given. Birge and Sponer1 showed, however, that in many cases this extrapolation, which at first sight seems unreliable, of a formula with two constants gives a value whose accuracy is within \(0.1\ \mathrm{V}\). This happens because, in a first approximation, when extrapolating, the effects of the higher terms of the serial formula cancel one another. In fact, one may successfully use the formula:2
\[ \nu_k=\nu_m+\frac{a^2}{4b}. \tag{32} \]
We shall confine ourselves here to only a few examples; as for the formulas, reference should be made to Tables 1 to 4.
For oxygen, extrapolation of the ultraviolet spectrum shows that the dissociation energy in the ground state of the molecule is \(11.7\ \mathrm{V}\); extrapolation of the first positive group leads to the value \(11.9\); on the other hand, Sponer, from the afterglow spectrum of nitrogen (see below) and from its excitation potential measured earlier, obtained the value \(11.4\ \mathrm{V}\). Further, one can also make use of the ionized molecule. For the latter a somewhat uncertain value of \(9.5\ \mathrm{V}\) is obtained as the dissociation potential. With the aid of the measured ionization potentials of the molecule (\(16.5\ \mathrm{V}\)) and of the atom (\(14.5\ \mathrm{V}\)), for the neutral molecule the calculation by means of equation (31) gives the value \(11.5\ \mathrm{V}\). Thus, on the average, an energy of \(11.6\ \mathrm{V}=268\) kg cal is obtained, and this is precisely the energy required in order to decompose the nitrogen molecule into its atoms. A known check on this
one may also use the value obtained for two spectra of nitrogen oxide. In the latter case, with the aid of a somewhat uncertain extrapolation, for the ground state of the NO molecule one obtains a dissociation energy of \(7.9\ \mathrm{V} = 182\ \text{kg cal}\). From the heat effect of the reaction
\[ \mathrm{N}_2 + \mathrm{O}_2 = 2\mathrm{NO} - 43.1\ \text{kg cal} \]
one may then, by equation (24), calculate the dissociation energy of nitrogen, since the dissociation energy of oxygen is already known. In this way one obtains \(246\ \text{kg cal}\)—a value which, only as a consequence of the unreliability of the extrapolation for NO, proves to be somewhat low.
In the case of carbon monoxide, which we shall consider as the last example of this type, for the ground state one obtains the value \(11.2\ \mathrm{V} = 254\ \text{kg cal}\). Here too a check is possible by means of the reaction
\[ \mathrm{C}_{\text{solid}} + \frac{1}{2}\mathrm{O}_2 = \mathrm{CO} + Q, \]
if for the heat of sublimation of carbon one uses the value \(141\ \text{kg cal}\) reported by Kohn and Guckel.^1 The calculation has already been given on p. 753 \((\mathrm{CO} = 249\ \text{kg cal})\).
The method considered above for determining the dissociation energy from the point of convergence of bands, which is determined by extrapolation, is applicable only to homeopolar compounds, i.e. to compounds which, upon dissociation, do not give ions. In attempts at the optical determination of dissociation energies also in the case of a polar bond, it turned out above all that the number of truly polar molecules in the gaseous state (spectra in these cases are observed only for gaseous molecules) is apparently smaller than had been thought before, and that in general one cannot sharply distinguish between polar and nonpolar bonds. Thus, for example, in the case of hydrogen halide compounds in gaseous form, also on the basis of other criteria (for example, molar refraction), we must conclude that the bond is nonpolar, although these compounds in solution immediately dissociate into ions.
^1 H. Kohn und M. Guckel. Naturwiss. 12, 139, 1924.
In calculating the dissociation energy from the long-wavelength limit of continuous absorption in hydrogen iodide \((\lambda\ 3200)\), after subtracting the excitation energy of the resulting iodine atom \((0.91\ \mathrm{V})\), one obtains \(D = 67\ \mathrm{kg\ cal}\)—a value in good agreement with that calculated from equation (24) from \(D\) for \(\mathrm{H}_2\) and \(\mathrm{J}_2\) and the thermal effect. Likewise, the very crude extrapolation of the infrared absorption bands of \(\mathrm{HCl}\) gives for \(D\) an acceptable value, namely \(111\ \mathrm{kg\ cal}\), as compared with the calculated value \(101\ \mathrm{kg\ cal}\). In exactly the same way, Kuhn and Franck1 showed that in the case of silver halide compounds the bond is apparently nonpolar, and they determined the dissociation energies by the method of band convergence. For the polar-bound alkali halide compounds, according to Franck, Kuhn, and Rollefson,2 the circumstances are such that, although decomposition in the ground state gives ions, photochemical decomposition by absorption of light is also possible. In this case, upon excitation of the molecule, in contrast to nonpolar molecules, dissociation gives two neutral and normal atoms, since the excitation must bring about the transfer of an electron from one partner to the other. Here continuous absorption spectra have been studied—and at high temperatures, moreover, diffuse band sequences on the long-wavelength side—which consist of two absorption regions: the long-wavelength limit of one of them corresponds to dissociation into two normal atoms, whereas the limit of the other corresponds to dissociation into a normal atom of the alkali metal and an excited \((^2P_1)\) atom of the halogen. In the case of chlorine compounds, owing to the small energy difference, these two absorption regions are indistinguishable; for the other halides the frequency differences approximately coincide with the excitation energy of the corresponding halogen. However, the limits can be established only very roughly. For numerical values see Table 11.
In conclusion I shall also indicate, for completeness, two opti—
TABLE 11.
Dissociation energies.
| Molecule | Upper predissociation limit | \(\nu_k\), in volts | Excitation state | Dissociation energy | \(D\), chemical |
|---|---|---|---|---|---|
| \(\mathrm{H_2}\) | \(849.4\,\text{Å}\) | 14.51 | \({}^{2}P_i = 10.15\,\mathrm{V}\) | \(4.36 \pm 0.01\,\mathrm{V} = 100.5\,\text{kg cal}\) | \(70\text{—}100\,\text{kg cal}\) |
| \(\mathrm{O_2}\) | 1751 | 7.05 | \({}^{1}D = 1.4\,\mathrm{V}\) | \(5.6 \pm 0.1 = 128\) | — |
| \(\mathrm{O_2^{+}}\) | — | 6.5 | \(\sim 0.01\) | \(6.5 \pm 0.3 = 150\) | — |
| \(\mathrm{Cl_2}\) | 4785 | 3.58 | \({}^{2}P_1 = 0.11\) | \(2.47 \pm 0.02 = 57.0\) | 57 |
| \(\mathrm{Br_2}\) | 5107 | 2.41 | \({}^{2}P_1 = 0.45\) | \(1.96 \pm 0.02 = 45.2\) | 46 |
| \(\mathrm{J_2}\) | 4995 | 2.47 | \({}^{2}P_1 = 0.94\) | \(1.53 \pm 0.01 = 35.2\) | 34.5 |
| \(\mathrm{Na_2}\) | (4000) | 3.1 | \({}^{2}P_i = 2.10\) | \(1.0 \pm 0.1 = 23\) | — |
| \(\mathrm{N_2}\) | — | — | — | \(11.6 \pm 0.2 = 268\) | — |
| \(\mathrm{N_2^{+}}\) | — | — | — | \(9.1 \pm 1 = 210\) | — |
| \(\mathrm{NO}\) | — | — | — | \(7.9 \pm 0.5 = 182\) | \(191\,\mathrm{N_2} + \mathrm{O_2}\) |
| \(\mathrm{CO}\) | — | — | — | \(11.0 \pm 0.5 = 254\) | 250 from \(\mathrm{O_2}\) |
| \(\mathrm{CO^{+}}\) | — | — | — | \(9.6 \pm 0.3 = 222\) | — |
| \(\mathrm{TlCl}^{*}\) | \(<1850\) | 6.7 | \(\mathrm{Tl}\,\lambda 3776;\ 3.28\,\mathrm{V}\) | \(3.4 = >78\) | 87 |
| \(\mathrm{TlBr}^{*}\) | 1915 | 6.5 | \(\mathrm{Tl}\,\lambda 3776;\ 3.28\,\mathrm{V}\) | \(3.2 \pm 0.1 = 74\) | 73 |
| \(\mathrm{TlJ}^{*}\) | 2085 | 5.9 | \(\mathrm{Tl}\,\lambda 3776;\ 3.28\,\mathrm{V}\) | \(3.6 \pm 0.3 = 61\) | 58 |
| \(\mathrm{NaJ}^{*}\) | 2460 | 5.0 | \(\mathrm{Na}\,\lambda 5890/96\) | \(2.9 \pm 0.1 = 69\) | 69 |
| \(\mathrm{CsJ}^{*}\) | 2085 | 5.9 | \(\mathrm{Cs}\,\lambda 4555/93\) | \(3.2 \pm 0.2 = 74\) | 75 |
| \(\mathrm{CaBr}\) | 3300 | 3.8 | — | \(3.8 \pm 1 = 88\) | 104 |
| \(\mathrm{KJ}\) | 3800 | 3.2 | — | \(3.2 \pm 1 = 75\) | 84 |
| \(\mathrm{KBr}\) | 3100 | 3.9 | — | \(3.9 \pm 1 = 90\) | 100 |
| \(\mathrm{KTl}\) | 2800 | 4.5 | — | \(4.5 \pm 1 = 105\) | 103 |
| \(\mathrm{NaBr}\) | 3100 | 3.9 | — | \(3.9 \pm 1 = 90\) | 84 |
| \(\mathrm{AgCl}\) | — | (3.1) | \(\mathrm{Cl}\ 0.11\,\mathrm{V}\) | \(3.0 \pm 0.5 = 70\) | — |
| \(\mathrm{AgBr}\) | — | (3.1) | \(\mathrm{Br}\ 0.45\) | \(2.7 \pm 0.5 = 62\) | — |
| \(\mathrm{AgJ}\) | — | (3.2) | \(\mathrm{J}\ 0.94\) | \(2.3 \pm 0.5 = 53\) | 47 |
| \(\mathrm{Cd_2}\) | 2561 | 4.82 | \({}^{3}P_i = 3.78\) | \(1.03 \pm 0.03 = 23.8\) | — |
* Determined by Terenin’s method (fluorescence method).
