Which formula for entropy is correct? (OR Is the fundamental thermodynamic equality always right?) Quick question, in the lecture notes to a thermodynamic course I'm taking,
$$d\underline{S}=\frac{d\underline{U}}{T}+\Big(\frac{\partial P}{\partial T}\Big)_{\underline{V}} d\underline{V}$$
But everywhere else I've looked I've found (by rearranging the fundamental thermodynamic relation:
$$d\underline{S}=\frac{d\underline{U}}{T}+\frac{P}{T}d\underline{V}$$
The two are obviously extremely similar, however I don't believe they're equivalent in general (I believe you'd have to make further assumptions about the system, particularly $P=kT$ at constant specific volume, to go from one to the other.)
Right now I'm just asking if anyone knows which relation is the more general. I was of the opinion that the fundamental thermodynamic relation was always correct, but now I'm doubting myself. Thanks.
 A: Both seem a little strange. So let us see where they are coming from. What we know is from the laws of thermodynamics that the total differential of the internal energy $U$ can be written as $$dU = \delta Q + dW$$
Here $\delta Q$ is an infinitesimal amound of thermal energy and $dW$ an infinitesimal amount of work. The work is the product of a generalized displacement times a generalized force, which are together a conjugated pair of variables. One such pair is pressure and volume, $P\, dV$. For a reversible process $\delta Q = T dS$, so that we get
$$dU = T\,dS - P\,dV$$
From here we easily arrive at your second equation:
$$ \frac{dU}{T} + \frac{P}{T} dV = dS$$
Now we compare with your first equation, which can only be true if$$\frac{P}{T} = \left(\frac{\partial P}{\partial T}\right)_V$$
which  certainly does not hold for all thermodynamic systems. So your first equation is more general and the second one only correct in specific circumstances. We rearrange the condition $$P(T,V) = \left(\frac{\partial P}{\partial T}\right)_V T$$ which can only be satisfied by a general function $f(V)$ and $$P(T,V) = f(V) \,T$$
