The Gibbs Free Energy

In Section 13.5, we showed that the change in entropy of a system plus its surroundings (that is, the total change of entropy, AStoJ provides a criterion for deciding whether a process is spontaneous, reversible, or impossible  

FIGURE 13.9 After the constraint between phases A and B is removed, matter can flow spontaneously between phases Inside a system held at constant T (temperature) and P (pressure) by Its surroundings. 

For the special case of processes at constant temperature and pressure, the most important in chemistry, such a state function exists. It is called the Gibbs free energy and is denoted by G. We start the discussion of G with a qualitative examination of spontaneous laboratory processes at fixed T and F. Then we define G and develop its properties. Finally, we apply AG to determine spontaneity in phase transitions and chemical reactions. 

Consider a system enclosed in a piston-cylinder assembly, which constrains pressure at the value F. The assembly is immersed in a heat bath, which constrains temperature at the value T. Experience shows that spontaneous processes under these conditions consist of spontaneous flow of molecules across a boundary completely internal to the system, separating different regions (called phases) of the system (Fig. 13.9). 

Whether molecules flow spontaneously from phase A to B, or vice versa, is determined by the associated change in Gibbs free energy, as we show in the following subsection. 

Note that one can propose using (9.13) as an alternative definition of the chemical potential, that is, 

Both definitions (9.9) and (9.14) are equivalent. Equation (9.13) is the basis for chemical thermodynamics. 

Equation (9.18) applies in general and provides additional significance to the concept of chemical potential. The Gibbs free energy of a system containing k chemical compounds 

This relation, known as the Gibbs-Duhem equation, shows that when the temperature and pressure of a system change there is a corresponding change of the chemical potentials of the various compounds. 

Similarly we consider a system held at constant temperature and constant pressure, where the process of interest involves adding an amount of heat q to the system and also an amount of work 

Note that AH is the heat absorbed by the system in a process carried out at constant T and P. 

Let us investigate free energies a bit further by writing relevant expressions for the differentials AA and AG employing the definitions 4.13 and 4.20. With use of the rules of differential calculus 

4 Isotope Effects on Equilibrium Constants of Chemical Reactions 

Thus the differential of A has been obtained in terms of the changes in volume and temperature. The partial derivatives of A are given by 

---

The relations which permit us to express equilibria utilize the Gibbs free energy, to which we will give the symbol G and which will be called simply free energy for the rest of this chapter. This thermodynamic quantity is expressed as a function of enthalpy and entropy. This is not to be confused with the Helmholtz free energy which we will note sF (L" j (j, > )... 

For spontaneous processes at constant temperature and pressure it is the Gibbs free energy G that decreases, while at equilibrium under such conditions dG = 0. 

Equation ( A2.1.39) is the generalized Gibbs-Diihem equation previously presented (equation (A2.1.27)). Note that the Gibbs free energy is just the sum over the chemical potentials. 

We have seen that equilibrium in an isolated system (dt/= 0, dF= 0) requires that the entropy Sbe a maximum, i.e. tliat dS di )jjy = 0. Examination of the first equation above shows that this can only be true if. p. vanishes. Exactly the same conclusion applies for equilibrium under the other constraints. Thus, for constant teinperamre and pressure, minimization of the Gibbs free energy requires that dGId Qj, =. p. =... 

Since equation (A3.6.4) is equal to the difference between the Gibbs free energy of... 

Within the framework of the same dielectric continuum model for the solvent, the Gibbs free energy of solvation of an ion of radius and charge may be estimated by calculating the electrostatic work done when hypothetically charging a sphere at constant radius from q = 0 q = This yields the Bom equation [13]... 

Kirkwood generalized the Onsager reaction field method to arbitrary charge distributions and, for a spherical cavity, obtained the Gibbs free energy of solvation in tenns of a miiltipole expansion of the electrostatic field generated by the charge distribution [12, 1 3]... 