of the methods of determining or estimating the dissociation energy, without dwelling on them in detail. Terenin’s fluorescence method consists in illuminating the corresponding gas with light of a definite wavelength, i.e. of definite energy, and observing whether the atoms produced by photochemical decomposition make themselves known by the emission of one or another spectral line. The energy reserve of the longest-wavelength absorption light that already gives such an effect, after subtracting the energy emitted after the breakup of the molecule as an atomic line, evidently gives the dissociation energy of the molecule. Thus here the point of convergence is determined experimentally with the aid of a sensitive reagent (emission of a spectral line). Terenin investigated in this way the molecules NaJ, CsJ, CuJ, TlJ, TlCl, TlBr; in the case of TlJ, for example, it was established that the thallium line \(\lambda 2776\) (and \(\lambda 5351\)) is emitted already upon absorption of the wavelength \(\lambda 2080 \pm 20\). The latter corresponds to an energy of \(136\ \mathrm{kg\,cal/mol}\), the thallium line to an energy of \(75\ \mathrm{kg\,cal}\), so that the dissociation energy must be
\[ 136 - 75 = 61 \pm 1\ \mathrm{kg\,cal}. \]
The chemiluminescence method (Haber and Zisch) is based on the reverse process: the optical utilization of the heat of combination that is liberated. The phenomenon of chemiluminescence consists in the fact that atoms, combining into molecules, liberate energy which goes into the optical excitation of the atoms or molecules of the participants in the reaction, or of an admixed foreign gas. The highest degree of excitation thereby obtained, established from the emission of the corresponding lines, gives in such a case the upper limit of the bond or dissociation energy. In this way Spohrer1 determined, for example, from the glow of active nitrogen, the dissociation energy of the nitrogen molecule.
In the case of homologous compounds one may further assume that the dissociation energies are related approximately as the fourth powers of the bonding force. As a result of this (compare p. 663)
from the magnitude of the observed vibrations of the nuclei one can determine the dissociation energy by the formula
\[ \nu^{2}\mu=\text{const} \]
(\(\mu\)—the reduced mass, see p. 633). This formula has proved very suitable for many compounds (halides and halogen compounds).
On the basis of the material obtained concerning the optical determination of dissociation energies, we are now in a position, using equation (24), to calculate the dissociation and cleavage energies of a number of other compounds from their calorimetrically determined heat effects. Such, for example, are the decomposition energies of some simple hydrogen compounds (Table 12). Likewise, for very
TABLE 12.
Dissociation energies of hydrides.
| 2 atoms | 2 atoms | 3 atoms | 3 atoms | 4 to 6 atoms | 4 to 6 atoms |
|---|---|---|---|---|---|
| HF | 185 kg cal | H\(_2\)O | 240 kg cal | NH\(_3\) | 294 kg cal |
| HCl | 101 | H\(_2\)S | 175 | CH\(_4\) | 362 |
| HBr | 86 | H\(_2\)Se | 145 | C\(_2\)H\(_4\) | 464 |
| HJ | 68 | H\(_2\)Te | (130) | C\(_2\)H\(_2\) | 329 |
many organic compounds, the dissociation energies are easily determined from calorimetrically measured heats of combustion. We know, indeed, that the combustion of solid carbon (diamond) to carbon dioxide requires, in round numbers, 94 kg cal, and the combustion of hydrogen to liquid water—34.5 kg cal; if this is referred to the atomic quantities C and H [equation (24)], one obtains \(141+94=235\) kg cal or \(50+34.5=84.5\) kg cal. If, for example, a hydrocarbon compound consists of \(x\) atoms of carbon and \(y\) atoms of hydrogen, then the dissociation energy according to equation (24) will be
\[ D=235x+84.5y-Q. \]
But for many such compounds the dissociation energies of complex molecules are often additively composed of the works
cleavage of individual atoms; the work of cleavage of such bonds, consequently, is independent within wide limits of neighboring bonds. Thus, for example, in the aliphatic hydrocarbon series the bond energy increases as a result of the addition of the group CH₂ each time by 247 kg cal. With each such addition one C—C bond and one C—H bond are added, so that, using also the values of \(D\) for CH₄, C₂H₆, etc., one can calculate the bond energies C—H and C—C. In an analogous way, from C₂H₂ and C₂H₄, or from C₆H₆, one can calculate the energies of cleavage of double (C=C), triple (C≡C), and cyclic C—bonds (Table 13). Further it turns out—
TABLE 13.
Bond energies.
| Bond | Energy | Bond | Energy |
|---|---|---|---|
| C—H | 90.5 kg cal | C—O—C (ether) | 171 kg cal |
| C—C (aliphatic) | 86 | C=O (aldehydes) | 171 |
| C=C (aromatic) | 98 | C=O (carbon oxide) | 248 |
| C=C | 102 | —C≡N (cyanogen) | 206 |
| C≡C | 148 |
—that the bond energy of oxygen, both with two and with one (4-valent) carbon atom, is each time, in round numbers, 171 kg cal, whereas the carbon oxide molecule (divalent carbon) requires 248 kg cal for this. We shall confine ourselves to these few examples.
VI. General isotopy of the elements and its spectroscopic discovery.
The very discovery of radioactive substances already showed that the old classical notions of the indivisibility of the elements were an illusion. This discovery naturally led to the concept of the isotopy of the radio-elements, i.e. to the concept of such elements which, in all their chemical properties, must be assigned to one and the same place in the periodic system, but which differ from one another in a very important quantity, namely atomic weight.
i.e. by the very quantity which until then had been inseparably connected with a definite place in the periodic system. From this it was but one step to assuming a similar general isotopy for the other non-radioactive elements as well, and at the same time to returning to the views first developed by Prout. The first proof of this general isotopy was given by Aston, with the aid of his mass spectrograph, based, as is known, on the magnetic and electric deflection of a beam of canal rays, i.e. a beam of positively charged material particles. Such an apparatus, in order to be suitable for the investigations named, must above all satisfy two conditions: 1) it must separate a mixture of atoms and molecules of different weights by a method that excludes the use of any chemical properties of the elements under investigation, i.e. by a method based exclusively on differences in the inertial masses of the atoms; 2) it must make it possible to determine the atomic weights of the individual particles with an accuracy of at least 1 per mille.
How well Aston’s mass spectrograph satisfies these conditions is well known; equally well known are his results, so important for atomic physics. It should not, however, be forgotten what experimental difficulties Aston had to overcome in order to obtain such results. And soon after the publication of his investigations it was rightly pointed out that the chemist has, in spectral analysis, a method which can satisfy both of the above-mentioned conditions and which, with respect to sensitivity and the possibility of separating atomic and molecular mixtures, as well as with respect to the accuracy of measurements, is in no way inferior to the mass spectrograph, although the course of analysis in this case may be longer and more laborious. At the same time it was expected that isotopy would also be found in those elements for which the mass spectrograph, for technical reasons, had hitherto been inapplicable—in particular in elements of high atomic weight, and also in those from which it is difficult to obtain volatile compounds.
Initially, attempts were made to detect the isotope effect in line spectra, i.e., spectra that are attributed to luminous atoms. However, these attempts were not crowned with indisputable success. Experiments were carried out with isotopic lead, which had been isolated from radioactive decay products, and subsequently also with mercury and chlorine. Only later was it apparently possible, for the lightest lithium isotope (6 and 7), to interpret the hitherto unobserved companion of the red line \(\lambda\,6708\) as a line caused by isotopy.¹ In fact, the atomic spectrum is of little use for detecting isotopes. This spectrum owes its origin to the motion of the electrons of the atom, and therefore it can readily provide information about the chemical properties of the atom, but not about the magnitude of the atomic mass. Thus it does not satisfy our first condition. To be sure, Bohr’s theory of the hydrogen atom requires a dependence of the Rydberg constant on the mass, owing to the so-called “proper” motion of the nucleus, and moreover in the ratio of the electron mass to the atomic mass, \(m/M\). For hydrogen this dependence causes a difference of frequencies of \(59.4\ \mathrm{cm}^{-1}\) \((R = 109\,737.1;\ R_H = 109\,677.7)\), i.e. the red hydrogen line \(\lambda\,6560\), as a result of this proper motion of the nucleus, must be shifted to the red side by \(3.56\ \text{Å}\). Assuming the applicability of the theory also to a system of several electrons, one should expect that the Rydberg constants for the two lightest isotopes, namely Li 6 and 7, should nevertheless differ by \(1/42\) of this quantity \((1.4\ \mathrm{cm}^{-1})\). In the case of the already mentioned red lithium line \(\lambda\,6708\), this difference entails a difference in wavelengths of \(0.086\ \text{Å}\), which lies precisely at the limit of measurability. Quite apart from the difficulties in the theory of systems with several electrons,² it is clear that line spectra are of little use for such a generalization.
Much greater hopes for success are offered by band spectra
¹ H. Schüler und H. Wurm. Naturwiss. 15, 971, 1922.
² An attempt to construct such a theory was recently made by G. Joos. Ann. d. Phys. 83, 1054, 1927.
spectra. Here, in fact, in a number of cases, which will be considered in more detail below, it has been possible to detect an isotope effect in the sense of the requirements 1 and 2 given earlier. We have already seen that in molecular spectra the motion of the electrons is accompanied also by the rotation of the atoms about their common center of gravity and by the oscillation of the nuclei relative to the equilibrium position. In both cases we are dealing with the motion of heavy nuclei, and therefore we must expect two isotope effects: the effect of rotation, observed on the individual lines of a band, and the effect of vibrations of the nuclei, observed on the individual bands of a system.
