<figure><img src="image_1.png" /></figure>

HAL
open science

# A review of hydrogen, carbon, nitrogen, oxygen, sulphur, and chlorine stable isotope fractionation among gaseous molecules

Pascal Richet, Yan Bottinga, Marc M. Javoy

► To cite this version:

Pascal Richet, Yan Bottinga, Marc M. Javoy. A review of hydrogen, carbon, nitrogen, oxygen, sulphur, and chlorine stable isotope fractionation among gaseous molecules. Annual Review of Earth and Planetary Sciences, 1977, 5, pp.65-110. 10.1146/annurev.ea.05.050177.000433. insu-01442296

HAL Id: insu-01442296

https://insu.hal.science/insu-01442296v1

Submitted on 7 Oct 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_3.png" /></figure>

# A REVIEW OF HYDROGEN, CARBON, NITROGEN, OXYGEN, SULPHUR, AND CHLORINE STABLE ISOTOPE FRACTIONATION AMONG GASEOUS MOLECULES<fnref n="1" />

P. Richet, Y. Bottinga, and M. Javoy

Laboratoire de Géochimie et Cosmochimie,<fnref n="2" /> Universités de Paris 6 et 7, 4 Place Jussieu, 75230 Paris Cedex 05, France

## 1 INTRODUCTION

Fairly soon after the discovery of the existence of stable isotopes, it was noticed that significant stable isotope fractionation occurred in nature for the isotopes of the light elements (Maillard 1936, Nier & Gulbransen 1939, Murphey & Nier 1941). But the real development of stable isotope geochemistry started when Urey (1947) published his paper on the thermodynamic properties of isotopic substances, and McKinney et al (1950) described a mass spectrometer with which one could measure on a routine basis, with high precision, the small differences in isotopic composition of naturally occurring substances. Soon after these developments a series of basic papers on stable isotope geochemistry was published by Silverman (1951) for oxygen, Friedman (1953) for deuterium, Craig (1953) for carbon, Tudge & Thode (1950) for sulphur and Epstein & Mayeda (1953) for water. In these early papers the isotopic composition of naturally occurring substances was surveyed and discussed, and with the theoretically calculated fractionation factors published by Urey (1947) and Tudge & Thode (1950), it was possible to study quantitatively the observed isotopic variations in certain cases.

In geochemistry the light stable isotopes are useful as tracers as well as palaeothermometers. Their palaeothermometric characteristics are a result of the fact that the isotopic composition of two cogenetic phases depends on the temperature at which these phases were or are in equilibrium. To use effectively the tracer and thermometric properties of stable isotope distributions one needs to know the isotopic fractionation factors and their temperature dependence. As is virtually

<footnote n="1">¹ Contribution I. P. G. NS 213</footnote>

<footnote n="2">² Laboratory associated with CNRS No. 196</footnote>
always the case the measurement of these fractionation factors is very tricky. Fortunately isotopically substituted molecules form perfect solutions; it is this aspect that makes it possible to calculate without too much difficulty stable isotope fractionation factors. The light stable isotopes are very useful as tracers and palaeo-temperature indicators, because they are ubiquitous in nature, very often in coexisting phases, which frequently are simple enough that one may calculate isotopic fractionation factors for them.

The theory of stable isotope fractionation probably starts with the work by Lindemann (1919) and Lindemann & Aston (1919), who applied statistical thermodynamics to calculate the difference in vapor pressure among lead isotopes. The method used by Lindemann has remained essentially the same since then. Following Lindemann, Urey & Rittenberg (1933) calculated equilibrium constants for exchange reactions involving hydrogen, deuterium and hydrogen and deuterated halogen compounds. Farkas & Farkas (1934) did the same for hydrogen-deuterium exchange with water. This work was followed by the well-known contribution of Urey & Greiff (1935) who did calculations for hydrogen, lithium, carbon, nitrogen, oxygen, and halogen isotope exchanges between various diatomic and polyatomic molecules. In these early days such calculations were very tedious because of the lack of modern computing facilities. Waldmann (1943) developed probably the first convenient equations for isotopic exchange calculations. Bigeleisen & Mayer (1947) derived in part (independently) the same equations as Waldmann. Urey (1947) in his classical survey of the stable isotope geochemistry developed a somewhat different formulation. Since then many authors have used the methods of Urey (1947) or Waldmann (1943) and Bigeleisen & Mayer (1947). These methods have been refined and applied in different disciplines.

It is correct to conclude that the theory of stable isotope fractionation is “old hat.” What is then the reason to review it once more? It is our experience that geochemists, even those who specialize in stable isotope geochemistry, are often badly informed on the theory of this subject. Another reason is that in geochemistry, more than in nearly any other discipline where stable isotopes are used, one insists on the availability of equilibrium constants that are as accurate as possible. This is of course due to the fact that the naturally observed fractionations are frequently very small. For these reasons we give only a condensed review of the theory but discuss in detail possible sources of errors in the calculations. We also discuss briefly the spectroscopic data, which are the input for the calculations.

We limit ourselves to a discussion of isotopic fractionation between gaseous substances. Various reviews have been published that deal with the isotopic fractionation among solids (see, for example, Taylor 1968, Bottinga & Javoy 1975). Fractionation factors for the exchange of hydrogen, carbon, nitrogen, oxygen, sulphur and chlorine isotopes among gases of geochemical interest have been calculated for this contribution and are given in the form of tables at convenient temperature intervals. One may wonder why it is necessary to calculate these fractionation factors in certain cases, because some of them have already been published previously, notably by Urey (1947), Tudge & Thode (1950, Saxena et al (1962), and Bottinga (1968a, 1969). Our reasons for doing this have been several:
1. We wanted to obtain a uniform set of fractionation factors, all calculated in the same way, in order to prevent errors that may result from combining calculated data from different authors.

2. Since the heretofore mentioned calculations, theoretical developments have taken place. Their importance has been evaluated for several cases.

3. New spectroscopic data, as well as more sophisticated analyses of old spectroscopic data, have been published, making more precise calculation possible.

4. Certain simplifying but also error-introducing approximations in previously published calculations are avoided.

5. In several of the older compilations of calculated fractionation factors the temperature intervals for which results are given are too small for modern geochemical needs.

The calculated results are compared with the available laboratory-measured fractionation factors. Lack of space prevents us from a discussion of naturally occurring isotopic fractionation in the light of our data. However, we hope that a uniform and readily available compilation of fractionation factors will stimulate geochemists to measure isotopic ratios in volcanic, hydrothermal, and bacterial gases, and most interestingly perhaps in the gases occurring in inclusions in many minerals.

## 2 CALCULATIONS—DIATOMIC MOLECULES

In this section we give an outline of the theory, in order that the general reader may obtain an idea of how these calculations are done and be aware of the approximations that are applied. For a full account of the theory we refer the reader to the papers by Mayer (1958) or Vojta (1960).

The simplest possible exchange reaction is between diatomic molecules and atoms. Such systems serve very well to illustrate the physical principles involved in isotopic fractionation. For reaction (1),

$$ZY + Y' = ZY' + Y, \qquad (1)$$

where the element Y has two isotopes Y and Y', the equilibrium constant $K_1$ is given by

$$K_1 = \frac{[ZY']}{[ZY]} \cdot \frac{[Y']}{[Y]}, \qquad (2)$$

where [ZY] is the concentration of ZY. A well-known result is (see, e.g., Denbigh 1971, Chap. 12)

$$K_1 = \left( \frac{Q'}{Q} \right)_{ZY} \cdot \left( \frac{Q'}{Q} \right)_Y, \qquad (3)$$

where $(Q'/Q)_{ZY}$ stands for the molecular partition-function ratio of molecules ZY' and ZY. The molecular partition function Q is defined as

$$Q = \sum_i g_i \exp(-E_i/kT), \qquad (4)$$
where the summation is over all quantum states $i$ accessible to the system. $E_i$ is the energy of state $i$ and $g_i$ is the degeneracy of state $i$. To evaluate Equation 4 one has to know $E_i$ and $g_i$. This information is contained in the Schrödinger equation

$$H\psi = E\psi$$

for the constituents of reaction (1). $H$ is the Hamiltonian operator and $\psi$ is the wave function.

The energy of a molecule may be split up into two components,

$$E = E_{tr} + E_{int}, \qquad (5)$$

where $E_{tr}$ is the energy associated with the movements of the center of mass of the molecule and $E_{int}$ is the internal energy. We do not take into consideration the potential energy the molecule may have due to the earth's gravity field; because of the weakness of the interaction this energy can be safely ignored. Hence the $E_{int}$ will be independent of the location of the molecule and thus we have that the wave function $\psi = \psi_{tr}\psi_{int}$. Consequently the translations of the molecule are governed by the translational Schrödinger equation

$$-\frac{\hbar^2}{2M} \nabla^2 \psi_{tr} = E_{tr} \psi_{tr}, \qquad (6)$$

where $M$ is the mass of the molecule and $\hbar$ is the Planck constant divided by $2\pi$. Solution of Equation 6 gives

$$E(n_x, n_y, n_z) = \frac{\hbar^2}{8M\aa^2} (n_x^2 + n_y^2 + n_z^2), \qquad (7)$$

where $n_x, n_y, n_z$ are zero or positive integers, the quantum numbers for translations in the $X, Y$, or $Z$ directions, and $\aa$ is the dimension of the cubic box in which the molecule is placed. To calculate $E_{int}$ one has to solve the Schrödinger equation

$$\sum_{n} \frac{\hbar^2}{2M_n} \nabla_n^2 \psi_{int} + \sum_{r} \frac{\hbar^2}{2m} \nabla_r^2 \psi_{int} + (E_{int} - V_{kk} - V_{ee} - V_{ke}) \psi_{int} = 0, \qquad (8)$$

where $M_n$ is the mass of nucleus $n$ and $m$ is the mass of an electron. $V_{kk}$ is the potential for the interaction between nuclei, $V_{ee}$ for electrons, and $V_{ke}$ for electron-nuclei interaction. For nearly all molecules of interest in geochemistry Equation 8 cannot be solved, and approximations have to be applied to obtain values for $E_{int}$.

The general practice is to apply the Born-Oppenheimer approximation (Born & Oppenheimer 1927). In this approximation one considers the nuclei to be fixed in space. The justification of this method is that the movements of the heavy nuclei are very small with respect to the rapid movements of the light electrons. As a result of this we can write

$$\psi_{int} = \psi_e \psi_k.$$

where $\psi_e$ is the wave function for the electron system and $\psi_k$ is the wave function for nuclei. Equation 8 can now be split into Equations 9 and 10
$$ \sum_{r} \frac{\hbar^2}{2m} \nabla_r^2 \psi_e + (E_e - V_{ee} - V_{ke}) \psi_e = 0 \qquad (9) $$

and

$$ \sum_{n} \frac{\hbar^2}{2M_n} \nabla_n^2 \psi_k + (E_{\text{int}} - E_e - V_{kk}) \psi_k = 0, \qquad (10) $$

where because of the Born-Oppenheimer approximation we have neglected derivatives like $\partial\psi_e/\partial x_n$. In Equations 9 and 10, we have $E_e$ = energy eigenvalue of the electron system. Ideally one would like to solve Equation 9 for $E_e$ and subsequently solve Equation 10 for $E_{\text{int}}$. However, in general the molecule with which one is concerned contains too many electrons for Equation 9 to be solved. $E_e$ will depend on the distance between the nuclei only.

## 2.1 Vibrational and Rotational Molecular Energy

In order to solve Equation 10 without having a solution for $E_e$ available, one assumes that the term $E_e + V_{kk}$ in Equation 10 may be represented by a potential like, for instance, the Morse potential (Morse 1929),

$$ V(r - r_e) = D_e[1 - \exp\{-a(r - r_e)\}]^2, \qquad (11) $$

where $r$ is the internuclear distance, $r_e$, the equilibrium internuclear distance, $D_e$, the dissociation energy of the molecule measured with respect to the minimum of the potential energy curve (i.e. when $r = r_e$), and $a$, a constant. Solving Equation 10 for $E_{\text{int}}$, while making use of the Morse potential, one obtains as a solution for the energy of the molecule in excess of the minimum of the electronic energy ($T_e$)

$$ E_{v,J}/hc = \omega_e(v+1/2) - \omega_e x_e (v+1/2)^2 + B_e J(J+1) - D J^2 (J+1)^2 - \alpha_e (v+1/2) J(J+1) \qquad (12) $$

for the energy level $E_{v,J}$ characterized by the vibrational quantum number $v$ and the rotational quantum number $J$. The constants in Equation 12, $\omega_e$, $x_e$, $B_e$, $D$, and $\alpha_e$, are functions of the parameters $D_e$, $r_e$, and $a$ in the Morse potential and the masses of the two nuclei in the diatomic molecule. We further define $\omega_e$ as $\omega_e = v/c$, where $v$ is the vibrational frequency and $c$ is the speed of light; $v$ can be expressed as $v = a/2\pi\sqrt{2D_e/\mu}$, where $\mu = M_1 M_2 / (M_1 + M_2)$, the reduced mass of the molecule with nuclei 1 and 2. We also have $B_e = h/(8\pi^2 I_e c)$, where $I_e$ is the moment of inertia; $D = 4B_e^3/\omega_e^2$; and

$$ \alpha_e = 3\hbar^2\omega_e\left[\frac{1}{ar_e} - \left(\frac{1}{ar_e}\right)^2\right] / 4\mu r_e^2 D_e. $$

The first and third terms on the right-hand side (RHS) of Equation 12 would have been obtained if a harmonic potential $V(r) = \frac{1}{2}K(r-r_e)^2$ with $r-r_e \ll r_e$ was used instead of the Morse potential; these terms correspond to a rigid rotator executing simple harmonic oscillations. The second RHS term is an anharmonicity correction to the first RHS term. The fourth RHS term is due to the nonrigidity of the rotator and represents rotational stretching of the molecule. The last RHS term
describes the interaction between vibration and rotation. By fitting Equation 12 to the observed vibrational and rotational spectrum of a diatomic molecule, the constants occurring in Equation 12 and hence the Morse potential parameters for the molecule can be obtained. These aspects have been extensively described by Herzberg (1950).

As is shown later, in the calculation of isotope effects, the zero-point energy, i.e. the energy difference between the minimum of the potential energy curve and the level $E_{0,0}$, plays a predominant role. For Equation 12 $E_{0,0} = \frac{1}{2}\omega_e - \frac{1}{4}\omega_e x_e$. This energy is not observable in spectroscopy, where one can measure only energy differences between excited states or between an excited state and the ground state. Hence the calculation of the zero-point energy amounts to an extrapolation of the observed spectrum. To arrive at the correct value of $E_{0,0}$ it is of great importance that Equation 12, and thus also the Morse potential, are valid expressions. It has been found that to describe adequately the vibrational part of diatomic spectra, the first two RHS terms of Equation 12 are not always sufficient. To obtain a good fit to the observed vibrational energies expression 13 has also been used:

$$E_v/hc = \omega_e(v+1/2) - \omega_e x_e(v+1/2)^2 + \omega_e y_e(v+1/2)^3 + \omega_e z_e(v+1/2)^4 \dots \quad (13)$$

In order to have an idea of the importance of the various terms in Equations 12 and 13, we give the molecular constants for H₂ and ¹⁶O₂ in Table 1.

As suggested by the diminishing values for the constants $\omega_e$, $\omega_e x_e$, and $\omega_e y_e$ in Table 1, the term $\omega_e z_e$ is not important. The data of Table 1 show also that vibrational energy is significantly greater than rotational energy. The term D (Table 1) is much smaller than the other constants in Equation 12, meaning that the energy associated with the rotational-stretching effect is at best only of minor importance.

In general (see e.g. Figure 1), the Morse potential reproduces the observations quite well; however, there remain a few discrepancies. In order to resolve these, Dunham (1932) has attempted to solve Equation 10 with a more general potential. For a general potential expanded about the equilibrium internuclear distance $r_e$ one can write

$$V(r) = V(r_e) + \left. \frac{dV}{dr} \right|_{r=r_e} \begin{cases} 1 & d^2V \\ 2 & r \end{cases} \begin{aligned}[t] & + \frac{1}{6} \left. \frac{d^3V}{dr^3} \right|_{r=r_e} (r-r_e)^3 + \frac{1}{24} \left. \frac{d^4V}{dr^4} \right|_{r=r_e} (r-r_e)^4 + \dots \end{aligned} \quad (14)$$

By definition we have $V(r_e) = 0$ and $(dV/dr)_{r=r_e} = 0$. Hence Equation 14 may be written as

$$V(r) = K_2(r - r_e)^2 + K_3(r - r_e)^3 + K_4(r - r_e)^4 \quad (15)$$

Using the WKB method, Dunham (1932) solved Equation 10 with the potential (15) obtaining Equation (16),

$$E_{v,J} = hc \sum_{j=0} \sum_{k=0} Y_{jk}(v+1/2)^j [J(J+1)]^k. \quad (16)$$
<table><caption>**Table 1** Molecular constants (cm⁻¹)</caption><thead><tr><th></th><th>H<sub>2</sub></th><th><sup>16</sup>O<sub>2</sub></th></tr></thead><tbody><tr><td>ω<sub>e</sub></td><td>4401.118</td><td>1580.360</td></tr><tr><td>ω<sub>e</sub>x<sub>e</sub></td><td>121.284</td><td>12.073</td></tr><tr><td>ω<sub>e</sub>y<sub>e</sub></td><td>0.804</td><td>0.047</td></tr><tr><td>B<sub>e</sub></td><td>60.8380</td><td>1.4456</td></tr><tr><td>α<sub>e</sub></td><td>3.0258</td><td>0.1593</td></tr><tr><td>D</td><td>0.0463</td><td>5 × 10<sup>-6</sup></td></tr></tbody></table>

The coefficients $Y_{jk}$ were calculated to second order. The resulting zero-point energy is

$$E_{0,0} = hc \left( G_0 + \frac{\omega_e}{2} - \frac{\omega_e x_e}{4} \right) \qquad (17)$$

with

$$G_0 = \frac{B_e}{4} + \frac{\alpha_e \omega_e}{12 B_e} + \left( \frac{\alpha_e \omega_e}{12 B_e} \right)^2 / B_e - \frac{\omega_e x_e}{4} \qquad (18)$$

<figure><img src="image_4.png" /><figcaption>**Figure 1** Potential energy curves and vibrational levels for the electronic ground state of H₂ (Herzberg 1950, modified): —, real potential; ---, Morse potential; ., harmonic potential.</figcaption></figure>
Equations 17 and 18 can be obtained from Dunham (1932), and they are also given in Herzberg (1950). Wolfsberg (1969b) has given numerical values for $G_0$ for several diatomic hydrides.

More recently Huffaker (1976a, b) has solved Equation 10, using a perturbed Morse oscillator,

$$V(r) = V_e \left( F^2 + \sum_{n=4} b_n F^n \right), \qquad (19)$$

as potential. In Equation 19 we have $F = 1 - \exp(a(r - r_e))$. The important aspect for us is that Huffaker (1976a) derived the same expression (16) as Dunham (1932) and that he continued the evaluation of the coefficients $Y_{jk}$ to a higher perturbation order (Huffaker 1976b) than Dunham. Huffaker expressed the coefficients $Y_{jk}$ as

$$Y_{jk} = \sum_{r=0} Y_{jk}^{2r}. \qquad (20)$$

The importance of the various terms for the zero-point energy is illustrated by the data listed in Table 2 for HF and CO. The zero-point energy in terms of Dunham coefficients ($Y_{jk}$) is given by

$$E_{0,0} = hc \sum_{r=0} \sum_{j=0} Y_{j0}^{2r} / 2^j. \qquad (21)$$

<table><caption>**Table 2** Molecular constants and Dunham coefficients ($Y_{jk}^{2r}$) in cm$^{-1}$ for CO and HF<sup>a</sup></caption><thead><tr><th></th><th>CO</th><th>HF</th></tr></thead><tbody><tr><td>$Y_{00}^0$</td><td>0.</td><td>0.</td></tr><tr><td>$Y_0^0$</td><td>1.9004 × 10<sup>-1</sup></td><td>3.90777</td></tr><tr><td>$Y_{00}^4$</td><td>8.5781 × 10<sup>-7</sup></td><td>4.5212 × 10<sup>-2</sup></td></tr><tr><td>$Y_{10}^0 = \omega_e$</td><td>2169.8181</td><td>4139.357</td></tr><tr><td>$Y_{10}^2$</td><td>-4.4372 × 10<sup>-4</sup></td><td>-3.2532 × 10<sup>-1</sup></td></tr><tr><td>$Y_{10}^4$</td><td>7.5284 × 10<sup>-6</sup></td><td>-6.4393 × 10<sup>-3</sup></td></tr><tr><td>$Y_{20}^0 = -\omega_e x_e$</td><td>-13.2906</td><td>-90.5660</td></tr><tr><td>$Y_{20}^2$</td><td>-3.9577 × 10<sup>-5</sup></td><td>1.3529 × 10<sup>-1</sup></td></tr><tr><td>$Y_{20}^4$</td><td>-3.0584 × 10<sup>-6</sup></td><td>-8.5241 × 10<sup>-3</sup></td></tr><tr><td>$Y_3^0$</td><td>1.0964 × 10<sup>-2</sup></td><td>1.2001</td></tr><tr><td>$Y_{30}^2$</td><td>1.3416 × 10<sup>-5</sup></td><td>-2.2417 × 10<sup>-2</sup></td></tr><tr><td>$Y_4^0$</td><td>2.29622 × 10<sup>-5</sup></td><td>-7.32256 × 10<sup>-2</sup></td></tr><tr><td>$Y_{40}^2$</td><td>-2.6547 × 10<sup>-8</sup></td><td>9.5086 × 10<sup>-4</sup></td></tr><tr><td>$B_e$</td><td>1.931265</td><td>20.94948</td></tr><tr><td>$D$ (Eq. 12)</td><td>6.1198 × 10<sup>-6</sup></td><td>2.1461 × 10<sup>-3</sup></td></tr><tr><td>$\alpha_e$</td><td>1.7505 × 10<sup>-2</sup></td><td>0.7961</td></tr><tr><td>$E_{0,0}/hc$ (Eq. 12)</td><td>1081.59</td><td>2047.04</td></tr><tr><td>$E_{0,0}/hc$ (Eq. 17)</td><td>1081.78</td><td>2050.94</td></tr><tr><td>$E_{0,0}/hc$ (Eq. 21)</td><td>1081.78</td><td>2051.00</td></tr></tbody></table>

<footnote><sup>a</sup> After Huffaker (1976b), with molecular constants slightly different from those we use (cf Table 5).</footnote>
We have gone into more detail than is customary to discuss the zero-point energy. The reasons for this are that the zero-point energy plays an important role in the calculation of isotopic exchange equilibrium constants and that the evaluation and importance of $G_0$ for isotopic calculations is somewhat controversial (Richet & Bottinga 1976, Bottinga 1968b).

