# All possible choices for two independent thermodynamic variables for a one-component, one-phase system

I am confused about the independent variables in thermodynamics. I know that for a one-component, one-phase system, there are only two independent intensive variables that can be chosen and which values will determine all other ones. I would like to know which variables we can choose for the intensive variables. As I understand, it must not be pressure and temperature (the natural variables of the internal energy) but can be any intensive variable, so two from a pool of variables like:

$$I = \{p, T, \rho = m/V, M = m/n, V_m, \dots\}$$

Are we really free to choose any 2 of those? I can e.g. not imagine choosing the molar mass $$M$$ and $$p$$ to describe all properties of an ideal gas, as the ideal gas equation can be reformulated as:

$$pV=nRT \Leftrightarrow pm/\rho =nRT \Leftrightarrow pM/\rho=RT$$

which means that for any choice of $$M$$ and $$p$$, $$T$$ is not determined, since another variable $$\rho$$ needs to be chosen.

Edit:

I think my question was too specific on the molar mass. Actually, I wanted to quite generally know how we can choose them. For example, if we include also extensive variables, such as conjugates of the intensive ones, or even $$U$$, $$H$$, $$G$$ and $$A$$ can the independent variables still be chosen completely freely?

• Could we e.g. take $$U$$ and $$H$$ as our two independent variables?
• Could we take $$p$$ and $$V$$ as out two independent variables?

In addition to p and T, the third variable is either molar density $$\rho=\frac{m}{MV}=\frac{n}{V}$$ or its reciprocal, molar volume $$V_m=\frac{1}{\rho}=\frac{MV}{m}=\frac{V}{n}$$.
Similarly, if you wish so, you can ignore the total amount of mass/volume of the system and look only at its local thermal pressure/response but then you are ignoring, for example, surface effects. In general, you need two equations of state to describe a thermostatic system: (1) thermal equation of state that is a relationship among the directly measurable but not necessarily controllable variables including temperature $$f(T, p,\rho, V, \mu, \mathcal E,\mathcal M, \mathcal B,...)=0$$ (2) and a caloric equation of state that describes how energy/entropy/work are related: $$g_1(U, T, p,\rho, V, \mu, \mathcal E,\mathcal M, \mathcal B,...)=0$$ or $$g_2(S, T, p,\rho, V, \mu, \mathcal E,\mathcal M, \mathcal B,...)=0.$$ For the simplest case the equations can be $$T=T(p,V)$$ or $$f=f(T,p,\rho), ...$$ and $$U=U(T,p)$$, or $$U=U(p, v)$$, or $$S=S(T,\rho)$$, $$g_1(U,p,v)=0$$, etc.$• Thanks. Of course, the choice of variables should be made regarding the system. But is it true that mathematically we are really free to choose any 2 variables from the set of conjugate thermodynamic variables (p/V, T/S, \dots) as the independent variables of a specific coordinate patch of the thermodynamic manifold? Let's say we have an open system, for which we need to define 3 independent variables and we choose$\rho=m/V$,$V$and$m\$, would these not be linearly dependent? Commented Mar 8, 2021 at 1:59