As with SCRF-PCM only macroscopic electrostatic contribntions to the Gibbs free energy of solvation are taken into account, short-range effects which are limited predominantly to the first solvation shell have to be considered by adding additional tenns. These correct for the neglect of effects caused by solnte-solvent electron correlation inclnding dispersion forces, hydrophobic interactions, dielectric saturation in the case of... 

Finally, exchange is a kinetic process and governed by absolute rate theory. Therefore, study of the rate as a fiinction of temperature can provide thennodynamic data on the transition state, according to equation (B2.4.1)). This equation, in which Ids Boltzmaim s constant and h is Planck s constant, relates tlie observed rate to the Gibbs free energy of activation, AG. ... 

This shows that Eqs. (1) and 2) are basically relationships between the Gibbs free energies of the reactions under consideration, and explains why such relationships have been termed linear free energy relationships (LEER). 

Our discussion so far has considered the calculation of Helmholtz free energies, which a obtained by performing simulations at constant NVT. For proper comparison with expe inental values we usually require the Gibbs free energy, G. Gibbs free energies are obtaini trorn a simulation at constant NPT. 

The chemical potential p, of the adsorbate may be defined, following standard practice, in terms of the Gibbs free energy, the Helmholtz energy, or the internal energy (C/,). Adopting the last of these, we may write... 

A quantity of great importance in chemical thermodynamics is the Gibbs free energy G. The latter is defined in terms of enthalpy H as... 

Figure 4.3a shows schematically how the Gibbs free energy of liquid (subscript 1) and crystalline (subscript c) samples of the same material vary with temperature. For constant temperature-constant pressure processes the criterion for spontaneity is a negative value for AG, where the A signifies the difference final minus initial for the property under consideration. Applying this criterion to Fig. 4.3, we conclude immediately that above T , AGf = Gj - G. is negative... 

As in the qualitative discussion above, let 7 be the Gibbs free energy per unit area of the interface between the crystal and the surrounding hquid. This is undoubtedly different for the edges of the plate than for its faces, but we... 

There are two ways in which the volume occupied by a sample can influence the Gibbs free energy of the system. One of these involves the average distance of separation between the molecules and therefore influences G through the energetics of molecular interactions. The second volume effect on G arises from the contribution of free-volume considerations. In Chap. 2 we described the molecular texture of the liquid state in terms of a model which allowed for vacancies or holes. The number and size of the holes influence G through entropy considerations. Each of these volume effects varies differently with changing temperature and each behaves differently on opposite sides of Tg. We shall call free volume that volume which makes the second type of contribution to G. 

To describe the state of a two-component system at equilibrium, we must specify the number of moles nj and na of each component, as well as—ordinarily- the pressure p and the absolute temperature T. It is the Gibbs free energy that provides the most familiar access to a discussion of equilibrium. The increment in G associated with increments in the independent variables mentioned above is given by the equation... 

Next we consider how to evaluate the factor 6p. We recognize that there is a local variation in the Gibbs free energy associated with a fluctuation in density, and examine how this value of G can be related to the value at equilibrium, Gq. We shall use the subscript 0 to indicate the equilibrium value of free energy and other thermodynamic quantities. For small deviations from the equilibrium value, G can be expanded about Gq in terms of a Taylor series ... 

Physical Equilibria and Solvent Selection. In order for two separate Hquid phases to exist in equiHbrium, there must be a considerable degree of thermodynamically nonideal behavior. If the Gibbs free energy, G, of a mixture of two solutions exceeds the energies of the initial solutions, mixing does not occur and the system remains in two phases. Eor the binary system containing only components A and B, the condition (22) for the formation of two phases is... 

The temperature is expressed ia degrees Celsius. The empirical equation for the Gibbs free energy change was found to be linear with temperature for AG° ia kJ/mol, Tia Kelvin. 

The relationship between the chemical equihbrium constant K and the Gibbs free energy is... 

---