PHYSICAL CHEMISTRY

I. B. Rabinovich and Z. V. Volokhova

THE EFFECT OF REPLACING HYDROGEN BY DEUTERIUM ON THE POLARIZABILITY OF MOLECULES

(Presented by Academician A. N. Frumkin, 27 VI 1958)

In a number of works ((^{1,18})) it was shown that gaseous deuterium compounds have a lower polarizability than their hydrogen analogues. It was also found that replacement of hydrogen by deuterium causes a decrease in the polarizability of certain liquids: water, hydrogen peroxide ((^{2,10})), benzene, cyclohexane ((^3)), and dibromomethane ((^4)).

In the present work, the dispersion of light and the density were studied, and the static polarizability ((\alpha_0)) was calculated, for 12 liquid deuterium compounds and their hydrogen analogues listed in Table 1. The synthesis of (\mathrm{CDCl_3}) is described in ((^5)), and that of the deuterioalcohols in ((^6)). (\mathrm{C_6D_6}) and (\mathrm{C_6D_5CH_3}) were prepared as described in ((^7)). To obtain (\mathrm{CD_3NO_2}), nitromethane was repeatedly exchanged with a (0.02\,M) solution of (\mathrm{NaOD}) in (\mathrm{D_2O}) at (110^\circ). The substances were carefully purified and dried. The density ((\rho^{20}_4)) and refractive index ((n^{20}_D)) of the hydrogen compounds agreed with reliable literature data ((^8)) with accuracies, respectively, of (1 \cdot 10^{-4}\ \mathrm{g/cm^3}) and (1 \cdot 10^{-4}).

The refractive index was measured with an IRF-23 Pulfrich-type refractometer, with a relative accuracy of (2 \cdot 10^{-5}), at (293 \pm 0.05^\circ \mathrm{K}), for the lines: (\mathrm{H}\alpha), D, (\mathrm{Hg_3}), (\mathrm{H}\beta), and (\mathrm{Hg}_{c-\phi}). The static polarizability was obtained by extrapolating to (\nu = 0) the dependence of ((n^2 + 2)/(n^2 - 1)) on (\nu^2), where (\nu) is the frequency of light. As Wulf ((^9)) showed, for colorless substances and when experimental data for the visible region of the spectrum are used, such extrapolation is theoretically justified. For all the substances we studied, the indicated dependence in the investigated frequency range was linear within the error of measurement of (n).

The density was determined with an accuracy of (1 \cdot 10^{-4}\ \mathrm{g/cm^3}), every (5^\circ), over the range (20—70^\circ) (for chloroform: (10—40^\circ); for methanol: (10—60^\circ)). The molar volume ((V_x)) of a compound with (x\%\ \mathrm{D}) was calculated on the basis of additivity. Table 1 gives values of (V_x) only for (20^\circ), but the relative difference in the molar volumes of isotopic analogues, with an accuracy of 0.01%, is the same throughout the indicated temperature range.

All 12 deuterium compounds we studied have a lower refractive index and a smaller polarizability than the corresponding hydrogen compounds (Table 1).

The decrease in polarizability upon replacement of hydrogen by deuterium can be explained by the accompanying decrease in the zero-point energy ((\varepsilon_0)) of the atomic vibrations, while the potential curve of the electronic energy and the force constants of the bonds ((f)) remain practically unchanged ((^{10})). In the simplest case of a diatomic molecule, by making certain approximations, it can be shown that the lowering of the vibrational levels of the electronic spectra causes an increase in the energy of the electronic transitions ((\varepsilon_{\mathrm{el}})) from the ground ((0)) level to an excited ((i)) level. Thus, neglecting the difference in electronic energy between the H- and D-compound at identical (i)-levels and also not taking into account the dif-

Table 1

Isotopic effect in the static polarizability ((\alpha_0)) and intermolecular dispersion energy ((\varepsilon_{\mathrm{D}})), 293 °K

difference in rotational energy, in view of the fact that in absolute magnitude it is small in comparison with the difference in vibrational energy, we obtain:

