5. Thermodynamic potentials and equilibrium

Extensive and intensive variables

All quantities in thermodynamics fall into two types: extensive and intensive. If we consider two independent systems, quantities which add up to give the corresponding quantities for the complete system are characterized as extensive quantities. The volume , the internal energy , the enthalpy, and as we will see later on, the entropy are extensive quantities. If we divide a system into subsystems, those quantities which remain unaltered are called intensive variables. The pressure 𝒫, the temperature , the surface tension are examples of intensive variables.

In any thermodynamic system, there is a natural pairing between extensive and intensive variables. For example, pressure and volume go together as in the formula. Temperature is paired with the entropy, surface tension with the area, etc.

5.1 Thermodynamic potentials

The second law leads to the result , so that, for a gaseous system, the first law may be written as

(5.1)

More generally, we have

(5.2)

where α is the surface tension, is the area, is the magnetization, is the magnetic field, etc.

Returning to the case of a gaseous system, we now define a number of quantities related to . These are called thermodynamic potentials and are useful when considering different processes. We have already seen that the enthalpy is useful for considering processes at constant pressure. This follows from

(5.3)

so that for processes at constant pressure, the inflow or outflow of heat may be seen as changing the enthalpy.

The Helmholtz free energy is defined by

(5.4)

Taking differentials and comparing with the formula for , we get

(5.5)

The Gibbs free energy is defined by

(5.6)

Evidently,

(5.7)

Notice that by construction, , and are functions of the state of the system. They may be expressed as functions of 𝒫 and , for example. They are obviously extensive quantities.

So far we have considered the system characterized by the pressure and volume. If there are a number of particles, say, which make up the system, we can also consider the - dependence of various quantities. Thus we can think of the internal energy as a function of , and ,so that

(5.8)

The quantity which is defined by

(5.9)

Is called the chemical potential. It is obviously an intensive variable. The corresponding equations for , , and are

(5.10)

Thus the chemical potential may also be defined as

(5.11)

Since is an extensive quantity and so are and , the internal energy has the general functional form

(5.12)

where depends only on and which are intensive variables. In a similar way, we can write

(5.13)

The last equation is of particular interest. Taking its variation, we find

(5.14)

Comparing with (5.10), we get

(5.15)

The quantity is identical to the chemical potential, so that we may write

(5.16)

We may rewrite the other two relations as

(5.17)

Further, using , we can rewrite the equation for as

(5.18)

This essentially combines the previous two relations and is known as the Gibbs-Duhem relation. It is important in that it provides a relation among the intensive variables of a thermodynamic system.

5.2 Thermodynamic equilibrium

The second law of thermodynamics implies that entropy does not decrease in any natural process. The final equilibrium state will thus be the state of maximum possible entropy. After attaining this maximum possible value, the entropy will remain constant. The criterion for equilibrium may thus be written as

(5.19)

We can take to be a function of , and . The system, starting in an arbitrary state, adjusts , and among its different parts and constitutes itself in such a way as to maximize entropy. Consider the system subdivided into various smaller subsystems, say, indexed by . The thermodynamic quantities for each such unit will be indicated by a subscript . For an isolated system, the total internal energy, the total volume and the total number of particles will be fixed, so that the the changes in the subsystems must be constrained as

(5.20)

Since is extensive, where . We can now maximize entropy subject to the constraints (5.20) by considering the maximization of

(5.21)

where the Lagrange multipliers enforce the required constraints. The variables , and can now be freely varied. Thus, for the condition of equilibrium, we get

(5.22)

Since the variations are now independent, this gives, for equilibrium,

(5.23)

This can be written as

(5.24)

where we used

(5.25)

Equation (5.24) tells us that, for equilibrium, the temperature of all subsystems must be the same, the pressure in different subsystems must be the same and the chemical potential for different subsystems must be the same.

Reaction equilibrium

Suppose we have a number of constituents , at constant temperature and pressure which undergo a reaction of the form

(5.26)

The entropy of the system is of the form

(5.27)

where the summation covers all ’s and ’s. Since the temperature and pressure are constant, reaction equilibrium is obtained when the change so as to maximize the entropy. This gives

(5.28)

The quantities are not independent, but are restricted by the reaction. When the reaction happens, of -particles must be destroyed, of -particles must be destroyed, etc., while of particles are produced, etc. Thus we can write

(5.29)

where is arbitrary. The condition of equilibrium thus reduces to

(5.30)

With the understanding that the 𝒱’s for the reactants will carry a minus sign while those for the products have a plus sign, we can rewrite this as

(5.31)

This condition of reaction equilibrium can be applied to chemical reactions, ionization and dissociation processes, nuclear reactions, etc.

5.3 Phase transitions

If we have a single constituent, for thermodynamic equilibrium, we should have equality of , and for different subparts of the system. If we have different phases of the system, such as gas, liquid, or solid, in equilibrium, we have

(5.32)

where the subscripts refer to various phases.

We will consider the equilibrium of two phases in some more detail. In this case, we have

(5.33)

If equilibrium is also obtained for a state defined by , then we have

(5.34)

These two equations yield

(5.35)

This equation will tell us how 𝒫 should change when is altered (or vice versa) so as to

preserve equilibrium. Expanding to first order in the variations, we find

(5.36)

where we have used

(5.37)

Equation (5.36) reduces to

(5.38)

Where is the latent heat of the transition. This equation is known as the

Clausius-Clapeyron equation. It can be used to study the variation of saturated vapor pressure with temperature (or, conversely, the variation of boiling point with pressure). As an example, consider the variation of boiling point with pressure, when a liquid boils to form gaseous vapor. In this case, we can take . Further, if we assume, for the sake of the argument, that the gaseous phase obeys the ideal gas law, , then the Clausius-Clapeyron equation (5.38) becomes

(5.39)

Integrating this from one value of to another,

(5.40)

Thus for , must be larger than ; this explains the increase of boiling point with pressure.

If and are continuous at the transition, and . In this case we have to expand to second order in the variations. Such a transition is called a second order phase transition. In general, if the first derivatives of are continuous, and the 𝓃 ^-th derivatives are discontinuous at the transition, the transition is said to be of the 𝓃 ^-th order. Clausius-Clapeyron equation, as we have written it, applies to first order phase transitions. These have a latent heat of transition.