partial derivative (Thermodynamics)

Question

Please help with this problem, I don't understand how to approach it or how to interpret the partial derivatives

This is an introductory math class, not a thermo class.

Answer 1

You have a bunch of physical quantities, and they are related to each other somehow. Whenever confusion arises in such a situation, I recommend using separate names for the functions that describe these relationships, rather than reusing the names of the quantities as the names of those functions too.

So instead of writing $U=U(T,V,n)$ like physicists usually do, be more of a fundamentalist mathematician for a moment, and write $$ U = f(T,V,n) $$ with a mathematical function $f$ which tells you how the physical quantity $U$ depends on the physical quantities $T$, $V$ and $n$.

And similarly write $$ U = g(T,P,n) $$ where the function $g$ tells you how $U$ depends on $T$, $P$ and $n$.

These functions $f$ and $g$ are of course not independent, since there is also a relation $$ P = h(T,V,n) $$ given by some function $h$, and the second expression for $U$ has to give the same value as the first expression when you let the value of $P$ be given by $T$, $V$ and $n$ like that. In other words, $$ U = f(T,V,n) = g(T,\underbrace{h(T,V,n)}_{=P},n) . \tag{$*$} $$

Now, the notation $\left( \frac{\partial U}{\partial T} \right)_{V,n}$ on the left-hand side of your equation means the partial derivative of $U$ where you let $T$ vary while keeping $V$ and $n$ constant; in our notation this is nothing but the partial derivative of the function $f$ with respect to the variable $T$: $$ \left( \frac{\partial U}{\partial T} \right)_{V,n} = \frac{\partial f}{\partial T}(T,V,n) . $$ And similarly the partial derivatives on the right-hand side are really partial derivatives of the functions $g$ and $h$ (where it's understood that after you have differentiated, the value of $P$ is given by $P=h(T,V,n)$). For example, $$ \left( \frac{\partial U}{\partial T} \right)_{P,n} = \frac{\partial g}{\partial T}(T,P,n) = \frac{\partial g}{\partial T}(T,\underbrace{h(T,V,n)}_{=P},n) . $$ Can you see now how your formula follows from taking the partial derivative with respect to $T$ of equation ($*$) using the multivariable chain rule? $$ \begin{aligned} f(T,V,n) &= g(T,h(T,V,n),n) \\[1em] \implies \frac{\partial f}{\partial T}(T,V,n) &= \cdots \end{aligned} $$ (I'm leaving this last step for you, so that I'm not spoiling the whole exercise!)