# 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!

Delta2