The data of Table 2 show clearly that for our purposes Equation 12 plus the term $G_0$ (Equation 18) are adequate. The higher perturbation contributions to the zero-point energy are insignificant. This will become clearer somewhat later on when we discuss the importance of the uncertainty in the spectroscopic data and evaluate the consequences of a $1 \text{ cm}^{-1}$ error in zero-point energy.

## 2.2 Born-Oppenheimer Approximation, Electronic Energy

The Born-Oppenheimer (B-O) approximation has been recently discussed in many papers. An excellent review of the current literature is given by Kolos (1970). The

<figure><img src="image_5.png" /><figcaption>Figure 2 Potential energy curves and vibrational levels for various electronic states of $O_2$, plotted versus internuclear distance (Herzberg 1950).</figcaption></figure>
<table><caption>**Table 3** Electronic states and molecular constants for $^{16}\text{O}_2$, in cm$^{-1}$ (Herzberg 1950)</caption><thead><tr><th>Electronic state</th><th>T<sub>e</sub><sup>a</sup></th><th>ω<sub>e</sub></th><th>B<sub>e</sub></th><th>N(T<sub>e</sub>)/N(0)<sup>b</sup></th></tr></thead><tbody><tr><td>b<sup>1</sup>Σ<sup>+</sup><sub>g</sub></td><td>13195</td><td>1433</td><td>1.40</td><td>7 × 10<sup>-6</sup></td></tr><tr><td>a<sup>1</sup>Δ<sub>g</sub></td><td>7918</td><td>1509</td><td>1.43</td><td>8 × 10<sup>-4</sup></td></tr><tr><td>X<sup>3</sup>Σ<sup>-</sup><sub>g</sub></td><td>0</td><td>1580</td><td>1.45</td><td>—</td></tr></tbody></table>

<sup>a</sup> Minimum electronic energies, measured with respect to the minimum of the electronic ground state.

<sup>b</sup> Fraction of the molecules $^{16}\text{O}_2$, which is in the indicated state at 1600°K.

importance of the B-O approximation is that it is related to the assumption that the potential energy curve for the electronic ground state, as it is given for instance in Figure 1, is the same for isotopically substituted molecules. This assumption is of great importance because spectroscopic data are usually available only for the most abundant isotopic molecule (for this molecule one knows thus the constants in Equation 12 and the values of $E_{v,0}$), but for the rare isotopic molecule these data must be usually calculated with the parameters of the potential (i.e. Equations 11, 15, or 19) known from the spectrum of the most abundant isotopic molecule.

As already discussed, Equation 9 is supposed to be solved in the B-O approximation for $E_e$ and subsequently one solves Equation 10. This means that Equation 10 has to be solved for the different excited electronic states of the molecule, and that the vibrational and rotational constants of the molecule must depend on its electronic state (see Figure 2). For all molecules of interest to us the energy gap between the electronic ground state and first excited state is large enough that even at 1600°K these molecules will be overwhelming in the ground state (see Table 3). In Table 3 we have given the data for oxygen because, of the molecules we discuss in this review, it has the lowest first excited electronic state. We have also listed the vibrational ($\omega_e$) and rotational ($B_e$) constants for the molecule $^{16}\text{O}_2$ when it is in an electronic excited state.

To improve on the B-O approximation, one applies the adiabatic approximation (Van Vleck 1936, Born 1951, Born & Huang 1954, Chap. IV). In the B-O approximation the nuclei are in a fixed position while the electrons move; in the adiabatic approximation the nuclei move and the electrons move too, but their movement is as if the nuclei were fixed in instantaneous positions. This means that the electronic state of the molecule changes because of the nuclear displacements, but the electrons do not make a transition from one state to the next; in other words, the electronic motion is adiabatic. Adiabatic calculations (see Kolos 1970) predict accurately the dissociation energy of molecular hydrogen, and nonadiabatic effects are believed to be very minor in comparison with adiabatic effects (see Kolos 1970). The possible consequences of the B-O approximation are discussed in a later section.

## 2.3 Evaluation of the Partition Function

As shown in the previous sections the energy $E_i$ in Equation 4 for the partition function is made up of a series of terms
$$E = E_{tr} + E_{0} + E_{vib} + E_{anh} + E_{rot} + E_{rot-str} + E_{rot-vib} + T_{e}, \quad (22)$$

where $E_{tr}$ stands for translational energy (see Equation 7); $E_0$, zero-point energy (for instance, Equation 17); $E_{vib}$, vibrational energy, from which is subtracted the zero-point energy, in the harmonic approximation; $E_{anh}$, anharmonic vibrational energy, without zero-point energy; $E_{rot}$, rotational energy (third term, RHS, Equation 12); $E_{rot-vib}$, energy associated with the rotation-vibrational interaction (fifth term, RHS, Equation 12); and $T_e$, electronic energy.

Combining Equations 22 and 4, and neglecting the term $T_e$, because all of the molecules we are dealing with are overwhelmingly in their ground state, we can write

$$Q = Q_{tr}Q_{0}Q_{vib}Q_{anh}Q_{rot}Q_{rot-vib}. \quad (23)$$

In Equation 23 we have neglected the contribution to the partition function of the rotational stretching term. Later on when we evaluate the isotopic exchange equilibrium constant $K_1$ for Equation 1 we justify this omission. From Equations 4 and 7 we obtain the translational partition function

$$Q_{tr} = (2\pi M k T / h^2)^{3/2} V, \quad (24)$$

where $V$ is the volume in which the molecule is enclosed. $Q_0$ is given by

$$Q_0 = \exp \left[ - \left( G_0 + \frac{\omega_e}{2} - \frac{\omega_e x_e}{4} \right) \frac{hc}{kT} \right] \quad (25)$$

and

$$Q_{vib} = 1/(1 - e^{-u}), \quad (26)$$

where $u = hc\omega_e/kT$. A reader unfamiliar with Equations 24 and 26 is advised to consult Denbigh (1971, Chap. 12). We also have

$$Q_{anh} = 1 - 2x_e u e^{-u} / (1 - e^{-u})^2 \quad (27)$$

as derived by Vojta (1961). The rotational partition function is given by

$$Q_{rot} = \sum_{J=0} (2J + 1) \exp[-B_e J(J+1)hc/kT]. \quad (28)$$

The factor $2J+1$ in Equation 28 is the degeneracy for the rotational state $J$. When $B_e$ is small or when the temperature is high the rotational levels will be closely spaced, and in that case the summation in Equation 28 may be replaced by an integration, and the rotational partition function will have its classical value (Equation 29):

$$(Q_{rot})_{\text{class}} = 1/s\sigma, \quad (29)$$

where $s$ is the symmetry number (see Mayer & Mayer 1940) and $\sigma = B_e hc/kT$. In order to include the rotation-vibration interaction for the vibrational ground state one can replace $B_e$ in Equation 28 by $B_0$, where $B_0 = B_e - \alpha_e/2$. The rotation-vibration interaction when the molecule is not in the vibrational ground state, i.e. $v \neq 0$ in Equation 12, gives rise to the factor

$$Q_{rot-vib} = 1 + \delta/(e^u - 1), \quad (30)$$
where $\delta = \alpha_e/B_0$. When $B_e$ is too large or $T$ too small in Equation 28, the rotational partition function will deviate significantly from its classical value (Equation 29). In that case one can either sum Equation 28 directly or use Equation 31, derived by Mulholland (1928) and Viney (1933):

$$Q_{\text{rot}} = \frac{1}{s\sigma} (1 + \sigma/3 + \sigma^2/15 + 4\sigma^3/315 + \dots). \quad (31)$$

The term between brackets at the RHS of Equation 31 is a quantum mechanical correction to Equation 29.

## 2.4 Partition Function Ratios

In order to calculate $K_1$ (Equation 2) we evaluate the ratio $(Q'/Q)_{ZY}$ of Equation 23:

$$\left(\frac{Q'}{Q}\right)_{ZY} = \left(\frac{Q'}{Q}\right)_{tr} \left(\frac{Q'}{Q}\right)_{0} \left(\frac{Q'}{Q}\right)_{vib} \left(\frac{Q'}{Q}\right)_{anh} \left(\frac{Q'}{Q}\right)_{rot} \left(\frac{Q'}{Q}\right)_{rot-vib}. \quad (32)$$

The ratio $(Q'/Q)_Y$ of Equation 3 is simply given by the ratio of translational partition functions, because the atoms have only translational energy. Hence, using Equation 24 as an analogy, we find that

$$\left(\frac{Q'}{Q}\right)_Y = (m'/m)^{3/2}, \quad (33)$$

where $m'$ and $m$ are the atomic masses of $Y'$ and $Y$ respectively. Similarly, the translational term in Equation 32 is given by

$$\left(\frac{Q'}{Q}\right)_{tr} = (M'/M)^{3/2}, \quad (34)$$

where now the $M'$ and $M$ are the molecular masses of $ZY'$ and $ZY$ respectively. From Equation 25 we find that

$$\left(\frac{Q'}{Q}\right)_0 = [\exp(-(G'_0 - G_0)hc/kT)][\exp(-(u'-u)/2)][\exp(u'x'_e - ux_e)/4], \quad (35)$$

and from Equation 26 we obtain

$$\left(\frac{Q'}{Q}\right)_{\text{vib}} = (1 - e^{-u})/(1 - e^{-u}). \quad (36)$$

The contribution to the partition function ratio of the anharmonicity of vibrationally excited molecules is given by

$$\left(\frac{Q'}{Q}\right)_{\text{anh}} = \frac{1 - 2x'_{e}u' e^{-u}/(1 - e^{-u})^2}{1 - 2x_{e}u e^{-u}/(1 - e^{-u})^2}. \quad (37)$$

To evaluate the rotational partition function ratio for the diatomic molecules, except for CS and $S_2$ we summed Equation 28 directly for the normal and isotopically substituted molecule and then calculated the ratio. The summation was continued until the fourth figure after the decimal point did not change with increasing $J$. For CS and $S_2$ we used Equation 38:
$$ \left(\frac{Q'}{Q}\right)_{\text{rot}} = \left(\frac{s\sigma}{s'\sigma'}\right) \left(\frac{1 + \sigma'/3 + \sigma'^2/15 + \dots}{1 + \sigma'/3 + \sigma'^2/15 + \dots}\right). \quad (38) $$

In both cases we replaced $B_e$ by $B_0$ and multiplied the obtained rotational partition function ratio by the factor

$$ \left(\frac{Q'}{Q}\right)_{\text{rot-vib}} = \frac{[1 + \delta'/(e^{u'} - 1)]/[1 + \delta/(e^{u'} - 1)]}{(39)} $$

to take into account the rotation-vibration interaction.

Of course we are well aware that for many calculations simpler expressions for the partition function ratios would be adequate. However, with modern computing equipment available it is no longer difficult to evaluate Equations 32-39. Moreover, for certain of the molecules treated, less complete equations would not suffice. The importance of the various terms in Equation 32 is illustrated by the calculated results given in Table 4, where we list the various contributions to the ratios $Q(HD)/Q(H_2)$, $Q(DF)/Q(HF)$, and $Q(C^{18}O)/Q(C^{16}O)$ at 0°C and 1000°C. It is evident from the values listed in Table 4 that the zero-point energy contribution (line 5 of Table 4, Equation 35) is very important in the calculation of the partition function ratio (line 12, Table 4). It was for that reason that in Section 2.2 so much attention was payed to the correct evaluation of the zero-point energy. Of the terms making up the zero-point energy, the harmonic vibration term (cf line 3, Table 4) is considerably greater than the other terms. However, its importance diminishes with increasing temperature. Under all conditions the $G_0$ term in the zero-point energy is of the least significance (see Table 4, line 2). A geochemist would like to know stable-isotope equilibrium constants with the same relative accuracy as he can measure isotopic ratios, which is often better than 0.5°/°. A 1 cm⁻¹ error in the difference $|\omega'_e - \omega_e|$ will already cause an uncertainty of 2.6°/° in the value of the partition function ratio at 273°K and 0.6°/° at 1273°K. Such uncertainties in the vibrational frequency differences are not excluded. Inspection of line 2, Table 4, shows that the $G_0$ effect is often comparable to the uncertainty introduced by the limits on the accuracy of the difference $|\omega'_e - \omega_e|$. This feature is also evident from the $G_0$ values computed by Wolfsberg (1969b). Calculations of $G_0$ for diatomic molecules not containing H show that the $G_0$ term can be safely neglected in such cases (see, for instance, the values on line 4, columns 3 and 7 of Table 4). In the calculation we report in this paper the $G_0$ effect has been ignored, because it appears only to be of interest in H exchange, but in that case the fractionations are so large that the $G_0$ effect becomes relatively unimportant in geochemistry. Moreover, $G_0$ cannot in general be calculated for polyatomic molecules, and thus for reasons of homogeneity we have also ignored the $G_0$ contribution in diatomic calculations.

The values of line 6, Table 4, indicate that the vibrational partition function ratio becomes significantly different from unity only at higher temperatures when the vibrational excited states become occupied. As shown by line 7, the importance of the anharmonic correction increases with temperature. Lines 8, 9 and 10 (Table 4) give calculated results for $(Q'/Q)_{\text{rot}}$. For hydrogen-bearing diatomic molecules it is recommended that Equation 28 be used rather than the classical ratio or the quantum
<table><caption>**Table 4** Contributions to the partition function ratio</caption><thead><tr><th rowspan="2"></th><th rowspan="2">Eq.</th><th colspan="4">0°C</th><th colspan="4">1000°C</th></tr><tr><th>1<br>HD/H<sub>2</sub></th><th>2<br>DF/HF</th><th>3<br>C<sup>18</sup>O/C<sup>16</sup>O</th><th>4<sup>b</sup><br>2/1</th><th>5<br>HD/H<sub>2</sub></th><th>6<br>DF/HF</th><th>7<br>C<sup>18</sup>O/C<sup>16</sup>O</th><th>8<sup>b</sup><br>6/5</th></tr></thead><tbody><tr><td>1</td><td>(M/M)<sup>3/2</sup></td><td>34</td><td>1.83561</td><td>1.07638</td><td>1.10929</td><td>0.58639</td><td>1.83561</td><td>1.07638</td><td>1.10929</td><td>0.58639</td></tr><tr><td>2</td><td>exp-(G<sub>0</sub>-G<sub>0</sub>)hc/kT</td><td>35</td><td>1.00910</td><td>1.01192</td><td>1.00005</td><td>1.00280</td><td>1.00194</td><td>1.00255</td><td>1.00001</td><td>1.00061</td></tr><tr><td>3</td><td>exp-1/2(u'-u)</td><td>35</td><td>4.71546</td><td>20.1428</td><td>1.14804</td><td>4.27165</td><td>1.39477</td><td>1.90456</td><td>1.03006</td><td>1.36550</td></tr><tr><td>4</td><td>exp 1/4(x'<sub>e</sub>u'-x<sub>e</sub>u)</td><td>35</td><td>0.96079</td><td>0.94356</td><td>0.99917</td><td>0.98207</td><td>0.99146</td><td>0.98761</td><td>0.99982</td><td>0.99612</td></tr><tr><td>5</td><td>(Q'/Q)<sub>0</sub></td><td>35</td><td>4.57179</td><td>19.2325</td><td>1.14714</td><td>4.20677</td><td>1.38554</td><td>1.88576</td><td>1.02988</td><td>1.36103</td></tr><tr><td>6</td><td>(Q'/Q)<sub>vib</sub></td><td>36</td><td>1.00000</td><td>1.00000</td><td>1.00000</td><td>1.00000</td><td>1.00663</td><td>1.02531</td><td>1.00578</td><td>1.01856</td></tr><tr><td>7</td><td>(Q'/Q)<sub>anh</sub></td><td>37</td><td>1.00000</td><td>1.00000</td><td>1.00000</td><td>1.00000</td><td>0.99908</td><td>0.99818</td><td>0.99993</td><td>0.99910</td></tr><tr><td>8</td><td>(sσ/s's')</td><td>38</td><td>2.65666</td><td>1.89381</td><td>1.05001</td><td>2.85142</td><td>2.65666</td><td>1.89381</td><td>1.05001</td><td>0.71285</td></tr><tr><td>9</td><td>(Q'/Q)<sub>rot</sub></td><td>38</td><td>2.58778</td><td>1.86165</td><td>1.04985</td><td>2.87758</td><td>2.64196</td><td>1.88691</td><td>1.04997</td><td>0.71421</td></tr><tr><td>10</td><td>(Q'/Q)<sub>rot</sub></td><td>28</td><td>2.59246</td><td>1.86165</td><td>1.04985</td><td>2.87234</td><td>2.64196</td><td>1.88680</td><td>1.04997</td><td>0.71417</td></tr><tr><td>11</td><td>(Q'/Q)<sub>rot-vib</sub></td><td>39</td><td>1.00000</td><td>1.00000</td><td>1.00000</td><td>1.00000</td><td>1.00025</td><td>1.00058</td><td>1.00003</td><td>1.00033</td></tr><tr><td>12</td><td>(Q'/Q)<sup>a</sup></td><td>32</td><td>21.7560</td><td>38.5389</td><td>1.33595</td><td></td><td>6.75935</td><td>3.92187</td><td>1.20640</td><td></td></tr><tr><td>13</td><td>K = [DF]/[HD]</td><td></td><td></td><td></td><td></td><td>1.77141</td><td></td><td></td><td></td><td>0.58021</td></tr></tbody></table>

<sup>a</sup> The product of lines 1, 5, 6, 7, 10 and 11; the values given are not the same as in Table 9 because, as explained in the text, in the tables the G<sub>0</sub> effect (line 2) was not included.

<sup>b</sup> In columns 4 and 8 we give the quotient of values listed in columns 2 and 1, and columns 6 and 5, respectively.
mechanically corrected classical ratio (Equation 38). However, for the non-hydrogen-bearing diatomic molecules Equation 38 and even the classical ratio are usually adequate. The rotation-vibration interaction is always of only marginal interest. As is to be anticipated, this term increases in significance with temperature. We have investigated numerically the rotational stretching effect which increases with temperature. Though it may be not completely negligible when the absolute value of the partition function ratio is calculated, its contribution to the partition function ratio is too small to be of interest.

The importance of the B-O approximation for isotopic calculations for diatomic molecules has been discussed in a series of papers by Kleinman & Wolfsberg (1973, 1974a, 1974b). These authors followed the procedure developed by Van Vleck (1936), Born (1951), and Born & Huang (1954, Chap. IV) and did an approximate calculation within the framework of the adiabatic approximation. They evaluated the energy difference (C) for the minimum of the potential-energy function, such as is shown in Figure 1, between the adiabatic approximation and the B-O approximation for various diatomic molecules. For reaction 1 it is thus important to know the quantity $\Delta(\text{B-O}) = C(ZY') - C(ZY)$. The quantity $\Delta(\text{B-O})$, which is zero in the B-O approximation but may be nonzero in a more exact treatment, enters the equilibrium constant calculation just as the term $G'_{0} - G_{0}$ enters Equation 35. Unfortunately, to calculate the terms $C(ZY')$ and $C(ZY)$ one needs the wave functions for these molecules, and they can only be approximated for most of the molecules that are of interest to us. This limits severely the accuracy to which one can calculate the $C(ZY)$. The $\Delta(\text{B-O})$ for an exchange reaction is always very small in comparison with the C terms for reactants and reaction products for such a reaction.

