Why in thermodynamics it is sufficient to know only two intensive parameters of state to derive all other information? I have encountered many times in thermodynamics books and lectures, information that it is sufficient to know only two independent parameters of state, to derive all other parameters of a given thermodynamic system. Unfortunately, it always was written without any proof. Can someone show me how to prove this statement?
 A: Two parameters are sufficient to describe a closed system on which one can perform one type of work (e.g., pressure–volume work). There are two ways one can add energy $U$ to such a system: by heating it or by doing work on it. The former is driven by a temperature difference and results in the distribution of particle energies being broadened. The latter is driven by a nonthermal gradient (in pressure, for example), and results in the particle energies being elevated together in concert. The corresponding extensive parameters that are shifted during these processes are the entropy $S$ and volume $V$. Thus, we can write the so-called fundamental relation as
$$dU=\left(\frac{\partial U}{\partial S}\right)_VdS+\left(\frac{\partial U}{\partial V}\right)_SdV.$$
Other parameters can be calculated from the relationships among the energy $U$ and the so-called natural variables $S$ and $V$. For example, temperature $T$ and pressure $P$ are defined as $T\equiv \left(\frac{\partial U}{\partial S}\right)_V$ and $P\equiv -\left(\frac{\partial U}{\partial V}\right)_S$, respectively, where the minus sign appears because pressurizing a system reduces its volume.
As another example, for an open two-phase system that involves only electrical work, we would write
$$dU=\left(\frac{\partial U}{\partial S}\right)_{Q,N_1,N_2}dS+\left(\frac{\partial U}{\partial Q}\right)_{S,N_1,N_2}dQ+\left(\frac{\partial U}{\partial N_1}\right)_{S,Q,N_2}dN_1+\left(\frac{\partial U}{\partial N_2}\right)_{S,Q,N_1}dN_2,$$
where $Q$ is charge and $N_1$ and $N_2$ are the amounts of the two phases. Now there are four natural variables, namely, $S$, $Q$, $N_1$, and $N_2$, because we could heat the system, do electrical work, add phase 1, or add phase 2. Defining some more intensive variables, we could rewrite that expression as
$$dU=T\,dS+E\,dQ+\mu_1\,dN_1+\mu_2\,dN_2,$$
where $E$ is the electric field and $\mu$ is the chemical potential.
More generally, the number of natural variables is $m+n+1$ when the fundamental relation is
$$dU=T\,dS+\sum_{i=1}^mF_i\,dq_i+\sum_{j=1}^n\mu_jN_j,$$
where $F_i$ and $q_i$ are a set of $m$ generalized forces $F$ and generalized displacements $q$, corresponding to $m$ types of work, and where there are $n$ phases.
