1.9.3: Entropy and Spontaneous Reaction
It is often stated that the entropy of a system is a maximum at equilibrium. This is not generally true and is certainly not the case for closed systems at either (a) fixed \(\mathrm{T}\) and \(\mathrm{p}\), or (b) fixed \(\mathrm{T}\) and \(\mathrm{V}\).
We rewrite the Master Equation in the following way:
\[\mathrm{dS}=(1 / \mathrm{T}) \, \mathrm{dU}+(\mathrm{p} / \mathrm{T}) \, \mathrm{dV}+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq \text { zero } \nonumber \]
Temperature \(\mathrm{T}\) is positive and non-zero. At constant energy and constant volume (i.e. isoenergetic and isochoric), spontaneous processes are accompanied by an increase in entropy. This statement is important in statistical thermodynamics where the condition, ‘constant \(\mathrm{U}\) and constant \(\mathrm{V}\)’ is important.
The following equation defines the enthalpy \(\mathrm{H}\) of a closed system.
\[\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V} \nonumber \]
Then
\[\mathrm{dU}=\mathrm{dH}-\mathrm{p} \, \mathrm{dV}-\mathrm{V} \, \mathrm{dp} \nonumber \]
From equation (a),
\[\mathrm{dS}=(1 / \mathrm{T}) \, \mathrm{dH}-(\mathrm{p} / \mathrm{T}) \, \mathrm{dV}-(\mathrm{V} / \mathrm{T}) \, \mathrm{dp}+(\mathrm{p} / \mathrm{T}) \, \mathrm{dV}+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi \text { with } \mathrm{A} \, \mathrm{d} \xi \geq \text { zero } \nonumber \]
Hence,
\[\mathrm{dS}=(1 / \mathrm{T}) \, \mathrm{dH}-(\mathrm{V} / \mathrm{T}) \, \mathrm{dp}+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq \text { zero } \nonumber \]
Temperature \(\mathrm{T}\) is always positive. Hence at constant enthalpy and pressure (i.e. iso-enthalpic and isobaric) all spontaneous processes produce an increase in entropy.
We have identified two sets of conditions under which an increase in entropy accompanies a spontaneous process. If we follow through a similar argument with respect to the Gibbs energy, the outcome is not straightforward. By definition,
\[\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S} \nonumber \]
Then
\[\mathrm{dG}=\mathrm{dH}-\mathrm{T} \, \mathrm{dS}-\mathrm{S} \, \mathrm{dT} \nonumber \]
Or,
\[\mathrm{S}=-\mathrm{dG} / \mathrm{dT}+\mathrm{dH} / \mathrm{dT}-\mathrm{T} \, \mathrm{dS} / \mathrm{dT} \nonumber \]
But from equation (e)
\[\mathrm{dH} / \mathrm{dT}=\mathrm{T} \, \mathrm{dS} / \mathrm{dT}+(\mathrm{V} / \mathrm{T}) \, \mathrm{dp} / \mathrm{dT}-(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi / \mathrm{dT} \nonumber \]
Hence,
\[\mathrm{S}=-(\mathrm{dG} / \mathrm{dT})+\mathrm{V} \,(\mathrm{dp} / \mathrm{dT})-\mathrm{A} \,(\mathrm{d} \xi / \mathrm{dT}) \text { with } \mathrm{A} \, \mathrm{d} \xi \geq \text { zero } \nonumber \]
Clearly no definite conclusions can be drawn about changes in entropy S under isobaric - isothermal conditions. We stress these points because again it is often tempting to link, misguidedly, entropies to the degree of ‘muddled-up-ness’. This is the basis of many explanations of entropy. For example, neither the volume nor energy of a deck of cards change on shuffling. Whether what actually happens on shuffling a new well-ordered deck of cards clarifies the meaning of entropy seems doubtful.