PHYSICAL CHEMISTRY

I. B. Rabinovich and V. A. Gorbushenkov

ISOTOPE EFFECT IN THE CRITICAL TEMPERATURE

(Presented by Academician A. N. Frumkin on 19 II 1958)

Literature data on the influence of isotopic substitution on critical quantities are very limited (Table 1), and the regularities of this influence

Table 1

Critical parameters of isotopic compounds

* This value refers to deuterium containing 0.022 $n$-D₂ and 0.978 $o$-D₂.

are still not clear. It was shown (⁹) that in the series H₂—HD—D₂, as the molecular weight $(M)$ increases, the critical temperature $(T_{\mathrm{k}})$ increases, and it depends approximately linearly on $1/\sqrt{M}$. However, the replacement of hydrogen by deuterium in H₂O (⁷), PH₃, or HBr (⁸) produces the opposite effect—a decrease in $T_{\mathrm{k}}$.

In the present work, $T_{\mathrm{k}}$ values of eleven deuterated compounds (Table 2) and their hydrogen analogues were compared. The synthesis of deuterochloroform is described in (¹⁴), and that of deuterated alcohols in (¹⁶). To obtain deuteronitromethane, ordinary nitromethane was repeatedly exchanged with a 0.02 M solution of NaOD in D₂O at 110°C in sealed ampoules. Deuterobenzene C₆D₆ and methyldeuterobenzene C₆D₅CH₃ were obtained as a result of prolonged repeated exchange with a 52 mol.% solution of D₂SO₄ in D₂O. The substances were carefully purified and dehydrated. The density $(\rho_4^{20})$, refractive index $(n_D^{20})$, and boiling temperature of the hydrogen compounds agreed with reliable literature data (¹⁷) with accuracies, respectively, of $1 \cdot 10^{-4}$ g/cm³; $1 \cdot 10^{-4}$; and 0.1–0.2°.

$T_{\mathrm{k}}$ was determined from the disappearance and appearance of the meniscus. The procedure and apparatus were similar to those described in (⁷). Three capillary ampoules were filled with each substance. The capillaries with isotopic analogues were kept in the thermostat under identical conditions. For each pair of such capillaries, 20 measurements were made. Along with measurements of the differences $\Delta T_{\mathrm{k}} = T_{\mathrm{k,H}} - T_{\mathrm{k,D}}$, determinations of the absolute values of $T_{\mathrm{k}}$ were carried out in

Table 2

Isotope effect in the critical temperature

* The product studied did not exactly correspond to the formula, and also contained isotopic analogues with a smaller number of deuterium atoms.
* In the group or ring in which it was substituted.
* With linear extrapolation to 100% D, $\Delta T_k = 1.6^\circ$.
*** At 100% D, $\Delta T_k = 1.5^\circ$.

in cases where it did not exceed 300°C. The results (Table 2) for hydrogen-containing compounds agree, to within 0.1–0.2°, with the values of $T_k$ given in the review (15), as the most reliable among the data of a number of authors. Above 300°C only $\Delta T_k$ was measured, while the values of $T_{k,\mathrm{H}}$ were taken from (15). For all substances the root-mean-square error in determining $\Delta T_k$ is about 0.1°, which for the butanols and isopropanol corresponds to 20–30%, and for the remaining substances to about 10% of the value of $\Delta T_k$.

In all cases we found that replacement of a light isotope by a heavy one lowers $T_k$. From the data of Table 1 it is evident that, with the exception of helium and hydrogen, an isotope effect in $T_k$ of the same sign is also observed for the substances studied earlier. Thus, for substances of various classes of inorganic and organic compounds in the region of moderate temperatures, the relation

[
T_{k,\mathrm{D}} < T_{k,\mathrm{H}},
\tag{1}
]

apparently is a general regularity. Applying the known equalities $a = 27R^2T_k^2/64P_k$ and $a = 9RT_kV_k/8$, where $a$ is the van der Waals constant, we have:

[
\frac{a_{\mathrm{D}}}{a_{\mathrm{H}}}
=
\frac{T_{k,\mathrm{D}}^2 P_{k,\mathrm{H}}}{T_{k,\mathrm{H}}^2 P_{k,\mathrm{D}}},
\tag{2}
]

[
\frac{a_{\mathrm{D}}}{a_{\mathrm{H}}}
=
\frac{(T_k V_k){\mathrm{D}}}{(T_k V_k).}}
\tag{3}
]

For the isotopic analogues of water, according to (3), $a_{\mathrm{D}}/a_{\mathrm{H}} = 0.989$. For the deuterium compounds listed in Table 2, $V_k$ or $P_k$ were not studied. An approximate idea of the difference between $V_{\mathrm{D}}$ and $V_{\mathrm{H}}$ in the region of $T_k$ can be obtained

according to the values of the volumes and the mean coefficients of thermal expansion (\bar\alpha_{\mathrm D}) and (\bar\alpha_{\mathrm H}) at lower (T). We measured the density of the indicated substances in the interval (293\text{—}343^\circ\mathrm K), at intervals of (5^\circ), and calculated (V) and (\alpha).

As is seen from Table 2, at (293^\circ) for chloroform, nitromethane, and acetic acid, within the limits of the determination error ((0.05\%)), (V_{\mathrm D}=V_{\mathrm H}). For (\mathrm{C_6D_6}) and (\mathrm{C_6D_5CH_3}), (V_{\mathrm D}) is less than (V_{\mathrm H}) by (0.1\text{—}0.2\%). This is explained by the fact that, at room temperature, in these compounds the molar volume is still appreciably affected by the decrease in the amplitude of zero-point molecular vibrations when H is replaced by D ((^{21})). In alcohols, on the contrary, (V_{\mathrm D}) exceeds (V_{\mathrm H}) by (0.1\text{—}0.2\%), which is probably a consequence of an increase in the degree of association by means of hydrogen bonding when hydroxyl hydrogen is replaced by deuterium ((^{13})). However, at (T) close to (T_{\mathrm k}) of the substances listed, when the quantities (h\nu^*) are small in comparison with (kT), and there is almost no association, the influence of both indicated factors should be insignificant. In the region studied ((293\text{—}343^\circ\mathrm K)), in all cases (\bar\alpha_{\mathrm D}>\bar\alpha_{\mathrm H}), but their difference is so small that it can give a difference in the isothermal values (V_{\mathrm D}) and (V_{\mathrm H}) of the order of (1\cdot10^{-4}\%) per degree. Moreover, with increasing (T) up to (T_{\mathrm k}), the values (\bar\alpha_{\mathrm D}) and (\bar\alpha_{\mathrm H}), owing to the weakening of intermolecular interaction, will become much closer. Consequently, there are grounds to suppose that for substances liquid at intermediate temperatures, in the region of their (T_{\mathrm k}), and with an accuracy up to (0.1\%),

[
V_{\mathrm D}=V_{\mathrm H}.
\tag{4}
]

The condition (T_{\mathrm{k,D}}<T_{\mathrm{k,H}}) can only give a decrease of (V_{\mathrm{k,D}}) relative to (V_{\mathrm{k,H}}), but not the opposite result. Thus, in view of (4), probably,

[
V_{\mathrm{k,D}}\ll V_{\mathrm{k,H}}.
\tag{4'}
]

Then, for compounds in which (\Delta T_{\mathrm k}) is about (1^\circ), from (3), (1), and (4) we find:

[
a_{\mathrm D}<a_{\mathrm H},
\tag{5}
]

whence, in view of (4), it follows that

[
\pi_{\mathrm D}<\pi_{\mathrm H},
\tag{6}
]

where (\pi=a/V^2) is the internal pressure. Thus, at intermediate (T), substitution of hydrogen by deuterium apparently leads to a decrease in the van der Waals interaction.