Van Vleck (1936) has shown that for the reaction $H_2 + D_2 = 2HD$, $\Delta(\text{B-O}) = 0$; Kleinman & Wolfsberg (1974a) showed that this is always the case for internal equilibria involving diatomic molecules. The mass dependence of the C terms (sce Bardo & Wolfsberg 1975) indicates also that a deviation from the B-O approximation has the most serious consequences for the case of deuterium-hydrogen exchange. A comparison of the rigorously derived C terms by Kolos & Wolniewicz (1964) for molecular hydrogen and its isotopically substituted varieties and those calculated by Kleinman & Wolfsberg (1973, 1974a) suggests that the uncertainty in the Kleinman & Wolfsberg results for the individual C terms is comparable to the $\Delta(\text{B-O})$ for a given exchange reaction. Hence in conclusion the effect may be of importance for H-D exchange, but otherwise it will be probably minor.

### 3 CALCULATIONS—POLYATOMIC MOLECULES

A possible exchange reaction involving polyatomic molecules is

$$ZY_n + nY' = ZY'_n + nY. \quad (40)$$

We have now

$$K_{40} = \left(\frac{Q'}{Q}\right)_{ZY_n} \left(\frac{Q'}{Q}\right)_Y^n \quad (41)$$
which is analogous to Equation 3. The partition function $Q$ is given by Equation 4. As in the diatomic case (Equation 22) the energy in Equation 4 is given by

$$E = E_{tr} + E_{0} + E_{vib} + E_{anh} + E_{rot} + \dot{E}_{rot-vib}. \quad (42)$$

We right away neglect the term $T_e$ (Equation 22) because all polyatomic molecules we are discussing in this review will be in the ground state for the temperature interval 273 to 1600°K. $E_{rot-str}$ is not appearing in Equation 42, because we have already found that in the diatomic case this energy term did not contribute to the isotopic fractionation. Moreover in most cases the polyatomic spectra are not well enough known to determine the needed constants.

The equation for the translational energy is the same as for the diatomic case and hence the translational partition function ratio is given by Equation 24.

A diatomic molecule has only one mode of vibration but a nonlinear polyatomic molecule has 3n-6 modes of vibration, where n is the number of atoms in the molecule. Linear polyatomic molecules have 3n-5 modes of vibration. The vibrational energy is the sum total of the vibrational energies associated with the different modes of vibration. We do not dwell on these aspects because they do not introduce uncertainties in the calculation of $K_{40}$ (Equation 41), and excellent discussions on these topics are available (see for instance Mayer & Mayer 1940). For the zero-point energy one has

$$E_0 = hc \left( G_0 + \frac{1}{2} \sum_i d_i \omega_i + \frac{1}{4} \sum_{i \le j} \sum_{i \le j} d_i d_j x_{ij} \right). \quad (43)$$

As before, the zero-point energy is not observable in molecular spectroscopy and to evaluate it correctly one should solve the polyatomic equivalents of Equations 10 and 15. This subject has been treated by Nielsen (1951), but while he mentions the existence of a term like $G_0$ (Equation 43), he does not give an expression for it. For the time being we will neglect $G_0$. The summations $i$ and $j$ are over the normal modes of vibration of the molecule. The coefficients $d_i$ and $d_j$ are the degeneracies of the modes of vibration $i$ and $j$. The coefficient $x_{ij}$ is due to anharmonicity and should be compared with the coefficient $-x_e\omega_e$ in Equation 17. The anharmonicity of normal mode $i$ is expressed by $x_{ii}$, while $x_{ij}$ is due to coupling between the modes $i$ and $j$, as a result of the anharmonicity of modes $i$ and $j$.

The vibrational energy associated with the excited vibrational states in the harmonic approximation is given by

$$E_{vib} = hc \sum_i d_i v_i \omega_i, \quad (44)$$

where the $\omega_i$ are the zero-order frequencies and are comparable to $\omega_e$ for the diatomic molecule. The anharmonicity correction to (44) is

$$E_{anh} = hc \sum_{i \le j} \sum_{i \le j} d_i d_j x_{ij} [v_i v_j + \frac{1}{2}(v_i + v_j)]. \quad (45)$$

Since molecular rotations are in most cases classical at room temperature and above, and the quantum mechanical expression for $E_{rot}$ is rather complicated (see for instance Nielsen 1951), we only give the classical expression for the
asymmetric top

$$E_{\text{rot}} = \frac{I_A \Omega_A^2}{2} + \frac{I_B \Omega_B^2}{2} + \frac{I_C \Omega_C^2}{2}, \qquad (46)$$

where $I_A$, $I_B$, and $I_C$ are the three principal moments of inertia and $\Omega_A$, $\Omega_B$, and $\Omega_C$ are the x, y, and z components of the angular velocity.

By means of Equations 43, 44, 45, and 46 we can evaluate the partition function 4 and calculate the following partition function ratios. For polyatomic molecules, $(Q'/Q)_0$ is given by Equation 32. The ratio $(Q'/Q)_0$ resembles very much Equation 35.

$$\left(\frac{Q'}{Q}\right)_0 = \left[ \exp(-(G'_0 - G_0)\frac{hc}{kT}) \left[ \exp - \frac{1}{2} \sum_i (u'_i - u_i)d_i \right] \right. \\ \left. \cdot \left[ \exp - \frac{1}{4kT} \sum_{i \le j} \sum d_i d_j (x'_{ij} - x_{ij}) \right] \right]^{3} \quad (47)$$

Likewise, the harmonic vibrational partition function ratio for polyatomic molecules,

$$\left(\frac{Q'}{Q}\right)_{\text{vib}} = \prod_i \frac{(1 - e^{-u_i})^{d_i}}{(1 - e^{-u'_i})^{d'_i}}, \qquad (48)$$

resembles Equation 36. The expression for the contribution due to the anharmonicity of the excited vibrational states is quite complicated; it is expressed by Vojta (1961) as

$$\left(\frac{Q'}{Q}\right)_{\text{anh}} = 1 + \frac{hc}{kT} \sum_i d_i (d_i + 1) \left\{ x_{ii} e^{u_i} / (e^{u_i} - 1)^2 - x'_{ii} e^{u'_i} / (e^{u'_i} - 1)^2 \right\} \\ + \frac{hc}{2kT} \sum_{i<j} \sum d_i d_j \left\{ x_{ij} (e^{u_i} + e^{u'_i}) / [(e^{u_i} - 1)(e^{u'_i} - 1)] - x'_{ij} (e^{u_i} + e^{u'_i}) / [(e^{u_i} - 1)(e^{u'_i} - 1)] \right\} \quad (49)$$

The rotational partition function ratio is virtually equal to the classical ratio. It is given by

$$\left(\frac{Q'}{Q}\right)_{\text{rot}} = \frac{s}{s'} \left[ \frac{I'_A I'_B I'_C}{M} \right]^{1/2} \qquad (50)$$

where as before $s$ and $s'$

substituted molecules. Common practice is to apply the Teller-Redlich product rule, which gives a relation between the normal-mode frequencies and the moments of inertia:

$$\left[ \frac{I'_A I'_B I'_C}{I_A I_B I_C} \right]^{1/2} \left[ \frac{M'}{M} \right]^{3/2} \left[ \frac{m}{m'} \right]^{3n/2} \prod_i \frac{u_i}{u'_i} = 1, \qquad (51)$$

where $n$ is the number of exchangeable atoms.

<footnote n="3">³ As is generally done, we have in writing Equations 47–49 assumed that the normal modes of the normal and isotopically substituted molecules have the same degeneracies.</footnote>
For the calculations reported in this paper we have not made use of Equation 51; for all nonlinear molecules not involved in hydrogen-deuterium exchange, Equation 52 was used. For polyatomic molecules the quantum-mechanical correction to the classical rotational partition function given by Stripp & Kirkwood (1951) was used:

$$Q_{\text{rot}} = (Q_{\text{rot}})_{\text{class}} Q_{\text{corr}}, \qquad (52)$$

where

$$Q_{\text{corr}} = 1 + \left[ 2\sigma_A + 2\sigma_B + 2\sigma_C - \frac{\sigma_A\sigma_B}{\sigma_C} - \frac{\sigma_B\sigma_C}{\sigma_A} - \frac{\sigma_A\sigma_C}{\sigma_B} \right] / 12 + \dots \qquad (53)$$

and

$$\sigma_A = \frac{\hbar^2}{2I_A k T}.$$

For linear polyatomic molecules Equation 38 was used.

The evaluation of the rotation-vibration interaction is analogous to the diatomic case (Equation 30). In Equations 50 and 53, the moments of inertia ($I_A$, $I_B$, $I_C$) or the coefficients $\sigma_A$, $\sigma_B$, $\sigma_C$, derived for the rotating molecule in the vibrational ground state, are used, and Equation 52 is multiplied by the polyatomic equivalent of Equation 30 (see Vojta 1960). We then have

$$Q_{\text{rot-vib}} = \prod_i \left\{ 1 + \frac{1}{2} \frac{\delta_i}{\exp(u_i) - 1} \right\}^{d_i} \qquad (54)$$

The multiplication over $i$ is over the normal modes of vibration and the $\delta_i$ are the rotation-vibration interaction constants. For all molecules $Q_{\text{rot-vib}}$ was calculated. In general one can safely ignore it; only in the case of hydrogen isotope exchange did this term become limited in interest.

The energy effects discussed in this section on polyatomic molecules are the only ones of importance, sometimes only marginally, in the calculation of the needed partition function ratio. Splitting of energy levels of polyatomic molecules such as may be caused by Fermi resonance in CO₂ or Darling-Dennison resonance in H₂O (see Herzberg 1945), does not affect our calculations in a measurable way. Similarly, the quantification of the rotation, which results from the composite motion of the degenerate modes of vibration, such as in a linear triatomic molecule, has no observable consequences for our calculations. Also in the temperature interval we are interested in, a phenomenon like ortho- and para-hydrogen has no measurable influence on our calculated results. The $G_0$ term occurring in Equation 43 is discussed in Section 6; in our calculations it has been neglected. The B-O approximation for polyatomic molecules has been treated by Bardo & Wolfsberg (1975); it does not seem to cause observable consequences in the exchanges discussed in this review. As in diatomic molecules the B-O approximation does not affect internal isotopic equilibria (Bardo & Wolfsberg 1975).

## 4 MOLECULAR CONSTANTS

In the discussion of the Born-Oppenheimer approximation mention was made that frequently not all molecular constants are available for all the isotopic molecules. If
<table><caption>Table 5 Molecular constants of diatomic molecules (in cm⁻¹)</caption><thead><tr><th></th><th>ω<sub>e</sub></th><th>ω<sub>e</sub>x<sub>e</sub></th><th>B<sub>0</sub></th><th>α<sub>e</sub></th><th>Ref.</th><th></th><th>ω<sub>e</sub></th><th>ω<sub>e</sub>x<sub>e</sub></th><th>B<sub>0</sub></th><th>Ref.</th></tr></thead><tbody><tr><td>H<sub>2</sub></td><td>4401.118</td><td>121.28</td><td>59.3251</td><td>3.0258</td><td>a</td><td><sup>14</sup>N<sup>16</sup>O</td><td>1904.087</td><td>14.044</td><td>1.696284</td><td>j</td></tr><tr><td>HD</td><td>3812.293</td><td>90.908</td><td>44.6613</td><td>2.0034</td><td>b</td><td><sup>15</sup>N<sup>16</sup>O</td><td>1869.997</td><td>13.535</td><td>1.636369</td><td>j</td></tr><tr><td>D<sub>2</sub></td><td>3113.259</td><td>60.687</td><td>29.9069</td><td>1.0711</td><td>c</td><td><sup>14</sup>N<sup>18</sup>O</td><td>1853.941</td><td>13.314</td><td>1.608332</td><td>c</td></tr><tr><td>HF</td><td>4138.32</td><td>89.88</td><td>20.5567</td><td>0.798</td><td>d</td><td><sup>12</sup>C<sup>16</sup>O</td><td>2169.817</td><td>13.290</td><td>1.922529</td><td>k</td></tr><tr><td>DF</td><td>2998.192</td><td>45.761</td><td>10.8546</td><td>0.2907</td><td>e</td><td><sup>13</sup>C<sup>16</sup>O</td><td>2121.430</td><td>12.698</td><td>1.837973</td><td>l</td></tr><tr><td>H<sup>35</sup>Cl</td><td>2990.946</td><td>52.819</td><td>10.44020<sup>g</sup></td><td>0.30718</td><td>f</td><td><sup>12</sup>C<sup>18</sup>O</td><td>2117.400</td><td>12.655</td><td>1.830967</td><td>c</td></tr><tr><td>D<sup>35</sup>Cl</td><td>2145.115</td><td>27.169</td><td>5.392271<sup>g</sup></td><td>0.11332</td><td>c</td><td><sup>16</sup>O<sub>2</sub></td><td>1580.361</td><td>12.073</td><td>1.438225<sup>g</sup></td><td>m</td></tr><tr><td>H<sup>37</sup>Cl</td><td>2988.681</td><td>52.738</td><td>10.42451<sup>g</sup></td><td>0.30668</td><td>c</td><td><sup>16</sup>O<sup>18</sup>O</td><td>1535.737</td><td>11.401</td><td>1.358365</td><td>c</td></tr><tr><td>D<sup>37</sup>Cl</td><td>2141.957</td><td>27.089</td><td>5.376489<sup>g</sup></td><td>0.11282</td><td>c</td><td><sup>18</sup>O<sub>2</sub></td><td>1489.777</td><td>10.729</td><td>1.278475</td><td>c</td></tr><tr><td><sup>35</sup>Cl<sub>2</sub></td><td>559.72</td><td>2.67</td><td>0.24325</td><td></td><td>h</td><td><sup>12</sup>C<sup>32</sup>S</td><td>1285.08</td><td>6.46</td><td>0.81708</td><td>o</td></tr><tr><td><sup>37</sup>Cl<sub>2</sub></td><td>544.37</td><td>2.53</td><td>0.23001</td><td></td><td>c</td><td><sup>13</sup>C<sup>32</sup>S</td><td>1248.51</td><td>6.10</td><td>0.77133</td><td>c</td></tr><tr><td><sup>14</sup>N<sub>2</sub></td><td>2358.027</td><td>14.135</td><td>1.98914</td><td></td><td>i</td><td><sup>12</sup>C<sup>34</sup>S</td><td>1274.74</td><td>6.36</td><td>0.80401</td><td>c</td></tr><tr><td><sup>14</sup>N<sup>15</sup>N</td><td>2318.512</td><td>13.665</td><td>1.92325</td><td></td><td>c</td><td><sup>32</sup>S<sub>2</sub></td><td>725.680</td><td>2.852</td><td>0.2948</td><td>p</td></tr><tr><td><sup>15</sup>N<sub>2</sub></td><td>2278.313</td><td>13.196</td><td>1.85721</td><td></td><td>c</td><td><sup>34</sup>S<sub>2</sub></td><td>704.040</td><td>2.684</td><td>0.2775</td><td>c</td></tr></tbody></table>

a Buijs & Gush (1971).

b Durie & Herzberg (1960).

c Calculated with Equations 55–58.

d Webb & Rao (1968).

e Spanbauer & Rao (1965).

f Rank et al (1965).

g De Lucia et al (1971).

h Clyne & Coxon (1970).

i Benesh et al (1965).

j Olman et al (1964).

k Roh & Rao (1974).

l Johns et al (1974).

m Babcock & Herzberg (1948).

n Amano & Hirota (1974).

o Barrow et al (1960).

p Herzberg (1950).
the B-O approximation is valid and the isotopic molecules have the same potential-energy function then it is simple to calculate the constants needed for the isotopically substituted molecule from those known for the normal molecule. For instance, Dunham (1932) has shown that for diatomic molecules

$$ \omega'_{e} = \omega_{e}\rho, \qquad (55) $$

where $\rho = \sqrt{\mu/\mu'}$, and $\mu$ is the reduced mass. Furthermore, we have

$$ \omega'_{e}x'_{e} = \omega_{e}x_{e}\rho^{2}, \qquad (56) $$

$$ \omega'_{e}y'_{e} = \omega_{e}y_{e}\rho^{3}, \\ B'_{e} = B_{e}\rho^{2}, \qquad (57) $$

and

$$ \alpha'_{e} = \alpha_{e}\rho^{3}. \qquad (58) $$

These equations were employed to complete the data set needed for the calculations. The molecular constants for diatomic molecules are listed in Table 5. Measurements for certain diatomic molecules such as H₂ have been repeated many times, but sometimes the agreement among these measurements is poor. In our selection of the data given in Tables 5–8 we have favored the most complete and most recent observations over incomplete and older ones, and we preferred data that obeyed Equations 55 to 58.

For polyatomic molecules the molecular constants are less well known than for the diatomic ones. In particular, measurements for isotopically substituted molecules are relatively rare when the number of atoms in a molecule becomes larger than three. To complete the data, force fields derived from the normal molecules were used to calculate zero-order frequencies for the isotopically substituted molecules. The semi-empirical Darling-Dennison rule (see Burneau 1974),

$$ x'_{ij} = x_{ij}\omega'_{i}\omega'_{j}/\omega_{i}\omega_{j}, \qquad (59) $$

was employed to calculate anharmonicity constants for the isotopically substituted molecules. For isotopic molecules with the same symmetry Equation 59 works well. Because of lack of anything better we have also used Equation 59 when the symmetry of the isotopic molecules is not the same. Pragmatically, we observed that the agreement between measured and calculated isotopic equilibrium constants does not seem to be affected by this possible misuse of Equation 59. The empirical relation

$$ \delta'_{i} = \delta_{i}\omega'_{i}/\omega_{i} \qquad (60) $$

introduced by Haar et al (1955) was used. However, the reduced significance of rotation-vibration interaction in isotopic fractionation calculations has a consequence that this will not be an important error source.

In the following paragraphs we furnish, briefly, detailed information on how the calculations indicated in Tables 5–8 were made.

**Hydrogen** There are no recent observations of the spectrum of D₂. We have used the data of Buijs & Gush (1971) to calculate these constants with Equations 55–58.
The resulting constants give for the internal equilibrium H₂ + D₂ = 2HD and G₀ = 2G₀(HD) - G₀(H₂) - G₀(D₂) = 0.86 cm⁻¹. This suggests a certain lack of internal consistence in the constants. This is also observable when one compares the data for H₂ (Buijs & Gush 1971) and those for HD (Durie & Herzberg 1960) with the predictions of Equations 55–58.

*Other diatomic molecules* Equations 55–58 are well obeyed.

*Carbon dioxide* The constants for ¹²C¹⁸O₂ were calculated from the data of Courtoy (1957, 1959) for ¹²C¹⁶O₂ by means of the Urey-Bradley force field equations given in Herzberg (1945). These equations were also used to calculate the vibrational frequencies for ¹²C³⁴S₂ from the data for ¹²C³²S₂ of Giguère et al (1973).

*Bent triatomic molecules* X Y₂ Zero-order frequencies for the isotopically substituted molecules were calculated by means of the equations given by Shaffer & Schuman (1944).

*Ammonia* The equations of Rosenthal (1935) and the force constants deduced by Benedict & Plyler (1957) for NH₃ were used to calculate the frequencies for the isotopic molecules of ammonia. Internuclear distances for NH₃ from Benedict & Plyler (1957) were used to calculate the rotational constants for the isotopic molecules of ammonia. The anharmonicity constants x₁₁ and x₂₂ of NH₃ have not been observed, but values of -30 and -51.8 cm⁻¹ were assumed, respectively, because they are in agreement with the observed fundamental frequencies. Equation 59 was used to calculate the remaining unknown anharmonicity constants.

*Sulphur-trioxide* The data of Dorney et al (1973) for ³²S¹⁶O₃ were used with the equations of Anderson et al (1936) to calculate the vibrational constants for ³⁴S¹⁶O₃ and ³⁴S¹⁸O₃.

*Methane* Zero-order frequencies for the isotopic methane molecules were calculated as in Bottinga (1969). Anharmonicity constants were calculated by means of Equation 59, and experimental rotational constants for CH₄, CH₃D, CH₂D₂, and CD₄ were employed.

