Thermodynamic Potentials and Legendre Transforms

The Logic of Variable Exchange

Imagine you have a function defined in terms of position and velocity. If you want to describe the system in terms of position and momentum, you do not discard the original function; you transform it. Thermodynamics works similarly. The internal energy \( U \) is defined by its differential \( \mathrm{d}U = T\mathrm{d}S - P\mathrm{d}V \). Here, \( S \) and \( V \) are the independent variables. However, in a beaker on a bench, the temperature \( T \) is set by the room, and the pressure \( P \) is set by the atmosphere. We need a potential whose natural variables are \( T \) and \( P \). A Legendre transform achieves this by subtracting the product of the variable we want to remove and its conjugate. For example, to remove \( V \) and replace it with \( P \), we add \( PV \) to \( U \). To remove \( S \) and replace it with \( T \), we subtract \( TS \). This is not an approximation; it is an exact mathematical identity that preserves all thermodynamic information while changing the perspective to match the experimental conditions.

Constructing the Four Potentials

Starting from the fundamental relation for a simple compressible system, we derive the other potentials by successive Legendre transforms. The internal energy \( U(S,V) \) is the starting point. To obtain the enthalpy \( H \), which is useful for constant pressure processes, we transform out \( V \). The definition is \( H = U + PV \). Taking the differential, we use the product rule \( \mathrm{d}(PV) = P\mathrm{d}V + V\mathrm{d}P \):

Substituting \( \mathrm{d}U = T\mathrm{d}S - P\mathrm{d}V \), the \( P\mathrm{d}V \) terms cancel, leaving \( \mathrm{d}H = T\mathrm{d}S + V\mathrm{d}P \). The natural variables for \( H \) are now \( S \) and \( P \). Next, to obtain the Helmholtz free energy \( F \), which is useful for constant temperature processes, we transform out \( S \) from \( U \). The definition is \( F = U - TS \). The differential is:

Substituting \( \mathrm{d}U \) again, the \( T\mathrm{d}S \) terms cancel, yielding \( \mathrm{d}F = -S\mathrm{d}T - P\mathrm{d}V \). The natural variables are \( T \) and \( V \). Finally, the Gibbs free energy \( G \) is constructed by transforming out both \( S \) and \( V \) from \( U \), or equivalently from \( H \) or \( F \). The definition is \( G = U - TS + PV \). Its differential is:

The natural variables are \( T \) and \( P \). This set of four potentials covers all combinations of the two pairs of conjugate variables \( (S,T) \) and \( (V,P) \).

Interrelations and Maxwell Relations

Because these potentials are derived from the same fundamental relation, they are not independent. They are linked by algebraic identities. For instance, since \( H = U + PV \) and \( F = U - TS \), we can relate \( H \) and \( F \) via \( G \). The most common relation is \( G = H - TS = F + PV \). These identities allow us to switch between potentials when solving problems. More profoundly, because \( U, H, F, \) and \( G \) are all state functions, their second mixed partial derivatives must be equal. This mathematical requirement generates the Maxwell relations. For example, from \( \mathrm{d}F = -S\mathrm{d}T - P\mathrm{d}V \), we identify \( \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V \) (the two minus signs in \( \mathrm{d}F \) cancel). This relation allows us to calculate entropy changes, which are difficult to measure directly, using pressure, volume, and temperature data, which are easy to measure. Each potential yields a different Maxwell relation, providing a powerful toolkit for thermodynamic analysis.

Physical Interpretation of the Potentials

Each potential has a distinct physical meaning related to the maximum useful work obtainable from a system under specific constraints. The internal energy \( U \) represents the total energy of the system. The enthalpy \( H \) is not a stored "heat content", but the change in enthalpy \( \Delta H \) equals the heat exchanged in a constant pressure process. The Helmholtz free energy \( F \) represents the maximum work that can be extracted from a system at constant temperature and volume. The Gibbs free energy \( G \) represents the maximum non-expansion work that can be extracted from a system at constant temperature and pressure. In chemical reactions, which typically occur at constant \( T \) and \( P \), the change in Gibbs free energy \( \Delta G \) determines spontaneity. If \( \Delta G < 0 \), the process is spontaneous. If \( \Delta G > 0 \), the process is non-spontaneous. If \( \Delta G = 0 \), the system is at equilibrium. This makes \( G \) the most frequently used potential in chemistry and materials science.