What Formula Should Be Used for Thermodynamics of Two Gases?

[fluidistic]

Homework Statement

Two particular systems separated by a diathermic wall have the following equations of state:
$\frac{1}{T^{(1)}}=\frac{3}{2}R \frac{N^{(1)}}{U^{(1)}}$
$\frac{1}{T^{(2)}}=\frac{5}{2}R \frac{N^{(1)}}{U^{(2)}}$ where R=1.986 cal/mol K, $N^{(1)}=2$ and $N^{(1)}=3$.

Homework Equations

Euler relation in entropy representation: $S=\sum _j F_j X_j$. In energy representation: $U=TS+\sum _j P_j X_j$.
Gibbs-Duhem relation under the entropy and energy form respectively: $\sum _j X_j dF_j=0$, $SdT+\sum _j X_j dP_j =0$.

The Attempt at a Solution

I simply don't know what formula to use and how exactly. Here they don't give the pressure so this seems really hard to use any formula. I don't know if I can use the ideal gas relations $PV=nRT$ and $U=KT+U_0$, they don't say anything about the gases.
I realize that V is constant.
Any help on getting me started will be appreciated!

 

[mark726]

Hello,

Thank you for your post. I understand your confusion, as the given equations of state do not mention any specific gases or pressures. However, there are a few things we can deduce from the equations that can help us solve the problem.

Firstly, we can see that the systems are separated by a diathermic wall, which means that heat can freely flow between them. This implies that the systems are in thermal equilibrium with each other, and therefore they must have the same temperature (T) at all times.

Secondly, we can see that the equations of state involve two different gases, with different values of N (number of moles) and U (internal energy). This suggests that the two systems are not identical, but rather have different properties.

Now, let's take a closer look at the equations of state. We can rewrite them as:

T^{(1)}=\frac{2}{3}R \frac{U^{(1)}}{N^{(1)}}
T^{(2)}=\frac{2}{5}R \frac{U^{(2)}}{N^{(2)}}

From these equations, we can see that the temperature of each system is directly proportional to its internal energy and inversely proportional to its number of moles. This is similar to the ideal gas law, where PV=nRT, but here we are dealing with internal energy instead of pressure and volume.

Using the ideal gas law, we can also write the following equation for each system:

P^{(1)}V^{(1)}=n^{(1)}RT^{(1)}
P^{(2)}V^{(2)}=n^{(2)}RT^{(2)}

Since the systems are separated by a diathermic wall, we can also assume that their volumes are constant. This means that the pressures must be equal:

P^{(1)}=P^{(2)}=P

Now, we can use the Gibbs-Duhem relation in the energy representation to relate the internal energies of the two systems:

U^{(1)}=TS^{(1)}+\sum _j P_j X_j^{(1)}
U^{(2)}=TS^{(2)}+\sum _j P_j X_j^{(2)}

Since the systems are in thermal equilibrium, their temperatures are the same. Also, since the systems are separated by a diathermic wall, the only extensive variables that