Translational Contribution

Translational Contribution.—Since for an ideal gas the energy of a molecule does not depend on its position in space, the total energy can be expressed as the sum of independent translational and internal energies 

The translational partition function of an ideal gas occupying volume Fis 

The partition function in equation (12) gives the translational contribution for all molecules, and the practical equations are given in Table 1. 

To derive the partition function for an ideal gas as defined in Equation 5.13 we should know the discrete spectrum, at least at a low energy. Imagine the molecules to be moving freely in a cube with the side equal to L. In this three-dimensional box model, the potential is equal to zero inside the cube and infinitely large outside the cube. Since P = 0 outside the cube, all wave functions tend to zero at the edge of the cube. Inside the cube, the Schrddinger equation (SE) is 

The origin is in one comer of the cube and the coordinate axes along the edges. It is easily seen that sin(kx) is a solution of Equation 5.47. This function also satisfies the boundary condition at x = 0. To satisfy the boundary condition at x = L, we must have 

Since we prefer to put zero at the lowest energy level. Equation 5.49 is written in simplified form as 

Since the particles are molecules with M IGOOm, and since L is of macroscopic dimensions, the number a, which determines the distance between the energy levels, will be extremely small compared with that in the case of electrons. If the temperature is not too small, the sum may be replaced by an integral  

If we return to the three-dimensional Equation 5.45, the total energy is E = Ex -I- Ey -I- E. The partition function is a product of the partition functions for X, y, and z. Consequently, the partition function for the motion of the molecule in three dimensions is 

---

Translational Contribution to Entropy We start with equation (10.82)... 

Again, we note that the RT term is not added in changing from (Am - Uo.m) to (Gm - H0.m). This term is added only in the translational contribution. 

Calculation of Thermodynamic Properties We note that the translational contributions to the thermodynamic properties depend on the mass or molecular weight of the molecule, the rotational contributions on the moments of inertia, the vibrational contributions on the fundamental vibrational frequencies, and the electronic contributions on the energies and statistical weight factors for the electronic states. With the aid of this information, as summarized in Tables 10.1 to 10.3 for a number of molecules, and the thermodynamic relationships summarized in Table 10.4, we can calculate a... 

Solution Ar is a monatomic gas with Zm.eica = 1. The translational contribution is the only one we need to consider. For Ar, M = 0.039948 kg-mol-1. We want the standard state entropy when p = 1.000 x 105 Pa. Substituting into the equation in Table 10.4 gives... 

Table A4.1 summarizes the equations needed to calculate the contributions to the thermodynamic functions of an ideal gas arising from the various degrees of freedom, including translation, rotation, and vibration (see Section 10.7). For most monatomic gases, only the translational contribution is used. For molecules, the contributions from rotations and vibrations must be included. If unpaired electrons are present in either the atomic or molecular species, so that degenerate electronic energy levels occur, electronic contributions may also be significant see Example 10.2. In molecules where internal rotation is present, such as those containing a methyl group, the internal rotation contribution replaces a vibrational contribution. The internal rotation contributions to the thermodynamic properties are summarized in Table A4.6.

The first term on the right is the formula for the chemical potential of component a at density pa = na/V in an ideal gas, as would be the case if interactions between molecules were negligible, fee is Boltzmann s constant, and V is the volume of the solution. The other parameters in that ideal contribution are properties of the isolated molecule of type a, and depend on the thermodynamic state only through T. Specifically, V/A is the translational contribution to the partition function of single a molecule at temperature T in a volume V... 

At high T (low X,), reduces to Il,(l/X,). Because the potential energies of molecules dilfering only in isotopic constituents are alike, one can define a separative elfect based on the partition function ratio / of isotopically heavy and light molecules ((T and Q°, respectively) in such a way that the rotational and translational contributions to the partition function cancel out ... 

The ratio of symmetry numbers s s° in equation 11.40 merely represents the relative probabilities of forming symmetrical and unsymmetrical molecules, and ni and nf are the masses of exchanging molecules (the translational contribution to the partition function ratio is at all T equal to the power ratio of the inverse molecular weight). Denoting as AX, the vibrational frequency shift from isotopically heavy to light molecules (i.e., AX, = X° — X ) and assuming AX, to be intrinsically positive, equation 11.40 can be transated into... 

Because the partition function ratio / is defined in such a way that the classical rotational and translational contributions are canceled, equations 11.40, 11.41 and 11.43 must be modified by introducing the ratio of the deviations from classical rotational behavior of heavy and light hydrogen molecules. For small values of [Pg.779]

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... 

The translational contribution to the heat capacity ClMrans is found using Eq. 8.124, and the rotational contribution CUjrot by Eq. 8.127 or 8.128. The vibrational contribution C ,vib... 

Bagchi and co-workers [47-50] have explored the role of translational diffusion in the dynamics of solvation by employing a Smoluchowski-Vlasov equation (see also Calef and Wolyness [37] and Nichols and Calef [42]). A significant contribution to polarization relaxation is observed in certain cases. It is found that the Onsager inverted snowball model is correct only when the rotational diffusion mechanism of solvation dominates the polarization relaxation. The Onsager model significantly breaks down when there is an important translational contribution to the polarization relaxation [47-50]. In fact, translational effects can rapidly accelerate solvation near the probe. In certain cases, the predicted behavior can actually approach the uniform continuum result that rs = t,. 

However, for long times the translational contribution is again larger than the rotational contribution in this simulation. 

For monatomic gases, we have the complete translational contribution at all temperatures. 

Since the translational energy is proportional to the square of the velocity and the energy associated with internal degrees of freedom is approximately independent of the velocity of the molecule (see Section D.2), the translational and internal energies should be transported differently, leading to different constants of proportionality between X and pic for each contribution. Equation (44) should be valid for the translational contribution, but the internal part will be transported more like momentum, whence X = may be approximately true for the internal contribution. Adding the two contributions therefore yields... 

Figure 1.16. Molar entropy (in units of R) of a localized (Langmuir) and a mobile fVolmer) adsorbate. For the former, S is symmetrical in 0, In the latter it is asymmetrical and contains a translational contribution 01n 2KmkTa Jh ). Parameter values for the mobile isotherms a ...

In these equations the quantity N arises because in the summation process the molecules are treated as distinguishable from one another, whereas in fact they are indistingnishable. The translational contribution comes from kinetic theory, whereas the intramolecnlar contributions derive from quanhim mechanics, with the qnantizedenergy levels deteiminedfrom appropriate spectroscopic measurements. 

---