Helmholtz free energy condensed phases

The standard state Helmholtz free energy difference, 8AA°, was introduced in Equations 5.9 and 5.11 to show the connection between VPIE and molecular structure and dynamics. Molecular properties are conveniently expressed using standard state canonical partition functions for the condensed and vapor phases, Qc° and Qv° remember A0 = —RT In Q°. The Q s are 3nN dimensional, n is the number of atoms per molecule and N is Avogadro s number. For convenience we have now dropped the superscript o s on the Q s. The o s specify standard state conditions, now to be implicitly understood. For VPIE and a respectively, not too close to the critical region,... 

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 thermodynamics, we focus on the most important variables needed to describe a system. Although we are interested in the size of a system (or of a phase), we usually do not concern ourselves with the shape of the system. One way in which the shape of a system does influence its thermodynamic properties is through its surface area. The surface of a phase is a different environment than its bulk region. Molecules on the surface of a material do not experience attractive interactions to other molecules in all directions and, therefore, have higher energy than molecules in the bulk of the material. Energy is increased when the surface area of a condensed system (usually a liquid) is increased at constant volume and temperature. Because the Helmholtz free energy has T and V as its natural variables, we can immediately write... 

Islands of uniform radius R of condensed phase are assumed to be distributed in a continuous LE phase. Of course, this assumption is no longer valid for > 0.6, because transitions to stripe configurations and further to islands of LE in LC phases occur in such cases,1121 It is convenient to write the Helmholtz free energy as the sum... 

Neglecting the small difference between the Helmholtz and the Gibbs free energies for a condensed phase, one readily obtains the electrostatic (excess) enthalpy of the polyelectrolyte solution ... 

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