## 5 CALCULATED RESULTS

Normal practice in geochemistry is to report the isotopic composition of a substance *V*<sub>n</sub> in the delta notation,

$$ \delta(V Y_n) = \left| \frac{R_Y(V Y_n)}{R_Y(ST)} - 1 \right| \times 1000, $$

where R<sub>Y</sub> stands for the atomic ratio Y'/Y and ST refers to a standard. In case there are two substances V*Y*<sub>n</sub> and W*Y*<sub>n</sub> that are in isotopic exchange equilibrium one can define a fractionation factor

$$ \alpha(V Y_n, W Y_m) = R(V Y_n)/R(W Y_m). \quad (61) $$
<table><caption>**Table 6** Molecular constants of linear triatomic molecules (in cm⁻¹)</caption><thead><tr><th></th><th>H<sup>12</sup>C<sup>14</sup>N</th><th>D<sup>12</sup>C<sup>14</sup>N</th><th>H<sup>13</sup>C<sup>14</sup>N</th><th>H<sup>12</sup>C<sup>15</sup>N</th><th>Ref.</th><th><sup>14</sup>N<sub>2</sub><sup>16</sup>O</th><th><sup>15</sup>N<sub>2</sub><sup>16</sup>O</th><th><sup>14</sup>N<sub>2</sub><sup>18</sup>O</th><th>Ref.</th></tr></thead><tbody><tr><td>ω<sub>1</sub></td><td>2128.67</td><td>1952.12</td><td>2094.16</td><td>2095.12</td><td></td><td>1298.66</td><td>1280.50</td><td>1255.30</td><td></td></tr><tr><td>ω<sub>2</sub></td><td>727.10</td><td>579.85</td><td>720.71</td><td>726.01</td><td>a</td><td>596.36</td><td>579.09</td><td>591.65</td><td>b</td></tr><tr><td>ω<sub>3</sub></td><td>3441.16</td><td>2703.34</td><td>3422.04</td><td>3439.85</td><td></td><td>2282.00</td><td>2209.00</td><td>2275.60</td><td></td></tr><tr><td>x<sub>11</sub></td><td>-10.45</td><td>-6.84</td><td>-10.27</td><td>-10.13</td><td></td><td>-4.30<sup>d</sup></td><td>-4.18</td><td>-4.02</td><td></td></tr><tr><td>x<sub>12</sub></td><td>-3.61</td><td>3.01</td><td>-4.13</td><td>-3.46</td><td></td><td>0.20</td><td>0.19</td><td>0.19</td><td></td></tr><tr><td>x<sub>13</sub></td><td>-14.61</td><td>-32.40</td><td>-12.90</td><td>-14.08</td><td>a</td><td>-25.56</td><td>-24.39</td><td>-24.63</td><td>c</td></tr><tr><td>x<sub>22</sub></td><td>-2.44</td><td>-2.08</td><td>-2.31</td><td>-2.46</td><td></td><td>-0.27</td><td>-0.25</td><td>-0.26</td><td></td></tr><tr><td>x<sub>23</sub></td><td>-18.98</td><td>-15.80</td><td>-18.38</td><td>-18.91</td><td></td><td>-14.63</td><td>-13.75</td><td>-14.47</td><td></td></tr><tr><td>x<sub>33</sub></td><td>-51.71</td><td>-20.50</td><td>-51.87</td><td>-51.88</td><td></td><td>-15.19</td><td>-14.24</td><td>-15.10</td><td></td></tr><tr><td>B<sub>0</sub></td><td>1.47822</td><td>1.20775</td><td>1.43999</td><td>1.43527</td><td>a</td><td>0.419011</td><td>0.404859</td><td>0.395577</td><td>e</td></tr><tr><td></td><td><sup>16</sup>O<sup>12</sup>C<sup>32</sup>S</td><td><sup>16</sup>O<sup>13</sup>C<sup>32</sup>S</td><td><sup>18</sup>O<sup>12</sup>C<sup>32</sup>S</td><td><sup>16</sup>O<sup>12</sup>C<sup>34</sup>S</td><td></td><td><sup>12</sup>C<sup>32</sup>S<sub>2</sub></td><td><sup>13</sup>C<sup>32</sup>S<sub>2</sub></td><td><sup>12</sup>C<sup>34</sup>S<sub>2</sub></td><td>Ref.</td></tr><tr><td>ω<sub>1</sub></td><td>875.70</td><td>871.15</td><td>854.13</td><td>863.89</td><td></td><td>671.89</td><td>671.89</td><td>651.39<sup>f</sup></td><td></td></tr><tr><td>ω<sub>2</sub></td><td>523.62</td><td>508.03</td><td>518.09</td><td>522.83</td><td>f</td><td>398.38</td><td>385.22</td><td>396.52</td><td>h</td></tr><tr><td>ω<sub>3</sub></td><td>2092.46</td><td>2037.45</td><td>2055.86</td><td>2091.77</td><td></td><td>1558.73</td><td>1507.25</td><td>1551.48</td><td></td></tr><tr><td>x<sub>11</sub></td><td>-4.32<sup>f</sup></td><td>-4.28</td><td>-4.11</td><td>-4.20</td><td></td><td>-0.83</td><td>-0.83</td><td>-0.78<sup>e</sup></td><td></td></tr><tr><td>x<sub>12</sub></td><td>-7.16</td><td>-6.91</td><td>-6.91</td><td>-7.05</td><td></td><td>-1.75</td><td>-1.68</td><td>-1.64</td><td></td></tr><tr><td>x<sub>13</sub></td><td>-2.97</td><td>-2.88</td><td>-2.85</td><td>-2.93</td><td>c</td><td>-7.59</td><td>-7.36</td><td>-7.32</td><td>h</td></tr><tr><td>x<sub>22</sub></td><td>1.78</td><td>1.68</td><td>1.74</td><td>1.77</td><td></td><td>0.82</td><td>0.77</td><td>0.81</td><td></td></tr><tr><td>x<sub>23</sub></td><td>-6.53</td><td>-6.17</td><td>-6.35</td><td>-6.52</td><td></td><td>-6.49</td><td>-6.04</td><td>-6.43</td><td></td></tr><tr><td>x<sub>33</sub></td><td>-10.83</td><td>-10.27</td><td>-10.45</td><td>-10.82</td><td></td><td>-6.54</td><td>-6.10</td><td>-6.48</td><td></td></tr><tr><td>B<sub>0</sub></td><td>0.202856</td><td>0.202202</td><td>0.190192</td><td>0.197897</td><td>g</td><td>0.109092</td><td>0.109092</td><td>0.102686</td><td>i</td></tr></tbody></table>
<table><caption>**Table 6 (continued)**</caption><thead><tr><th></th><th><sup>12</sup>C<sup>16</sup>O<sub>2</sub></th><th><sup>13</sup>C<sup>16</sup>O<sub>2</sub></th><th>Ref.</th><th><sup>12</sup>C<sup>16</sup>O<sub>2</sub></th><th><sup>12</sup>C<sup>18</sup>O<sub>2</sub></th><th>Ref.</th></tr></thead><tbody><tr><td>ω<sub>1</sub></td><td>1353.637</td><td>1353.680</td><td></td><td>1354.91<sup>1</sup></td><td>1277.25</td><td></td></tr><tr><td>ω<sub>2</sub></td><td>672.625</td><td>653.771</td><td>j</td><td>673.00</td><td>662.70</td><td>m</td></tr><tr><td>ω<sub>3</sub></td><td>2396.269</td><td>2328.037</td><td></td><td>2396.49</td><td>2359.81</td><td></td></tr><tr><td>x<sub>11</sub></td><td>-2.885</td><td>-2.885</td><td></td><td>-3.75<sup>1</sup></td><td>-3.33</td><td></td></tr><tr><td>x<sub>12</sub></td><td>-5.416</td><td>-5.247</td><td></td><td>3.62</td><td>3.39</td><td></td></tr><tr><td>x<sub>13</sub></td><td>-19.205</td><td>-18.731</td><td></td><td>-19.37</td><td>-17.98</td><td>c</td></tr><tr><td>x<sub>22</sub></td><td>1.642</td><td>1.546</td><td>j</td><td>-0.63</td><td>-0.61</td><td></td></tr><tr><td>x<sub>23</sub></td><td>-12.515</td><td>-11.728</td><td></td><td>-12.53</td><td>-12.15</td><td></td></tr><tr><td>x<sub>33</sub></td><td>-12.564</td><td>-11.797</td><td></td><td>-12.63</td><td>-12.25</td><td></td></tr><tr><td>B<sub>o</sub></td><td>0.390210</td><td>0.390250</td><td>k</td><td>0.390210</td><td>0.346820</td><td>k</td></tr></tbody></table>

<sup>a</sup> Nakagawa & Morino (1969).

<sup>b</sup> Jones (1970).

<sup>c</sup> Calculated with Equation 59.

<sup>d</sup> Pliva (1968).

<sup>e</sup> Griggs et al (1968).

<sup>f</sup> Foord et al (1975).

<sup>g</sup> Morino & Nakagawa (1968).

<sup>h</sup> Giguère et al (1973).

<sup>i</sup> Foss Smith & Overend (1971).

<sup>j</sup> Jobard & Chedin (1975).

<sup>k</sup> Courtoy (1959).

<sup>l</sup> Courtoy (1957).

<sup>m</sup> Calculated (see text).
<table><caption>**Table 7** Molecular constants of bent triatomic molecules (in cm⁻¹)</caption><thead><tr><th></th><th>H<sub>2</sub><sup>16</sup>O</th><th>HD<sup>16</sup>O</th><th>D<sub>2</sub><sup>16</sup>O</th><th>H<sub>2</sub><sup>18</sup>O</th><th>Ref.</th><th>H<sub>2</sub><sup>32</sup>S</th><th>HD<sup>32</sup>S</th><th>D<sub>2</sub><sup>32</sup>S</th><th>H<sub>2</sub><sup>34</sup>S</th><th>Ref.</th></tr></thead><tbody><tr><td>ω<sub>1</sub></td><td>3835.37</td><td>2823.19</td><td>2762.84</td><td>3827.59</td><td></td><td>2722.28</td><td>1958.34</td><td>1952.27</td><td>2720.07</td><td></td></tr><tr><td>ω<sub>2</sub></td><td>1647.59</td><td>1444.53</td><td>1206.72</td><td>1640.62</td><td>a</td><td>1214.38</td><td>1057.29</td><td>872.10</td><td>1213.24</td><td>d</td></tr><tr><td>ω<sub>3</sub></td><td>3938.74</td><td>3888.63</td><td>2885.99</td><td>3922.69</td><td></td><td>2733.75</td><td>2727.91</td><td>1964.21</td><td>2731.20</td><td></td></tr><tr><td>x<sub>11</sub></td><td>-44.18</td><td>-41.51</td><td>-21.94</td><td>-44.01</td><td></td><td>-25.09<sup>e</sup></td><td>-12.98</td><td>-12.91</td><td>-25.05</td><td></td></tr><tr><td>x<sub>12</sub></td><td>-16.20</td><td>-16.93</td><td>-8.77</td><td>-16.11</td><td></td><td>-19.69</td><td>-12.33</td><td>-10.14</td><td>-19.65</td><td></td></tr><tr><td>x<sub>13</sub></td><td>-164.49</td><td>-12.91</td><td>-85.76</td><td>-163.49</td><td></td><td>-94.68</td><td>-67.96</td><td>-48.80</td><td>-94.51</td><td></td></tr><tr><td>x<sub>22</sub></td><td>-17.22</td><td>-11.90</td><td>-9.46</td><td>-17.07</td><td>a</td><td>-5.72</td><td>-4.34</td><td>-2.95</td><td>-5.71</td><td>f</td></tr><tr><td>x<sub>23</sub></td><td>-20.11</td><td>-20.08</td><td>-10.17</td><td>-19.94</td><td></td><td>-21.09</td><td>-18.32</td><td>-10.88</td><td>-21.05</td><td></td></tr><tr><td>x<sub>33</sub></td><td>-45.12</td><td>-82.34</td><td>-24.99</td><td>-44.74</td><td></td><td>-24.00</td><td>-23.90</td><td>-12.39</td><td>-23.95</td><td></td></tr><tr><td>A<sub>o</sub></td><td>27.88039</td><td>23.41107</td><td>15.41988</td><td>27.53087</td><td></td><td>10.34656</td><td>9.75179</td><td>5.48950</td><td>10.44493<sup>h</sup></td><td></td></tr><tr><td>B<sub>o</sub></td><td>14.51317</td><td>9.09802</td><td>7.27059</td><td>14.51234</td><td>b</td><td>9.03572</td><td>4.93214</td><td>4.51580</td><td>9.13083</td><td>g</td></tr><tr><td>C<sub>o</sub></td><td>9.28551</td><td>6.41288</td><td>4.48756</td><td>9.24608</td><td></td><td>4.72680</td><td>3.22570</td><td>2.44316</td><td>4.87189</td><td></td></tr><tr><td>δ<sub>1</sub></td><td>0.065</td><td>0.050</td><td>0.045</td><td></td><td></td><td>0.028<sup>e</sup></td><td>0.020</td><td>0.020</td><td></td><td></td></tr><tr><td>δ<sub>2</sub></td><td>-0.102</td><td>-0.082</td><td>-0.077</td><td></td><td>c</td><td>-0.033</td><td>-0.029</td><td>-0.024</td><td></td><td></td></tr><tr><td>δ<sub>3</sub></td><td>0.066</td><td>0.062</td><td>0.055</td><td></td><td></td><td>0.094</td><td>0.093</td><td>0.067</td><td></td><td></td></tr></tbody></table>
<table><thead><tr><th></th><th><sup>14</sup>N<sup>16</sup>O<sub>2</sub></th><th><sup>15</sup>N<sup>16</sup>O<sub>2</sub></th><th>Ref.</th><th><sup>14</sup>N<sup>16</sup>O<sub>2</sub></th><th><sup>14</sup>N<sup>18</sup>O<sub>2</sub></th><th>Ref.</th><th><sup>32</sup>S<sup>16</sup>O<sub>2</sub></th><th><sup>32</sup>S<sup>18</sup>O<sub>2</sub></th><th>Ref.</th><th><sup>32</sup>S<sup>16</sup>O<sub>2</sub></th><th><sup>34</sup>S<sup>16</sup>O<sub>2</sub></th><th>Ref.</th></tr></thead><tbody><tr><td>ω<sub>1</sub></td><td>1355.90</td><td>1338.50</td><td></td><td>1357.78</td><td>1306.71</td><td></td><td>1167.60</td><td>1113.98</td><td></td><td>1166.92</td><td>1160.37</td><td></td></tr><tr><td>ω<sub>2</sub></td><td>756.80</td><td>747.10</td><td></td><td>756.90</td><td>728.73</td><td></td><td>526.27</td><td>505.30</td><td>n</td><td>523.39</td><td>518.55</td><td>p</td></tr><tr><td>ω<sub>3</sub></td><td>1663.50</td><td>1628.30</td><td></td><td>1665.82</td><td>1633.96</td><td></td><td>1380.91</td><td>1335.18</td><td></td><td>1380.96</td><td>1363.52</td><td></td></tr><tr><td>x<sub>11</sub></td><td>-8.1</td><td>-7.4</td><td></td><td>-8.1<sup>j</sup></td><td>-7.5</td><td></td><td>-3.99</td><td>-3.15</td><td></td><td>-3.99<sup>n</sup></td><td>-3.94</td><td></td></tr><tr><td>x<sub>12</sub></td><td>-9.7</td><td>-9.5</td><td></td><td>-9.7</td><td>-9.0</td><td></td><td>-2.05</td><td>-1.20</td><td></td><td>-2.05</td><td>-2.02</td><td></td></tr><tr><td>x<sub>13</sub></td><td>-28.7</td><td>-27.7</td><td></td><td>-28.7</td><td>-27.1</td><td></td><td>-13.71</td><td>-12.85</td><td>n</td><td>-13.71</td><td>-13.46</td><td>f</td></tr><tr><td>x<sub>22</sub></td><td>-0.5</td><td>-0.4</td><td></td><td>-0.5</td><td>-0.5</td><td></td><td>-3.00</td><td>-3.20</td><td></td><td>-3.00</td><td>-2.94</td><td></td></tr><tr><td>x<sub>23</sub></td><td>-2.7</td><td>-2.6</td><td></td><td>-2.7</td><td>-2.5</td><td></td><td>-3.90</td><td>-3.10</td><td></td><td>-3.90</td><td>-3.81</td><td></td></tr><tr><td>x<sub>33</sub></td><td>-15.6</td><td>-15.3</td><td></td><td>-15.6</td><td>-15.0</td><td></td><td>-5.17</td><td>-4.65</td><td></td><td>-5.17</td><td>-5.04</td><td></td></tr><tr><td>A<sub>o</sub></td><td>8.00251</td><td>7.63062</td><td></td><td></td><td></td><td></td><td>2.03735</td><td>1.91414</td><td></td><td>2.02735</td><td>1.96773</td><td></td></tr><tr><td>B<sub>o</sub></td><td>0.43367</td><td>0.43372</td><td>k</td><td></td><td></td><td></td><td>0.34417</td><td>0.30589</td><td>o</td><td>0.34417</td><td>0.34418</td><td>o</td></tr><tr><td>C<sub>o</sub></td><td>0.41049</td><td>0.40949</td><td></td><td></td><td></td><td></td><td>0.29353</td><td>0.26317</td><td></td><td>0.29353</td><td>0.29225</td><td></td></tr></tbody></table>

a Khachkuruzov (1959).

b Cook et al (1974).

c Benedict et al (1956).

d Calculated (see text) with force constants derived by Gamo (1967).

e Allen & Plyler (1956).

f Calculated with Equation 59.

g H<sub>2</sub>S: Cook et al (1973); HDS: Helminger et al (1971); D<sub>2</sub>S: Helminger et al (1972).

h Burrus & Gordy (1953); their constants for H<sub>2</sub><sup>32</sup>S (A<sub>o</sub> = 10.55676, B<sub>o</sub> = 9.22344, C<sub>o</sub> = 4.92127) were used for sulphur exchange

i Calculated with Equation 60.

j Blanck & Hause (1970).

k Olman & Hause (1968).

l Calculated (see text) with force constants derived by Papousek & Pliva (1964).

m β(NO<sub>2</sub>) for nitrogen exchange was computed using the Teller-Redlich product rule.

n S<sup>16</sup>O<sub>2</sub>: Shelton et al (1953); S<sup>18</sup>O<sub>2</sub>: Barbe & Jouve (1971).

o Steenbeckeliers (1968) for <sup>32</sup>S<sup>16</sup>O<sub>2</sub>; De Hemptinne et al (1963) for <sup>32</sup>S<sup>18</sup>O<sub>2</sub>; Van Riet (1964) for <sup>34</sup>S<sup>16</sup>O<sub>2</sub>.

p Calculated (see text) with force constants derived by Barbe & Jouve (1971).
When the delta values are small we have

$$1000 \ln \alpha(V_{Y_n}, W_{Y_m}) \approx \delta(V_{Y_n}) - \delta(W_{Y_m}) = \Delta(V_{Y_n}, W_{Y_m}).$$

In the case of simple molecules, when *n* and *m* are unity, α(*V*<sub>Yn</sub>, *W*<sub>Ym</sub>) is equal to the equilibrium constant for the exchange reaction,

$$VY + WY' = VY' + WY,$$

but when the element *V* has one or more isotopes and *n* and/or *m* are not unity, the fractionation factor is not any longer equal to an equilibrium constant. Examples of this are carbon dioxide and methane discussed by Urey (1947) and Bottinga (1969), respectively.