Let us first consider the first effect and confine ourselves here to diatomic molecules, which, however, in no way affects the generality of the formulas given below. The rotational energy of such a molecule, consisting of two nuclei, is, as is known, determined by its moment of inertia, namely
\[ W_{\mathrm{rot}}=\frac{m^2h^2}{8\pi^2J}=Bm^2, \]
and the angular velocity of rotation is expressed by the corresponding formula
\[ \omega_{\mathrm{rot}}=\frac{mh}{2\pi J}. \]
Since the distance between the nuclei is conditioned only by the arrangement of the electrons and consequently does not depend on the masses of the nuclei, the heavier isotope must always rotate more slowly than the lighter one. In order to be able to calculate the splitting of the lines resulting from this, let us assume that one atom possesses only two isotopes with masses \(\mu_1\) and \(\mu_1+\Delta\), while the other is simple; further, since \(M=\mu_1+\Delta+\mu_2\) is the molecular weight of the heavier compound, the difference in the energy of the two kinds of molecules, caused by the unequal speed of rotation, will be
\[ \Delta W=\frac{m^2h^2}{8\pi^2r^2}\left(\frac{1}{\mu_1}-\frac{1}{\mu_1+\Delta}\right) =\frac{\Delta}{M}\cdot\frac{\mu_2}{\mu_1}W_{\mathrm{rot}}. \tag{34} \]
Let us recall further that, according to Bohr’s frequency condition, each spectral line is represented by the difference of two similar
…energies; therefore, if isotopy is present, the line must exhibit a splitting having the magnitude
\[ \Delta \nu = \left(\frac{A}{M}\frac{\mu_2}{\mu_1}\right) \left(\frac{W'_{\mathrm{rot}}}{h}-\frac{W''_{\mathrm{rot}}}{h}\right) = \delta \cdot \nu_{\mathrm{rot}} \qquad \left[\delta=\frac{A}{M}\frac{\mu_2}{\mu_1}\right], \tag{35} \]
where \(\nu_{\mathrm{rot}}\), according to equation (15) (p. 647), has a very clear meaning: it is the distance of the corresponding spectral line from the zero position \(m=0\) (Fig. 18). Consequently, the splitting increases directly proportionally to this distance, and it follows from this that in a branch going toward the band edge and then returning back, the splitting disappears again as soon as the branch passes through the zero position a second time. In other words: the splitting of lines caused by isotopy is completely symmetric with respect to the zero position, and in this it differs very substantially from the similar splitting of lines which is proportional to the number \(m\) and is caused by the interaction of rotation with the motion of the electrons (see p. 679).
Fig. 18.
Since we can often follow the lines of one band as far as \(100\) Å from the zero position, \(\delta\) has an order of magnitude of \(0.01\); therefore the splittings may increase to several Angstrom units.
It may also be noted here that, counting from the zero position toward longer wavelengths, the heavier isotope produces the component of the doublet with the shorter wavelength, while on the side of shorter wavelengths the same isotope produces the component with the longer wavelength.
Band Spectra and Their Significance for Chemistry
A considerably larger magnitude of the splitting is given by the second effect—the effect of vibrations of the nuclei. The vibrational energy, as is known, is expressed as
\[ W_s=an-bn^2 \]
or, if we denote the factors of the vibrational constants \(a\) and \(b\), which do not depend on the mass of the nucleus, by \(f_a\) and \(f_b\), then the energy is expressed as
\[ W_s=\frac{f_a}{\sqrt{\mu}}\,n-\frac{f_b}{\mu}\,n^2, \tag{36} \]
therefore in this case the difference of the energies of the two isotopic compounds will be
\[ \Delta W_s=f_a\cdot n\left[\frac{1}{\sqrt{\mu}}-\frac{1}{\sqrt{\mu'}}\right] -f_b n^2\left[\frac{1}{\mu_1}-\frac{1}{\mu_1+\Delta}\right]. \tag{37} \]
But taking into account that the difference of the isotope masses \(\Delta\), in comparison with the atomic weight, is in most cases very small, one may expand the expression
\[ \sqrt{\frac{1}{\mu'}}=\sqrt{\frac{1}{\mu_1+\Delta}+\frac{1}{\mu_2}} \]
in a series
\[ \sqrt{\frac{1}{\mu'}}=\sqrt{\frac{1}{\mu}}\left(1-\frac{1}{2}\delta\right), \tag{38} \]
As a result, completely analogously to equation (34), the splitting obtained is
\[ \Delta\nu_s=\frac{1}{2}\cdot\frac{\Delta}{M}\cdot\frac{\mu_2}{\mu_1} \left[(a'n'-2b'n'^2)-(a''n''-2b''n''^2)\right] \tag{39} \]
or, more simply, with the same abbreviations as in equation (35),
\[ \Delta\nu_s=\frac{1}{2}\delta\nu_s. \tag{40} \]
If one disregards the factor 2 before \(bn^2\), which has little significance, then \(\nu_s\) again has the value of the distance from the zero vibrational position \((n', n''=0)\), i.e. every band is split into a double band, the distance of which, in the first approximation, is proportional to this distance. This splitting by itself would be half as large as the rotational effect in the lines of the band.
But since the bands of the band system extend over much larger wavelength intervals (approximately 1000 Å), this splitting may assume considerably greater dimensions than the first, and therefore it can easily be detected even with small spectral apparatus.
Here also the band of the heavier isotope is always displaced toward the zero position, i.e. it corresponds to shorter waves on the long-wave side and to longer waves on the short-wave side (Fig. 19).
Fig. 19.
Before I pass to the consideration of individual results, it is necessary also to emphasize certain circumstances. As a rule both effects, of course, are superposed, but then the total splitting is equal to the sum of the separate effects; consequently
\[ \Delta \nu = \delta \left( \frac{1}{2}\nu_s + \nu_{\mathrm{rot}} \right); \]
further, it is easy to see that the detection of isotopy is the easier, the lighter the isotopes are and the greater the difference of their masses. As for the other element of the chemical compound \(\mu_2\), it should be chosen as heavy as possible in order to make the splitting large.^1 Therefore hydride compounds are of little use for determining isotopes. In those cases where such determinations are carried—
^1 A small transformation gives:
\[ \delta = \frac{\Delta}{N_1}\left(\frac{1}{1 + \frac{\mu_2 + \Delta}{\mu_2}}\right). \]
were taken (CuH, MgH, HCl), the splittings observed lay at the limit of the resolving power of the spectral apparatus. It is more convenient to use oxides and nitrides, but most convenient of all are the halogen compounds, especially iodine compounds, since iodine itself has no isotopes. Finally, it must be mentioned that in molecules with two identical atoms, even in the case of only two isotopes (for example, in Cl$_2$), the splitting will have three different magnitudes corresponding to the three kinds of molecules:
\[ \mathrm{Cl}^{35}\mathrm{Cl}^{35},\quad \mathrm{Cl}^{35}\mathrm{Cl}^{37},\quad \mathrm{Cl}^{37}\mathrm{Cl}^{37}. \]
VII. SPECTROSCOPIC DETERMINATION OF ISOTOPES.
H, He. Hydrogen and helium have also proved, spectroscopically, to be completely devoid of isotopes. Given the considerable magnitude of the effect expected here (for hydride compounds \(\delta \sim 1\) (!); for He$_2$, \(\delta = 0.12\)), these isotopes could hardly have remained unnoticed in the numerous precise investigations of the spectra of hydrides, as well as of the band spectrum of helium.
However, in not a single case was a splitting discovered that could have been interpreted as an effect of isotopy.
Li. Lithium is the first element which, according to Aston and other investigators, certainly possesses isotopes. Two band spectra of lithium, Li$_2$ and LiH, are known; however, they have not been investigated sufficiently to make it possible to discover isotopes. Thus the investigation of the band spectra of lithium from the point of view of isotopy is still a task for the future.
Be, B. The next element—Be—must be simple. For this element only the spectrum of its oxide, BeO, is known, and from it no precise conclusions about isotopy can be drawn. Boron, on the other hand, possesses two isotopes, B$^{10}$ and B$^{11}$, and thus is the first element in which isotopes have been detected spectroscopically. This was done precisely on the basis of
spectrum of its oxide compound. In this spectrum, three band systems were known in all, of which only the two most intense are suitable for investigation from the point of view of isotopy. The distribution of the bands and the exact numerical reduction of the observations led Mulliken¹ in this case to the following vibration formulae:
\[ \begin{aligned} \mathrm{B}^{11}\mathrm{O}:\ \nu&=23\,960{,}2 \\ \nu&=23\,834{,}0 \end{aligned} \left\} +(1258{,}5\,n' - 10{,}6\,n'^2) - (1884{,}9\,n'' - 11{,}68\,n''^2)\right. \]
\[ \begin{aligned} \mathrm{B}^{10}\mathrm{O}:\ \nu&=23\,956{,}6 \\ \nu&=23\,833{,}4 \end{aligned} \left\} +(1297{,}3\,n' - 11{,}7\,n'^2) - (1939{,}0\,n'' - 12{,}21\,n''^2)\right. \]
\[ \mathrm{B}^{11}\mathrm{O}:\ \nu=43\,166{,}2+(1280{,}3\,n' - 10{,}07\,n'^2) - (1885{,}7\,n'' - 11{,}77\,n''^2) \]
\[ \mathrm{B}^{10}\mathrm{O}:\ \nu=43\,168{,}6+(1316{,}7\,n' - 19{,}53\,n'^2) - (1941{,}5\,n'' - 12{,}58\,n''^2). \]
From these four pairs of frequency values one immediately obtains the splitting factors given in Table 15, whose mean value agrees excellently with the theoretical one.
TABLE 15.
