Find the equation of state of each gas

[curious_mind]

Homework Statement

Consider three gases with $(P_1,V_1),(P_2,V_2)$ and $(P_3,V_3)$. It is found that when the first two are in equilibrium the following condition is satisfied: $P_1V_1 =\left(P_2 +\frac{a}{V_2^2}\right)(V_2 −b)$, while the equation satisfied when the first and the last are in equilibrium is $P_3(V_3 −c)=P_1V_1 e^{\frac{−d}{V_3P_1V_1}}$. Find the respective equations of state and identify them.

Relevant Equations

Equation of states of gas at temperature T$f(P,V,T)=0$

The problem is from the book "The Principles of Thermodynamics" by ND Hari dass.

It looks trivial problem, but I am not able to form logical arguements for going into next step.

For example, It seems like first gas has equation of state $PV =nRT$ and second has $\left( P_2 +\frac{a}{V_2^2} \right) (V_2 −b) = nRT$
But I cannot straightforward assume Right hand side of equation of state to be simply $nRT$ in general right ? So what could be valid way to proceed from thermodynamical laws ?

Thanks.

 

[haruspex]

For the relationships to hold across all T, each relationship must be of the form (first expression =second expression = some function of T).

 

[curious_mind]

If we equate all three relations, then it will be valid only if all three gases in equilibrium, which is not required to be found. We require to find all three gases equation of state separately, at different temperatures.

Also, how can we say that it individual $f(P,V)$ is some function of $T$ ONLY?. Right hand side of equation of state might containt terms like $\cos (TVe^P)$ etc etc or something, in general - right m? Or am I missing something very fundamental?

 

[haruspex]

If we equate all three relations

I did not say that. The "some function of T" does not have to be the same for each.

Also, how can we say that it individual $f(P,V)$ is some function of $T$ ONLY?.

It is the same principle as "separation of variables" in PDEs.
We know that $T_1=g_1(P_1,V_1)$ and $T_2=g_2(P_2,V_12)$ for some functions $g_1, g_2$. So at any given temperature T we know $g_1(P_1,V_1)=g_2(P_2,V_12)$. And these are the forms you are given.

 

[Chestermiller]

It seems to me your original approach was correct, except I would assume the V is molar volume rather than volume itself, so you would get rid of the n's in the equations.