[
\Delta \varepsilon_{\mathrm{el}}
=
\varepsilon_{\mathrm{el,D}}^{0,i}
-
\varepsilon_{\mathrm{el,H}}^{0,i}
=
(\varepsilon_{0,\mathrm{H}}-\varepsilon_{i,\mathrm{D}})^0
-
(\varepsilon_{0,\mathrm{H}}-\varepsilon_{0,\mathrm{D}})^i .
\tag{1}
]

Since, neglecting terms of higher powers, we have ((^{10})):

[
\varepsilon_{0,\mathrm{H}}-\varepsilon_{0,\mathrm{D}}
=
\frac{1}{2}h\omega_{0,\mathrm{H}}(1-\zeta)
-
\frac{1}{4}h\omega_{0,\mathrm{H}}\chi_0(1-\zeta^2)
=
]

[

\frac{h}{4\pi}\mu_{\mathrm{H}}^{-1/2} f^{1/2}
\left(
1-\zeta-\frac{\chi_0}{2}+\frac{\chi_0}{2}\zeta^2
\right),
]

where (\mu) is the reduced mass, (\chi_0) is the anharmonicity coefficient, and (\zeta=(\mu_{\mathrm{H}}/\mu_{\mathrm{D}})^{1/2}), then from (1) we find:

[
\Delta \varepsilon_{\mathrm{el}}
=
\frac{h}{4\pi}\mu_{\mathrm{H}}^{-1/2}
\left[
(f^0)^{1/2}-(f^i)^{1/2}
\right]
\left[
(1-\zeta)-\frac{\chi_0}{2}(1-\zeta^2)
\right].
\tag{1′}
]

Since (f^0>f^i); ((1-\zeta)) and ((1-\zeta^2)) are positive quantities of the same order, while (\chi_0) is of the order of (0.01) ((^{10})), then, according to (1′), we have

[
\Delta \varepsilon_{\mathrm{el}}>0
\quad \text{or} \quad
\nu_{i,\mathrm{D}}>\nu_{i,\mathrm{H}},
\tag{2}
]

where (\nu_i) is the frequency of the electronic transition.

The consequences of (2) are probably also valid for a polyatomic molecule. This opinion is apparently held by Ingold, Raisin, and Wilson ((^{11})), since, in explaining the decrease in the polarizability of benzene upon replacement of hydrogen by deuterium, they present, in a more approximate form, essentially the same arguments as those given above for a diatomic molecule. Urey and Teal ((^{12})) explain the greater polarizability of heavy water, as compared with ordinary water, also by the fact that (\nu_{i,\mathrm{D}}>\nu_{i,\mathrm{H}}).

The validity of (2) is confirmed by experimental data. Thus, Ingold and Wilson ((^{13})) found for (\mathrm{C_6D_6}) and (\mathrm{C_6H_6}) in the fluorescence spectrum of benzene that (\nu_{i,\mathrm{D}}) is greater than (\nu_{i,\mathrm{H}}) by approximately (200\ \mathrm{cm}^{-1}). Herzberg and Schmid ((^{14})), on the basis of predissociation phenomena in the CH and CD spectra, came to the conclusion that the energy levels of combinations of atomic terms in CD lie approximately (350\ \mathrm{cm}^{-1}) higher than in CH. Franck and Wood ((^{15})) established that the long-wavelength absorption limit of (\mathrm{D_2O}) lies farther from the ultraviolet region than that of (\mathrm{H_2O}), and consequently (\nu_{i,\mathrm{D}}>\nu_{i,\mathrm{H}}). It was also shown for methane ((^{16})) and ammonia ((^{17})) that (J_{\mathrm{D}}>J_{\mathrm{H}}), where (J) is the ionization potential.