Isotopy of boron.
| \(\frac{1}{2}\delta\) | \(\delta\) | |
|---|---|---|
| \(n'\) (α) | 0,0308 | 0,104 |
| \(n''\) (α) | 0,0287 | 0,045 |
| \(n'\) (β) | 0,0284 | 0,046 |
| \(n''\) (β) | 0,0296 | 0,069 |
| Mean | 0,0293 | 0,062 |
| ± 0,0005 | ± 0,009 | |
| BO (theoretical) | 0,0292 | 0,059 |
| BN (theoretical) | 0,0276 | 0,056 |
The rotational effect was also investigated by Jenkins² and was found to be in complete agreement with theory. He found \(\delta = 0{,}0595\), as compared with \(\delta_{\mathrm{theor.}} = 0{,}0593\).
Thus we see from this example that our two conditions 1 and 2 for the detection of isotopy are well fulfilled in the present case. Conversely, from these values one can
¹ R. Mulliken, Phys. Rev. 25, 259, 1925.
² F. A. Jenkins, Proc. Nat. Acad. 13, 496, 1927.
can be calculated the difference in the masses of the two isotopes, for which the following value is obtained: \(\Delta = 1.004 \pm 0.01\). The accuracy obtained is hardly inferior to the accuracy of Aston’s mass spectrograph. The spectrum of boron is, furthermore, a classic example of how, on the basis of the isotope effect, one can reliably draw a conclusion about the carrier of a spectrum. Indeed, in this case the carrier might also have been boron nitride, BN, which for a long time was considered such, since active nitrogen was used to excite the spectrum.1 However, Table 14 shows that the splitting factor calculated for BN cannot be reconciled with observation. The same spectrum also yielded the important conclusion already mentioned earlier. Namely: Mulliken showed that the isotope effect disappears not at \(n'\) and \(n'' = 0\), but at \(n'\) and \(n'' = -\dfrac{1}{2}\); therefore, even before Schrödinger, he introduced half-integral values for the quantum numbers of vibration.
C, N, O, F. For the first three elements such a large number of band spectra is known, and they have been studied so thoroughly (for example, the spectra \(C_2\), CN, CO, CH, \(N_2\), \(N_2^+\), NO, CO), that the isotope effect could not have been overlooked.1 We therefore conclude, together with Aston, that these elements in fact have no isotopes.2 The same is true, with certain limitations, for fluorine as well. From this point of view only CuF has been investigated.
Mg. For this element the hydride compound MgH was successfully investigated, and specifically the effect of rotation on the band \(\lambda 5211\) was studied.^1 Although in the present case, owing to the fact that we are dealing here with a hydride compound, the splittings are very small \((\delta = 0.0016)\), nevertheless on this band it was possible to establish with certainty the existence of three magnesium isotopes 24, 25, 26. The quite clearly observed residual of \(0.073\ \mathrm{cm}\) (0.029), which should be attributed to the energy of vibrations, again shows that the band corresponds to the quantum numbers \(n', n''\) \(1/2, 1/2\), and not \(0,0\), as was required by the old quantum theory. In the spectrum of magnesium oxide—MgO—also studied, the isotope has not yet been established; the same applies to the spectra of numerous compounds of the other alkaline-earth elements, chiefly the halide compounds of Ca, Sr, and Ba, all of which should exhibit an effect of easily measurable magnitude (for example: CaJ (40,44): \(\delta = 0.0744\); SrJ (86,88): \(\delta = 0.0137\)).
Al. The spectrum of aluminum oxide is known well enough that, on the basis of it, one could draw a conclusion about the isotope effect. However, in agreement with Aston’s result, no splittings have been found here. Consequently aluminum is a simple element. Little is known about the spectra of the remaining elements of this group.
Si. For silicon, in SiN, three isotopes 28, 29, and 30 have been established with certainty on the basis of the isotope effect in the vibrational spectrum,^2 \(\delta = 0.0116\) and \(0.0227\).
K. In order finally to establish the carrier of the band spectrum, Krchuhl and Villars^3 found in it a vibrational effect corresponding to the molecule \(K_2\). According to Aston, potassium in fact has isotopes 39 and 41.
Cu and the halides F, Cl, Br, J. The most extensive and fruitful investigations of the isotope effect have been carried out on the halide compounds of copper. Therefore the listed
^1 W. W. Watson and Ph. Rudnik. Astr. Journ., 63, 20, 1926.
^2 R. S. Mulliken. Phys. Rev., 26, 319, 1925.
^3 R. Ritsch und D. Villars. Naturwiss., 16, 219, 1928.
five elements are considered together. Copper hydride makes it possible to detect the isotopy that was to be expected for copper (here the rotational effect was investigated). The extremely small splitting (\(\delta = 0.0005\)) is, however, insufficient for establishing the isotopes of copper. The particularly well investigated halide compounds reveal isotopy with absolute certainty and, in addition, make it possible simultaneously to carry out tests for the isotopes of the halides. In this case only the vibrational effect is taken into account, since the resolving power of spectral apparatus does not permit an investigation of the rotational effect. Here we have to do with the following types of molecules:
\[ \begin{gathered} \mathrm{Cu}^{63}\mathrm{F},\qquad \mathrm{Cu}^{65}\mathrm{F}\\ \mathrm{Cu}^{63}\mathrm{Cl}^{35},\ \mathrm{Cu}^{63}\mathrm{Cl}^{37}\qquad \mathrm{Cu}^{65}\mathrm{Cl}^{35}\ \mathrm{Cu}^{65}\mathrm{Cl}^{37}\\ \mathrm{Cu}^{63}\mathrm{Br}^{79},\ \mathrm{Cu}^{63}\mathrm{Br}^{81}\qquad \mathrm{Cu}^{65}\mathrm{Br}^{79}\ \mathrm{Cu}^{65}\mathrm{Br}^{81}\\ \mathrm{Cu}^{63}\mathrm{J}^{122}\qquad \mathrm{Cu}^{65}\mathrm{J}^{127} \end{gathered} \]
on which the values of \(\delta\) are plotted. The splitting of copper increases with the increase of the atomic weight of the halide, but the greatest width of splitting is given by copper chloride quartet. All the spectra investigated are extremely rich in quanta.^1 Thus the effect in CuF could be established on three different band systems, in CuCl—on four, in CuBr—on three, and finally in CuJ—even on five systems. Thus altogether there were almost 1000 measurements of quanta on 44 band systems, which gave reliable confirmation of isotopy. It would be superfluous to give here the quantum formulae of these 44 systems. Jevons^2 was able to detect the isotopy of chlorine also in the spectrum of stannous chloride. For the corresponding detection of the isotopy of tin [according to Aston, tin has 8 isotopes: 116, 117, 118, 119, 120 (121), 122, 124] the dispersion of his spectral apparatus was insufficient (\(\delta = 0.015\)); however, the blurriness of the bands indicated the existence of numerous isotopes of tin. In the same way
^1 R. Ritschl. Z. Physik. 42, 172, 1927.
^2 W. Jevons. Proc. Roy. Soc. 110, 365, 1926.
in chloroauric gold1 it proved possible to carry out the detection of chlorine isotopes. Similarly, in iodine monochloride as well.2
Thus, from these investigations we see that isotopy has been verified spectroscopically for at least 15 elements.3 Of these, 5 turned out to be simple (H, He, F, Al, J), while the rest (C, N, O, B, Mg, Si, Cl, Br, K, and Cu) turned out to be complex. I have already sufficiently indicated the substantial assistance rendered by the spectroscopic detection of isotopes in establishing the half-integral quantum numbers of vibration and in determining the carriers of the band spectrum. As for the latter point, quite recently there still existed doubt as to the correctness of the identification—for example, with respect to the spectroscopic possibility of the existence of diatomic hydride compounds of metals. The magnitude of the splitting due to isotopy in a number of examples (CuH, MgH, CdH, ZnH, HgH) has finally removed this doubt. Although the hope of going further in the search for new isotopes, as compared with investigations by means of the mass spectrograph,4 has not yet been realized, the spectroscopic method could nevertheless provide very substantial support for the latter in one respect. The point is the as yet unperformed determination of the proportion in which the isotopes of a given element are mixed, by measuring intensity. Spectroscopic measurements of intensity are not in themselves too difficult, so that such measurements would make it possible to answer the question—whether the chemically established atomic weight corresponds to the actual percentage content of the isotope. As is known, Aston had already established that the integrality of atomic weights is not strictly fulfilled, but that the true atomic weights differ from whole numbers by insignificant amounts, which—
which could be interpreted energetically as mass defects. The aforementioned study of the isotope effect from the point of view of intensity might here open the way for further investigation of this interesting discovery.
TABLE 16.