In general we have

$$\alpha(V_{Y_n}, W_{Y_m}) = K(V_{Y_n}, W_{Y_m})^{1/mn} \frac{X(V_{Y_n})}{X(W_{Y_m})}, \quad (62)$$

where

$$K(V_{Y_n}, W_{Y_m}) = \left[ \frac{(V_{Y_n})^m}{(V_{Y_m})} \right] / \left[ \frac{(W_{Y_m})^n}{(W_{Y_m})} \right]$$

is the equilibrium constant for the reaction

$$mV_{Y_n} + nW_{Y_m} = mV_{Y_n} + nW_{Y_m}.$$

<table><caption>**Table 8** Molecular constants of tetra- and pentatomic molecules (in cm⁻¹)ᵃ</caption><thead><tr><th></th><th><sup>32</sup>S<sup>16</sup>O<sub>3</sub></th><th><sup>32</sup>S<sup>18</sup>O<sub>3</sub></th><th><sup>34</sup>S<sup>16</sup>O<sub>3</sub></th><th>Ref.</th><th><sup>14</sup>NH<sub>3</sub></th><th><sup>14</sup>ND<sub>3</sub></th><th><sup>15</sup>NH<sub>3</sub></th><th>Ref.</th></tr></thead><tbody><tr><td>ω<sub>1</sub></td><td>1064.89</td><td>1003.85</td><td>1064.89</td><td></td><td>3505.7</td><td>2498.9</td><td>3503.9</td><td></td></tr><tr><td>ω<sub>2</sub></td><td>505.97</td><td>494.58</td><td>496.97</td><td>b</td><td>1022.0</td><td>778.4</td><td>1016.4</td><td>b</td></tr><tr><td>ω<sub>3</sub></td><td>1410.00</td><td>1368.08</td><td>1390.33</td><td></td><td>3573.1</td><td>2630.0</td><td>3563.4</td><td></td></tr><tr><td>ω<sub>4</sub></td><td>535.62</td><td>508.69</td><td>533.53</td><td></td><td>1689.7</td><td>1226.5</td><td>1686.4</td><td></td></tr><tr><td>x<sub>11</sub></td><td>-1.4<sup>d</sup></td><td>-1.2</td><td>-1.4</td><td></td><td>-30.0<sup>f</sup></td><td>-15.2</td><td>-30.0</td><td></td></tr><tr><td>x<sub>12</sub></td><td>5.1</td><td>4.7</td><td>5.0</td><td></td><td>20.6</td><td>11.2</td><td>20.5</td><td></td></tr><tr><td>x<sub>13</sub></td><td>-6.9</td><td>-6.3</td><td>-6.8</td><td></td><td>-92.0</td><td>-48.3</td><td>-91.7</td><td></td></tr><tr><td>x<sub>14</sub></td><td>10.3</td><td>9.2</td><td>10.2</td><td></td><td>-6.7</td><td>-3.5</td><td>-6.7</td><td></td></tr><tr><td>x<sub>22</sub></td><td>-3.2</td><td>-3.1</td><td>-3.1</td><td>c</td><td>-51.8</td><td>-30.0</td><td>-51.2</td><td>c</td></tr><tr><td>x<sub>23</sub></td><td>-2.4</td><td>-2.2</td><td>-2.3</td><td></td><td>32.4</td><td>18.2</td><td>32.1</td><td></td></tr><tr><td>x<sub>24</sub></td><td>-2.2</td><td>-2.0</td><td>-2.1</td><td></td><td>-10.7</td><td>-5.9</td><td>-10.6</td><td></td></tr><tr><td>x<sub>33</sub></td><td>-4.6</td><td>-4.4</td><td>-4.5</td><td></td><td>-18.5</td><td>-10.0</td><td>-18.4</td><td></td></tr><tr><td>x<sub>34</sub></td><td>-2.6</td><td>-2.4</td><td>-2.6</td><td></td><td>-17.3</td><td>-9.2</td><td>-17.2</td><td></td></tr><tr><td>x<sub>44</sub></td><td>-2.6</td><td>-2.4</td><td>-2.6</td><td></td><td>-8.0</td><td>-4.6</td><td>-8.8</td><td></td></tr><tr><td>A<sub>o</sub></td><td>0.1746</td><td>0.1551</td><td>0.1746</td><td>e</td><td>9.9444</td><td>5.1426</td><td>9.9209</td><td>b</td></tr><tr><td>B<sub>o</sub></td><td>0.1746</td><td>0.1551</td><td>0.1746</td><td></td><td>9.9444</td><td>5.1426</td><td>9.9209</td><td>b</td></tr><tr><td>C<sub>o</sub></td><td>0.3491</td><td>0.3103</td><td>0.3491</td><td></td><td>6.1960</td><td>3.1170</td><td>6.1960</td><td></td></tr><tr><td>δ<sub>1</sub></td><td></td><td></td><td></td><td></td><td>0.040</td><td>0.028</td><td></td><td></td></tr><tr><td>δ<sub>2</sub></td><td></td><td></td><td></td><td></td><td>0.019</td><td>0.014</td><td></td><td></td></tr><tr><td>δ<sub>3</sub></td><td></td><td></td><td></td><td></td><td>0.034</td><td>0.025</td><td></td><td>f</td></tr><tr><td>δ<sub>4</sub></td><td></td><td></td><td></td><td></td><td>-0.036</td><td>-0.026</td><td></td><td></td></tr></tbody></table>
<table><caption>**Table 8** (continued)</caption><thead><tr><th></th><th><sup>14</sup>NH<sub>2</sub>D</th><th><sup>14</sup>NHD<sub>2</sub></th><th>Ref.</th><th><sup>12</sup>CH<sub>4</sub></th><th><sup>12</sup>CD<sub>4</sub></th><th><sup>13</sup>CH<sub>4</sub></th><th><sup>12</sup>CH<sub>3</sub>D</th><th>Ref.</th></tr></thead><tbody><tr><td>ω<sub>1</sub></td><td>2587.9</td><td>2544.1</td><td></td><td>3143.0</td><td>2223.3</td><td>3143.0</td><td>2306.1</td><td></td></tr><tr><td>ω<sub>2</sub></td><td>943.5</td><td>857.2</td><td></td><td>1573.0</td><td>1112.7</td><td>1573.0</td><td>1512.1</td><td></td></tr><tr><td>ω<sub>3</sub></td><td>3528.0</td><td>3550.2</td><td>b</td><td>3154.0</td><td>2333.0</td><td>3143.0</td><td>3146.0</td><td>b</td></tr><tr><td>ω<sub>4</sub></td><td>3572.8</td><td>2630.1</td><td></td><td>1357.0</td><td>1026.8</td><td>1348.5</td><td>3153.6</td><td></td></tr><tr><td>ω<sub>5</sub></td><td>1657.2</td><td>1279.2</td><td></td><td></td><td></td><td></td><td>1352.2</td><td></td></tr><tr><td>ω<sub>6</sub></td><td>1439.1</td><td>1513.3</td><td></td><td></td><td></td><td></td><td>1197.3</td><td></td></tr><tr><td>x<sub>11</sub></td><td>-16.3</td><td>-15.8</td><td></td><td>-64.6</td><td>-32.3</td><td>-64.6</td><td>-34.8</td><td></td></tr><tr><td>x<sub>12</sub></td><td>14.0</td><td>12.5</td><td></td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td></td></tr><tr><td>x<sub>13</sub></td><td>-67.1</td><td>-66.3</td><td></td><td>-65.0</td><td>-34.0</td><td>-64.8</td><td>-47.6</td><td></td></tr><tr><td>x<sub>14</sub></td><td>-67.9</td><td>-49.1</td><td></td><td>0.0</td><td>0.0</td><td>0.0</td><td>-47.7</td><td></td></tr><tr><td>x<sub>15</sub></td><td>-4.9</td><td>-3.7</td><td></td><td></td><td></td><td></td><td>0.0</td><td></td></tr><tr><td>x<sub>16</sub></td><td>-4.2</td><td>-4.4</td><td></td><td></td><td></td><td></td><td>0.0</td><td></td></tr><tr><td>x<sub>22</sub></td><td>-44.1</td><td>-36.4</td><td></td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td></td></tr><tr><td>x<sub>23</sub></td><td>29.5</td><td>27.0</td><td></td><td>-15.0</td><td>-7.8</td><td>-14.9</td><td>-14.4</td><td></td></tr><tr><td>x<sub>24</sub></td><td>29.9</td><td>20.0</td><td></td><td>-11.2</td><td>-6.0</td><td>-11.1</td><td>-14.4</td><td></td></tr><tr><td>x<sub>25</sub></td><td>-9.7</td><td>-6.8</td><td></td><td></td><td></td><td></td><td>-10.7</td><td></td></tr><tr><td>x<sub>26</sub></td><td>-8.4</td><td>-8.0</td><td></td><td></td><td></td><td></td><td>-9.5</td><td></td></tr><tr><td>x<sub>33</sub></td><td>-18.0</td><td>-18.3</td><td>c</td><td>-17.5</td><td>-9.6</td><td>-17.4</td><td>-17.4</td><td>b</td></tr><tr><td>x<sub>34</sub></td><td>-18.3</td><td>-13.5</td><td></td><td>-12.0</td><td>-6.7</td><td>-11.9</td><td>-17.5</td><td></td></tr><tr><td>x<sub>35</sub></td><td>-16.8</td><td>-13.0</td><td></td><td></td><td></td><td></td><td>-11.9</td><td></td></tr><tr><td>x<sub>36</sub></td><td>-14.5</td><td>-15.4</td><td></td><td></td><td></td><td></td><td>-10.6</td><td></td></tr><tr><td>x<sub>44</sub></td><td>-18.5</td><td>-10.0</td><td></td><td>-6.0</td><td>-3.4</td><td>-5.9</td><td>-17.5</td><td></td></tr><tr><td>x<sub>45</sub></td><td>-17.0</td><td>-9.6</td><td></td><td></td><td></td><td></td><td>-12.0</td><td></td></tr><tr><td>x<sub>46</sub></td><td>-14.7</td><td>-11.4</td><td></td><td></td><td></td><td></td><td>-10.6</td><td></td></tr><tr><td>x<sub>55</sub></td><td>-8.5</td><td>-5.0</td><td></td><td></td><td></td><td></td><td>-6.0</td><td></td></tr><tr><td>x<sub>56</sub></td><td>-7.4</td><td>-6.0</td><td></td><td></td><td></td><td></td><td>-5.3</td><td></td></tr><tr><td>x<sub>66</sub></td><td>-6.4</td><td>-7.1</td><td></td><td></td><td></td><td></td><td>-4.7</td><td></td></tr><tr><td>A<sub>0</sub></td><td>9.6805</td><td>7.4460</td><td></td><td>5.2406</td><td>2.6330</td><td>5.2406</td><td>5.2507</td><td></td></tr><tr><td>B<sub>0</sub></td><td>6.3936</td><td>5.3231</td><td>b</td><td>5.2406</td><td>2.6330</td><td>5.2406</td><td>3.8801</td><td>g</td></tr><tr><td>C<sub>0</sub></td><td>4.6620</td><td>3.7179</td><td></td><td>5.2406</td><td>2.6330</td><td>5.2406</td><td>3.8801</td><td></td></tr><tr><td>δ<sub>1</sub></td><td>0.029</td><td>0.029</td><td></td><td>0.025<sup>h</sup></td><td>0.017<sup>h</sup></td><td></td><td>0.018</td><td></td></tr><tr><td>δ<sub>2</sub></td><td>0.017</td><td>0.016</td><td></td><td>-0.050</td><td>-0.035</td><td></td><td>-0.048</td><td></td></tr><tr><td>δ<sub>3</sub></td><td>0.033</td><td>0.034</td><td>i</td><td>0.015</td><td>0.011</td><td></td><td>0.015</td><td>i</td></tr><tr><td>δ<sub>4</sub></td><td>0.034</td><td>0.025</td><td></td><td>0.043</td><td>0.032</td><td></td><td>0.015</td><td></td></tr><tr><td>δ<sub>5</sub></td><td>-0.035</td><td>-0.027</td><td></td><td></td><td></td><td></td><td>0.043</td><td></td></tr><tr><td>δ<sub>6</sub></td><td>-0.030</td><td>-0.032</td><td></td><td></td><td></td><td></td><td>0.038</td><td></td></tr></tbody></table>

<sup>a</sup> ω<sub>3</sub> and ω<sub>4</sub> are doubly degenerate vibrations for X Y<sub>3</sub> molecules; ω<sub>2</sub> is doubly, ω<sub>3</sub> and ω<sub>4</sub> are triply degenerate for X Y<sub>4</sub> molecules; ω<sub>2</sub>, ω<sub>4</sub>, and ω<sub>6</sub> are doubly degenerate for X Y<sub>3</sub>Y<sup>+</sup> molecules.

<sup>b</sup> Calculated (see text).

<sup>c</sup> Calculated with Equation 59.

<sup>d</sup> Dorney et al (1973).

<sup>e</sup> Kaldor et al (1973).

<sup>f</sup> Benedict & Plyler (1957).

<sup>g</sup> CH<sub>4</sub>: Thomas & Welsh (1960); CD<sub>4</sub>: Olafson et al (1961); CH<sub>3</sub>D: Olson (1972).

<sup>h</sup> Calculated with the data of Ref. g and those of Kutchisu & Bartell (1962).

<sup>i</sup> Calculated with Equation 60.
We call the factors *X* in Equation 62 excess factors. They are functions of the internal equilibrium constants involving the molecule *V*Y*n* for *X*(*V*Y*n*) and *W*Y*m* for *X*(*W*Y*m*) and the natural abundance ratio *Y*/*Y*', and if *V* has isotopes the natural abundances of these isotopes enter also into the expression for *X*(*V*Y*n*). The dependence of the excess factors on the natural abundance ratios is weak. To tabulate our results we have made use of the beta factor defined as

$$ \beta(V_{Y_n}, Y) = K(V_{Y_n}, Y)X(V_{Y_n}), $$

where *K* is the equilibrium constant for the reaction

$$ \frac{1}{n} V Y_n + Y' = \frac{1}{n} V Y'_n + Y. $$

Physically, the *β* factor is the fractionation factor between *V*Y*n* and *Y*, i.e.

$$ \beta = \left[ \frac{Y'}{-Y} \right]_{VY_n} \left[ \frac{Y'}{-Y} \right]. $$

<table><caption>**Table 9** *β* factors for hydrogen exchange</caption><thead><tr><th>T °C</th><th>NH<sub>3</sub></th><th>H<sub>2</sub>O</th><th>CH<sub>4</sub><sup>a</sup></th><th>CH<sub>4</sub><sup>b</sup></th><th>HF</th><th>HCN</th><th>H<sub>2</sub>S</th><th>HCl</th><th>H<sub>2</sub></th></tr></thead><tbody><tr><td>0</td><td>16.1336</td><td>15.1307</td><td>14.0578</td><td>12.8201</td><td>13.4815</td><td>11.0964</td><td>6.7076</td><td>6.3512</td><td>3.8161</td></tr><tr><td>10</td><td>14.3395</td><td>13.5308</td><td>12.5241</td><td>11.4585</td><td>12.1573</td><td>9.9831</td><td>6.1713</td><td>5.8795</td><td>3.6197</td></tr><tr><td>20</td><td>12.8490</td><td>12.1930</td><td>11.2473</td><td>10.3214</td><td>11.0407</td><td>9.0497</td><td>5.7107</td><td>5.4716</td><td>3.4460</td></tr><tr><td>30</td><td>11.5980</td><td>11.0635</td><td>10.1737</td><td>9.3626</td><td>10.0907</td><td>8.2597</td><td>5.3120</td><td>5.1163</td><td>3.2915</td></tr><tr><td>40</td><td>10.5385</td><td>10.1014</td><td>9.2629</td><td>8.5470</td><td>9.2755</td><td>7.5852</td><td>4.9644</td><td>4.8046</td><td>3.1533</td></tr><tr><td>50</td><td>9.6336</td><td>9.2753</td><td>8.4838</td><td>7.8475</td><td>8.5707</td><td>7.0045</td><td>4.6594</td><td>4.5295</td><td>3.0290</td></tr><tr><td>75</td><td>7.8753</td><td>7.6572</td><td>6.9666</td><td>6.4801</td><td>7.1754</td><td>5.8625</td><td>4.0416</td><td>3.9671</td><td>2.7674</td></tr><tr><td>100</td><td>6.6177</td><td>6.4873</td><td>5.8783</td><td>5.4943</td><td>6.1520</td><td>5.0319</td><td>3.5749</td><td>3.5370</td><td>2.5594</td></tr><tr><td>125</td><td>5.6866</td><td>5.6129</td><td>5.0708</td><td>4.7597</td><td>5.3775</td><td>4.4076</td><td>3.2128</td><td>3.1994</td><td>2.3905</td></tr><tr><td>150</td><td>4.9775</td><td>4.9414</td><td>4.4546</td><td>4.1970</td><td>4.7759</td><td>3.9255</td><td>2.9254</td><td>2.9288</td><td>2.2508</td></tr><tr><td>175</td><td>4.4242</td><td>4.4135</td><td>3.9733</td><td>3.7559</td><td>4.2981</td><td>3.5446</td><td>2.6930</td><td>2.7080</td><td>2.1336</td></tr><tr><td>200</td><td>3.9835</td><td>3.9903</td><td>3.5895</td><td>3.4033</td><td>3.9115</td><td>3.2377</td><td>2.5021</td><td>2.5249</td><td>2.0340</td></tr><tr><td>250</td><td>3.3326</td><td>3.3598</td><td>3.0221</td><td>2.8800</td><td>3.3285</td><td>2.7773</td><td>2.2088</td><td>2.2404</td><td>1.8741</td></tr><tr><td>300</td><td>2.8810</td><td>2.9177</td><td>2.6282</td><td>2.5152</td><td>2.9136</td><td>2.4519</td><td>1.9961</td><td>2.0311</td><td>1.7516</td></tr><tr><td>350</td><td>2.5534</td><td>2.5942</td><td>2.3424</td><td>2.2497</td><td>2.6059</td><td>2.2120</td><td>1.8363</td><td>1.8718</td><td>1.6550</td></tr><tr><td>400</td><td>2.3072</td><td>2.3492</td><td>2.1278</td><td>2.0497</td><td>2.3702</td><td>2.0293</td><td>1.7128</td><td>1.7474</td><td>1.5770</td></tr><tr><td>450</td><td>2.1169</td><td>2.1585</td><td>1.9622</td><td>1.8950</td><td>2.1848</td><td>1.8863</td><td>1.6152</td><td>1.6481</td><td>1.5128</td></tr><tr><td>500</td><td>1.9665</td><td>2.0069</td><td>1.8313</td><td>1.7725</td><td>2.0359</td><td>1.7721</td><td>1.5366</td><td>1.5674</td><td>1.4591</td></tr><tr><td>600</td><td>1.7458</td><td>1.7826</td><td>1.6398</td><td>1.5928</td><td>1.8131</td><td>1.6023</td><td>1.4189</td><td>1.4454</td><td>1.3748</td></tr><tr><td>700</td><td>1.5938</td><td>1.6269</td><td>1.5083</td><td>1.4690</td><td>1.6560</td><td>1.4837</td><td>1.3362</td><td>1.3586</td><td>1.3117</td></tr><tr><td>800</td><td>1.4842</td><td>1.5138</td><td>1.4138</td><td>1.3798</td><td>1.5405</td><td>1.3972</td><td>1.2757</td><td>1.2946</td><td>1.2632</td></tr><tr><td>900</td><td>1.4024</td><td>1.4289</td><td>1.3436</td><td>1.3133</td><td>1.4529</td><td>1.3320</td><td>1.2301</td><td>1.2461</td><td>1.2249</td></tr><tr><td>1100</td><td>1.2902</td><td>1.3120</td><td>1.2477</td><td>1.2221</td><td>1.3308</td><td>1.2419</td><td>1.1670</td><td>1.1785</td><td>1.1689</td></tr><tr><td>1300</td><td>1.2185</td><td>1.2370</td><td>1.1869</td><td>1.1638</td><td>1.2515</td><td>1.1837</td><td>1.1265</td><td>1.1348</td><td>1.1308</td></tr></tbody></table>

<footnote>a The excess factor needed to calculate *β* was computed in the harmonic approximation, as explained in the text (see Section 7).</footnote>

<footnote>b The excess factor was calculated anharmonically.</footnote>
In terms of $\beta$ factors expression 62 becomes

$$ \alpha(V_{Y_n}, W_{Y_m}) = \beta(V_{Y_n}, Y)/\beta(W_{Y_m}, Y). \qquad (63) $$

In terms of partition functions $\beta$ may be expressed as

$$ \begin{aligned} \beta(V_{Y_n}, Y) &= \left[ \frac{Q(V_{Y'_n})}{Q(V_{Y_n})} \right]^{1/n} \left[ \frac{m'}{m} \right]^{3/2} X(V_{Y_n}) \\ &= f^{1/n}(V_{Y'_n}, V_{Y_n}) X(V_{Y_n}), \end{aligned} $$

where $f$ is the reduced partition function ratio (Urey 1947, Bigeleisen & Mayer 1947). The need for this terminology arises from the fact that stable isotope geochemists measure atomic isotopic ratios instead of isotopic molecular abundances. It should be appreciated that often the excess factors are very nearly unity.

In Tables 9-14, we list our calculated $\beta$ factors for hydrogen, carbon, nitrogen, oxygen sulphur and chlorine isotope exchange. Geochemical applications of these data are the subject of a forthcoming publication. The extrapolation of these results to low or high temperatures is beset with problems. Frequently the $\beta$ factors and the fractionation factors that one may calculate from them do not linearly decrease

<table><caption>**Table 10** $\beta$ factors for carbon exchange</caption><thead><tr><th>T°C</th><th>CO<sub>2</sub></th><th>OCS</th><th>HCN</th><th>CS<sub>2</sub></th><th>CH<sub>4</sub></th><th>CO</th><th>CS</th></tr></thead><tbody><tr><td>0</td><td>1.2171</td><td>1.1695</td><td>1.1431</td><td>1.1302</td><td>1.1262</td><td>1.1095</td><td>1.0692</td></tr><tr><td>10</td><td>1.2060</td><td>1.1608</td><td>1.1363</td><td>1.1232</td><td>1.1204</td><td>1.1045</td><td>1.0657</td></tr><tr><td>20</td><td>1.1958</td><td>1.1527</td><td>1.1301</td><td>1.1168</td><td>1.1150</td><td>1.0999</td><td>1.0624</td></tr><tr><td>30</td><td>1.1864</td><td>1.1453</td><td>1.1244</td><td>1.1109</td><td>1.1100</td><td>1.0957</td><td>1.0593</td></tr><tr><td>40</td><td>1.1777</td><td>1.1384</td><td>1.1190</td><td>1.1055</td><td>1.1053</td><td>1.0917</td><td>1.0565</td></tr><tr><td>50</td><td>1.1697</td><td>1.1321</td><td>1.1140</td><td>1.1005</td><td>1.1009</td><td>1.0880</td><td>1.0538</td></tr><tr><td>75</td><td>1.1519</td><td>1.1180</td><td>1.1030</td><td>1.0893</td><td>1.0912</td><td>1.0796</td><td>1.0480</td></tr><tr><td>100</td><td>1.1370</td><td>1.1062</td><td>1.0936</td><td>1.0800</td><td>1.0829</td><td>1.0725</td><td>1.0430</td></tr><tr><td>125</td><td>1.1243</td><td>1.0962</td><td>1.0854</td><td>1.0720</td><td>1.0758</td><td>1.0663</td><td>1.0387</td></tr><tr><td>150</td><td>1.1133</td><td>1.0875</td><td>1.0784</td><td>1.0651</td><td>1.0696</td><td>1.0609</td><td>1.0350</td></tr><tr><td>175</td><td>1.1038</td><td>1.0800</td><td>1.0722</td><td>1.0592</td><td>1.0641</td><td>1.0561</td><td>1.0318</td></tr><tr><td>200</td><td>1.0954</td><td>1.0734</td><td>1.0667</td><td>1.0540</td><td>1.0593</td><td>1.0519</td><td>1.0290</td></tr><tr><td>250</td><td>1.0815</td><td>1.0625</td><td>1.0576</td><td>1.0455</td><td>1.0512</td><td>1.0447</td><td>1.0244</td></tr><tr><td>300</td><td>1.0705</td><td>1.0538</td><td>1.0502</td><td>1.0388</td><td>1.0446</td><td>1.0389</td><td>1.0207</td></tr><tr><td>350</td><td>1.0615</td><td>1.0468</td><td>1.0441</td><td>1.0334</td><td>1.0393</td><td>1.0341</td><td>1.0178</td></tr><tr><td>400</td><td>1.0541</td><td>1.0410</td><td>1.0391</td><td>1.0290</td><td>1.0348</td><td>1.0302</td><td>1.0155</td></tr><tr><td>450</td><td>1.0480</td><td>1.0363</td><td>1.0349</td><td>1.0254</td><td>1.0311</td><td>1.0268</td><td>1.0135</td></tr><tr><td>500</td><td>1.0428</td><td>1.0323</td><td>1.0313</td><td>1.0225</td><td>1.0279</td><td>1.0240</td><td>1.0119</td></tr><tr><td>600</td><td>1.0346</td><td>1.0260</td><td>1.0255</td><td>1.0178</td><td>1.0228</td><td>1.0195</td><td>1.0094</td></tr><tr><td>700</td><td>1.0285</td><td>1.0213</td><td>1.0212</td><td>1.0145</td><td>1.0190</td><td>1.0161</td><td>1.0076</td></tr><tr><td>800</td><td>1.0238</td><td>1.0178</td><td>1.0179</td><td>1.0119</td><td>1.0160</td><td>1.0135</td><td>1.0063</td></tr><tr><td>900</td><td>1.0202</td><td>1.0151</td><td>1.0153</td><td>1.0100</td><td>1.0137</td><td>1.0114</td><td>1.0053</td></tr><tr><td>1100</td><td>1.0150</td><td>1.0110</td><td>1.0115</td><td>1.0072</td><td>1.0103</td><td>1.0085</td><td>1.0038</td></tr><tr><td>1300</td><td>1.0115</td><td>1.0086</td><td>1.0089</td><td>1.0054</td><td>1.0079</td><td>1.0066</td><td>1.0029</td></tr></tbody></table>
with $T^{-2}$, where T is the absolute temperature (see Stern et al 1968, Spindel et al 1970; Bottinga 1968a and 1969).

