Chemistry 351

Deriving expressions for Partial Derivatives

When approaching a partial derivative derivation in thermodynamics, there are three common stating places.

1. Choose two independent variables on which a thermodynamic function depends. For example, U(V,T) or H(p, T). From this, you can write the total differential and go from there. For example \[ dU = \left( \frac{\partial U}{\partial V} \right)_T dV + \left( \frac{\partial U}{\partial T} \right)_V dT \]
2. Start from the definiition of a thermodynamic function, such as H - U + pV, and differentiate. \[ dH = dU + pdV + Vdp \]
3. Start with the combined statement of the 1^st and 2^nd Laws: \[ dU = TdS - pdV \] The thermodynamic variable \( S \) will be introduced in Chapter 5. So for now, you can just consider the first two.

Many derivations will require multiple steps, where the result of one gets substituted into another.

Example: Rewriting \(\left( \frac{\partial H}{\partial V} \right)_T\)

Suppose we need an expression for

\[ \left( \frac{\partial H} {\partial V} \right)_T \]

Begin with the definition of enthalpy:

\[ H = U + pV \]

Differentiate with respect to \(V\) at constant \(T\):

\[ \left( \frac{\partial H} {\partial V} \right)_T = \left( \frac{\partial U} {\partial V} \right)_T + p + V \left( \frac{\partial p} {\partial V} \right)_T \]

Now use the relationship

\[ \left( \frac{\partial U} {\partial V} \right)_T = T \left( \frac{\partial p} {\partial T} \right)_V - p \]

Substitution gives

\[ \left( \frac{\partial H} {\partial V} \right)_T = T \left( \frac{\partial p} {\partial T} \right)_V + V \left( \frac{\partial p} {\partial V} \right)_T \]

Substituting Measurable Properties

From the definitions of \(\alpha\) and \(\kappa_T\), we can show that

\[ \left( \frac{\partial p} {\partial T} \right)_V = \frac{\alpha} {\kappa_T} \]

and

\[ \left( \frac{\partial p} {\partial V} \right)_T = - \frac{1} {V\kappa_T} \]

Substituting these into the expression above:

\[ \left( \frac{\partial H} {\partial V} \right)_T = T \left( \frac{\alpha} {\kappa_T} \right) + V \left( - \frac{1} {V\kappa_T} \right) \]

\[ \left( \frac{\partial H} {\partial V} \right)_T = \frac{T\alpha} {\kappa_T} - \frac{1} {\kappa_T} \]

\[ \boxed{ \left( \frac{\partial H} {\partial V} \right)_T = \frac{T\alpha-1} {\kappa_T} } \]

Big picture: The derivative \(\left( \partial H/\partial V \right)_T\) is not something we would usually measure directly. Thermodynamics allows us to rewrite it in terms of \(\alpha\), \(\kappa_T\), and \(T\), which are measurable or tabulated quantities.