Spectroscopic discovery of isotopes.
| Elements | Atomic masses | Discovery |
|---|---|---|
| H | 1 | Hydrides |
| He | 4 | He₂ |
| B | 10, 11 | BO: \(\delta = 0.0593\) |
| C | 12, 13 | C₂, CN, CO, CH |
| N | 14, 15 | CN, N₂, NO |
| O | 16, 17, 18 | CO, O₂, NO, etc. |
| F | 19 | Cu F |
| Mg | 24, 25, 26 | MgH: \(\delta = 0.0016\) and \(0.0031\) |
| Al | 27 | AlO |
| Si | 28, 29, 30 | SiN: \(\delta = 0.0116\) and \(0.0227\) |
| Cl | 35, 37 | HCl: \(\delta = 0.0015\), Cu³⁵Cl: \(\delta = 0.0360\), SnCl: \(\delta = 0.0436\) |
| K | 39, 41 | K₂: \(\delta = 0.025\) and \(0.050\) |
| Cu | 63, 65 | CuH: \(\delta = 0.0005\) Cu-Hal. See Fig. 20 |
| Br | 79, 81 | Cu⁶³Br: \(\delta = 0.0111\) |
| I | 127 | CuJ |
| Pb | 206, 208 | PbO: \(\delta = 0.00069\) |
VIII. The chemical constant.
The band spectra of polyatomic gases make it possible to test the theory of this constant, which is especially important for thermochemistry. As is known, from the heat theorem of Nernst—on the disappearance of the entropy of solid and liquid bodies at the point of absolute zero—it follows that this constant \(i\) enters, as a term independent of temperature, into the formula for vapor pressure:
\[ R \ln p = - \frac{\lambda_0}{T} + \frac{F_g - F_k}{T} + i \tag{42} \]
In this formula \(\lambda_0\) denotes the heat of vaporization at absolute zero, and \(F_g\) and \(F_k\)—the free energy of the gas and of the condensate (depending on temperature and referred to
to constant pressure). Both energies can, however, be calculated by double integration from the formula for the temperature dependence of the specific heats
\[ \frac{F}{T}=-R\ln \Sigma ge^{-\frac{E}{kT}} =-\int_{0}^{T}\frac{dT}{T^{2}}\int_{0}^{T} C_{p}\,dT . \tag{43} \]
Thus, if the specific heats and vapor pressures have been measured over as wide a range of temperatures as possible (and usually it is sufficient to measure the specific heat of the condensate; on the calculation of the specific heat of gases, see below), then the determination of the chemical constant presents no further experimental difficulties.
It is also possible to determine the constant from the equations of chemical equilibria, if the temperature dependence of the equilibrium constant \(K_{p}\) in the law of mass action has been determined by measurements. The completely analogous formula in this case is
\[ R\ln K_{p}=-\frac{Q_{0}}{T}-\frac{1}{T}\Sigma F+\Sigma i \tag{44} \]
only in place of the heat of vaporization there now stands the thermal effect \(Q_{0}\) of the reaction at absolute zero. The possibility of determining the chemical constant from this formula presents only the inconvenience that here, when several gaseous members of the reaction are taken into account, we obtain an algebraic sum of chemical constants:\(^{1}\) \(i_{R}=\Sigma i\). Often, however, many terms of this sum are already known from other reactions, which contain only one gaseous component; for example, \(i_{\mathrm{H_{2}O}}\) can be found from the reaction \(\mathrm{CaO}+\mathrm{H_{2}O}=\mathrm{Ca(OH)_{2}}\); or \(\mathrm{CuSO_{4}}+\mathrm{H_{2}O}=\mathrm{CuSO_{4}\cdot H_{2}O}\), so that here too the calculation presents no difficulties. The theory of Nernst makes the prerequi—
\(^{1}\) For example, in the reaction of formation of water \(2\mathrm{H}_{2}+\mathrm{O}_{2}=2\mathrm{H}_{2}\mathrm{O}+i_{R}\), \(i_{R}=2i_{\mathrm{H_{2}O}}-2i_{\mathrm{H_{2}}}-i_{\mathrm{O_{2}}}\); the constants must be introduced into the formula with the corresponding “weight” and the correct sign.
reference that the chemical constants \(i_D\), determined directly from the vapor-pressure curve, must coincide with the values obtained from the equations of gaseous reactions, i.e., that in general the relation always holds:
\[ i^R=\sum i_D \tag{45} \]
To what extent this premise is justified under the present simple formulation of the question, we shall now examine. Proceeding by a purely thermodynamic route, nothing can be said about the magnitude of the constant \(i\); but on the basis of kinetic considerations, using the quantum theory for monatomic gases possessing only translational energy, various investigators were able to determine \(i\) from quantities known to us, \(m\) (mass of the atom), \(h\), and \(k\). The constant \(i\) is then determined as
\[ i_1=\ln\left(\frac{\sqrt{2\pi mk}}{h}\right)^3\cdot k. \tag{46} \]
If \(i\) is referred to the gram-molecule, the pressure measured, as is customary, in atmospheres, and Briggs logarithms are introduced, then \(i_1\) is obtained equal to
\[ i_1=-1.587+\frac{3}{2}\log M \tag{46a} \]
\(i_1\) therefore depends—apart from universal constants—only on the atomic weight of the gas \(M\).
Experiments have, in general, confirmed these considerations well, although it cannot be passed over in silence that in some cases (Cl, Br, J, Na, K) discrepancies were obtained between the observed and calculated values which exceeded the limits of error and have not yet found their explanation. By an analogous route one can calculate the chemical constant for diatomic and polyatomic gases, for which to the energy of translational motion there is added also the energy of rotation. We obtain the following formulas:
\[ \begin{aligned} i_2&=\ln\left(\frac{8\pi^2 J}{h^3}\right)k+\ln\frac{g}{s}\quad \text{(diatomic gases)}\\ i_2&=\ln\left(\frac{8\pi^2 \sqrt[3]{J_1\cdot J_2\cdot J_3}}{h^2}\right)k+\frac{3}{2}\ln\frac{g}{s}\quad \text{(polyatomic gases)} \end{aligned} \tag{47} \]
or, substituting the numerical data:
\[ i_2 = 38.40 + \log J + \log \frac{g}{s} \]
\[ i_2 = 57.60 + \frac{3}{2}\log J + \frac{3}{2}\log \frac{g}{s}; \quad J = \sqrt[3]{J_1 \cdot J_2 \cdot J_3} \tag{47a} \]
Thus \(i\) is equal to the sum \(i_1 + i_2\), and the constant, along with the molecular weight \(M\), here also includes the moment of inertia of the molecule, or, for polyatomic molecules, the geometric mean \(J\) of the three principal moments of inertia. As for the additional term \(\log \frac{g}{s}\), which, it is true, is not written by all authors but is theoretically quite well justified, in it \(s\) represents the so-called symmetry number of the molecule. If the molecule consists of unlike atoms, then \(s\) is, of course, equal to 1; but if the molecule, as in some of the cases given below, contains two identical atoms, then \(s = 2\). The same will be the case, for example, for triatomic molecules (\(\mathrm{CO_2}\) and \(\mathrm{H_2O}\)). For the methane molecule \(\mathrm{CH_4}\), \(s\) even turns out to be equal to 12. That molecular symmetry plays a certain role in the quantum theory of band spectra we could already see when examining the question of the successively varying intensity of the lines composing the bands; moreover, the appearance of lines of alternating intensity could even serve as a criterion for the presence of such symmetric molecules. It is therefore quite plausible that the quantity \(s\) also enters into the Nernst constant. Table 16 will confirm this for us. The quantity \(g\) represents the “statistical weight of the lowest quantum state” of the gas (or condensate). The expediency and significance of this additional term in equation (47), however, have not yet been fully clarified. Its introduction into our formula would mean that in this case (and only in this case) for different states of the system the energy at the point of absolute zero and the chemical constants for all states will be the same, if the statistical weight \(g\) for the lowest quantum state is one and the same for all participants in the reaction.
More will be said about this below.
The formulas given above for diatomic and polyatomic gases retain their validity only so long as the vibrational frequencies of the nucleus are not excited. If, however, the latter are excited—for example, for \(J_2\) above \(100^\circ\)—then to the parts depending on the translational energy \(L_1\) and on the rotational energy \(i_2\) there is added still another part, expressed as follows:
\[ \left. \begin{aligned} i_3 &= \ln \frac{k g}{h\nu} = -0.155 - \log \nu - \log g \quad \text{(diatomic molecule)} \\ i_3 &= \ln \frac{k^f (g_1 \cdot g_2 \cdots g_f)} {h^f (\nu_1 \cdot \nu_2 \cdots \nu_f)} = -0.155 \cdot f - \Sigma \log \nu_f - \Sigma \log g_f \end{aligned} \right\} \tag{48} \]
(polyatomic molecule with \(f\) degrees of freedom of vibration)
\(\nu_1, \nu_2, \ldots \nu_f\) denote the vibrational frequencies of the molecule’s nucleus expressed in \(\mathrm{cm}^{-1}\), that is, again, quantities which we can obtain directly by means of band spectra. We thus see that, in order to test the formulas given above, it is necessary to know the moments of inertia and the vibrational frequencies; and these can be determined with the required accuracy only by optical means. Such determinations for the molecules of interest to us have become possible only recently. Let us therefore examine the available material in somewhat greater detail, in order to establish to what extent experiment, in conjunction with data obtained from molecular spectroscopy, confirms the theory. I shall follow closely the critical survey and new treatment of the material carried out several years ago by Eucken, Karwat, and Fried.^1
In Table 16 there are first given the values of \(i\), calculated from formulas (46) and (47) (taking into account the number of symmetries \(s\), given in the second column). In this calculation the moments of inertia for HCl, HBr, NO, \(\mathrm{H_2O_2}\), and \(J_2\) could be taken directly from the analysis of bands (see the table), and, of course, the moments of inertia were taken for the normal state of the molecule, so that the constants are obtained with accu-
^1 A. Euken, E. Karwat, E. Fried. Z. Physik, 29, 1, 1924; cf. also Geiger und Scheel. Handb. d. Phys. X, p. 387.
ness is in no case less than 0.01. The moments of inertia of HJ, Cl₂, Br₂ can be determined with great accuracy (see p. 861), whereas the moments of inertia of CO and CO₂, for which an analysis of the bands of the normal state has not yet been carried out, can be calculated only indirectly, from the distances between double bands in the infrared region. However, even in the latter case the accuracy hardly falls below 0.01.
TABLE 17.