The polarizability of a molecule is expressed by the dispersion formula ((^{22}))

[
\alpha=\frac{2}{h}\sum \frac{\nu_i P_i^2}{\nu_i^2-\nu^2},
\tag{3}
]

where (P_i^2) are the probabilities of electronic transitions, and (\nu) is the frequency of the incident light. In view of the identity of the electron shells of isotopic compounds, (P_{i,\mathrm{D}}^2 \approx P_{i,\mathrm{H}}^2). Therefore, according to (3), for the polarizability (\alpha_i) at level (i) we obtain:

[
\frac{\alpha_{i,\mathrm{D}}}{\alpha_{i,\mathrm{H}}}
=
\frac{\nu_{i,\mathrm{H}}-\nu^2/\nu_{i,\mathrm{H}}}
{\nu_{i,\mathrm{D}}-\nu^2/\nu_{i,\mathrm{D}}}.
\tag{4}
]

Since (\nu_{i,\mathrm{H}}<\nu_{i,\mathrm{D}}), and for colorless substances (\nu_i>\nu), it follows from (4) that (\alpha_{i,\mathrm{D}}<\alpha_{i,\mathrm{H}}). Hence, probably, in general

[
\alpha_{\mathrm{D}}<\alpha_{\mathrm{H}}
\tag{5}
]

in agreement with the experimental data ((^{1})).

Substituting into (4) the expansion of the quantity ((\nu_{i,\mathrm{D}}-\nu^2/\nu_{i,\mathrm{D}})^{-1}) in a series, we obtain:

[
\frac{\alpha_{i,\mathrm{D}}}{\alpha_{i,\mathrm{H}}}
=
\frac{\nu_{i,\mathrm{H}}}{\nu_{i,\mathrm{D}}}
-\nu^2
\left(
\frac{1}{\nu_{i,\mathrm{H}}\nu_{i,\mathrm{D}}}
-
\frac{\nu_{i,\mathrm{H}}}{\nu_{i,\mathrm{D}}^3}
\right)
-\nu^4
\left(
\frac{1}{\nu_{i,\mathrm{H}}\nu_{i,\mathrm{D}}^3}
-
\frac{\nu_{i,\mathrm{H}}}{\nu_{i,\mathrm{D}}^5}
\right)
-
]

[
-\nu^6
\left(
\frac{1}{\nu_{i,\mathrm{H}}\nu_{i,\mathrm{D}}^5}
-
\frac{\nu_{i,\mathrm{H}}}{\nu_{i,\mathrm{D}}^7}
\right)
-\ldots
-\nu^{2n}
\left(
\frac{1}{\nu_{i,\mathrm{H}}\nu_{i,\mathrm{D}}^{2n-1}}
-
\frac{\nu_{i,\mathrm{H}}}{\nu_{i,\mathrm{D}}^{2n+1}}
\right)
-\ldots
\tag{6}
]

In (6) all factors multiplying (\nu^{2n}) have a common form, and, since (\nu_{i,\mathrm{D}}>\nu_{i,\mathrm{H}}), these factors are positive. In view of this, and since in the right-hand side of (6) all terms beginning with the second are subtracted from ((\nu_{i,\mathrm{H}}/\nu_{i,\mathrm{D}})), the difference between (\alpha_{\mathrm{D}}) and (\alpha_{\mathrm{H}}) increases with increasing (\nu). This also agrees with experimental literature data and with our data.

For the static polarizability ((\nu=0)), from (4) we have:

[
\left(
\frac{\alpha_{i,\mathrm{D}}}{\alpha_{i,\mathrm{H}}}
\right)0
=
\frac{\nu.}}}{\nu_{i,\mathrm{D}}
\tag{7}
]

At moderate temperatures, a consequence of relation (5) is a decrease in the intermolecular dispersion energy ((\varepsilon_{\mathrm{D}})) when hydrogen is replaced by deuterium.* Thus, according to the formula of Slater and Kirkwood ((^{23}))

[
|\varepsilon_{\mathrm{D}}|
=
\frac{3eh}{8\pi r^6}
\left(
\frac{n\alpha_0^3}{m}
\right)^{1/2},
\tag{8}
]

where (n) is the total number of electrons in the outer shells of the atoms comprising the molecule, (m) and (e) are the mass and charge of the electron, and (r) is the distance between

