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PHYSICAL CHEMISTRY
A. M. ROZEN, A. I. MIKHAILICHENKO
ON THE PERIODIC REGULARITY OF ISOTOPE EXCHANGE IN DIATOMIC MOLECULES
(Presented by Academician V. N. Kondrat’ev on 26 VII 1962)
In the statistical calculation of isotope-separation coefficients in isotope-exchange reactions it is convenient to use the ratio of sums over states, from which the symmetry numbers are excluded \((^1)\). Ya. M. Varshavsky and S. E. Vaisberg \((^{2-4})\), who called the indicated quantities β-factors,* generalized these quantities to the case of polyatomic molecules and widely applied them in calculations. These authors discovered a very interesting periodic dependence of the β-factors of diatomic hydrides \( \mathrm{H}Э_i \) on the atomic number (or, what is the same, on the degree of filling of the outer electron shells) of the element \(Э_i\) bonded to hydrogen (Fig. 1). The purpose of the present article is to explain this regularity and to derive certain quantitative relations.
Fig. 1. Dependence of β-factors (solid line) and force constants \(k_e\) (dashed line) of diatomic hydrides on the atomic number \(N\) of the element bonded to hydrogen
The β-factor of a diatomic molecule is determined by the vibrational frequencies of the isotopic molecules (see, for example, \((^5)\))
\[ \beta = \left[ \frac{u^*}{u}\, e^{-(u^*-u)/2} \frac{1-e^{-u}}{1-e^{u^*}} \right]^{1/n}, \tag{1} \]
* It can be shown \((^5)\) that the β-factor is expressed in the following way through the vibrational sums over states:
\[ \beta = \left[ \left( \frac{Z^*}{Z_{\text{class}}} \right) \Big/ \left( \frac{Z}{Z_{\text{class}}} \right) \right]^{1/n}, \]
where \(Z_{\text{class}}\) and \(Z\) are the values of the statistical sum given, respectively, by classical and quantum mechanics.
where \(u = hc\omega_e/kT\); \(\omega_e\) is the vibrational frequency of the isotopic molecule; \(c\) is the speed of light; \(h\) is Planck’s constant; \(k\) is Boltzmann’s constant; \(T\) is the temperature (°K); \(n\) is the number of isotope atoms being substituted in the molecule; the asterisk indicates that the given quantity refers to the heavy isotope. The vibrational frequency, in turn, is determined by the relation
\[ \omega_e=\frac{1}{2\pi c}\sqrt{\frac{k_e}{\mu}}, \tag{2} \]
where \(k_e\) is the force constant, \(\mu\) is the reduced mass of the molecule, \(\mu=mm_0/(m+m_0)\), and \(m\) and \(m_0\) are, respectively, the masses of the isotopic atom and the second atom.
Since the reduced mass of the molecule depends monotonically on the mass of the second atom, if the force constant were constant or a monotonic function of atomic number, no periodic dependence of the \(\beta\)-factors could exist. Consequently, the periodic character of the variation of the \(\beta\)-factors must be due to the periodic character of the variation of the force constants \(\left(^{6}\right)\).
To verify this conclusion, Fig. 1 plots the values of the \(\beta\)-factors and the force constants \(k_e\) of diatomic hydrides as functions of the atomic number of the element bonded to hydrogen (the values of \(k_e\) were calculated from equation (2), on the basis of vibrational frequencies, or taken from Refs. \(\left(^{7,8}\right)\)). As is seen from Fig. 1, the quantities \(\beta\) and \(k_e\) are approximately proportional. Thus, it may be regarded as established that the periodic dependence of the \(\beta\)-factors is caused by an analogous regularity in the variation of the force constants.