Chemical constants.
| No. | Molecule | $s$ | $i$ calc. | $i$ vapor pressure | $i$ equilibrium constant | $\Delta$ |
|---|---|---|---|---|---|---|
| 1 | H₂O | 2 | −2.31 | −1.94 ± 0.03 | −1.99 ± 0.06 | +0.33 |
| 2 | CO₂ | 2 | +0.66 | +0.91 ± 0.06 | +0.90 ± 0.15 | +0.25 |
| 3 | HCl | 1 | −0.45 | −0.26 ± 0.04 | −0.18 ± 0.03 | +0.29 |
| 4 | NBr | 1 | +0.18 | +0.53 ± 0.07 | +0.53 ± 0.1 | +0.29 |
| 5 | HJ | 1 | +0.57 | +0.90 ± 0.15 | +0.86 ± 0.2 | +0.29 |
| 6 | H₂ | 2 | −3.36 | −3.88 ± 0.03 | −3.71 ± 0.05 | −0.33 |
| 7 | O₂ | 2 | +0.05 | +0.54 ± 0.05 | −0.58 ± 0.1 | +0.50 |
| 8 | Cl₂ | 2 | +1.31 | +1.67 ± 0.16 | +1.99 ± 0.2 | +0.2 |
| 9 | Br₂ | 2 | +2.34 | +2.55 ± 0.10 | +2.97 ± 0.4 | +0.61 |
| 10 | J₂ | 2 | +2.99 | +3.12 ± 0.20 | +3.59 ± 0.2 | +0.61 |
| 11 | NO | 1 | +0.25 | +0.52 ± 0.06 | +0.83 ± 0.1 | +0.27 |
| 12 | CO | 1 | +0.15 | −0.05 ± 0.1 | +0.29; +0.03 ± 0.1 | −0.2 |
| 13 | N₂ | 2 | −0.15 | −0.11 ± 0.05 | +0.21 ± 0.05 | −0.04 |
| 14 | CH₂ | 12 | −2.65 | −2.30 ± 0.08 | −2.25 ± 0.3 | +0.35 |
With somewhat less accuracy one can estimate the moment of inertia of N₂, since the spectrum of the normal state of nitrogen could not be analyzed, for it lies in the far ultraviolet region.
The moment of inertia of methane can also be calculated in the following way. From the analysis of the band spectrum we know the distance between the nuclei in the simple molecule CH ($1.1 \cdot 10^{-8}$ cm). There are a number of grounds (which we shall not examine more closely here) to suppose that even in the presence of four hydrogen atoms ...
for the atoms this distance does not change appreciably. Taking into account also the tetrahedral form of the methane molecule, we can easily calculate the mean moment of inertia of the molecule and obtain \(J_1=J_2=J_3=5.7\cdot 10^{-40}\). This calculation is important because it fully justifies the introduction into the chemical constant of the symmetry number \(s\), which in the present case has a very large value, equal to 12 \(\left({}^{3}/_{2}\log 12=1.62\right)\). Failure to take this number into account, even with so rough an estimate of the moment of inertia, would give for the chemical constant a value in no way agreeing with the results of measurements.
In the next column of the table are placed the values obtained directly experimentally from the vapor-pressure curve. The last column contains directly the reaction constants. A few words should be said about how they were obtained. The value for \(\mathrm{CO}_2\) was derived from a reaction with one gaseous component: \(\mathrm{CaO}+\mathrm{CO}_2=\mathrm{CaCO}_3\); the reaction therefore gives the constant directly, and moreover in good agreement with the value obtained from the vapor-pressure curve. For \(\mathrm{H}_2\mathrm{O}\) one may use two one-gas reactions \([\mathrm{CaO}+\mathrm{H}_2\mathrm{O}=\mathrm{Ca(OH)}_2\) and \(\mathrm{CuSO}_4+\mathrm{H}_2\mathrm{O}=\mathrm{CuSO}_4\cdot\mathrm{H}_2\mathrm{O}]\), and also a reaction involving three gases—the reaction of formation of water vapor \(2\mathrm{H}_2+\mathrm{O}_2=2\mathrm{H}_2\mathrm{O}\), which, together with \(\mathrm{H}_2=-3.36\), is perhaps still more suitable for determining \(\mathrm{O}_2\); here it can be used as a check. For \(\mathrm{HCl}\), as also for \(\mathrm{HBr}\) and \(\mathrm{HJ}\), one may use the reactions of formation from the elements: \(\mathrm{H}_2+\mathrm{Cl}_2=2\mathrm{HCl}\). Since in these reactions, in which three gases participate, the values for \(\mathrm{Cl}_2\) and \(\mathrm{I}_2\) can also be determined independently, and the value for \(\mathrm{H}_2\) is known (see p. 661), from these reactions one can calculate the chemical constant. The constants calculated in this way for \(\mathrm{HCl}\) (and \(\mathrm{HJ}\)) agree well with the values obtained from the vapor-pressure curves; the same agreement may therefore be assumed for all three hydrogen halides. For \(\mathrm{HCl}\) there is also a control reaction \(4\mathrm{HCl}+\mathrm{O}_2=2\mathrm{Cl}_2+2\mathrm{H}_2\mathrm{O}\), into which the not entirely reliable value of the constant for oxygen enters only to the fourth part
This value for oxygen, independently of other gas reactions, could be determined from the dissociation reaction of mercuric oxide: \(\mathrm{Hg}\) (vapor) \(+\mathrm{O}_2=2\mathrm{HgO}\). It proved to be in good agreement with the value obtained from the vapor-pressure curve. For determining \(\mathrm{H}_2\) there is likewise a whole series of reactions to choose from, including reactions with one gaseous phase:
\(\mathrm{H}_2+\mathrm{HgO}=\mathrm{Hg}+\mathrm{H}_2\mathrm{O}\) (solid),
\(2\mathrm{C}_6\mathrm{H}_4\mathrm{O}_2\) (quinone) \(+\mathrm{H}_2=\mathrm{C}_6\mathrm{H}_4\mathrm{O}_2\cdot\mathrm{C}_6\mathrm{H}_4(\mathrm{OH})_2\) (quinhydrone), and
\(\mathrm{C}_6\mathrm{H}_4\mathrm{O}_2\cdot\mathrm{C}_6\mathrm{H}_4(\mathrm{OH})_2\) (quinhydrone) \(+\mathrm{H}_2=2\mathrm{C}_6\mathrm{H}_4(\mathrm{OH})_2\) (hydroquinone),
all of which give values coinciding with the values from the vapor-pressure curve.
These first six constants, obtained from reactions, within the limits of error excellently satisfy the requirement of agreement between the data computed in this way and from the vapor-pressure curves. The same cannot be said of the following ones. Thus the reactions of the halides (for example, chlorine in computing the electromotive force of a cell with the participation of one gaseous component:
\(2\mathrm{Ag}+\mathrm{Cl}_2=2\mathrm{AgCl}\), \(2\mathrm{Hg}+\mathrm{Cl}_2=2\mathrm{HgCl}\), \(\mathrm{Pb}+\mathrm{Cl}_2=\mathrm{PbCl}_2\),
the same is obtained for \(\mathrm{J}_2\); for \(\mathrm{Br}_2\) reaction No. 4 was used) give values which, rather consistently, turn out to be larger by approximately \(0.48\). For \(\mathrm{N}_2\), the basis of the calculation is the ammonia reaction
\(\mathrm{N}_2+3\mathrm{H}_2=2\mathrm{NH}_3\),
where, by way of exception, the value for \(\mathrm{NH}_3\) was taken from the vapor-pressure curve, since there is no reaction with one gaseous phase. Here the difference is \(+0.32\). Likewise for \(\mathrm{NO}\) there is only one reaction—the reaction into which three gases enter,
\(\mathrm{N}_2+\mathrm{O}_2=2\mathrm{NO}\),
where the values for \(\mathrm{N}_2\) and \(\mathrm{O}_2\) are determined in another way. The difference obtained is \(0.31\). For \(\mathrm{CO}\) one can again carry out a check by several reactions, which, however, do not give homogeneous results. From reactions into which two gases enter,
\(2\mathrm{CO}=\mathrm{C}+\mathrm{CO}_2\),
one obtains, with the value of the constant for \(\mathrm{CO}_2=+0.91\), a number that coincides with the constant calculated from the vapor-pressure curve \((+0.03\pm0.1)\); conversely, both reactions
\(2\mathrm{CO}+\mathrm{O}_2=2\mathrm{CO}_2\) and \(\mathrm{CO}+\mathrm{H}_2\mathrm{O}=\mathrm{CO}_2+\mathrm{H}_2\)
(the reaction of formation of water vapor serves as a check for four gases) give values larger by approximately \(0.30\) \((+0.29\pm0.1)\). For methane as well, the results are not quite unambiguous—
... Direct formation of methane from carbon and hydrogen,
$C + 2H_2 = CH_4$,
does, it is true, give a value that agrees well with that obtained from vapor pressure; but the two other reactions, in which the initial product is carbon monoxide or carbon dioxide,
$CO + 3H_2 = CH_4 + H_2O$
and
$CO_2 + 4H_2 = CH_4 + 2H_2O$,
lead, while agreeing rather closely with one another, to a much lower value, \(i = -1.58\).
Having reviewed the available numerical material, we cannot recognize the results as especially satisfactory. It is true that, for a number of molecules, the required agreement is obtained between the values of the chemical constant derived from vapor-pressure curves and from chemical reactions; but for other molecules, and indeed for almost half of them, the discrepancies obtained are far beyond the limits of observational error, which here were probably taken not too small. At the same time it is striking that the constants obtained from reactions are each time greater than those obtained from vapor-pressure curves; the difference is approximately from \(+0.3\) to \(0.4\), so that from this one might draw the conclusion that the statistical weights differ (in round numbers) from 2 to 3. It should be noted that the theoretical value (with the exception, perhaps, of nitrogen and \(CO\)) is never obtained. Most often the experimental values come out larger, and again by such amounts as can be taken as \(\log 2\) and \(\log 3\). The only exception is nitrogen, which from vapor pressure gives an approximately correct number, but from calculation from chemical reactions gives an incorrect one; values smaller than the calculated ones are obtained from vapor pressure for \(H_2\) \((0.32)\) and \(CO\) \((0.2)\). In the case of \(H_2\), however, one may think that this difference can be attributed to a not entirely correct calculation of the heat of rotation (see § 8), because at very low temperatures, where this heat need not be taken into account, the theoretically correct value for a monatomic gas is obtained.