From the standpoint of the nature of dispersion forces this follows from the Slater and Kirkwood equation ((^{18})). Applying the latter to isotopic analogs and replacing, in the first approximation, ((r_{\mathrm D}/r_{\mathrm H})^3), where (r) is the distance between molecules, by (V_{\mathrm D}/V_{\mathrm H}), we obtain:

[
\frac{\varepsilon_{\mathrm D}}{\varepsilon_{\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{7}
]

where (\varepsilon) is the intermolecular dispersion energy, and (\alpha_0) the static polarizability. The lowering of the zero-point energy of the corresponding atomic bonds upon replacement of hydrogen by deuterium causes an increase in the frequencies of electronic transitions. As follows from the dispersion equation ((^{19})), this leads to a decrease in polarizability, since, owing to the identity of the electron shells, the probabilities of electronic transitions in isotopic analogs should differ only insignificantly. A decrease in polarizability upon replacement of hydrogen by deuterium has also been established experimentally for water ((^{13,20})), hydrogen peroxide ((^{20})), benzene and cyclohexane ((^{21})), and deuteroethanes ((^{22})). Along with this, at intermediate temperatures in all the substances studied, except (\mathrm{C_6D_{12}}), (\mathrm{C_6D_6}), and (\mathrm{C_6D_5CH_3}), (V_{\mathrm D}>V_{\mathrm H}). In the latter, at room (T), (V_{\mathrm H}) exceeds (V_{\mathrm D}) by (0.1\text{—}0.2\%), but this difference in percentage rela-

* (\nu) is the frequency of intermolecular vibrations (tens of (\mathrm{cm}^{-1})).

is several times smaller than the isotope effect in (a_0) ({}^{(21)}). Hence, in view of (7), it follows that

[
|\varepsilon_D| < |\varepsilon_H|.
\tag{8}
]

The decrease in the van der Waals interaction upon replacing hydrogen by deuterium is also indicated by the fact that the compressibility then increases ({}^{(16)}). The lower strength of the van der Waals forces in deutero-compounds, as compared with their hydrogen analogues, can also explain the larger values of (\bar{\alpha}) in the former.

However, the isotope effects described in (a), (\pi), and (\varepsilon), and hence the conclusion that the van der Waals interaction decreases when a light isotope is replaced by a heavy one, cannot be extended to the region of low temperatures. In view of the fact that, at such temperatures, the amplitude of the zero molecular vibrations has a strong influence on the molar volume, the latter in liquid tritium, deuterium, and He(^4) is considerably smaller than in their light analogues at the same temperature. For example, for D(_2) and H(_2) at 19.5°K the corresponding difference is about 17% ({}^{(23)}). In these cases the difference in molar volumes prevails over the decrease in polarizability, and then it follows from (7) that, in liquid D(_2) and He(^4), the dispersion interaction is stronger than, respectively, in H(_2) and He(^3). The isotope effect in the critical parameters of D(_2) and He(^3) shows that this is indeed so. Substituting the values of (P_k) and (T_k) into (2), we obtain for D(_2) and H(_2): (a_D/a_H = 1.037), and for He(^4) and He(^3): (a_4/a_3 = 1.26); whereas the percentage difference in the molecular weights of the isotopes is three times greater for hydrogen than for helium, the percentage difference in (T_k), (P_k), and (a), on the contrary, is 3–7 times greater for the helium isotopes. This depends on the especially low (T_k) of the latter. Since, in the low-temperature region, the heavy isotope has a smaller molar volume than its light analogue, then, according to the relation (\pi_D/\pi_H = (a_D/a_H)(V_H/V_D)^2), for hydrogen and helium we have: (\pi_D > \pi_H) and (\pi_4 > \pi_3).

Thus, at temperatures in the region where liquid helium and hydrogen exist, replacement of a light isotope by a heavy one leads to an enhancement of the van der Waals interaction, i.e., to an inverse effect, opposite to the effect at intermediate temperatures.

We express our gratitude to Corresponding Member of the Academy of Sciences of the USSR A. I. Brodskii for discussion of the results and to S. D. Ravikovich for methodological advice.

Scientific Research Chemical Institute
of Gorky State University
named after N. I. Lobachevsky

Submitted
28 XII 1957

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