Vibrational-rotational partition functions

All terms have been defined previously except for Z r, which here stands for the vibration-rotation partition function for active degrees of freedom of the molecule. 

Calculated Vibrational-Rotational Partition Functions and Free-Energy Contributions of HC1 as a Function of Temperature1 ... 

Vibration-rotation partition function for HC1 obtained via standard Rayleigh-Ritz variational (var) basis-set methods from Topper et al. [46]. 

Vibration-rotation partition function for HC1 obtained via Fourier path-integral AOSS-U Monte Carlo calculations from Topper et al. [46]. Error bars are given at 95% confidence level (2w ). Unless otherwise noted, all calculations used = 128 Fourier coefficients per degree of freedom and n = 100000 Monte Carlo samples. 

But molecular gases also have rotation and vibration. We only make the correction for indistinguishability once. Thus, we do not divide by IV l to write the relationship between Zro[, the rotational partition function of N molecules, and rrol, the rotational partition function for an individual molecule, if we have already assigned the /N term to the translation. The same is true for the relationship between Zv,h and In general, we write for the total partition function Z for N units... 

MMl represents the mass and moment-of-inertia term that arises from the translational and rotational partition functions EXG, which may be approximated to unity at low temperatures, arises from excitation of vibrations, and finally ZPE is the vibrational zero-point-energy term. The relation between these terms and the isotopic enthalpy and entropy differences may be written... 

The translational contribution to the molecular partition function, which is calculated using Eq. 8.59, clearly makes the largest contribution. (In obtaining this value, we also made use of the ideal gas law to calculate the volume V = 0.02479 m3 of a mole of gas at this temperature and pressure.) The rotational partition function is evaluated via Eq. 8.67, and the vibrational partition function for each mode is found via Eq. 8.71. Only the very... 

For diatomic molecules, corrections can be made for the assumption used in the derivation of the rotational partition function that the rotational energy levels are so closely spaced that they can be considered to be continuous. The equations to be used in making these corrections are given in Appendix 6. Also given are the equations to use in correcting for vibrational anharmonicity and nonrigid rotator effects. These corrections are usually small.22... 

The vaporization of a pure liquid or the reverse process, the condensation of the liquid, provides an interesting test of this delayed equilibration hypothesis. Thus as a hydrogen-bonded molecule, which vibrates in the liquid, separates from the surface it frees itself from the potential energy restrictions which prevented rotation. However, in so far as the evaporating molecule has insufficient collisions with neighbors to equilibrate to the free rotational partition function, fgy of the gas, it will retain substantially the partition function, fb of the condensed phase even in the activated complex. Consequently for the condensation process the usual... 

The rotational partition function for an N2 molecule bound to a tungsten surface is more in doubt, but it would probably have very little temperature dependence if the molecule could turn or vibrate on the surface only with difficulty. 

Rotation-vibration interactions, if present, make the calculation more difficult, because then the vibration and rotation partition functions are coupled and cannot be separate factors. 

If we use the approximation of factoring these internal partition functions into electronic, vibrational, and rotational partition functions, Q(interual) = QeiecQvibQrot and wc Call write (Sec. IX) ... 

The rotational partition function is dependent upon the equilibrium geometry. Assuming separation of rotational and vibrational motions. 

Same as 1(b) when the /th-state vibrational partition function, Q [, and the /th-state rotational partition function, Q[y2Ltt equal to the respective ground-state partition functions. In this case the partition function... 

Here, q(Ps) denotes the rotational, vibrational, etc., partition function of a single s molecule at a specific conformation. Ps, -f (Ps) dPs is the mole fraction of s molecules at conformations between Ps and Ps + dPs. The final expression on the rhs of equation (7.167) is a double average quantity one, over all configurations of the N molecules (excluding the solvaton) and the second (the outer average), over all the conformations of the s molecule with distribution function )P(PS). 

Reactant vibrational and rotational partition functions are the same for both reactions (i.e., those for H2) and the ratio of reactant partition functions reduces to the ratio of translational partition functions, which depends only on the reduced masses for the relative motion of the reactants... 

---