How to find $dT$? Thermodynamics (equation of state)

Question

I recently had my first lesson thermodynamics, and there is an exercises that I don't quite understand how you have to get the answer. It goes as follows:

$$PV^{2}=nRe^{3T}$$

we are given that n en R are both constants. The question is what is $dT$?

I don't think I have the right mathematical background to solve this question. I really would love to know where I could find the name of such kind problems. I now how to derive (and partial derivation), but I'm used to working with $\frac{dy}{dx}$, but know it's only $dy$ or in my case $dT$.

If someone could help, solve this problem and where I can find theory to practice such mathematical problems that would be very helpful

Thanks in advance

Answer 1

I think the topic you want to review is the Chain Rule for partial derivatives, which seems simple but can get very tricky. First, solve your equation of state for T. With n and R constant you have T(P,V). So dT = ∂T/∂P dP + ∂T/∂V dV. Take the partial derivatives of your T(P,V) and plug them in.
I get T = $\frac{1}{3}(ln P + 2 ln V - ln(nR)$ So I get ∂T/∂P = $\frac{1}{3P}$ and $∂T/∂V = \frac{2}{3V}$ If you have a very hard time with differentials in particular, just divide all of them by ds or dt (small t) or some other variable not involved. Then they are in effect derivatives.

Answer 2

The problem is asking you to consider the differential form of the equation of state: \begin{align} PV^{2}&=nRe^{3T}\tag{1}\\ \implies 2PV\,dV + V^{2}\,dP &=3nRe^{3T}\,dT \end{align} where here we are thinking of $dV, dP, dT$ as infinitesimal changes in the quantities. So, we get $$dT = \frac{2PV\,dV + V^{2}\,dP}{3nRe^{3T}}.$$ To remove the dependence on $T$ on the right hand side above, we can use $(1)$ to obtain $$dT = \frac{2PV\,dV + V^{2}\,dP}{3PV^{2}} = \frac{2P\,dV + V\,dP}{3PV}.$$