* Bell ((^{18})) also proposes another approach to the consequence of (5). He qualitatively relates (5) to the fact that, owing to the anharmonicity of zero-point atomic vibrations, the increase in the mean internuclear distance, relative to its value at the minimum of the potential curve, is smaller in the case of a D compound than in an H compound.

molecules. In the first approximation, ((r_{\mathrm H}/r_{\mathrm D})^3 = V_{\mathrm H}/V_{\mathrm D}). Therefore

[
\frac{\varepsilon_{\mathrm{D},\mathrm{D}}}{\varepsilon_{\mathrm{D},\mathrm{H}}}
=
\left(\frac{V_{\mathrm H}}{V_{\mathrm D}}\right)^2
\left(\frac{\alpha_{0,\mathrm D}}{\alpha_{0,\mathrm H}}\right)^{3/2}.
\tag{9}
]

As is seen from Table 1, calculation by (9) showed that, for the substances studied,

[
|\varepsilon_{\mathrm{D},\mathrm D}| < |\varepsilon_{\mathrm{D},\mathrm H}|
\quad \text{(intermediate temperatures).}
\tag{10}
]

This is due to the fact that at intermediate temperatures the relative difference in the value of (\alpha^{3/2}) of isotopic analogues is greater than such a difference in the values of (V^2), so that, according to (9), the sign of the ratio ((\varepsilon_{\mathrm{D},\mathrm D}/\varepsilon_{\mathrm{D},\mathrm H})) is determined by the sign of the ratio ((\alpha_{0,\mathrm D}/\alpha_{0,\mathrm H})).

Consequence (10) is in agreement with the fact that replacement of hydrogen by deuterium causes a lowering of the critical temperature ((^{24})), if it is not in the region of low temperatures, and also with the fact that deuterium compounds have a greater compressibility than their hydrogen analogues ((^6)), if the low-temperature region is excluded.

However, at low temperatures the molar volume is sharply affected by the change in the amplitude of the zero-point vibrations of the molecules. Therefore, for (\mathrm D_2) and (\mathrm H_2) near (19.5^\circ\mathrm K), for example, (V_{\mathrm D}) is smaller than (V_{\mathrm H}) by 17% ((^{19})), and for (\mathrm{He}^4) and (\mathrm{He}^3) ((^{20})) at (2^\circ\mathrm K), (V_4) is smaller than (V_3) by 29%. In these cases the difference in the value of (\varepsilon_{\mathrm D}) for isotopic analogues, according to (9), is determined mainly by the difference in the values of (V), and

[
|\varepsilon_{\mathrm{D},\mathrm D}| > |\varepsilon_{\mathrm{D},\mathrm H}|
\quad \text{(low temperatures).}
\tag{11}
]

Thus, if the values of (V) and (\alpha_0) ((^{21})) for (\mathrm{He}^4) and (\mathrm{He}^3) corresponding to (2^\circ\mathrm K) are substituted in (9), we obtain that (|\varepsilon_{\mathrm{D},4}|) is greater than (|\varepsilon_{\mathrm{D},3}|) by 45%. It is interesting that the result of this very approximate calculation agrees well with the experimental data on the heat of vaporization of (\mathrm{He}^4) and (\mathrm{He}^3) ((^{20})): at (2^\circ\mathrm K), respectively, 22.2 cal/mol* and 11.1 cal/mol.

Consequence (11) also corresponds to the fact that (\mathrm D_2) and (\mathrm{He}^4) have a higher critical temperature than, respectively, (\mathrm H_2) and (\mathrm{He}^3) ((^{24})).

We express our gratitude to Prof. L. S. Mayants for discussion of the results.

Institute of Chemistry
at Gorky State University
named after N. I. Lobachevsky

Received
6 VI 1958

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* This value refers to HeI; however, in the present case this apparently has no substantial significance. As is known, the heat of the (\lambda)-transition in liquid helium is negligible, and between 1.75° and 3.5°K the heat of vaporization changes only within 4%.