Answer to Question #148392 in Molecular Physics | Thermodynamics for maxxam

Question

In thermodynamics the free energy is a function of the volume V and the temperature T. These two variables are called independent. On the other hand on can find the volume and the temperature as a part of the ideal gas equation, which means that they are related to each other. Therefore I dont understand, why they are named independent.

Answer 1

Let's imagine that we have a thermodynamical system. What do we need to describe it? The most common way is to use any two variables from $P, V, T, S$. All thermodynamic potentials are written in their native variables, namely $U(S, V) , H(S, P), G(P, T), F(V, T)$ . But only two (any two!) out of four variables are independent. If we change, for example, volume and temperature, we can get P and S from the known equations. In the same way we can change T and S, and then find P and V. What does it mean to be independent in mathematical sense? Recall $y=f(x)$ , y is dependent variable, x is independent. We can change x as much as we want and get y that correspond to it. Going into higher dimensions, consider $z=f(x,y)$. Again, x and y are independent variables and z depends on both x and y. For example, let z be a volume of cylinder, x - height, y - area of the base. If we change heigh making the cylinder higher, the volume will be bigger, but what happens to the area of the base? Nothing! It doesn't depend on change of the heigh. So x and y are independent.

In the same sense, from $pV = nRT$ we see that $P=f(V,T)$ , then V and T are independent, no matter how we change V, it doesn't affect T at all; only P is dependent variable. We can also express $V=f(P,T)$ or $T=f(V,P)$. In any case, there will be two independent variables.