Heat capacity at constant pressure from Helmholtz free energy

Question

From statistical mechanical theory, a simple model for a hypothetical hard-sphere liquid (spherical molecules of finite size without attractive intermolecular forces) gives the following expression for the Helmholtz free energy $A$ with its natural variables $T$, $V$, and $n$ as the independent variables:

$$ A = -nRT \ln{\left(cT^{\frac{3}{2}}\left( \dfrac{V}{b}-b \right)\right)}-nRT+na\text{.} $$

To calculate $C_P$ from Helmholtz free Energy, I followed the following approach:

Step 1: Calculate $S$ $$ \begin{align} S &= -\left( \dfrac{\partial A}{\partial T} \right)_{V,n}\\ &= nR \ln{\left(cT^{\frac{3}{2}}\left( \dfrac{V}{b}-b \right)\right)} + \dfrac{5}{2}nR\text{.} \end{align} $$

Step 2: Calculate $U$ $$ \begin{align} U &= A + TS\\ &= \dfrac{3}{2}nRT + na\text{.} \end{align} $$

Step 3: Calculate $C_V$ $$ \begin{align} C_V &= \left( \dfrac{\partial U}{\partial T} \right)_{V,n}\\ &= \dfrac{3}{2}nR\text{.} \end{align} $$

Step 4: Calculate $C_P$ $$ \begin{align} C_P &= C_V + nR\\ &= \dfrac{5}{2}nR \text{.} \end{align} $$

Is there a direct method to obtain $C_P$ from $A$.

Answer 1

Your working is already quite the optimal. It is however possible to avoid calculating $C_V$ and assuming ideal gas law: $$\mathrm d A = - S \mathrm d T - p \mathrm d V + \mu \mathrm d n \\H = U + p V = A + T S + p V \\C_p = \left ( \partial_T H \right )_{p,n} \\ -p = \left ( \partial_V A \right )_{T,n} = - \frac{n R T}{V - b^2} \\\therefore \qquad V = b^2 +\frac{n R T}p \\H = \frac32 n R T + n a + p b^2 + n R T = \frac52 n R T + n a + p b^2 \\\therefore \qquad C_p = \frac52 n R $$

Answer 2

Is there a direct method to obtain $C_P$ from $A$.

From $C_P\equiv\left(\frac{\partial H}{\partial T}\right)_P$ and $H=A+TS+PV$, for a closed system with only pressure–volume work, we always have

$$C_P=\left(\frac{\partial \left[A-T\left(\frac{\partial A}{\partial T}\right)_V-V\left(\frac{\partial A}{\partial V}\right)_T\right]}{\partial T}\right)_{\left(\frac{\partial A}{\partial V}\right)_T},$$

to which the relevant constitutive laws (and partial differential relations) can be applied as necessary.