Can these discrepancies, not especially satisfactory for the theory, be attributed wholly to the statistical weights of the gases, which in the future will be determined more accurately; or, possibly, were the limits of error taken...
too low, and that a better agreement may be obtained by a more correct adjustment of the results of the measurements—will be shown by the future. There are sufficient grounds for both possibilities. We know, for example, that in the optical dissociation of the halogens \((\mathrm{Cl}_2, \mathrm{Br}_2, \mathrm{J}_2)\), two normal atoms are never formed at first; instead, one normal atom is obtained (\({}^{2}P_2\)-term) and one excited atom (term \({}^{2}P_1\), see § 5). And this means that, in the optical equilibrium of the dissociation \(\mathrm{Cl}_2=\mathrm{Cl}+\mathrm{Cl}\), chlorine and the products of its dissociation possess different statistical weights with respect to the electronic terms. There is therefore nothing surprising in the fact that Wohl,\(^1\) using Bodenstein’s measurements for monatomic chlorine, bromine, and iodine and taking as his basis the values given above, obtains for the molecules values of \(i\) differing from the theoretical ones by approximately \(+0.62\;(=\log 4)\); all the more so since the experiments have to be carried out at comparatively high temperatures. In exactly the same way we now already know that, for example, in the reaction \(\mathrm{N}_2+\mathrm{O}_2=2\mathrm{NO}\), the ground state of NO is probably represented by the term \({}^{2}P_i\); the state of \(\mathrm{O}_2\) by the \({}^{3}S\)-term, and the state of \(\mathrm{N}_2\) by the \({}^{1}S\)-term, i.e. here there are three different electronic terms with entirely different statistical weights. Later data, which will be obtained from more accurate and more broadly planned measurements, will show whether it will actually be possible, as is done in the following paragraph, to identify the statistical weights of the electronic terms with the values of \(g\) given above. The considerations set forth here strongly support this, but they are still insufficient for a final decision. Moreover, for the present we know absolutely nothing about the statistical weights of solid and liquid bodies. As for the second possibility—improving the constant by means of a more correct adjustment of the measurement results—we refer the reader likewise to the following paragraph. If in this way the result obtained is not entirely satisfactory, nevertheless the investigation of band spectra—
\(^1\) K. Wohl. Z. Phys. Chem. 110, 166, 1924.
could resolve a number of particular questions. First of all, we see that chemically related compounds (for example, halides or hydrogen halides) always behave in the same way, i.e. they always give the same discrepancies between experiment and theory, and thereby show that the deviations do in fact exist and cannot be ascribed to errors of observation. Next, one may consider that the correctness of introducing the symmetry factor \(s\) (especially in the case of \(\mathrm{CH}_4\), but also for other molecules) is fully confirmed, while the distinction between statistical weights becomes very probable, at the very least. In conclusion we shall note also the circumstance that, for reactions in which hydrogen enters, in calculating the chemical constant one has to take for \(\mathrm{H}_2\) the theoretical value \((-3.36)\), in order to obtain the mutually comparable values given in Table 16.
It is precisely this circumstance that speaks strongly in favor of the supposition that the excessively low value of the constant \((-3.7)\) is explained by a not entirely correct calculation of the heat of rotation, giving an erroneous value of \(i\). Now that the heat of rotation has been theoretically clarified, this supposition could easily be tested.
IX. Band Spectra and Specific Heat
For the calculation of chemical constants it was necessary to know the free energy (at constant pressure). Therefore, for a condensate (also for solid and liquid bodies participating in a reaction in the study of equilibrium states), one first determines the temperature dependence of the specific heat \(c_p\), then constructs an exponential formula expressing the dependence of \(c_p\) on \(T\), and from it, by means of an easily performed double integration, determines the free energy according to formula (43). It is often possible even to expand the specific heat with the aid of the Debye function \(D\!\left(\frac{\Theta}{T}\right)\), into which, as is known, only one parameter \(\Theta\) enters, the so-called characteristic temperature, i.e. a certain vibrational frequency, expressed in \(\mathrm{cm}^{-1}\), multiplied by
the Planck radiation constant \(c_2=\dfrac{hc}{k}=1.43\ \text{cm deg}\).
In the latter case, this characteristic temperature is first determined by means of measurements, best of all by the “\(T^3\)” law at low temperatures, \(c_v=464\left(\dfrac{T}{\Theta}\right)^3\); it can also be calculated approximately from the elastic, thermal, or optical properties of the body—from the residual rays—and then, from tables1, one can immediately find the free energy as a function of temperature; here, however, it must be borne in mind that the Debye function refers to the specific heat at constant volume, so that here it is necessary to introduce also a small correction to \(c_p\).
In the case of gases it is more expedient to proceed not from the specific heats, but directly from the sum
\[ \sum \sum \sum g_e\cdot g_s\cdot g_r\cdot e^{-\frac{W_e+W_s+W_r}{kT}} = \sum \frac{g_e\cdot e^{-\frac{W_e}{kT}}}{} \cdot \sum g_s e^{-\frac{W_s}{kT}} \cdot \sum g_r e^{-\frac{W_r}{kT}}, \tag{49} \]
where the logarithm of this expression, multiplied by \(R\), according to equation (43)2 is directly equal to \(F/T\) (and here referred to constant volume; but for gases the conversion to constant pressure is easy to carry out, knowing the work of expansion). The advantage of this method of calculating the free energy consists in the fact that at the present time, with the aid of band spectra, we can determine with any desired accuracy both the magnitude of the separate kinds of energy \(W_e\) (electrons), \(W_s\) (nuclear vibrations), and \(W_r\) (rotation), and also their statistical weight \(g\); the energy can be calculated from the series formulas of the bands, and the statistical weight from the intensities of these bands. It was precisely the determination of the statistical weights from the intensities of the lines in the bands that has now made it possible, in determining the heat of rotation of hydrogen, to remove the long-noted discrepancy between experimental and theoretical data. All the former formulas for calcu-
the calculation of the heat of rotation did not take into account the circumstance that, in symmetrically constructed molecules, even quantum states have a different statistical weight than odd ones (in the case of hydrogen the corresponding weights are in the ratio \(1:3\)). Only the discovery of the alternating intensity of lines in the series of one band,\(^1\) encountered, as was indicated, only in symmetrically constructed molecules, filled this omission and led, as Hund\(^2\) and Dennison\(^3\) were able to show, to the correct formula and to exact agreement of theory with experiment.
Likewise, another assumption which, under the former theories of specific heat, was put forward simply ad hoc, in order better to reconcile the results of observations with theory, is now fully confirmed with the aid of the spectroscopy of band spectra and the new quantum theory. The point is the introduction of “half” quantum numbers: we must now, as has already been mentioned more than once, in the formulas expressing the energy of rotation and vibration of the nucleus, \(Bm^2\) and \(an - bn^2\), replace the quantum numbers \(m^2\) and \(n\) by \(m(m+1)\) and \(n+\frac{1}{2}\). In application, for example, to the Planck oscillator, equation (49) immediately leads to an energy at the absolute-zero point equal to \(\frac{1}{2}h\nu\), as was already required by Planck’s second hypothesis.
In gas molecules, at the “average” temperatures encountered in practice, of the three component parts of the energy \(W_e\), \(W_v\), and \(W_r\), the rotational energy turns out to be completely excited—all degrees of freedom are present (except, perhaps, in hydrogen), so that already here we obtain the classical value \(f/2R\) of the equipartition theory; the energy is excited only partially, and the more so the lower the vibrational frequencies, while the energy of electronic motion is still entirely absent. Therefore in the majority
\(^1\) R. Mecke. Phys. ZS. 25, 597, 1924; Z. Physik 31, 709, 1925.
\(^2\) F. Hund, Z. Physik, 42, 93, 1927.
\(^3\) D. M. Dennison, Proc. Roy. Soc. 115, 483, 1927; cf. also H. Boutler, Z. Physik 50, 581, 1928.
cases in the formulas giving the sum of the energies, it is possible to get by with a few terms and to simplify the calculation. And since, moreover, the first two parts of the energy, as is known, are quite closely expressed by the formulas \(Bm(m+1)\) and \(am-bm^2\), here too one may introduce for each part of the energy a “characteristic” temperature— for rotation
\[ \Theta_r=\frac{h^2}{8\pi^2 J k}, \]
and for vibrations of the nuclei
\[ \Theta_k=\frac{m-bm}{k}. \]
In Table 17 there is again given a summary of the material collected from the spectroscopic data from this point of view. It is self-evident that only the normal state of the molecules is involved, since thermal excitation of the electronic energy, as was already mentioned, is not taken into account. Therefore only those values were used for which absorption spectra had been investigated. In the fourth column are given the values which Eucken, Karwat, and Fries used at the time for calculating \(i\). At that time the greater part of the data had to be obtained indirectly, so that discrepancies were found in some of the data, and it was still necessary to check to what extent these discrepancies influence the determination of the chemical constant and whether they make it necessary to carry out all the calculations anew. In the last column are given the heats of vaporization at absolute zero, calculated from the vapor-pressure formula. They are given here because, in compounds analogous in chemical composition, there is observed a certain connection between the heats of vaporization and the frequency of the vibrations; moreover, the smaller the vibrational energy \(\Theta_\vartheta\), the greater is the heat of vaporization.
A few more words about the statistical weight. The values of \(g\) for vibrations of the nucleus, according to the theory, must always come out the same and, moreover, equal to 1. Difficulties should therefore arise only when vibrations of several nuclei are present, i.e. in polyatomic compounds, for which it is necessary to estimate the relative strength of the individual vibrations.