In the case of diatomic molecules, the force constant is the second derivative of the potential energy of the molecule with respect to distance (at the equilibrium point). Since a quantum-mechanical calculation of the potential energy of molecules is still impossible, a theoretical calculation of \(k_e\) is likewise impossible (the hydrogen molecule, as is well known, is an exception). It is known only that \(k_e\) depends to a considerable degree on the properties of the atoms forming the molecule, on the structure of their electron shell, and on the type of electrons forming the bond \(\left(^{9}\right)\).
For quantitative calculations one may use the empirical Guggenheimer formula \(\left(^{8}\right)\)
\[ \omega_e=c_1(z_1z_2)^{1/4}\mu^{-1/2}r_e^{-c_2}, \tag{3} \]
where \(z_1\) and \(z_2\) are the numbers of electrons in the outer shells of the two atoms forming the molecule; \(r_e\) is the equilibrium internuclear distance; \(c_1=(2.065\div 2.976)\cdot10^3\) and \(c_2=0.92\div1.23\) are constants found by the author from analysis of experimental data on the vibrational frequencies of more than 150 diatomic molecules.
From equations (2) and (3) we obtain
\[ k_e=4\pi^2c^2c_1^2(z_1z_2)^{1/2}r_e^{-2c_2}. \tag{4} \]
Hence it is clear that the periodicity in the variation of the force constants of diatomic hydrides, and consequently also of the \(\beta\)-factors, is due to the periodic character of the filling of the electron shells of the second atom \(z_2\) and of the equilibrium internuclear distance of the molecule \(r_e\). The proportionality of the force constant to the geometric mean of the numbers of electrons in the outer shells of the atoms is a remarkable fact. Some qualitative considerations explaining this fact are given in Ref. \(\left(^{8}\right)\).
Using relations (1) and (2), one can establish a connection between the \(\beta\)-factor and the force constant. In the general case, comparatively cumbersome expressions are obtained; however, they are simplified for \(u\ll2\) or for \(u\gg5\). In the first case (small quantum effects—small
frequencies or high temperatures), according to (1):
\[ \beta=\left(1+\bar u\,\Delta u/12\right)^{1/n}, \tag{5} \]
where \(\bar u=(u+u^*)/2,\ \Delta u=u-u^*\).
Substituting the values of \(u\) from equation (2), we obtain
\[ \beta^n\big|_{u<2}=1+\frac{1}{96\pi^2}\left(\frac{h}{kT}\right)^2 \left(\frac{1}{\mu}-\frac{1}{\mu^*}\right)k_e, \tag{6} \]
i.e., the quantity \((\beta^n-1)\) is proportional to the force constant. In the second case (sufficiently low temperatures or large frequencies, when the vibrations are not yet excited and the correction \(\left(\dfrac{1-e^{-u}}{1-e^{-u^*}}\right)\) in equation (1) may be neglected), we find
\[ \beta^n\big|_{u>5}=\sqrt{\mu/\mu^*}\,e^{b\sqrt{k_e}} \tag{7} \]
where*
\[ b=\frac{1}{4\pi}\frac{h}{kT}\left(\sqrt{\frac{1}{\mu}}-\sqrt{\frac{1}{\mu^*}}\right). \]
In both cases the \(\beta\)-factor increases monotonically with increasing \(k_e\) (the data for hydrides shown in Fig. 1 refer to the second case, when \(u\geq 5\)). The concerted change of \(\beta^n\) and \(k_e\) is seen from Figs. 1 and 2. Figure 2 shows the dependence of \(\beta^n\) (\(n=1\), except for the hydrogen molecule, where \(n=2\)) on \(k_e\) for hydrides of the first and second periods of the periodic table. The same figure also gives data for the lithium halides \(\mathrm{Li}^{6-7}\mathrm{F}\), \(\mathrm{Li}^{6-7}\mathrm{Cl}\), \(\mathrm{Li}^{6-7}\mathrm{Br}\), \(\mathrm{Li}^{6-7}\mathrm{J}\) (the \(\beta\)-factors were calculated by the authors). In this case as well, when \(2<u<5\), \(\beta^n\) and \(k_e\) are correlated.
