Deriving heat capacity in terms of internal energy U and natural variables S & V

My lecture notes have the following equation regarding heat capacities:

$$C_p = C_v + T\left(\frac{\partial p}{\partial T}\right)_v\left(\frac{\partial V}{\partial T}\right)_p$$

(just to be clear, all terms after the brackets in this question indicate a constant process of that term)

It then states that by using internal energy $U$, volume $V$ and entropy $S$, you can express heat capacity at constant pressure, $C_p$, just in terms of these 3 variables:

$$C_p = \frac {\left(\frac{\partial U}{\partial S}\right)_v\left(\frac{\partial ^2 U}{\partial V^2}\right)_s }{\left(\frac{\partial ^2 U}{\partial V^2}\right)_s\left(\frac{\partial ^2 U}{\partial S^2}\right)_v - \left(\frac{\partial ^2 U}{\partial S \cdot\partial V}\right)^2}$$

with no inbetween steps given.

From a previous derivation of $C_v$ I know that:

$$C_v = \frac{\left(\frac{\partial U}{\partial S}\right)_v}{\left(\frac{\partial ^2 U}{\partial S^2}\right)_v}$$

and I think I understand at the least how to express the $\left(\frac{\partial p}{\partial T}\right)_v$ term, because you can express $p$ and $T$ in terms of $U, S, V$ like so:

\begin{align*} -p &= \left(\frac{\partial U}{\partial V}\right)_s \\ T &= \left(\frac{\partial U}{\partial S}\right)_v \end{align*}

The real killer is the second term $\left(\frac{\partial V}{\partial T}\right)_p$, because the only thing I know I can do with this equation is use the Maxwell relation:

$$-\left(\frac{\partial V}{\partial T}\right)_p = \left(\frac{\partial S}{\partial p}\right)_T$$

but this still leaves a constant temperature term $T$ which is not present in the final equation. So, any help with this would be really appreciated.

• Suggested reading: Physical Chemistry, Peter Atkins and Julio de Paula. Apr 11 '18 at 15:22
• Read this Ref.1 and Ref.2 Apr 11 '18 at 20:20
• In the first equation you quoted, the natural variables are either $T$ and $P$ or $T$ and $V$. On the other hand, in the equation you are trying to derive, the natural variables are $S$ and $V$. It should in principle be possible to start from the former and arrive at the latter through a series of manipulations, but it should be much easier to simply start from scratch with the natural variables $S$ and $V$. Apr 15 '18 at 4:55

Usually this part of thermodynamics is not presented in the most efficient way. To derive the expression for, e.g., $C_P - C_V$, one goes through a series of steps involving Maxwell's relations and the triple product rule. Figuring out what to do in each step is done mostly by guesswork.

However, all relations of these sorts really come from just a single mathematical relation that more generally holds. That is to say,

Consider a twice-differentiable, two-variable function $f(x,y)$ and its Legendre transform with respect to $y$, defined by $g(x,f_y) := f - f_y y$. 1 Here, $f_y$ denotes the derivative of $f$ with respect of $y$ with other independent variables held fixed. Then, the following identity is satisfied: $$\boxed{g_{xx} - f_{xx} = - \frac{f_{xy}^2}{f_{yy}} = \frac{g_{x f_y}^2}{g_{f_y f_y}}}.$$ Proof. We have $$\bigg(\frac{\partial f}{\partial x} \bigg)_{f_y} = f_x + f_y \bigg(\frac{\partial y}{\partial x}\bigg)_{f_y} = f_x - f_y \frac{f_{xy}}{f_{yy}}.$$ On the above, the first equality holds because of the chain rule [for the change of variables $(x,y)$ $\to$ $(x, f_y)$], and the second equality because of the triple product rule. Then, $$g_x = \bigg(\frac{\partial f}{\partial x} \bigg)_{f_y} - f_y\bigg(\frac{\partial y}{\partial x}\bigg)_{f_y} =f_x,$$ and $$g_{xx} = \bigg(\frac{\partial f_x}{\partial x} \bigg)_{f_y}= f_{xx} + f_{xy}\bigg(\frac{\partial y}{\partial x}\bigg)_{f_y} = f_{xx} - \frac{f_{xy}^2}{f_{yy}}.$$ Also, using the chain rule for the change of variables $(x,y)$ $\to$ $(x, f_y)$, we get $$g_{f_y} = \bigg(\frac{\partial f}{\partial f_y} \bigg)_{x} -\bigg(\frac{\partial y}{\partial f_y} \bigg)_{x}f_y - y = -y.$$ Therefore, Legendre transform of $g$ with respect to $f_y$ is simply $g - g_{f_y} f_y = f$. Then, performing a similar analysis as above for $g$ gives $$f_{xx} = g_{xx} - \frac{g_{x f_y}^2}{g_{f_y f_y}}.$$

Now, all that remains to be done is to choose the variables and functions $x$, $y$, $f$ and $g$. Below are some examples.

