Molar Free-Energy Changes

2 MOLAR FREE-ENERGY CHANGES FOR CONSERVED AND NONCONSERVED ORDER PARAMETERS 

4The Helmholtz free energy, used here as an example of a molar free energy, is the appropriate minimizing functional for a system at fixed volume in equilibrium with a reservoir at fixed temperature. For different types of system constraints, F would be replaced with another appropriate molar free energy. 

Consider a general system at composition XB. If any of its subsystems (i.e., material portions) could transform to a new composition X B without affecting the rest of the system, the change in the system s total free energy T would be simply 

In a closed system, where the numbers of A and B atoms are conserved, a change in any subsystem must affect the rest of the system—atoms must be exchanged internally to accomplish the transformation. For each mole transformed, the change in F for the X B moles of the B component is FB(X B) — FB(XB)]X B, with a similar term for the (1 — X B) moles of the A component, 

Consider a small fluctuation of composition (or any other order parameter that is conserved for a system), SXb = XB — XB. Expanding F Xb) in SXB yields 

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The standard molar free energy change upon adsorption of the probe gas is thus given by... 

We know that in a closed system dng = -dnh If the temperature and pressure are changed by very small amounts and the system is allowed to come to equilibrium, the molar free energy changes of the two phases must be equal ... 

The molar free-energy change associated with the isothermal vapor-pressure change from Pq to P is given by... 

In thermodynamic considerations of A-B binary solution, if the standard state of B is clanged from the Raoultian standard state to the Henrian standard state, the standard molar free energy changes accordingly. 

The second term on the left-hand side represents the molar free energy change of formation of the solid from its elements hence, the left is equivalent to the free energy of formation of the solid in the saturated solution we therefore write, in conformity with Section 3.9,... 

In Eq. (96) A/y is the molar free energy change if cosphere solvent is transferred from the solvate state to the bulk solvent, and Vy is the molar volume of the pure solvent in the same units as the overlap volume Vmu The overlap volume is a function of the distance r, the ionic radii a/, and the cosphere radius w, which is usually calculated... 

Recognizing that the lumped standard-state chemical potential terms represent the standard-state molar free-energy change for the reaction, AG°, the equation can be simplified to a final form ... 

Now, to convert this molar free-energy change into a volumetric free-energy change, we need to divide by the molar volume of liquid water ... 

It is only under standard conditions (1 atm and 25°C) that A is sufficient to determine whether a reaction is spontaneous or not. We will now derive a relation for the molar free energy change AG under any conditions in terms of AG and the equilibrium constant for the chemical reaction. 

This AF° was estimated as follows For the reaction in question, aH may be taken as — 11.3 kcal/mole = —11300 cal/mole (Table V, data of Durell et al., 1962). The total entropy change was estimated as — 2.5 — 16.5 = — 19.0 entropy units. Therefore, AF° = AH - rAS = —11300—300 (—19) =-11300-1-5700=-5600 cal/mole. This molar free-energy change and others designated here by the superscript apply to a standard state with all reactants and products at a concentration of IM, with other conditions as defined above. If the standard state is taken as one in which the hy-dronium ion is at a concentration of 10 , as at pH 7.0, then the superscript will be used. 

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