Fig. 2. Dependence of \(\beta\)-factors on the force constants \(k_e\): 1 — diatomic hydrides of the first and second periods of the periodic table of elements; 2 — lithium halides \(\mathrm{Li}^{6-7}\mathrm{F}\), \(\mathrm{Li}^{6-7}\mathrm{Cl}\), \(\mathrm{Li}^{6-7}\mathrm{Br}\), \(\mathrm{Li}^{6-7}\mathrm{J}\).
Using relations (6) and (4), we obtain
\[ \beta^n\big|_{u<2}=1+\frac{1}{24}\left(\frac{hc}{kT}\right)^2 c_1^2(z_1z_2)^{1/2}r_e^{-2c_2}\frac{m^*-m}{m^*m}. \tag{8} \]
From (7) and (4) it follows that
\[ \beta^n\big|_{u>5}= \sqrt{\frac{\mu}{\mu^*}}\, \exp\left[ \frac{c_1}{2}\frac{hc}{kT} \left(\sqrt{\frac{1}{\mu}}-\sqrt{\frac{1}{\mu^*}}\right) (z_1z_2)^{1/4}r_e^{-c_2} \right]. \tag{9} \]
Equations (8) and (9) constitute a quantitative formulation of the periodic regularity in the variation of \(\beta\)-factors: (8) for the region of small quantum effects, when \(u\leq 2\), and (9) for the region of large quantum effects, when \(u\geq 5\) (the case of hydrides). They relate the \(\beta\)-factor to the filling of electron shells and, moreover, show that the \(\beta\)-factors depend not only on the filling of the electron shells but also on the internuclear distance.
* For small \(\Delta m/m\), we find
\[ b\approx \frac{h}{8\pi kT}\frac{\Delta m}{m^2}\sqrt{\mu}, \]
respectively, for \(m\gg m_0\),
\[ b\approx \frac{h}{8\pi kT}\times \frac{\Delta m}{m^2}\sqrt{m_0}, \]
and for \(m\ll m_0\),
\[ b\approx \frac{h}{8\pi kT}\frac{\Delta m}{m^{3/2}}. \]
As is known, the internuclear distance of diatomic hydrides also changes periodically (see Fig. 3, constructed from data \(^{6,7}\)); within a period (with increasing \(z_2\)), \(r_e\) decreases, while at constant \(z_2\), \(r_e\) increases (dashed curves in Fig. 2). An increase in the internuclear distance with increasing \(N\) leads to a decrease in the \(\beta\)-factors (at \(z_2=\mathrm{const}\)) as the atomic number of the element bonded to hydrogen increases (Fig. 1). The decrease of \(r_e\) within a period leads to a strengthening of the periodic dependence of the \(\beta\)-factor on the atomic number of the second (nonisotopic) atom.
Fig. 3. Dependence of the equilibrium internuclear distances \(r_e\) of diatomic hydrides on the atomic number \(N\) of the element bonded to hydrogen
Fig. 4. Dependence of the \(\beta\)-factors of diatomic lithium compounds (\(\mathrm{Li}^{6-7}\)) on the atomic number \(N\) of the element bonded to lithium. The lines connecting the points of maxima and minima are drawn, conventionally, as straight lines
Since the derived relations (8) and (9) are general in character, the regularities considered in the variation of the \(\beta\)-factors are valid not only for hydrogen, but also for any monovalent element. There are insufficient spectral data to construct the curve of the periodic regularity in the variation of \(\beta\)-factors for other elements. However, the available information on vibrational frequencies makes it possible to find extrema on the indicated curve for some monovalent elements, for example \(\mathrm{Li}^{6-7}\), as is shown in Fig. 4 (the \(\beta\)-factors of the corresponding diatomic lithium compounds were calculated by the authors).
Moscow Institute of Chemical Technology
named after D. I. Mendeleev
Received
20 VII 1962
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