There is another thermodynamic state function widely used in applications to atmospheric science, the Gibbs energy (sometimes called the Gibbs free energy or just the free energy). It proves to be useful for processes which occur at constant pressure and constant temperature. We can use the Gibbs energy to help us in deciding the direction of a chemical reaction and in determining the equilibrium phases or concentrations of chemical species in equilibrium. The Gibbs energy is particularly useful for open systems (those in which mass can enter or leave the system) and for systems in which the internal composition might change due to chemical reactions. We will take up some of these cases later in this chapter.

The Gibbs energy can be defined as

[definition of Gibbs energy] (4.94)

where H is enthalpy, T temperature and S is entropy. The differential of G can then be written:

Substituting for dH:

Along a reversible path we can take dQ = TdS which leads to dG = Vdp — SdT [differential for Gibbs energy]. (4.97)

This last expression (which combines the First and Second Laws) tells us that G is a natural function of T and p, G(T, p). Hence, in a change in which the mass, pressure and temperature are held fixed, the Gibbs energy will not change. This actually happens in a phase transition. For example, consider a chamber with a movable piston held at fixed temperature. Let the chamber contain a liquid with its own vapor in the volume above it. If the piston is withdrawn isothermally and quasi-statically some of the liquid will evaporate into the volume above the liquid surface. The pressure is just the vapor pressure and is constant since it depends only on the (fixed) temperature. We should note that the pressure in the liquid is the same as the pressure in the vapor (we ignore gravity here). Different positions of the piston (leading to different volumes of the vapor) correspond to the same temperature and the same pressure (in both liquid and vapor). Along this locus of points in the state space (say the V-p diagram) for this composite system the Gibbs energy is constant. We will return to the two-phase problem in Chapter 5.

Returning to the general problem we see from (4.97):

As indicated in an earlier chapter, the reactions of trace gases in the atmosphere occur at constant pressure and temperature. In this case the atmosphere which contains orders of magnitude more neutral background molecules (nitrogen, oxygen and argon) than the (usually trace) reactants acts as a massive thermal and pressure buffer holding the temperature and pressure constant. Hence, the reactions among trace gases in the atmosphere occur at fixed pressure (altitude) and temperature (that of the background gas). In these cases where T and p are held constant only the concentration of the species is allowed to change. This is the perfect setup for use of the Gibbs energy.

4.8.1 Gibbs energy for an ideal gas

Begin with the definition of G

Write H = McpT and the expression for entropy:

where k = R/cp as before. Then

To VPo

To Po

The specific Gibbs energy is G normalized by the mass, sometimes denoted g(T, p):

[ideal gases].

Another form that is useful especially in chemistry is the molar Gibbs energy. Instead of specifying the specific Gibbs energy as per unit mass it may be more convenient to express it as a per mole quantity. In the expression for G note that for heating at constant pressure:

where cp is in J kg 1 K 1, M is in kg, v is the number of moles, and cp is in J mol-1 K-1. Also recall that R* is the universal gas constant (8.3145 J mol-1 K-1).

Then

To po

In these expressions, k is the same dimensionless number since R/cp = R*/cp.

A composite system consisting of several distinct subsystems of ideal gases i = 1,..., n leads to an expression for the Gibbs energy:

4.8.2 Equilibrium criteria for the Gibbs energy

As with entropy and internal energy, there is a condition for equilibrium for the Gibbs energy and it is often more useful than the others:

which after dividing each side by dS simplifies to

This last is an important result, since so many processes take place at constant temperature and pressure. The inequalities derived earlier are somewhat less useful since in applications it is more difficult to control H, S or U. The last equation states that in a spontaneous process the (possibly composite) system will adjust its coordinates in such a way as to lower the value of the system's Gibbs energy; in this sense it behaves like a potential energy function in mechanics where a system tends toward minimum potential energy (see Figure 4.5). Equilibrium will establish itself at the minimum of G(T,p), much as it did in the last chapter for a maximum of S(U, V) only this time we have a function whose dependent variables are more under our control (or more to the point those found in naturally occurring circumstances).

Now for a process at constant temperature and pressure, dGT, p < 0 [equilibrium criterion for Gibbs energy].

State of system

Figure 4.5 In a spontaneous transition the Gibbs energy is a minimum.

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