(1) $x = S$, $y = V$, $f = U$, $g = H$, and $f_y = -P$: $$H_{SS} - U_{SS} = -\frac{U_{SV}^2}{U_{VV}}.$$ Note that $$U_{SS} = \bigg(\frac{\partial^2 U}{\partial S^2}\bigg)_V = \bigg(\frac{\partial T}{\partial S}\bigg)_V = \frac{T}{T(\partial S/\partial T)_V} = \frac{T}{C_V},$$ and that, similarly, $$H_{SS} = \bigg(\frac{\partial^2 H}{\partial S^2}\bigg)_P = \bigg(\frac{\partial T}{\partial S}\bigg)_P = \frac{T}{T(\partial S/\partial T)_P} = \frac{T}{C_P}.$$ Hence, we have $$\boxed{\frac{1}{C_P} - \frac{1}{C_V} = -\frac{U_{SV}^2}{T U_{VV}} = -\frac{\Big(\frac{\partial^2 U}{\partial S \partial V}\Big)^2}{\Big(\frac{\partial U}{\partial S}\Big)_V \Big(\frac{\partial^2 U}{\partial V^2}\Big)_S}}$$ which is equivalent to the relation OP wanted to derive.

(2) $x = T$, $y = V$, $f = A$, $g = G$, $f_y = -P$, and noting that $f_{-y} = -f_y$, $$G_{TT} - A_{TT} = \frac{G_{T,-P}^2}{G_{-P,-P}} = \frac{G_{TP}^2}{G_{PP}}.$$ Here, $$A_{TT} = \bigg(\frac{\partial^2 A}{\partial T^2}\bigg)_V = -\bigg(\frac{\partial S}{\partial T}\bigg)_V = - \frac{1}{T}C_V,$$ $$G_{TT} = \bigg(\frac{\partial^2 G}{\partial T^2}\bigg)_P = -\bigg(\frac{\partial S}{\partial T}\bigg)_P = - \frac{1}{T}C_P,$$ $$G_{TP} = \frac{\partial^2 G}{\partial T \partial P} = \bigg(\frac{\partial V}{\partial T}\bigg)_P = V \alpha,$$ where $\alpha := \frac{1}{V}\Big(\frac{\partial V}{\partial T}\Big)_P$ is the volume expansion coefficient, $$G_{PP} = \bigg(\frac{\partial^2 G}{\partial P^2}\bigg)_T = \bigg(\frac{\partial V}{\partial P}\bigg)_T = - V\kappa_T,$$ where $\kappa_T := -\frac{1}{V}\Big(\frac{\partial V}{\partial P}\Big)_T$ is the isothermal compressibility. Then, it follows that $$\boxed{C_p - C_V = \frac{TV\alpha^2}{\kappa_T}}.$$

(3) $x = V$, $y = S$, $f = H$, $g = G$, and $f_y = T$: $$G_{PP} - H_{PP} = \frac{G_{TP}^2}{G_{TT}}.$$ We have $$H_{PP} = \bigg(\frac{\partial^2 H}{\partial P^2}\bigg)_S = \bigg(\frac{\partial V}{\partial P}\bigg)_S = - V\kappa_S,$$ where $\kappa_S := -\frac{1}{V}\Big(\frac{\partial V}{\partial P}\Big)_S$ is the adiabatic compressibility. We have already shown that $$G_{PP} = -V\kappa_T, \quad G_{TP} = V \alpha, \quad G_{TT} = - \frac{C_P}{T},$$ whence we obtain $$\boxed{\kappa_T - \kappa_S = \frac{TV\alpha^2}{C_P}}.$$

1 To be more precise, one should first write down $f(x,y) - y f_y$ and then express $y$ as a function of $x$ and $f_y$. Also, for the Legendre transform to exist, $f(x,y)$ should be a convex or concave function of $y$.