When using the $\beta$ factors in Tables 9-14, the difference between equilibrium constants and $\beta$ factors should be kept in mind. For example, for the simple reaction

$$HD + HF = H_2 + DF, \qquad (64)$$

we have given on line 13 in Table 4 the values of the equilibrium constant $K_{64}$ at 0°C. A geochemist who has in a container a gaseous mixture of hydrogen and hydrogen fluoride measures the ratios $(D/H)_{hydrogen\ fluoride}$ and $(D/H)_{hydrogen}$. Assuming that the container was at 0°C and the gases were in isotopic exchange equilibrium, the fractionation factor $\alpha = (D/H)_{hydrogen\ fluoride}/(D/H)_{hydrogen}$ (Equation 61) should be equal to the ratio of $\beta$ factors for hydrogen fluoride over that of hydrogen (Equation 63), which at 0°C is equal to 3.533, while the equilibrium constant in Table 4 is 1.771 or roughly half of the fractionation factor. This is because we have

$$\alpha = \frac{[DF]/[HD] + 2[D_2]}{[HF]/2[H_2] + [HD]} = \frac{[DF]}{[HF]} \cdot \left\{ \frac{[HD] + 2[D_2]}{[HD]} \right\},$$

and since $D/H \le 1$, we also have $\alpha \approx 2K_{64}$.

<table><caption>**Table 11** $\beta$ factors for nitrogen exchange</caption><thead><tr><th>T°C</th><th>NO<sub>2</sub></th><th>N<sub>2</sub>O</th><th>N<sub>2</sub></th><th>HCN</th><th>NH<sub>3</sub></th><th>NO</th></tr></thead><tbody><tr><td>0</td><td>1.1224</td><td>1.1213</td><td>1.0902</td><td>1.0801</td><td>1.0778</td><td>1.0743</td></tr><tr><td>10</td><td>1.1162</td><td>1.1153</td><td>1.0862</td><td>1.0765</td><td>1.0744</td><td>1.0709</td></tr><tr><td>20</td><td>1.1105</td><td>1.1097</td><td>1.0825</td><td>1.0731</td><td>1.0713</td><td>1.0677</td></tr><tr><td>30</td><td>1.1052</td><td>1.1046</td><td>1.0791</td><td>1.0700</td><td>1.0683</td><td>1.0648</td></tr><tr><td>40</td><td>1.1003</td><td>1.0998</td><td>1.0759</td><td>1.0671</td><td>1.0656</td><td>1.0620</td></tr><tr><td>50</td><td>1.0957</td><td>1.0954</td><td>1.0729</td><td>1.0643</td><td>1.0631</td><td>1.0594</td></tr><tr><td>75</td><td>1.0856</td><td>1.0856</td><td>1.0662</td><td>1.0582</td><td>1.0574</td><td>1.0537</td></tr><tr><td>100</td><td>1.0770</td><td>1.0772</td><td>1.0604</td><td>1.0530</td><td>1.0525</td><td>1.0488</td></tr><tr><td>125</td><td>1.0696</td><td>1.0701</td><td>1.0554</td><td>1.0485</td><td>1.0482</td><td>1.0445</td></tr><tr><td>150</td><td>1.0632</td><td>1.0640</td><td>1.0510</td><td>1.0445</td><td>1.0446</td><td>1.0407</td></tr><tr><td>175</td><td>1.0577</td><td>1.0586</td><td>1.0471</td><td>1.0410</td><td>1.0413</td><td>1.0374</td></tr><tr><td>200</td><td>1.0528</td><td>1.0539</td><td>1.0437</td><td>1.0379</td><td>1.0384</td><td>1.0345</td></tr><tr><td>250</td><td>1.0448</td><td>1.0460</td><td>1.0378</td><td>1.0327</td><td>1.0336</td><td>1.0296</td></tr><tr><td>300</td><td>1.0384</td><td>1.0397</td><td>1.0330</td><td>1.0284</td><td>1.0297</td><td>1.0256</td></tr><tr><td>350</td><td>1.0333</td><td>1.0346</td><td>1.0291</td><td>1.0249</td><td>1.0264</td><td>1.0224</td></tr><tr><td>400</td><td>1.0291</td><td>1.0304</td><td>1.0258</td><td>1.0221</td><td>1.0237</td><td>1.0197</td></tr><tr><td>450</td><td>1.0257</td><td>1.0269</td><td>1.0230</td><td>1.0196</td><td>1.0215</td><td>1.0174</td></tr><tr><td>500</td><td>1.0228</td><td>1.0240</td><td>1.0206</td><td>1.0176</td><td>1.0195</td><td>1.0155</td></tr><tr><td>600</td><td>1.0183</td><td>1.0193</td><td>1.0168</td><td>1.0143</td><td>1.0165</td><td>1.0125</td></tr><tr><td>700</td><td>1.0150</td><td>1.0159</td><td>1.0140</td><td>1.0119</td><td>1.0141</td><td>1.0103</td></tr><tr><td>800</td><td>1.0125</td><td>1.0132</td><td>1.0117</td><td>1.0100</td><td>1.0123</td><td>1.0086</td></tr><tr><td>900</td><td>1.0106</td><td>1.0112</td><td>1.0100</td><td>1.0085</td><td>1.0108</td><td>1.0073</td></tr><tr><td>1100</td><td>1.0079</td><td>1.0083</td><td>1.0075</td><td>1.0064</td><td>1.0088</td><td>1.0054</td></tr><tr><td>1300</td><td>1.0061</td><td>1.0063</td><td>1.0058</td><td>1.0050</td><td>1.0074</td><td>1.0041</td></tr></tbody></table>
<table><caption>**Table 12** β factors for oxygen exchange</caption><thead><tr><th>T°C</th><th>CO<sub>2</sub></th><th>CO</th><th>OCS</th><th>SO<sub>3</sub></th><th>NO</th><th>N<sub>2</sub>O</th><th>SO<sub>2</sub></th><th>NO<sub>2</sub></th><th>O<sub>2</sub></th><th>H<sub>2</sub>O</th></tr></thead><tbody><tr><td>0</td><td>1.1333</td><td>1.1191</td><td>1.1296</td><td>1.1169</td><td>1.1118</td><td>1.1092</td><td>1.1053</td><td>1.1027</td><td>1.0874</td><td>1.0705</td></tr><tr><td>10</td><td>1.1265</td><td>1.1137</td><td>1.1225</td><td>1.1104</td><td>1.1066</td><td>1.1035</td><td>1.0997</td><td>1.0973</td><td>1.0831</td><td>1.0675</td></tr><tr><td>20</td><td>1.1203</td><td>1.1087</td><td>1.1168</td><td>1.1045</td><td>1.1017</td><td>1.0982</td><td>1.0945</td><td>1.0923</td><td>1.0791</td><td>1.0648</td></tr><tr><td>30</td><td>1.1145</td><td>1.1040</td><td>1.1111</td><td>1.0990</td><td>1.0972</td><td>1.0933</td><td>1.0896</td><td>1.0878</td><td>1.0754</td><td>1.0622</td></tr><tr><td>40</td><td>1.1091</td><td>1.0997</td><td>1.1059</td><td>1.0940</td><td>1.0930</td><td>1.0888</td><td>1.0852</td><td>1.0835</td><td>1.0719</td><td>1.0598</td></tr><tr><td>50</td><td>1.1041</td><td>1.0956</td><td>1.1010</td><td>1.0893</td><td>1.0891</td><td>1.0846</td><td>1.0810</td><td>1.0796</td><td>1.0687</td><td>1.0576</td></tr><tr><td>75</td><td>1.0931</td><td>1.0865</td><td>1.0902</td><td>1.0790</td><td>1.0803</td><td>1.0753</td><td>1.0719</td><td>1.0709</td><td>1.0614</td><td>1.0526</td></tr><tr><td>100</td><td>1.0837</td><td>1.0788</td><td>1.0811</td><td>1.0704</td><td>1.0728</td><td>1.0674</td><td>1.0642</td><td>1.0635</td><td>1.0553</td><td>1.0483</td></tr><tr><td>125</td><td>1.0757</td><td>1.0720</td><td>1.0734</td><td>1.0631</td><td>1.0663</td><td>1.0608</td><td>1.0577</td><td>1.0572</td><td>1.0500</td><td>1.0445</td></tr><tr><td>150</td><td>1.0688</td><td>1.0661</td><td>1.0667</td><td>1.0569</td><td>1.0607</td><td>1.0550</td><td>1.0521</td><td>1.0518</td><td>1.0453</td><td>1.0412</td></tr><tr><td>175</td><td>1.0628</td><td>1.0609</td><td>1.0609</td><td>1.0515</td><td>1.0557</td><td>1.0500</td><td>1.0472</td><td>1.0471</td><td>1.0413</td><td>1.0383</td></tr><tr><td>200</td><td>1.0575</td><td>1.0563</td><td>1.0558</td><td>1.0469</td><td>1.0513</td><td>1.0456</td><td>1.0430</td><td>1.0430</td><td>1.0378</td><td>1.0357</td></tr><tr><td>250</td><td>1.0488</td><td>1.0485</td><td>1.0474</td><td>1.0393</td><td>1.0439</td><td>1.0385</td><td>1.0361</td><td>1.0362</td><td>1.0318</td><td>1.0313</td></tr><tr><td>300</td><td>1.0419</td><td>1.0422</td><td>1.0408</td><td>1.0334</td><td>1.0380</td><td>1.0328</td><td>1.0307</td><td>1.0309</td><td>1.0271</td><td>1.0277</td></tr><tr><td>350</td><td>1.0363</td><td>1.0370</td><td>1.0354</td><td>1.0287</td><td>1.0331</td><td>1.0283</td><td>1.0264</td><td>1.0266</td><td>1.0234</td><td>1.0247</td></tr><tr><td>400</td><td>1.0318</td><td>1.0327</td><td>1.0310</td><td>1.0250</td><td>1.0291</td><td>1.0246</td><td>1.0230</td><td>1.0232</td><td>1.0203</td><td>1.0222</td></tr><tr><td>450</td><td>1.0280</td><td>1.0291</td><td>1.0273</td><td>1.0219</td><td>1.0258</td><td>1.0216</td><td>1.0202</td><td>1.0203</td><td>1.0177</td><td>1.0201</td></tr><tr><td>500</td><td>1.0249</td><td>1.0260</td><td>1.0243</td><td>1.0193</td><td>1.0230</td><td>1.0191</td><td>1.0178</td><td>1.0180</td><td>1.0156</td><td>1.0182</td></tr><tr><td>600</td><td>1.0200</td><td>1.0211</td><td>1.0195</td><td>1.0155</td><td>1.0185</td><td>1.0153</td><td>1.0143</td><td>1.0143</td><td>1.0123</td><td>1.0152</td></tr><tr><td>700</td><td>1.0163</td><td>1.0174</td><td>1.0160</td><td>1.0126</td><td>1.0152</td><td>1.0124</td><td>1.0117</td><td>1.0116</td><td>1.0098</td><td>1.0129</td></tr><tr><td>800</td><td>1.0136</td><td>1.0146</td><td>1.0133</td><td>1.0106</td><td>1.0127</td><td>1.0103</td><td>1.0097</td><td>1.0096</td><td>1.0080</td><td>1.0110</td></tr><tr><td>900</td><td>1.0115</td><td>1.0124</td><td>1.0112</td><td>1.0090</td><td>1.0107</td><td>1.0087</td><td>1.0083</td><td>1.0081</td><td>1.0066</td><td>1.0095</td></tr><tr><td>1100</td><td>1.0085</td><td>1.0092</td><td>1.0083</td><td>1.0067</td><td>1.0079</td><td>1.0064</td><td>1.0062</td><td>1.0059</td><td>1.0047</td><td>1.0073</td></tr><tr><td>1300</td><td>1.0065</td><td>1.0071</td><td>1.0063</td><td>1.0053</td><td>1.0061</td><td>1.0049</td><td>1.0049</td><td>1.0045</td><td>1.0034</td><td>1.0058</td></tr></tbody></table>

In Table 15, we have tabulated certain similarities and differences in the isotopic fractionation behavior of various molecules. Oxygen β values for CO₂, COS, and CO are very much alike, as are sulphur β values for CS₂, COS, and CS. Such similarities are due to the fact that by means of a valence force field one can adequately explain the observed vibrational frequencies for these molecules and that the valence force-field constant for a certain bond is virtually the same for each molecule that possesses this bond. For instance, the force constant for the vibrational frequency of CO is nearly the same as that for the stretching vibrational mode for CO₂. Oxygen in CO, COS, and CO₂ and sulphur in CS, COS, and CS₂ are in equivalent positions with the result that the isotopic fractionation for sulphur or oxygen among these molecules is very small. In contrast with this one notices a strong fractionation for carbon isotopes between CO and CO₂ and for nitrogen isotopes between NO and NO₂. This can be understood as being due to the fact that the position of N in NO and NO₂ is different, as is the position of C in CO and CO₂.

Table 15 also shows clearly that oxygen β factors for O₂, H₂O, CO₂, COS, and CO are larger than sulphur β factors for the corresponding series of molecules S₂, H₂S, CS₂, COS, and CS. This illustrates very well that stable isotope fractionation
diminishes with a diminishing relative mass difference between the stable isotopes in equivalent positions.

When the results are compared with previously made calculations, then one may notice a certain amount of agreement as well as disagreement. The differences are due mainly to the following causes: 1. More recent and presumably more accurate spectroscopic data were used. 2. The spectroscopic data available at the moment are more complete than before. 3. Only approximations were applied, which were shown to be harmless. 4. The use of the Teller-Redlich product rule was avoided. 5. Excess factors were calculated. 6. No use was made of the hydrogen isotope data by Haar et al (1961) as was done for example by Bottinga (1969). 7. Anharmonicity of methane was taken care of, but in a different way from Bottinga (1969).

In Table 16 a survey of comparisons with previously calculated results is given.

## 6 COMPARISON WITH EXPERIMENTAL DETERMINATIONS

In the internal equilibrium $H_2 + D_2 = 2HD$ the $G_0$ energy contribution should cancel. As remarked before, the constants available for hydrogen and its deuterated variations do not result in a zero $G_0$ term in the zero-point energy. The calculated

<table><caption>**Table 13** β factors for sulphur exchange</caption><thead><tr><th>T°C</th><th>SO<sub>3</sub></th><th>SO<sub>2</sub></th><th>OCS</th><th>CS<sub>2</sub></th><th>CS</th><th>S<sub>2</sub></th><th>H<sub>2</sub>S</th></tr></thead><tbody><tr><td>0</td><td>1.0928</td><td>1.0507</td><td>1.0211</td><td>1.0222</td><td>1.0192</td><td>1.0147</td><td>1.0125</td></tr><tr><td>10</td><td>1.0879</td><td>1.0480</td><td>1.0199</td><td>1.0209</td><td>1.0182</td><td>1.0138</td><td>1.0119</td></tr><tr><td>20</td><td>1.0834</td><td>1.0456</td><td>1.0188</td><td>1.0198</td><td>1.0173</td><td>1.0130</td><td>1.0114</td></tr><tr><td>30</td><td>1.0793</td><td>1.0434</td><td>1.0178</td><td>1.0188</td><td>1.0165</td><td>1.0122</td><td>1.0110</td></tr><tr><td>40</td><td>1.0754</td><td>1.0413</td><td>1.0169</td><td>1.0178</td><td>1.0157</td><td>1.0116</td><td>1.0105</td></tr><tr><td>50</td><td>1.0718</td><td>1.0393</td><td>1.0160</td><td>1.0169</td><td>1.0150</td><td>1.0109</td><td>1.0101</td></tr><tr><td>75</td><td>1.0640</td><td>1.0350</td><td>1.0141</td><td>1.0149</td><td>1.0134</td><td>1.0096</td><td>1.0092</td></tr><tr><td>100</td><td>1.0573</td><td>1.0314</td><td>1.0125</td><td>1.0133</td><td>1.0121</td><td>1.0085</td><td>1.0084</td></tr><tr><td>125</td><td>1.0516</td><td>1.0283</td><td>1.0112</td><td>1.0119</td><td>1.0109</td><td>1.0075</td><td>1.0077</td></tr><tr><td>150</td><td>1.0467</td><td>1.0256</td><td>1.0100</td><td>1.0107</td><td>1.0099</td><td>1.0067</td><td>1.0071</td></tr><tr><td>175</td><td>1.0425</td><td>1.0233</td><td>1.0091</td><td>1.0097</td><td>1.0090</td><td>1.0060</td><td>1.0066</td></tr><tr><td>200</td><td>1.0388</td><td>1.0212</td><td>1.0082</td><td>1.0089</td><td>1.0082</td><td>1.0054</td><td>1.0061</td></tr><tr><td>250</td><td>1.0327</td><td>1.0179</td><td>1.0069</td><td>1.0074</td><td>1.0069</td><td>1.0045</td><td>1.0053</td></tr><tr><td>300</td><td>1.0279</td><td>1.0152</td><td>1.0058</td><td>1.0063</td><td>1.0059</td><td>1.0038</td><td>1.0047</td></tr><tr><td>350</td><td>1.0241</td><td>1.0131</td><td>1.0050</td><td>1.0055</td><td>1.0051</td><td>1.0032</td><td>1.0041</td></tr><tr><td>400</td><td>1.0210</td><td>1.0114</td><td>1.0043</td><td>1.0048</td><td>1.0044</td><td>1.0028</td><td>1.0037</td></tr><tr><td>450</td><td>1.0185</td><td>1.0100</td><td>1.0037</td><td>1.0042</td><td>1.0038</td><td>1.0024</td><td>1.0033</td></tr><tr><td>500</td><td>1.0164</td><td>1.0088</td><td>1.0033</td><td>1.0037</td><td>1.0034</td><td>1.0021</td><td>1.0030</td></tr><tr><td>600</td><td>1.0132</td><td>1.0070</td><td>1.0026</td><td>1.0030</td><td>1.0027</td><td>1.0017</td><td>1.0024</td></tr><tr><td>700</td><td>1.0108</td><td>1.0057</td><td>1.0021</td><td>1.0024</td><td>1.0022</td><td>1.0013</td><td>1.0020</td></tr><tr><td>800</td><td>1.0091</td><td>1.0047</td><td>1.0018</td><td>1.0020</td><td>1.0018</td><td>1.0011</td><td>1.0017</td></tr><tr><td>900</td><td>1.0077</td><td>1.0039</td><td>1.0015</td><td>1.0017</td><td>1.0015</td><td>1.0009</td><td>1.0015</td></tr><tr><td>1100</td><td>1.0059</td><td>1.0029</td><td>1.0011</td><td>1.0013</td><td>1.0011</td><td>1.0007</td><td>1.0011</td></tr><tr><td>1300</td><td>1.0047</td><td>1.0022</td><td>1.0008</td><td>1.0010</td><td>1.0008</td><td>1.0005</td><td>1.0009</td></tr></tbody></table>
<table><caption>**Table 14** β factors for chlorine exchange</caption><thead><tr><th>T°C</th><th>0</th><th>25</th><th>50</th><th>100</th><th>200</th><th>400</th><th>700</th><th>1000</th><th>1300</th></tr></thead><tbody><tr><th>Cl<sub>2</sub></th><td>1.0086</td><td>1.0073</td><td>1.0063</td><td>1.0048</td><td>1.0031</td><td>1.0016</td><td>1.0008</td><td>1.0005</td><td>1.0003</td></tr><tr><th>HCl</th><td>1.0055</td><td>1.0050</td><td>1.0045</td><td>1.0038</td><td>1.0028</td><td>1.0017</td><td>1.0010</td><td>1.0006</td><td>1.0004</td></tr></tbody></table>

