Condensed-phase energy equation

With reference to Figs. 3.9 and 3.10, the energy and the species equations are condensed-phase energy equation ... 

The condensed phase energy equation, being decoupled from the species... 

The value of this standard molar Gibbs energy, p°(T), found in data compilations, is obtained by integration from 0 K of the heat capacity determined by the translational, rotational, vibrational and electronic energy levels of the gas. These are determined experimentally by spectroscopic methods [14], However, contrary to what we shall see for condensed phases, the effect of pressure often exceeds the effect of temperature. Hence for gases most attention is given to the equations of state. 

Equation A1.3 shows that isotope effects calculated from standard state free energy differences, and this includes theoretical calculations of isotope effects from the partition functions, are not directly proportional to the measured (or predicted) isotope effects on the logarithm of the isotopic pressure ratios. Rather they must be corrected by the isotopic ratio of activity coefficients. At elevated pressures the correction term can be significant, and in the critical region it may even predominate. Similar considerations apply in the condensed phase except the fugacity ratios which define Kf are replaced by activity ratios, a = Y X and a = y C , for the mole fraction or molar concentration scales respectively. In either case corrections for nonideality, II (Yi)Vi, arising from isotope effects on the activity coefficients can be considerable. Further details are found in standard thermodynamic texts and in Chapter 5. 

Equation 5.19 relates the molecular energy states of the primed and unprimed isotopomers in condensed and vapor phase to VPIE. The correction terms account for the difference between the Gibbs and Helmholtz free energies of the condensed phase, and vapor nonideality. The comparison is between separated isotopomers at a common temperature, each existing at a different equilibrium volume, V or V, and at a different pressure, P or P, although AV = (V — V) and AP = (P — P) are small. Still, because condensed phase Q s are functions of volume, Q = Q(T,V,N), rigorous analysis requires knowledge of the volume dependence of the partition function, and thus MVIE, since the comparisons are made at V and V. That point is developed later. 

In order to describe the energy transfer process in the condensed phase, several additional assumptions are applied to the above equations P-io] (xj j o endothernric or exothermic reaction is involved within the condensed phase (below the burning surface), (2) the lurrtinous flame zone does not contribute to the conductive heat... 

A simple diagram depicting the differences between these two complementary theories is shown in Fig. 1, which represents reactions at zero driving force. Thus, the activation energy corresponds to the intrinsic barrier. Marcus theory assumes a harmonic potential for reactants and products and, in its simplest form, assumes that the reactant and product surfaces have the same curvature (Fig. la). In his derivation of the dissociative ET theory, Saveant assumed that the reactants should be described by a Morse potential and that the products should simply be the dissociative part of this potential (Fig. Ib). Some concerns about the latter condition have been raised. " On the other hand, comparison of experimental data pertaining to alkyl halides and peroxides (Section 3) with equations (7) and (8) seems to indicate that the simple model proposed by Saveant for the nuclear factor of the ET rate constant expression satisfactorily describes concerted dissociative reductions in the condensed phase. A similar treatment was used by Wentworth and coworkers to describe dissociative electron attachment to aromatic and alkyl halides in the gas phase. ... 

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