Calculating work and heat transfer in this Carnot process

[approx12]

Homework Statement

I'm given the fundamental equation $UN^{1/2}V^{3/2}=\alpha (S-R)^3$, where $A=2*10^{-2} (K^3 m^{9/2} J^3)$. Two moles of this fluid are used as the auxiliary system in a Carnot cycle, operating between two reservoirs of $T_1=373K$ and $T_2=273K$ In the first isothermal expansion $10^6 J$ is extracted from the high-temperature reservoir.
Find the heat transfer and the work transfer for each of the four processes in the Carnot cycle.

Relevant Equations

$UN^{1/2}V^{3/2}=A(S-R)^3$

Hey guys! This is problem from Callens Thermodynamics textbook and I'm stuck with it.

My goal was to get a expression for the entropy $S$ which is dependent on $T$ so I can move into the $T-S$-plane to do my calculations:
I startet by expressing the fundamental equation as a function of $S(U,V,N)$ and then computing the partial derivative with respect to $U$: $$\frac{\partial S}{\partial U}=\frac{1}{T}=\frac{1}{3\alpha}U^{-2/3}N^{1/6}V^{1/2}$$ Through that I'm able to get a expression of $U(T,V,N)$ which helps me to express $S$ as a function of $T,V$ and $N$. Rearranging the partial derivative and plugging back into the original equation I get: $$S(T,V,N)=R+\sqrt{\frac{T}{3}}\frac{1}{\alpha^{3/2}}V^{3/4}N^{1/4}$$

Now it shouldn't be so hard to calculate the the heat and work transfer but what I'm missing is the volume $V$. It's not given in the problem statement and without it I can't calculate $S$ so I'm stuck with my calculations in the $T-S$-plane.

I must be missing something here but I can't see what it is. I would appreciate any guidance in the right direction!

 

[Chestermiller]

I would start out by deriving the equation of state.