6.2: Other Relations

The TdS Equations

The entropy \(S\) is a function of the state of the system. We can take it to be a function of any two of the three variables (\(p\), \(T\), \(V\)). Taking \(S\) to be a function of \(p\) and \(T\), we write

\[T dS = T \biggl( \frac{\partial S}{\partial T} \biggr)_p dT \;+\; T \biggl( \frac{\partial S}{\partial p} \biggr)_T dp \]

For the first term on the right hand side, we can use

\[T \biggl( \frac{\partial S}{\partial T} \biggr)_p dT \;=\; \biggl( \frac{\partial Q}{\partial T} \biggr)_p dT \;=\; C_p \label{6.2.2} \]

where \(C_p\) is the specific heat at constant pressure. Further, using the last of the Maxwell relations, we can now write Equation \ref{6.2.2} as

\[Tds \;=\; C_pdT \;-\;T\biggl( \frac{\partial V}{\partial T} \biggr)_p dp \label{6.2.3}\]

The coefficient of volumetric expansion (due to heating) is defined by

\[ \alpha \;=\; \frac{1}{V} \biggl( \frac{\partial V}{\partial T} \biggr)_p \]

Equation \ref{6.2.3} can thus be rewritten as

\[T dS \;=\; C_p dT \;−\; \alpha T V dp \label{6.2.5}\]

If we take \(S\) to be a function of \(V\) and \(T\),

\[T dS \;=\; T \biggl( \frac{\partial S}{\partial T} \biggr)_V dT \;+\; T \biggl( \frac{\partial S}{\partial V} \biggr)_T dV \]

Again the first term on the right hand side can be expressed in terms of Cv, the specific heat at constant volume, using

\[ T \biggl( \frac{\partial S}{\partial T} \biggr)_V \;=\; C_v \]

Further using the Maxwell relations, we get

\[T dS \;=\; C_vdT \;+\; T \biggl( \frac{\partial p}{\partial T} \biggr)_V dV \label{6.2.8} \]

Equations \ref{6.2.3} (or \ref{6.2.5} and \ref{6.2.8}) are known as the \(T dS\) equations.

Equations for Specific Heats

Equating the two expressions for \(T dS\), we get

\[(C_p \;−\; C_v) dT \;=\; T \biggl[ \biggl( \frac{\partial p}{\partial T} \biggr)_V dV \;+\; \biggl( \frac{\partial V}{\partial T} \biggr)_pdp \biggr] \label{6.2.9}\]

By the equation of state, we can write \(p\) as a function of \(V\) and \(T\), so that

\[dp \;=\; \biggl( \frac{\partial p}{\partial T} \biggr)_V dT \;+\; \biggl( \frac{\partial p}{\partial V} \biggr)_T dV\]

Using this in Equation \ref{6.2.9}, we find

\[(C_p \;−\; C_v) dT \;=\; T \biggl[ \biggl( \frac{\partial p}{\partial T} \biggr)_V \;+\; \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggl( \frac{\partial p}{\partial V} \biggr)_T \biggr]dV \;+\; T \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggl( \frac{\partial p}{\partial T} \biggr)_V dT \label{6.2.11}\]

However, using Equation 6.1.9, taking \(X \;=\; V\), \(Y \;=\; p\) and \(Z \;=\; T\), we have

\[ \biggl( \frac{\partial p}{\partial T} \biggr)_V \;+\; \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggl( \frac{\partial p}{\partial V} \biggr)_T \;=\; 0 \label{6.2.12}\]

Thus the coefficient of \(dV\) in Equation \ref{6.2.11} vanishes and we can simplify it as

\[C_p \;-\; C_v \;=\; T \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggl( \frac{\partial p}{\partial T} \biggr)_V \;=\; -T \biggl[ \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggr]^2 \biggl( \frac{\partial p}{\partial V} \biggr)_T \]

where we have used Equation \ref{6.2.12} again. We have already defined the coefficient of volumetric expansion \( \alpha \). The isothermal compressibility \( \kappa_T\) is defined by

\[ \frac{1}{ \kappa_T} \;=\; -V \biggl( \frac{\partial p}{\partial V} \biggr)_T \]

In terms of these we can express \(C_p \;−\; C_v\) as

\[ C_p \;-\; C_v \;=\; V \frac{ \alpha^2T}{ \kappa_T} \]

This equation is very useful in calculating \(C_v\) from measurements of \(C_p\) and \(\alpha\) and \( \kappa_T\). Further, for all substances, \( \kappa_T \;>\; 0\). Thus, we see from this equation that \(C_p ≥ C_v\). (The result \( \kappa_T \;>\; 0\) can be proved in statistical mechanics.)