<table><caption>**Table 15** Similarities in β factors for similar molecules at 0°C</caption><thead><tr><th>β(Hydrogen)</th><th>β(Carbon)</th><th>β(Nitrogen)</th><th>β(Oxygen)</th><th>β(Sulphur)</th></tr></thead><tbody><tr><td>HF</td><td>13.5</td><td></td><td></td><td></td></tr><tr><td>HCl</td><td>6.4</td><td></td><td></td><td></td></tr><tr><td>H<sub>2</sub>O</td><td>15.1</td><td></td><td>O<sub>2</sub></td><td>1.09</td><td>S<sub>2</sub></td><td>1.01</td></tr><tr><td>H<sub>2</sub>S</td><td>6.7</td><td></td><td>H<sub>2</sub>O</td><td>1.07</td><td>H<sub>2</sub>S</td><td>1.01</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>H<sub>2</sub>S</td><td>1.01</td></tr><tr><td></td><td>CO<sub>2</sub></td><td>1.22</td><td>NO<sub>2</sub></td><td>1.12</td><td>CO<sub>2</sub></td><td>1.13</td></tr><tr><td></td><td>COS</td><td>1.17</td><td></td><td></td><td>COS</td><td>1.12</td><td>COS</td><td>1.02</td></tr><tr><td></td><td>CS<sub>2</sub></td><td>1.13</td><td></td><td></td><td></td><td></td><td>CS<sub>2</sub></td><td>1.02</td></tr><tr><td></td><td>CO</td><td>1.11</td><td>NO</td><td>1.07</td><td>CO</td><td>1.12</td><td>CS</td><td>1.02</td></tr></tbody></table>

<table><caption>**Table 16** Comparison of calculated β factors at 25°C.</caption><caption>Authors</caption><tbody><tr><th colspan="8">Hydrogen exchange</th></tr><tr><td>Urey (1947)</td><td>H<sub>2</sub>O</td><td>HCl</td><td>H<sub>2</sub></td><td></td><td></td><td></td><td></td></tr><tr><td>This work</td><td>12.518</td><td>5.306</td><td>3.735</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>11.605</td><td>5.288</td><td>3.367</td><td></td><td></td><td></td><td></td></tr><tr><th colspan="8">Carbon exchange</th></tr><tr><td>Urey (1947)</td><td>CO<sub>2</sub></td><td>HCN</td><td>CO</td><td></td><td></td><td></td><td></td></tr><tr><td>Bottinga (1968a)</td><td>1.191</td><td>1.121</td><td>1.097</td><td></td><td></td><td></td><td></td></tr><tr><td>This work</td><td>1.192</td><td>—</td><td>—</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>1.191</td><td>1.127</td><td>1.098</td><td></td><td></td><td></td><td></td></tr><tr><th colspan="8">Nitrogen exchange</th></tr><tr><td>Urey (1947)</td><td>N<sub>2</sub></td><td>NH<sub>3</sub></td><td>NO</td><td>HCN</td><td></td><td></td><td></td></tr><tr><td>This work</td><td>1.081</td><td>1.069</td><td>1.066</td><td>1.066</td><td></td><td></td><td></td></tr><tr><td></td><td>1.082</td><td>1.070</td><td>1.066</td><td>1.072</td><td></td><td></td><td></td></tr><tr><th colspan="8">Oxygen exchange</th></tr><tr><td>Urey (1947)</td><td>CO<sub>2</sub></td><td>CO</td><td>SO<sub>2</sub></td><td>O<sub>2</sub></td><td>H<sub>2</sub>O</td><td></td><td></td></tr><tr><td>Bottinga (1968a)</td><td>1.117</td><td>1.105</td><td>1.089</td><td>1.082</td><td>1.067</td><td></td><td></td></tr><tr><td>Saxena et al (1962)</td><td>1.117</td><td>—</td><td>1.088</td><td></td><td>1.064</td><td></td><td></td></tr><tr><td>This work</td><td>1.117</td><td>1.106</td><td>1.092</td><td>1.077</td><td>1.069</td><td></td><td></td></tr><tr><td></td><td>1.117</td><td>1.106</td><td>1.092</td><td>1.077</td><td>1.064</td><td></td><td></td></tr><tr><th colspan="8">Sulphur exchange</th></tr><tr><td>Tudge & Thode (1950)</td><td>SO<sub>3</sub></td><td>SO<sub>2</sub></td><td>OCS</td><td>CS<sub>2</sub></td><td>S<sub>2</sub></td><td>H<sub>2</sub>S</td><td></td></tr><tr><td>Sakai (1957)</td><td>1.084</td><td>1.045</td><td>1.019</td><td>1.019</td><td>1.013</td><td>1.013</td><td></td></tr><tr><td>This work</td><td>1.081</td><td>1.044</td><td>1.019</td><td>1.019</td><td>1.013</td><td>1.011</td><td></td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>Figure 3 Equilibrium constant for hydrogen internal equilibrium. Observed values: ●, Rittenberg et al (1934); ■, Gould et al (1934); ▲, Niki et al (1965). Calculated curve: —, this review. Equilibrium constant for water vapor internal equilibrium is also shown. Observed values: ○, Pyper et al (1967a); □, Friedman & Shiner (1966). Calculated curves (this review): —, harmonic; —, anharmonic without $G_0$; ---, anharmonic with $G_0$.</figcaption></figure>

results are plotted on Figure 3 together with the experimental observations, and the agreement is good. Undoubtedly $H_2$ is spectroscopically one of the best known and most often looked at molecules. The molecular constants for $H_2$ derived in the past have shown a certain amount of divergence, which was well in excess of $1 \text{ cm}^{-1}$. At the moment the two most recent determinations are in almost perfect agreement (Foltz et al 1966, Buijs & Gush 1971); nevertheless, the difference in $G_0$ for these two sets amounts to $0.3 \text{ cm}^{-1}$ while the difference in zero-point energy without $G_0$ contribution amounts to only $0.03 \text{ cm}^{-1}$. This illustrates very well how sensitive the computation of $G_0$ is to the input data. This sensitivity stems from the fact that $G_0$ is a small difference of large numbers (see Equation 17). In Table 17 we give certain values for $G_0$ and $\omega_e x_e/4$.

<table><caption>Table 17 Values for $G_0$ and $\omega_e x_e/4$ in $\text{cm}^{-1}$</caption><thead><tr><th>Molecule</th><th>ω<sub>e</sub>x<sub>e</sub>/4</th><th>G<sub>0</sub></th></tr></thead><tbody><tr><td>H<sub>2</sub></td><td>30.32</td><td>8.60</td></tr><tr><td>HD</td><td>22.73</td><td>6.88</td></tr><tr><td>D<sub>2</sub></td><td>15.17</td><td>4.30</td></tr><tr><td>HF</td><td>22.47</td><td>4.13</td></tr><tr><td>DF</td><td>11.44</td><td>1.88</td></tr><tr><td>H<sup>35</sup>Cl</td><td>13.20</td><td>1.60</td></tr><tr><td>D<sup>35</sup>Cl</td><td>6.79</td><td>0.82</td></tr><tr><td><sup>14</sup>N<sub>2</sub></td><td>3.53</td><td>0.23</td></tr><tr><td><sup>15</sup>N<sub>2</sub></td><td>3.30</td><td>0.21</td></tr><tr><td><sup>16</sup>O<sub>2</sub></td><td>3.02</td><td>0.21</td></tr><tr><td><sup>18</sup>O<sub>2</sub></td><td>2.68</td><td>0.19</td></tr><tr><td>C<sup>16</sup>O</td><td>3.32</td><td>0.19</td></tr><tr><td>C<sup>18</sup>O</td><td>3.16</td><td>0.18</td></tr></tbody></table>
In Figure 3 we have also plotted our calculated results for the internal equilibrium

$$H_2O + D_2O \rightleftharpoons 2HDO. \qquad (65)$$

It is evident that the anharmonic calculation, without taking into consideration $G_0$, does not agree with the experimental observation. For this reason Hulston (1969) and Wolfsberg (1969a) proposed that in the calculation of isotope effects the term $G_0$, neglected until then, should be included. For the bent triatomic molecule Wolfsberg (1969a) gave the equation

$$G_0 = G_{0av} + G_{0rv} \qquad (66)$$

where

$$G_{0av} = \frac{1}{16} \left\{ \sum_{i} (6K_{iiii} - 7K_{iiij}^2/\omega_i) - 4\omega_1\omega_2\omega_3 K_{123}^2 \left[ (\omega_1 + \omega_2 + \omega_3) \times (\omega_1 - \omega_2 - \omega_3)(\omega_1 - \omega_2 + \omega_3)(\omega_1 + \omega_2 - \omega_3) \right] \sum_{i \neq j} 3K_{ij}^2 \omega_i \omega_j (4\omega_i^2 - \omega_j^2) \right\} \qquad (67)$$

and

$$G_{0rv} = -(A_e + B_e + 3C_e)/4. \qquad (68)$$

$G_{0av}$ depends on the zero-order frequencies and the force constants while $G_{0rv}$ depends on the rotational constants. Evaluation of Equation 68 does not pose any problem since the rotational constants are usually well known. Unfortunately it is the $G_{0av}$ contribution that is important and difficult to evaluate in most cases because of lack of input information. As shown in Figure 3 the calculation in which $G_0$ has been used is in much better agreement than the calculation in which it has been ignored.

For the similar internal equilibrium $H_2S + D_2S = 2HDS$ we calculate an equilibrium constant of 3.88 (25°C). This coincides with the observed value by Pyper & Newbury (1970). Wolfsberg et al (1970), who included $G_0$ in their computation, have calculated a value of 3.93, which is outside the limits of the experimental accuracy. Because of the lack of detailed information in the paper by Wolfsberg et al (1970), it is difficult to discuss their calculated value, which would be of interest since they calculated an equilibrium constant for the hydrogen-sulfide internal equilibrium, excluding the $G_0$ effect, of 3.69; this differs considerably from our analogous value of 3.88.

The uncertainty we noted already in the calculated $G_0$ values for $H_2$ is appreciably larger for the triatomic molecules we have listed in Table 18. The maximum difference in $\Delta G_0$ for water (Table 18) amounts to about $4 \text{ cm}^{-1}$, which means an uncertainty of 2% in the calculated equilibrium constant for reaction 65 at 25°C. Table 18 suggests that the uncertainty in the $G_0$ value of the individual molecules is important. $\Delta G_0$ for water is appreciable because of the “anomalous” value for $G_0$(HDO).
<table><caption>**Table 18** G₀ values for triatomic molecules (cm⁻¹)</caption><thead><tr><th></th><th>H₂O</th><th>HDO</th><th>D₂O</th><th>ΔG₀<sup>h</sup></th><th>H₂S</th><th>D₂S</th><th><sup>14</sup>NO₂</th><th><sup>15</sup>NO₂</th><th><sup>32</sup>S<sup>16</sup>O₂</th></tr></thead><tbody><tr><td>G₀<sub>trv</sub></td><td>-17.64</td><td>-12.87</td><td>-9.33</td><td>+1.23</td><td>-8.39</td><td>-4.33</td><td>-2.42</td><td>-2.32</td><td>-0.81</td></tr><tr><td rowspan="7">G₀<sub>av</sub></td><td>a</td><td>27.33</td><td>7.08</td><td>14.48</td><td>-27.65</td><td>16.93</td><td></td><td>7.66</td><td>2.37</td></tr><tr><td>b</td><td>21.46</td><td>3.15</td><td>10.99</td><td>-26.15</td><td>15.22</td><td></td><td>7.18</td><td>1.46</td></tr><tr><td>c</td><td>22.02</td><td></td><td>7.05</td><td></td><td>14.85</td><td></td><td></td><td>1.69</td></tr><tr><td>d</td><td>19.75</td><td></td><td>11.89</td><td></td><td></td><td></td><td></td><td>1.22</td></tr><tr><td>e</td><td>22.39</td><td></td><td>11.39</td><td></td><td></td><td></td><td></td><td>1.29</td></tr><tr><td>f</td><td>21.43</td><td>1.44</td><td>11.78</td><td>-30.33</td><td>11.92</td><td>6.05</td><td>5.05</td><td></td></tr><tr><td>g</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1.16</td></tr></tbody></table>

a, b, c Papousek & Pliva (1964), with force fields a, SUBFF, b, MUBFF, c, EXP.

d, e Kutchisu & Morino (1965), with force fields d, OBS and e, CALC.

f Foss Smith & Overend (1972).

g Saito (1969).

$$h \Delta G_0 = 2G_0(\text{HDO}) - G_0(\text{H}_2\text{O}) - G_0(\text{D}_2\text{O})$$
<table><caption>**Table 19** Ammonia internal equilibria</caption><thead><tr><th rowspan="2">Equilibrium constant</th><th rowspan="2">Temperature °C</th><th colspan="2">Experimental value</th><th colspan="2">Calculated values</th></tr><tr><th>Pyper et al (1967<sup>b</sup>)</th><th>Bron & Wolfsberg (1972) with G<sub>0</sub></th><th>Bron & Wolfsberg (1972) without G<sub>0</sub></th><th>This work without G<sub>0</sub></th></tr></thead><tbody><tr><td>K(69)</td><td>0</td><td>2.90 ± 0.06</td><td>2.89</td><td>3.48</td><td>2.70</td></tr><tr><td>K(69)</td><td>25</td><td>2.92 ± 0.08</td><td>2.91</td><td>3.45</td><td>2.73</td></tr><tr><td>K(70)</td><td>0</td><td>2.86 ± 0.09</td><td>2.90</td><td>3.01</td><td>2.85</td></tr><tr><td>K(70)</td><td>25</td><td>2.89 ± 0.09</td><td>2.92</td><td>3.02</td><td>2.87</td></tr></tbody></table>

The internal equilibria

$$NH_3 + NHD_2 = 2NH_2D \qquad (69)$$

and

$$ND_3 + NH_2D = 2NHD_2 \qquad (70)$$

have been measured by Pyper et al (1967<sup>b</sup>).

Observed and calculated results for these equilibria are given in Table 19. For K<sub>69</sub> the results by Bron & Wolfsberg (1972) with G<sub>0</sub> are in good agreement with the observations; the agreement with our calculations is less satisfactory. For K<sub>70</sub> the agreement with the experimentally determined values of the Bron & Wolfsberg results with G<sub>0</sub> and our results without G<sub>0</sub> is to within the experimental error given by Pyper et al (1967<sup>b</sup>). There is a large disagreement between Bron & Wolfsberg's calculations without G<sub>0</sub> and ours. This is due to the fact that these calculations are done quite differently.

In Figure 4 we have plotted our calculated results for the equilibrium HD + NH<sub>3</sub> = H<sub>2</sub> + NH<sub>2</sub>D and the results computed by Bron & Wolfsberg (1972) together with various experimental measurements. The figure shows well the spread in values

<table><caption>**Table 20** Deuterium fractionation among molecular hydrogen and hydrogen sulfide</caption><thead><tr><th>Temperature °C</th><th>K(experimental)</th><th>K(calculated)</th></tr></thead><tbody><tr><td>350</td><td>0.855</td><td>0.864</td></tr><tr><td>400</td><td>0.889</td><td>0.894</td></tr><tr><td>450</td><td>0.919</td><td>0.919</td></tr><tr><td>500</td><td>0.943</td><td>0.939</td></tr><tr><td>550</td><td>0.963</td><td>0.955</td></tr><tr><td>600</td><td>0.973</td><td>0.968</td></tr></tbody></table>

<footnote><sup>a</sup> Grafe et al (1939). $$K = \frac{[D_2]}{[H_2]} \frac{[D_2S]}{[H_2S]}$$</footnote>
resulting from the various force fields used by Bron & Wolfsberg. Our calculations without $G_0$ agree with the observations, as is the case for Bron & Wolfsberg (1972) with force field II.

The experimental measurements of the equilibrium $H_2O + HD = HDO + H_2$ are plotted in Figure 5 together with calculated results. Calculations with and without $G_0$ are in equally good agreement with the observations. The perfect agreement

<figure><img src="image_7.png" /></figure>

**Figure 4** Equilibrium constant for $NH_3 + HD = NH_2D + H_2$. Observed values: ◊, Herrick & Sabi (1943); Δ, Perlman et al (1953); ○, Ravoire et al (1963). Calculated values, Bron & Wolfsberg (1972): ■, anharmonic without $G_0$; ●, anharmonic with $G_0$ (force field II); ◊, anharmonic with $G_0$ (force field III). This review: —, harmonic; —, anharmonic without $G_0$.
<figure><img src="image_8.png" /><figcaption>Figure 5 Equilibrium constant for H₂O + HD = HDO + H₂. Observed values: ○, Suess (1949); ●, Cerrai et al (1954); Δ, Rolston et al (1975). Calculated values (this review): ——, harmonic; —, anharmonic without G₀; —, anharmonic with G₀.</figcaption></figure>

obtained with the calculations of Bottinga (1968b) is probably accidental; he used the now superseded free enthalpy tables of Haar et al (1961) for H₂ and HD in his calculations.

The equilibrium D₂S + H₂ = H₂S + D₂ has been measured indirectly by Grafe et al (1939). The measured values are compared with calculated values in Table 20, and the agreement is satisfactory. In Figure 6, we show experimental data for the equilibrium H₂O + HDS = HDO + H₂S obtained by Marx (1960) and our calculations. Again the difference between observations and calculations is negligible.

The equilibrium

$$ ^{32}\text{SO}_2 + H_2^{34}\text{S} = ^{34}\text{SO}_2 + H_2^{32}\text{S} \qquad (71) $$

has been studied experimentally (Grinenko & Thode 1970, Thode et al 1971) and theoretically (Sakai 1968, Thode et al 1971). Previously obtained results and our
<figure><img src="image_9.png" /><figcaption>Figure 6 Equilibrium constant for H₂O + HDS = HDO + H₂S. Observed values: ●, Marx (1960). Calculated values: —, anharmonic without G₀, this review.</figcaption></figure>

results are plotted in Figure 7. The experimental results show too much dispersion to allow a meaningful comparison with the calculations. Calculations by Thode et al and us are in good agreement. The small difference between these two calculations is due to the slightly different force fields used to compute the frequencies of the isotopically substituted molecules.

To compare our calculated results with equilibria involving liquid water instead of water vapor, we have made use of the most recent measurement of the fractionation factor α(steam-liquid water) = 1.00937 (Majoube 1971) at 25°C. In Table 21 we list the experimental values for the oxygen isotope exchange between CO₂ and liquid water and between NO and liquid water at 25°C. As is evident the calculations reproduce the observed results quite well. The difference between our result and the calculated α(CO₂–H₂O) = 1.0411 reported by Bottinga & Craig (1969) is mainly due to the use of the α(steam-liquid water) at 25°C of Majoube (1971), which is 0.2‰, larger than the corresponding α used by Bottinga & Craig.

# 7 DISCUSSION AND CONCLUSIONS

At present the main error source in the calculation of fractionation factors is the uncertainty in the vibrational molecular constants. Errors as small as 1 cm⁻¹ in the frequency difference between a vibrational mode of a normal and of an isotopically substituted molecule already have consequences in the evaluation of the β factor that are considerably larger than the accuracy limits with which one can
<figure><img src="image_10.png" /><figcaption>**Figure 7** Equilibrium constant for $ ^{32}SO_2 + H_2^{34}S = ^{34}SO_2 + H_2^{32}S $. Observed values: ●, Grinenko & Thode (1970); ○, Thode et al (1971). Calculated values: ···, Sakai (1968), - -, Thode et al (1971), ____, this review.</figcaption></figure>
<table><caption>**Table 21** Oxygen isotope exchange between carbon dioxide, nitric oxide, and liquid water at 25°C</caption><thead><tr><th colspan="2">α(CO<sub>2</sub>·H<sub>2</sub>O)</th><th></th></tr><tr><th>Observed</th><th>Calculated</th><th>Reference</th></tr></thead><tbody><tr><td>1.0407</td><td></td><td>Compston & Epstein (1958)</td></tr><tr><td>1.0417</td><td></td><td>Majoube (1966)</td></tr><tr><td>1.0407</td><td></td><td>O'Neil & Epstein (1966)</td></tr><tr><td>1.0409</td><td></td><td>Bottinga & Craig (1969)</td></tr><tr><td>1.0414</td><td></td><td>Horibe et al (1973)</td></tr><tr><td>1.0417</td><td></td><td>Blattner (1973)</td></tr><tr><td>1.0412</td><td></td><td>O'Neil et al (1975)</td></tr><tr><td></td><td>1.0408</td><td>This work</td></tr></tbody><tbody><tr><th colspan="2">α(NO—H<sub>2</sub>O)</th><th></th></tr><tr><td>1.025±0.003</td><td></td><td>Jordan & Bonner (1971)</td></tr><tr><td>1.024±0.003</td><td></td><td>Jordan & Bonner (1971)</td></tr><tr><td>1.027±0.004</td><td></td><td>Jordan & Bonner (1971)</td></tr><tr><td></td><td>1.024</td><td>This work</td></tr></tbody></table>

measure isotopic compositions. This is a very important restriction because spectral data of many molecules are only incompletely known, because of the complexity of the molecule or the fact that the spectrum is difficult to observe: such is the case for HDO and HDS.

