When deriving an expression describing how a thermodynamic function (U, H, A, or G) changes with p, V or T, there are three basic starting points from which you can commonly choose:
Other tools that may be useful include the Maxwell Relations, especially those on A and G:
\( \left( \frac{\partial p}{\partial T} \right)_V = \left( \frac{\partial S}{\partial V} \right)_T \) and \( \left( \frac{\partial V}{\partial T} \right)_p = - \left( \frac{\partial S}{\partial p} \right)_T \)
Derive an expression for \[ \left(\frac{\partial U}{\partial p}\right)_T \] in terms of measurable properties.
Start by treating internal energy as a function of \(V\) and \(T\):
\[ U = U(V,T) \]
Therefore,
\[ dU = \left( \frac{\partial U}{\partial V} \right)_T dV + \left( \frac{\partial U}{\partial T} \right)_V dT \]
Divide by \(dp\) and constrain the change to constant temperature:
\[ \left. \frac{\partial U}{\partial p} \right|_T = \left( \frac{\partial U}{\partial V} \right)_T \left. \frac{\partial V}{\partial p} \right|_T + \left( \frac{\partial U}{\partial T} \right)_V \left. \frac{\partial T}{\partial p} \right|_T \]
Since the temperature is held constant,
\[ \left. \frac{\partial T}{\partial p} \right|_T = 0 \]
so the second term vanishes. Converting to partial derivatives:
\[ \left( \frac{\partial U}{\partial p} \right)_T = \left( \frac{\partial U}{\partial V} \right)_T \left( \frac{\partial V}{\partial p} \right)_T \]
From the previous Maxwell-relation result,
\[ \left( \frac{\partial U}{\partial V} \right)_T = T \left( \frac{\partial p}{\partial T} \right)_V - p \]
Also, the isothermal compressibility is defined as
\[ \kappa_T = - \frac{1}{V} \left( \frac{\partial V}{\partial p} \right)_T \]
so
\[ \left( \frac{\partial V}{\partial p} \right)_T = -V\kappa_T \]
Finally, using
\[ \left( \frac{\partial p}{\partial T} \right)_V = \frac{\alpha}{\kappa_T} \]
gives
\[ \left( \frac{\partial U}{\partial V} \right)_T = T\frac{\alpha}{\kappa_T} - p \]
Therefore,
\[ \left( \frac{\partial U}{\partial p} \right)_T = \left( T\frac{\alpha}{\kappa_T} - p \right) (-V\kappa_T) \]
\[ \boxed{ \left( \frac{\partial U}{\partial p} \right)_T = V \left( p\kappa_T - T\alpha \right) } \]
Interpretation: This expression shows how internal energy changes with pressure at constant temperature. The result is written entirely in terms of measurable quantities: \(V\), \(p\), \(T\), \(\alpha\), and \(\kappa_T\).
Derive an expression for \[ \left(\frac{\partial G}{\partial V}\right)_T \] starting from \[ G = U+pV-TS. \]
Differentiate \(G\):
\[ dG = dU + p\,dV + V\,dp - T\,dS - S\,dT \]
Divide by \(dV\) and constrain to constant temperature:
\[ \left(\frac{\partial G}{\partial V}\right)_T = \left(\frac{\partial U}{\partial V}\right)_T + p + V\left(\frac{\partial p}{\partial V}\right)_T - T\left(\frac{\partial S}{\partial V}\right)_T \]
Now use the Maxwell-relation result
\[ \left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial p}{\partial T}\right)_V - p \]
and the Maxwell Relation on \(A\):
\[ \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial p}{\partial T}\right)_V \]
Substituting gives
\[ \left(\frac{\partial G}{\partial V}\right)_T = \left[ T\left(\frac{\partial p}{\partial T}\right)_V - p \right] + p + V\left(\frac{\partial p}{\partial V}\right)_T - T\left(\frac{\partial p}{\partial T}\right)_V \]
The entropy-related terms cancel, and the \(p\) terms cancel:
\[ \left(\frac{\partial G}{\partial V}\right)_T = V\left(\frac{\partial p}{\partial V}\right)_T \]
Using the definition of the isothermal compressibility,
\[ \kappa_T = - \frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_T \]
it follows that
\[ \left(\frac{\partial p}{\partial V}\right)_T = - \frac{1}{V\kappa_T} \]
Therefore,
\[ \boxed{ \left(\frac{\partial G}{\partial V}\right)_T = -\frac{1}{\kappa_T} } \]
Interpretation: At constant temperature, changing the volume changes the Gibbs function through the pressure response of the system. Since \(\kappa_T\) measures how compressible the system is, a less compressible substance has a larger magnitude for \(\left(\frac{\partial G}{\partial V}\right)_T\).