In the \(T dS\) equations, if \(dT\), \(dV\) and \(dp\) are related adiabatically, \(dS \;=\; 0\) and we get

\[ C_p \;=\; T \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggl( \frac{\partial p}{\partial T} \biggr)_S,\;\;\;\; C_v \;=\; -T \biggl( \frac{\partial p}{\partial T} \biggr)_V \biggl( \frac{\partial V}{\partial T} \biggr)_S \]

This gives

\[ \frac{C_p}{C_v} \;=\; - \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggl( \frac{\partial p}{\partial T} \biggr)_S \biggl[ \biggl( \frac{\partial p}{\partial T} \biggr)_V \biggl( \frac{\partial V}{\partial T} \biggr)_S \biggr]^{-1} \]

We have the following relations among the terms involved in this expression,

\[ \biggl( \frac{\partial p}{\partial T} \biggr)_S \;=\; \biggl( \frac{\partial V}{\partial T} \biggr)_S \biggl( \frac{\partial p}{\partial V} \biggr)_S \;=\; \biggl( \frac{\partial V}{\partial T} \biggr)_S \frac{1}{V \kappa_S} \\ \biggl( \frac{\partial V}{\partial T} \biggr)_p \;=\; \biggl( \frac{\partial p}{\partial T} \biggr)_V \biggl( \frac{\partial T}{\partial p} \biggr)_V \biggl( \frac{\partial V}{\partial T} \biggr)_p \;=\; -\biggl( \frac{\partial p}{\partial T} \biggr)_V \frac{1}{\bigl( \frac{\partial p}{\partial V} \bigr)_T} \;=\; \biggl( \frac{\partial p}{\partial T} \biggr)_V V \kappa_T \]

Using these we find

\[ \frac{C_p}{C_v} \;=\; \frac{\kappa_T}{\kappa_S}\]

Going back to the Maxwell relations and using the expressions for \(T dS\), we find

\[dU \;=\; C_vdT \;+\; \biggl[T \biggl( \frac{\partial p}{\partial T} \biggr)_V -p \biggr]dV \\ dH \;=\; C_pdT \;+\; \biggl[V- T \biggl( \frac{\partial V}{\partial T} \biggr)_p \biggr]dp \]

these immediately yield the relations

\[ C_v \;=\; \biggl( \frac{\partial U}{\partial T} \biggr)_V,\;\;\;\;C_p \;=\; \biggl( \frac{\partial H}{\partial T} \biggr)_p \\ \biggl( \frac{\partial U}{\partial V} \biggr)_T \;=\; T \biggl( \frac{\partial p}{\partial T} \biggr)_V -p \;\;\;\;\;\;\; \biggl( \frac{\partial H}{\partial p} \biggr)_T \;=\; V- T \biggl( \frac{\partial V}{\partial T} \biggr)_p \]

Gibbs-Helmholtz Relation

Since the Helmholtz free energy is defined as \(F = U − T S\),

\[dF \;=\; dU \;−\; T dS \;−\; SdT \;=\; −S dT \;−\; p dV\]

This gives immediately

\[S \;=\; -\biggl( \frac{\partial F}{\partial T} \biggr)_V,\;\;\;\; p \;=\; -\biggl( \frac{\partial F}{\partial V} \biggr)_T \]

Using this equation for entropy, we find

\[U \;=\; F \;+\; T S \;=\; F \;−\; T \biggl( \frac{\partial F}{\partial T} \biggr)_V \]

This is known as the Gibbs-Helmholtz relation. If \(F\) is known as a function of \(T\) and \(V\), we can use these to obtain \(S\), \(p\) and \(U\). Thus, all thermodynamic variables can be obtained from \(F\) as a function of \(T\) and \(V\).