The existence of a $G_0$ term in the zero-point energy is well established. However, its importance in isotopic calculations is less certain (Richet & Bottinga 1976a). The only exchange reaction for which we could not get agreement between the observations and the calculations was the internal equilibrium for water. Our imprecise knowledge of the HDO vibrational constants is here of importance and one also would like to see confirmed the experimental observations, which are quite difficult to perform. One notices (Figure 3) that even when $G_0$ is included in the calculation, the theoretical internal equilibrium constant falls outside the error limits of the experimentally determined one. In the case of the internal equilibrium of water it was observed by Wolfsberg (1969a) and Hulston (1969) that the contribution of the anharmonicity correction to the zero-point energy in the calculation of the equilibrium constant was comparable to the $G_0$ contribution. The data given by Wolfsberg et al (1970) suggest that this may be a general feature for internal equilibria involving hydrogen-deuterium exchange; see for instance on Figure 3 how relatively well the harmonic calculation of the internal equilibrium for water agrees with the measured equilibrium constant. Our calculated harmonic calculations for the internal equilibria of ammonia (Equations 69 and 70) are in agreement with the observations to within the experimental error limits. But for normal exchanges we find that our anharmonic calculation without $G_0$ is closer to the
observed values than the harmonically calculated value. Similarly, we find that for
hydrogen-deuterium fractionation between molecular hydrogen and water (Figure 4),
the anharmonic calculations are superior to the harmonic ones. Therefore it seems
advisable that when one cannot calculate G₀ for an internal equilibrium, one does
the calculation in the harmonic approximation, i.e. one ignores in that case the
terms on lines 2, 4, and 7 of Table 4 and does the same for polyatomic molecules.
In the evaluation of excess factors, internal equilibrium constants are needed; for
methane they were calculated in the anharmonic as well as in the harmonic
approximation for this reason. But for normal isotopic exchange reactions an-
harmonicity should be taken into account.

The computation of G₀ demands a very precise spectral knowledge, because it is
a small difference of large numbers. Often the required information is not available;
hence for reasons of homogeneity we have ignored the G₀ effect. Comparison of
our results with observations does not suggest that this neglect of G₀ has observable
consequences.

For this review we have calculated β factors with the most recent data available
and as accurately as is feasible at present. Comparison with previous calculations
shows good agreement as well as divergencies. One of the reasons for this lack of
agreement is that previous authors have sometimes applied approximations that
are clearly not justified. It should be evident that the calculation amounts to the
evaluation of a very small difference of two large energy terms. This is the free
enthalpy difference between a molecule and its isotopically substituted equivalent.
Such differences are far too small to be directly measurable with present day
techniques of calorimetry. Because this energy difference is so very small one should
be very careful when one applies approximations.

In the calculation of β factors there are no free parameters that one can change
in order to obtain agreement with observations. However, there is still a choice of
possible input data, i.e. the spectroscopic information. It was for this reason that
we used the water data by Khachkuruzov (1959), rather than the different sets
published by other authors [however, in this case, we have obtained with data by
Benedict et al (1956) β factors for hydrogen exchange which are very similar to
those listed in Table 9; see Richet & Bottinga (1976b)]. But once we made a choice,
which most often was only based on the criteria given in Section V, we used the
complete set of data for a given molecule, as given by the author. Notwithstanding
these restrictions we have found that our calculated results are in good agreement
with the observations, except for the internal equilibrium of water (Equation 65).
But for other hydrogen-deuterium exchange reactions involving water no such
problems were encountered.

To interpolate the values given in Tables 9–14, it is probably best to assume
that the β factors vary linearly with $T^{-2}$. However, this linearity should not be
relied upon to extrapolate the tabulated values. In many cases we noted that our
results do not follow a $T^{-2}$ law.

In general the value of the excess factors is very close to unity, except for the
exchange of hydrogen-deuterium; in particular when molecular hydrogen, water,
hydrogen sulphide, ammonia or methane are involved. This is caused by significant
deviations from the classical value of the concerned internal equilibrium constants.

As far as we are aware, our results form the most extensive and homogeneous set of stable isotope fraction factors that is available for gaseous substances. These data can be of use in many different fields. In volcanic gas research they may be applied in order to understand the genesis as well as the isotopic composition of the source region of these gases. In geothermal exploration the gas phase plays an important role; an understanding of its isotopic composition, without knowing the necessary fractionation factors, is not possible. In biochemistry the isotopic composition of gases produced by living organisms, such as CO₂ and CH₄ by bacteria, may be of interest. The listed fractionation factors serve to determine under which physical conditions these gases were formed. In cosmochemistry, one has measured the isotopic composition of certain interstellar gas clouds, stars, planets, and meteorites. Some of the observed variations may be explicable in terms of isotopic exchange reactions; in such case our listed fractionation factors should be of use.

### Literature Cited

Allen, H. C., Plyler, E. K. 1956. *J. Chem. Phys.* **25**: 1132–36

Amano, T., Hirota, E. 1974. *J. Mol. Spectrosc.* **53**: 346–63

Anderson, T., Lassettre, E. N., Yost, D. M. 1936. *J. Chem. Phys.* **4**: 703–06

Babcock, H. D., Herzberg, L. 1948. *Ap. J.* **108**: 167–90

Barbe, A., Jouve, P. 1971. *J. Mol. Spectrosc.* **38**: 273–80

Bardo, R. D., Wolfsberg, M. 1975. *J. Chem. Phys.* **62**: 4555–58

Barrow, R. F., Dixon, R. N., Lagerquist, A., Wright, C. V. 1960. *Ark. Fys.* **18**: 543–62

Benedict, W. S., Gailar, N., Plyler, E. K. 1956. *J. Chem. Phys.* **24**: 1139–65

Benedict, W. S., Plyler, E. K. 1957. *Can. J. Phys.* **35**: 1235–41

Benesh, W. B., Van der Slice, J. T., Tilford, S. G., Wilkinson, P. G. 1965. *Ap. J.* **142**: 1227–40

Bigeleisen, J., Mayer, M. G. 1947. *J. Chem. Phys.* **15**: 261–67

Blanck, C. E., Hause, C. D. 1970. *J. Mol. Spectrosc.* **34**: 478–86

Blattner, P. 1973. *Geochim. Cosmochim. Acta* **37**: 2691–93

Born, M. 1951. *Nachr. Akad. Wiss. Göttingen Math. Phys. Kl.* p. 1

Born, M., Huang, K. 1954. *Dynamical Theory of Crystal Lattices*. London: Oxford Univ. Press. 430 pp.

Born, M., Oppenheimer, R. 1927. *Ann. Phys.* **84**: 457–84

Bottinga, Y. 1968a. *J. Phys. Chem.* **72**: 800–08

Bottinga, Y. 1968b. *J. Phys. Chem.* **72**: 4338–40

Bottinga, Y. 1969. *Geochim. Cosmochim. Acta* **33**: 49–64

Bottinga, Y., Craig, H. 1969. *Earth Planet. Sci. Lett.* **5**: 285–95

Bottinga, Y., Javoy, M. 1975. *Rev. Geophys. Space Phys.* **13**: 401–18

Bron, J., Wolfsberg, M. 1972. *J. Chem. Phys.* **57**: 2862–69

Buijs, H. L., Gush, H. D. 1971. *Can. J. Phys.* **49**: 2366–75

Burneau, A. 1974. *Spectrochim. Acta* **30A**: 1861–71

Burrus, C. A., Gordy, W. 1953. *Phys. Rev.* **92**: 274–77

Cerrai, E., Marchetti, C., Renzoni, R., Rosco, L., Silvestri, M., Villari, S. 1954. *Chem. Eng. Prog. Symp. Ser.* **50**: 271–80

Clyne, M. A. A., Coxon, J. A. 1970. *J. Mol. Spectrosc.* **33**: 381–406

Compston, W., Epstein, S. 1958. *Trans. Am. Geophys. Union* **39**: 511

Cook, R. L., De Lucia, F. C., Helminger, P. 1973. *J. Chem. Phys.* **56**: 4581–84

Cook, R. L., De Lucia, F. C., Helminger, P. 1974. *J. Mol. Spectrosc.* **53**: 62–76

Courtoy, C. P. 1957. *Can. J. Phys.* **35**: 608–48

Courtoy, C. P. 1959. *Ann. Soc. Sci. Bruxelles* **73**: 5–230

Craig, H. 1953. *Geochim. Cosmochim. Acta* **3**: 53–92

De Hemptinne, M., Van Riet, R., Defossez, A., Bruyninckx, F., Dachelet, P. 1963. *Ann. Soc. Sci. Bruxelles* **77**: 163–71

De Lucia, F. C., Helminger, P., Gordy, W. 1971. *Phys. Rev. A* **3**: 1849–57

Denbigh, K. 1971. *The Principles of Chemical Equilibrium*. London: Cambridge Univ. Press. 494 pp. 3rd ed.
Dorney, A. J., Hoy, A. R., Mills, I. M. 1973. *J. Mol. Spectrosc.* 45: 253-60

Dunham, J. L. 1932. *Phys. Rev.* 41: 721-31

Durie, R. A., Herzberg, G. 1960. *Can. J. Phys.* 38: 806-18

Epstein, S., Mayeda, T. 1953. *Geochim. Cosmochim. Acta* 31: 181-24

Farkas, A., Farkas, L. 1934. *Proc. R. Soc. London Ser. A* 144: 467-80

Foltz, J. V., Rank, D. H., Wiggins, T. A. 1966. *J. Mol. Spectrosc.* 21: 203-16

Foord, A., Smith, J. G., Whiffen, D. H. 1975. *Mol. Phys.* 29: 1685-704

Foss Smith, D. Jr., Overend, J. 1971. *J. Chem. Phys.* 54: 3632-39

Foss Smith, D. Jr., Overend, J. 1972. *Spectrochim. Acta Part A.* 28: 471-83

Friedman, I. 1953. *Geochim. Cosmochim. Acta* 4: 89-103

Friedman, L., Shiner, V. J. Jr. 1966. *J. Chem. Phys.* 44: 4339-40

Gamo, I. 1967. *J. Mol. Spectrosc.* 23: 472-75

Giguère, J., Wang, V. K., Overend, J., Cabana, A. 1973. *Spectrochim. Acta Part A.* 29: 1197-1206

Gould, A. J., Bleakney, W., Taylor, H. S. 1934. *J. Chem. Phys.* 2: 362-73

Grafe, D., Clusius, K., Kruis, A. 1939. *Z. Phys. Chem. Abt.* B. 43: 1-19

Griggs, J. L. Jr., Rao, K. N., Jones, L. H., Potter, R. M. 1968. *J. Mol. Spectrosc.* 25: 34-61

Grinenko, V. A., Thode, H. G. 1970. *Can. J. Earth Sci.* 7: 1402-09

Haar, L., Bradley, J. C., Friedman, A. S. 1955. *J. Res. Nat. Bur. Stand.* 55: 285-90

Haar, L., Friedman, A. S., Beckett, C. W. 1961. *Nat. Bur. Stand.* U.S. Monogr. 20

Helminger, P., Cook, R. L., De Lucia, F. C. 1971. *J. Mol. Spectrosc.* 40: 125-36

Helminger, P., Cook, R. L., De Lucia, F. C. 1972. *J. Mol. Spectrosc.* 41: 123-36

Herrick, C. E., Sabi, N. 1943. *SAM Rep. A* 765

Herzberg, G. 1945. *Molecular Spectra and Molecular Structure*. Vol. II, Infrared and Raman Spectra. New York: Van Nostrand, 632 pp.

Herzberg, G. 1950. *Molecular Spectra and Molecular Structure*. Vol. I, Spectra of Diatomic Molecules. New York: Van Nostrand, 658 pp. 2nd ed.

Horibe, Y., Shigehara, K., Takakuwa, Y. 1973. *J. Geophys. Res.* 78: 2625-29

Huffaker, J. N. 1976a. *J. Chem. Phys.* 64: 3175-81

Huffaker, J. N. 1976b. *J. Chem. Phys.* 64: 4564-70

Hulston, J. R. 1969. *J. Chem. Phys.* 50: 1483-84

Jobard, I., Chedin, A. 1975. *J. Mol. Spectrosc.* 57: 464-79

Johns, J. W. C., McKellar, A. R. W., Weitz, D. 1974. *J. Mol. Spectrosc.* 51: 539-45

Jones, L. H. 1970. *J. Mol. Spectrosc.* 34: 108-12

Jordan, S., Bonner, F. T. 1971. *J. Chem. Phys.* 54: 4963-64

Kaldor, A., Maki, A. G., Dorney, A. J., Mills, I. M. 1973. *J. Mol. Spectrosc.* 45: 247-52

Khachkuruzov, G. A. 1959. *Gos. Inst. Prikl. Khim.* 42: 51-131

Kleinman, L. I., Wolfsberg, M. 1973. *J. Chem. Phys.* 59: 2043-53

Kleinman, L. I., Wolfsberg, M. 1974a. *J. Chem. Phys.* 60: 4740-48

Kleinman, L. I., Wolfsberg, M. 1974b. *J. Chem. Phys.* 60: 4749-54

Kolos, W. 1970. *Adv. Quantum Chem.* 5: 98-133

Kolos, W., Wolniewicz, L. 1964. *J. Chem. Phys.* 41: 3663-73

Kutchisu, K., Bartell, L. S. 1962. *J. Chem. Phys.* 36: 2478-81

Kutchisu, K., Morino, Y. 1965. *Bull. Chem. Soc. Jpn.* 38: 814-24

Lindemann, F. A. 1919. *Philos. Mag.* 38: 173-81

Lindemann, F. A., Aston, F. W. 1919. *Philos. Mag.* 37: 523-34

McKinney, C. R., McCrea, J. M., Epstein, S., Allen, H. A., Urey, H. C. 1950. *Rev. Sci. Instrum.* 21: 724-30

Maillard, A. 1936. *C.R. Acad. Sci.* 203: 804-06

Majoube, M. 1966. *J. Chim. Phys.* 63: 563-68

Majoube, M. 1971. *J. Chim. Phys.* 68: 1423-36

Marx, D. 1960. *C.E.A. Rep.* No. 1382. 58 pp.

Mayer, J. E. 1958. *Handb. Phys.* 12: 73-204

Mayer, J. E., Mayer, M. G. 1940. *Statistical Mechanics*. New York: Wiley, 495 pp.

Morino, Y., Nagakawa, T. 1968. *J. Mol. Spectrosc.* 26: 496-523

Morse, P. M. 1929. *Phys. Rev.* 34: 57-64

Mulholland, H. P. 1928. *Proc. Camb. Philos. Soc.* 24: 280-89

Murphey, B. F., Nier, A. O. 1941. *Phys. Rev.* 59: 771-72

Nagakawa, T., Morino, Y. 1969. *Bull. Chem. Soc. Jpn.* 42: 2212-19

Nielsen, H. H. 1951. *Rev. Mod. Phys.* 23: 90-136

Nier, A. O., Gulbransen, E. A. 1939. *J. Am. Chem. Soc.* 61: 697-98

Niki, H., Rousseau, Y., Mains, G. J. 1965. *J. Phys. Chem.* 69: 45-52

Olafson, R. A., Thomas, M. A., Welsh, H. L. 1961. *Can. J. Phys.* 39: 419-36

Olman, M. D., Hause, C. D. 1968. *J. Mol. Spectrosc.* 26: 241-53

Olman, M. D., McNelis, M. D., Hause, C. D.
1964. *J. Mol. Spectrosc.* **14**: 62–78

Olson, W. B. 1972. *J. Mol. Spectrosc.* **43**: 190–98

O'Neil, J. R., Epstein, S. 1966. *J. Geophys. Res.* **71**:4955–61

O'Neil, J. R., Adami, L. H., Epstein, S. 1975. *J. Res. US Geol. Survey* **3**: 623–24

Papousek, D., Pliva, J. 1964. *Collect. Czech. Chem. Commun.* **29**: 1973–97

Perlman, M., Bigeleisen, J., Elliott, N. 1953. *J. Chem. Phys.* **21**: 70–72

Pliva, J. 1968. *J. Mol. Spectrosc.* **27**: 461–88

Pyper, J. W., Newbury, R. S. 1970. *J. Chem. Phys.* **52**: 1966–71

Pyper, J. W., Newbury, R. S., Barton, G. W. Jr. 1967a. *J. Chem. Phys.* **46**: 2253–57

Pyper, J. W., Newbury, R. S., Barton, G. W. Jr. 1967b. *J. Chem. Phys.* **47**: 1179–82

Rank, D. H., Rao, B. S., Wiggins, T. A. 1965. *J. Mol. Spectrosc.* **17**: 122–30

Ravoire, J., Grandcollot, P., Dirian, G. 1963. *C.E.A. Rep. No.* 2244, 32-pp.

Richet, P., Bottinga, Y. 1976a. *C.R. Acad. Sci. Ser. B* **282**: 425–28

Richet, P., Bottinga, Y. 1976b. *C.R. Acad. Sci. Ser. D* **283**: 299–302

Rittenberg, D., Bleakney, W., Urey, H. C. 1934. *J. Chem. Phys.* **2**: 48–49

Roh, W. B., Rao, K. N. 1974. *J. Mol. Spectrosc.* **49**: 317–21

Rolston, J. H., Den Hartog, J., Butter, J. B. 1975. *A.E.C.L. Rep.* **5025**, 25 pp.

Rosenthal, J. E. 1935. *Phys. Rev.* **47**: 235–37

Saito, S. 1969. *J. Mol. Spectrosc.* **30**: 1–16

Sakai, H. 1957. *Geochim. Cosmochim. Acta* **12**: 150–69

Sakai, H. 1968. *Geochim. Cosmochim. Acta* **2**: 29–49

Saxena, S. C., Bhatnagar, D. N., Ramaswamy, S. 1962. *J. Chem. Eng. Data* **7**: 240–42

Shaffer, W. H., Schuman, R. P. 1944. *J. Chem. Phys.* **12**: 504–13

Shelton, R. D., Nielsen, H. H., Fletcher, W. H. 1953. *J. Chem. Phys.* **21**: 2178–83 and 1954. *J. Chem. Phys.* **22**: 1791 (correction)

Silverman, S. R. 1951. *Geochim. Cosmochim. Acta* **2**: 26–42

Spanbauer, R. N., Rao, K. N. 1965. *J. Mol. Spectrosc.* **16**: 100–02

Spindel, W., Stern, M. J., Monse, E. U. 1970. *J. Chem. Phys.* **52**: 2022–35

Steenbeckeliers, G. 1968. *Ann. Soc. Sci. Bruxelles* **82**: 331–404

Stern, M. J., Spindel, W., Monse, E. U. 1968. *J. Chem. Phys.* **48**: 2908–19

Stripp, K. F., Kirkwood, J. G. 1951. *J. Chem. Phys.* **19**: 1131–33

Suess, H. 1949. *Z. Naturforsch. Teil A* **4**: 328–32

Taylor, H. P. Jr. 1968. *Contrib. Mineral. Petrol.* **19**: 1–71

Thode, H. G., Cragg, C. B., Hulston, J. R., Rees, C. E. 1971. *Geochim. Cosmochim. Acta* **35**: 35–45

Thomas, M. A., Welsh, H. L. 1960. *Can. J. Phys.* **38**: 1291–1303

Tudge, A. P., Thode, H. G. 1950. *Can. J. Res. Sect. B* **28**: 567–78

Urey, H. C. 1947. *J. Chem. Soc.* pp. 562–81

Urey, H. C., Greiff, L. J. 1935. *J. Am. Chem. Soc.* **57**: 321–27

Urey, H. C., Rittenberg, D. 1933. *J. Chem. Phys.* **1**: 137–43

Van Riet, R. 1964. *Ann. Soc. Sci. Bruxelles* **78**: 97–104

Van Vleck, J. H. 1936. *J. Chem. Phys.* **4**: 327–38

Viney, I. E. 1933. *Proc. Camb. Philos. Soc.* **29**: 142–48

Vojta, G. 1960. *Kernenergie* **3**: 917–27

Vojta, G. 1961. *Ann. Phys.* **7**: 397–402

Waldmann, L. 1943. *Naturwissenschaften* **31**: 205–06

Webb, D. U., Rao, K. N. 1968. *J. Mol. Spectrosc.* **28**: 121–24

Wolfsberg, M. 1969a. *J. Chem. Phys.* **50**: 1484–86

Wolfsberg, M. 1969b. *Adv. Chem.* **89**: 185–91

Wolfsberg, M., Massa, A. A., Pyper, J. W. 1970. *J. Chem. Phys.* **53**: 3138–46