5.1: Thermodynamic Potentials

The second law leads to the result \(dQ = TdS\), so that, for a gaseous system, the first law may be written as

\[dU = T dS − p dV\]

More generally, we have

\[dU = T dS − p dV − σ dA − M dB\]

where \(σ\) is the surface tension, \(A\) is the area, \(M\) is the magnetization, \(B\) is the magnetic field, etc.

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

\[dH = d(U + pV) = dQ + V dp\]

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

\[F = U − TS\]

Taking differentials and comparing with the formula for \(dU\), we get

\[dF = −SdT − pdV\]

The Gibbs free energy \(G\) is defined by

\[G = F + pV = H − TS = U − TS + pV\]

Evidently,

\[dG = −SdT + Vdp\]

Notice that by construction, \(H\), \(F\) and \(G\) are functions of the state of the system. They may be expressed as functions of \(p\) and \(V\), for example. They are obviously extensive quantities.

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

\[dU = \biggl(\frac{∂U}{∂S}\biggr)_{V,N}dS + \biggl(\frac{∂U}{∂V}\biggr)_{S,N}dV + \biggl(\frac{∂U}{∂N}\biggr)_{S,V}dN ≡ TdS − pdV + \mu dN\]

The quantity \(\mu\) which is defined by

\[\mu=\biggl(\frac{∂U}{∂N}\biggr)_{S,V}\]

is called the chemical potential. It is obviously an intensive variable. The corresponding equations for \(H\), \(F\) and \(G\) are

\[dH = T dS + V dp + \mu dN \\ dF = −S dT − p dV + \mu dN \\ dG = −S dT + V dp + \mu dN \label{5.1.10}\]

Thus the chemical potential µ may also be defined as

\[\mu=\biggl(\frac{∂H}{∂N}\biggr)_{S,p} = \biggl(\frac{∂F}{∂N}\biggr)_{T,V} = \biggl(\frac{∂G}{∂N}\biggr)_{T,p}\]

Since \(U\) is an extensive quantity and so are \(S\) and \(V\), the internal energy has the general functional form

\[U = N u\biggl(\frac{S}{N},\frac{V}{N}\biggr)\]

where \(u\) depends only on \(\frac{S}{N}\) and \(\frac{V}{N}\) which are intensive variables. In a similar way, we can write

\[H = N\; h\biggl(\frac{S}{N}, p\biggr) \\ F = N\; f\biggl(T,\frac{V}{N}\biggr) \\ G = N\; g(T, p)\]

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

\[dG = N\;\biggl(\frac{∂g}{∂T}\biggr)_pdT + N\;\biggl(\frac{∂g}{∂p}\biggr)_TdT + g\;dN\]

Comparing with Equation \ref{5.1.10}, we get

\[S= −N\;\biggl(\frac{∂g}{∂T}\biggr)_p, \;\;\;V= N\;\biggl(\frac{∂g}{∂p}\biggr)_T, \;\;\; \mu = g\]

The quantity \(g\) is identical to the chemical potential, so that we may write

\[G = \mu N\]

We may rewrite the other two relations as

\[S= −N\;\biggl(\frac{∂ \mu}{∂T}\biggr)_pdT,\;\;\; V= N\;\biggl(\frac{∂ \mu}{∂p}\biggr)_T\]

Further, using \(G = \mu N\), we can rewrite the equation for \(dG\) as

\[N d \mu + S dT − V dp = 0\]

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.