In making this estimate there is at present still a known arbitrariness, because exact measurements of intensi-
there are almost no data for the corresponding spectra (the question concerns exclusively infrared absorption bands). Usually it is sufficient to include in the calculation the two most intense fundamental vibrations, which makes it possible to get by with a formula containing two constants. Thus, for example, Eucken and others use the following formulae:
\[ \begin{aligned} \mathrm{CO}_2 \ .\ .\ .\ .\ .\quad F &= 2F_1\left(\frac{960}{T}\right)+2F_2\left(\frac{3400}{T}\right) \\ \mathrm{H_2O} \ .\ .\ .\ .\quad F &= F_1\left(\frac{2300}{T}\right)+2F_2\left(\frac{5800}{T}\right) \\ \mathrm{CH}_4 \ .\ .\ .\ .\quad F &= 6F_1\left(\frac{2000}{T}\right)+3F_2\left(\frac{4350}{T}\right) \\ \mathrm{NH}_3 \ .\ .\ .\quad F &= 3F_1\left(\frac{2900}{T}\right)+3F_2\left(\frac{5000}{T}\right), \end{aligned} \]
where the admissibility of such a simplification should still be confirmed by more accurate optical measurements.
For the energy of rotation we can give the statistical weights with sufficient confidence. Numerically, a certain difference is obtained depending on which electronic term was taken as the basis of the calculation; in the simplest case of a \({}^1S\) term it can be shown (and this has also been confirmed experimentally) that the statistical weights are proportional to the quantum numbers \((2m+1)\), i.e. are in the ratio \(1:3:5\ldots\)
For more accurate calculations it would therefore be necessary to establish exactly which electronic term should be used. From Table 17, however, it is evident that the characteristic temperatures are very low; the heat of rotation therefore already attains the classical value \(f\frac{R}{2}\), and consequently the special statistical questions are of no interest here, at least insofar as the matter concerns only the course of the specific heat. For the calculation of the chemical constant, however, it is necessary to know whether the molecule ultimately possesses electronic angular momentum or not,—for only in the latter case is the statistical weight of the lowest quantum state equal to 1; here we come precisely to the question raised in the preceding paragraph concerning the influence of electronic energy on the magnitude of the chemical con-
constant. Since this third constituent part of the energy of our equation (43) proves to be unexcited, the “sum of states” reduces to the first term. This means that, in the expression for the free energy, the constant term \(\log g\) remains unchanged; it is added to the chemical constant; this term is precisely the statistical weight of our quantum state. From the series laws of line spectra we know that this statistical weight of an electronic term is equal to \((2j + 1)\), the total angular momentum of the term (i.e., equal to the number of possible orientations of \(j\) with respect to a certain fixed direction; see p. 674). For monatomic gases (for which the basic terms have been established with much greater precision), the discrepancies noted above between experiment and theory in fact disappear when we take into account the statistical weights of the corresponding terms. Although this is not, strictly speaking, relevant here, we nevertheless give these values in Table 18. This table contains the basic terms, their statistical weights, and also the uncorrected, corrected, and experimentally determined chemical constants. All deviations lie within the limits of observational error. For diatomic and polyatomic molecules—as has already been said—the agreement is not always so satisfactory. For NO and \(O_2\) we know with certainty that the basic term has total electronic angular momentum \(j\). We also know that this basic term for NO is the term \({}^{2}P_1\), for oxygen the term \({}^{3}S\), and for CO, \(H_2\), and \(N_2\) we assume that it is the term \({}^{1}S\). With the statistical weights obtained from spectroscopic data, the discrepancies between experiment and theory disappear for these gases, and also for hydrogen, since—as was already mentioned on p. 806—the deviation in Table 18 can be attributed to an incorrect calculation of the heat of rotation. Therefore the value of the constant at low temperatures, at which the heat of rotation “freezes out,” is given here. A further difficulty is presented by the determination of the constant of the halogens and hydrogen halides, for which the term \({}^{1}S\) must also be taken as the basis of the calculation; and also by the still very poorly known polyatomic gases. It is possible that for them
and the statistical weights of the solid constituent parts will turn out to be different from unity. However, we shall not examine this question further here. Summing up, we may say that our present, deeper acquaintance with the structure of band spectra has also made it possible to confirm the theory of chemical constants and specific heats, which until then had almost defied verification. The clarification of some essential questions, however, still awaits further investigation.
TABLE 18.
Characteristic temperatures
| \(\Theta_r\) | \(\Theta_s\) | \(\Theta^1_s\) | \(\Delta_0\) | |
|---|---|---|---|---|
| \(\mathrm{H_2}\) . . . | \(84,7^\circ\) | \(5930^\circ - 163(n + 1/2)\) | \(5000^\circ\) | \(0,184\) kg cal. |
| \(\mathrm{N_2}\) . . . | \(2,56\) | \(3330 - 21(n + 1/2)\) | \(3800\) | \(1,650\) |
| \(\mathrm{O_2}\) . . . | \(2,06\) | \(2220 - 16(n + 1/2)\) | \(3300\) | \(2,177\) |
| \(\mathrm{Cl_2}\) . . . | \(0,37^\circ\) | \(781^\circ - 3,7(n + 1/2)\) | \(830^\circ\) | \(7,220\) kg cal. |
| \(\mathrm{Br_2}\) . . . | \(0,12\) | \(464 - 1,7(n + 1/2)\) | \(510\) | \(10,950\) |
| \(\mathrm{J_2}\) . . . | \(0,05\) | \(304 - 0,8(n + 1/2)\) | \(350\) | \(15,435\) |
| \(\mathrm{HF}\) . . . | \(30,0^\circ\) | \(5650^\circ\) | — | — |
| \(\mathrm{HCl}\) . . . | \(15,0\) | \(4130^\circ - 76(n + 1/2)\) | \(4150^\circ\) | \(4,910\) kg cal. |
| \(\mathrm{HBr}\) . . . | \(12,0\) | \(3660^\circ\) | \(3700\) | \(5,640\) |
| \(\mathrm{HJ}\) . . . | \(9,0\) | — | — | \(6,300\) |
| \(\mathrm{NO}\) . . . | \(2,39^\circ\) | \(2680^\circ - 21(n + 1/2)\) | \(2720^\circ\) | \(3,715\) kg cal. |
| \(\mathrm{CO}\) . . . | \(2,89\) | \(3050 - 18(n + 1/2)\) | \(3100\) | \(2,070\) |
| \(\mathrm{CO_2}\) . . . | \(0,78\) | \(3340^\circ\) | \(3400\) | \(6313\) |
| \(\mathrm{CO_2}\) . . . | \(0,78\) | \(960\) | \(960\) | \(6313\) |
| \(\mathrm{CH_4}\) . . . | \(6,25\) | \(4350\) | \(4350\) | \(2,170\) |
| \(\mathrm{CH_4}\) . . . | \(6,25\) | \(1860\) | \(2000\) | \(2,170\) |
| \(\mathrm{H_2O}\) . . . | — | \(5360\) | \(5800\) | \(11,325\) |
| \(\mathrm{H_2O}\) . . . | — | \(2540\) | \(2300\) | \(11,325\) |
Table 19.
Chemical constants (corrected).
| Element | \(T\) | \(g = 2j + 1\) | \(i_{\mathrm{theor.}}\) | \(i_{\mathrm{corr.}}\) | \(i_{\mathrm{exp.}}\) | \(\Delta\) |
|---|---|---|---|---|---|---|
| A | \({}^{1}S\) | 1 | \(+0,81\) | \(+0,81\) | \(+0,79 \pm 0,04\) | \(-0,02\) |
| Hg | \({}^{1}S\) | 1 | \(+1,87\) | \(+1,87\) | \(+1,95 \pm 0,06\) | \(+0,08\) |
| Na | \({}^{2}S\) | 2 | \(+0,46\) | \(+0,76\) | \(+0,97 \pm 0,23\) | \(+0,19\) |
| K | \({}^{2}S\) | 2 | \(+0,80\) | \(+1,10\) | \(+1,13 \pm 0,32\) | \(+0,03\) |
| Cl | \({}^{2}P_{2}\) | 4 | \(+0,73\) | \(+1,33\) | \(+1,44 \pm 0,24\) | \(+0,11\) |
| Br | \({}^{2}P_{2}\) | 4 | \(+1,25\) | \(+1,85\) | \(+1,89 \pm 0,26\) | \(+0,10\) |
| T | \({}^{2}P_{2}\) | 4 | \(+1,56\) | \(+2,16\) | \(+2,08 \pm 0,23\) | \(-0,08\) |
| Pb | \({}^{3}P_{0}\) | 1 | \(+1,89\) | \(+1,89\) | \(+2,27 \pm 0,36\) | \(+0,42\) |
| W | \({}^{5}D\) | 9 | \(+1,81\) | \(+2,76\) | \(+3,7 \pm 0,3\) | \(+0,9\) |
| H\(_2\) | \({}^{1}S\) | 1 | \(-1,13\) | \(-1,13\) | \(-1,11 \pm 0,03\) | \(+0,02\) |
| O\(_2\) | \({}^{3}S\) | 3 | \(+0,05\) | \(+0,53\) | \(+0,55 \pm 0,06\) | \(+0,02\) |
| NO | \({}^{2}Pi\) | \(2(4)\) | \(+0,25\) | \(+0,55\) | \(+0,52 \pm 0,06\) | \(-0,03\) |
| NO | \({}^{2}Pi\) | \(2(4)\) | \(+0,25\) | \(+0,85\) | \(+0,83 \pm 0,1\) | \(-0,02\) |
| CO | \({}^{1}S\) | 1 | \(+0,15\) | \(+0,15\) | \(+0,29;\ +0,03 \pm 0,1\) | \(\pm 0,14\) |
| N\(_2\) | \({}^{1}S\) | 1 | \(-0,15\) | \(-0,15\) | \(-0,11 \pm 0,01\) | \(+0,04\) |