Gas-phase entropies

Term (2) calculated by the high-accuracy multistep CBS-APNO method (Section 5.5.2.2b) was 341.2 kcal mol-1 or 1426 kJ mol-1. The Sackur-Tetrode equation for the gas-phase entropy of the proton was mentioned in this regard, but in fact the algorithm automatically handles this. 

Rate constants and equilibrium constants should be checked for thermodynamic consistency if at all possible. For example, the heat of adsorption derived from the temperature dependence of should be negative since adsorption reactions are almost always exothermic. Likewise, the entropy change A5ads for nondissociative adsorption must be negative since every gas phase molecule loses translational entropy upon adsorption. In fact, AS < S (where Sg is the gas phase entropy) must also be satisfied because a molecule caimot lose more entropy than it originally possessed in the gas phase. A proposed kinetic sequence that produces adsorption rate constants and/or equilibrium constants that do not satisfy these basic principles should be either discarded or considered very suspiciously. 

The initial entropy of cumene adsorption on a commercial aluminosilicate was measured to be about —750JmoP K , whereas the gas-phase entropy of cumene at 298 K is 389 J moP K (709). This seemingly inconsistent result appears to be caused by the dissociative adsorption of cumene at low coverages on this catalyst. In this case, the measured heat corresponds to a combination of heats of adsorption and reaction. Higher coverages produced lower, nearly constant heats and entropies of adsorption. These entropies correspond to the loss of between two and three degrees of translational freedom. The adsorption of benzene on these samples did not show abnormally... 

Estimation of the entropy of solvation requires calculation of the entropy of the ion in the gas phase. For a monoatomic ion, the main contribution to the entropy comes from its translational energy. Simple ions formed from the main group elements have the electronic structure of an inert gas and therefore do not have an electronic contribution to the entropy. On the other hand, ions formed from transition metals may have an electronic contribution to the gas phase entropy, which depends on the electronic configuration of the ion s ground state and of any other electronic states which are close in energy to the ground state. The translational entropy is given by the Sackur-Tetrode equation, which is obtained from the solution of the SWE for a particle in a box (see section 2.2)... 

Pa (1 bar). See the reference for information on the dependence of gas-phase entropy on the choice of standard state pressure. 

The terms in equations 5-8 are defined as AH(gas) = gas phase enthalpy, AE(SCF) = electronic energy, AZPE = zero-point energy correction, AG(gas) = gas phase Gibbs free energy, AS(gas) = gas phase entropy, AG(sol) = solution phase free energy, AG(solv) = free energy of solvation, AG (sol) = solvated free energy of reduction, and E° = standard reduction potential. 

By evaluating the second partial derivatives of V at a local minimum, one can use the procedure of Section 15.13 to calculate molecular vibrational frequencies. Using the calculated structure and vibrational frequencies, one can estimate the gas-phase entropy S298 (Section 15.14). 

Entropies. Gas-phase entropies can be calculated from molecular properties as discussed in Section 15.14. Because semiempirical methods such as AMI and PM3 give inaccurate results for low-frequency torsional vibrations, which contribute significantly to the entropy at room temperature, they do not give accmate entropies for compounds with internal rotation. Ab initio methods can be used to calculate entropies rather reliably, as discussed in Section 15.14. 

Consider a pure ideal-gas phase. Entropy is an extensive property, so if we divide this phase into two subsystems with an internal partition, the total entropy remains unchanged. The reverse process, the removal of the partition, must also have zero entropy change. Despite the fact that the latter process allows the molecules in the two subsystems to intermingle without a change in T or />, it cannot be considered mixing because the entropy does not increase. The essential point is that the same substance is present in both of the subsystems, so there is no macroscopic change of state when the partition is removed. 

This approximation in the case of solvation entropies in water states that they can be approximated in qualitative discussions as 50% of the gas phase entropy, with the opposite sign. 

When a molecule adsorbs on the surface, it will lose a major part of its gas-phase entropy, as it loses the translational freedom from the gas phase. The translational and rotational degrees of freedom typically become constrained and turn into vibrational modes (at least at low temperatures—at higher temperatures, they might become frustrated translational or frustrated rotational). The total contribution of a vibrational mode with frequency, i/., to the standard Gibbs free energy is (except for the zero-point energy (ZPE) contributions that are discussed in Chapter 2 and 4)... 

The last piece of missing information comprises the entropy changes between initial and transition states, which are needed to calculate the pre-exponential factors. The entropy values of adsorbed species and transition states depend only weakly on the transition metal surface. Therefore, we may assume the entropy to be independent of the catalyst. For a rough approximation, we can further assume that an adsorbed species or transition state has lost all degrees of freedom (it is completely locked in place on the surface), and the entropy is therefore zero. The gas-phase entropies of N2 and H2 at standard state are A°(N2) = 192.77 J mol K and iS°(H2) 130.68 Jmol K , respectively, and we can now estimate the entropy changes between the initial states and the... 

---