5.7: Surface Work
Sometimes we need more than the usual two independent variables to describe an equilibrium state of a closed system of one substance in one phase. This is the case when, in addition to expansion work, another kind of work is possible. The total differential of \(U\) is then given by \(\dif U = T\dif S - p\dif V + Y\dif X\) (Eq. 5.2.7), where \(Y\dif X\) represents the nonexpansion work \(\dw'\).
A good example of this situation is surface work in a system in which surface area is relevant to the description of the state.
A liquid–gas interface behaves somewhat like a stretched membrane. The upper and lower surfaces of the liquid film in the device depicted in Fig. 5.1 exert a force \(F\) on the sliding rod, tending to pull it in the direction that reduces the surface area. We can measure the force by determining the opposing force \(F\subs{ext}\) needed to prevent the rod from moving. This force is found to be proportional to the length of the rod and independent of the rod position \(x\). The force also depends on the temperature and pressure.
The surface tension or interfacial tension, \(\g\), is the force exerted by an interfacial surface per unit length. The film shown in Fig. 5.1 has two surfaces, so we have \(\g = F/2l\) where \(l\) is the rod length.
To increase the surface area of the film by a practically-reversible process, we slowly pull the rod to the right in the \(+x\) direction. The system is the liquid. The \(x\) component of the force exerted by the system on the surroundings at the moving boundary, \(F_x\sups{sys}\), is equal to \(-F\) (\(F\) is positive and \(F_x\sups{sys}\) is negative). The displacement of the rod results in surface work given by Eq. 3.1.2: \(\dw' = -F_x\sups{sys}\dx = 2\g l\dx\). The increase in surface area, \(\dif A\subs{s}\), is \(2l\dx\), so the surface work is \(\dw' = \g \dif A\subs{s}\) where \(\g\) is the work coefficient and \(A\subs{s}\) is the work coordinate. Equation 5.2.7 becomes \begin{equation} \dif U=T\dif S -p\dif V+\g\dif A\subs{s} \tag{5.7.1} \end{equation} Substitution into Eq. 5.3.6 gives \begin{equation} \dif G = -S \dif T + V \difp + \g \dif A\subs{s} \tag{5.7.2} \end{equation} which is the total differential of \(G\) with \(T\), \(p\), and \(A\subs{s}\) as the independent variables. Identifying the coefficient of the last term on the right side as a partial derivative, we find the following expression for the surface tension: \begin{equation} \g = \Pd{G}{A\subs{s}}{T,p} \tag{5.7.3} \end{equation} That is, the surface tension is not only a force per unit length, but also a Gibbs energy per unit area.
From Eq. 5.7.2, we obtain the reciprocity relation \begin{equation} \Pd{\g}{T}{p,A\subs{s}} = -\Pd{S}{A\subs{s}}{T,p} \tag{5.7.4} \end{equation} It is valid to replace the partial derivative on the left side by \(\pd{\g}{T}{p}\) because \(\g\) is independent of \(A\subs{s}\). Thus, the variation of surface tension with temperature tells us how the entropy of the liquid varies with surface area.