Expanding state variables and state functions of a thermodynamic system

Question

In this Wikipedia article under the section "Heat capacities of a homogeneous system undergoing different thermodynamic processes" there is on line that says: $$ \delta Q=dU+pdV=\bigg(\frac{\partial U}{\partial p}\bigg)_Tdp+\bigg(\frac{\partial U}{\partial T}\bigg)_pdT+p\Bigg[\bigg(\frac{\partial V}{\partial T}\bigg)_pdT+\bigg(\frac{\partial V}{\partial p}\bigg)_Tdp\Bigg] $$ So $U$ and $V$ are assumed to be a function of $p,T$. I saw this step already a few times that a differential of a state function or state variable gets expanded like this. My question now would if, in general, every state function and state variable can get extended with every variable. So for example could I also expand $U$ with $S,V$? like this: $$ dU=\bigg(\frac{\partial U}{\partial S}\bigg)_VdS+\bigg(\frac{\partial U}{\partial V}\bigg)_SdV $$ Or maybe with 4 variables $S,V,p,T$? I know that one requirement is that they are not assumed to be constant. So here $dS,dV,dp,dT\neq0$. But are there any other things I have to keep in mind when expanding a state function like $U$ or a state Variable like $V$?

Answer 1

For a pure homogeneous phase, Gibbs's rule gives 2 independent variables: $$ (\text{# components})-(\text{# phases}) + 2 = 2 $$ This is the number of independent variables we need to describe the intensive properties of the mixture. In principle we can choose any two among $\{p, T, V, U, S \cdots\}$. Pressure and temperature is a very convenient pair because we can easily measure them and control them experimentally. However, any other set can be used. The equation you wrote, $$ dU = \left(\frac{\partial U}{\partial S}\bigg)_VdS + \right(\frac{\partial U}{\partial V}\bigg)_SdV $$ is correct and is equivalent to $$ dU = T dS - p dV $$

Mixtures The number of independent variables is always determined by Gibbs's rule. In a two-component one-phase system we have three independent variables. We normally choose the first two to come from $p, T, V, U, S \cdots$ and the third one is the mol fraction of one of the two components. Again, this is the number of variables we need to describe the intensive properties of the mixture. We may also write equations for the extensive properties of mixture but I will not go there unless there is a question about it.