Carnot cycle, fundamental equation of ideal gas

[fluidistic]

Homework Statement

Assuming that the auxiliary system in the Carnot cycle is a monoatomic ideal gas whose fundamental equation is $S=\frac{NS_0}{N_0} +NR \ln \left [ \left ( \frac{U}{U_0} \right ) ^{3/2} \left ( \frac{V}{V_0} \right ) \left ( \frac{N}{N_0} \right ) ^{-5/2} \right ]$.
1)Find the adiabats in a diagram T-V.
2)Sketch a P-V diagram of the cycle.
3)Describe the operation of a cycle as a refrigerator.
4)Describe the functionning of a Carnot cycle as a heat pump.

Homework Equations

$dS=0$.

The Attempt at a Solution

For 1) my strategy is to first use $dS=0= \frac{\partial S} {\partial N}dN+\frac{\partial S} {\partial U}dU+\frac{\partial S} {\partial V}dV$ then isolate V in function of T. But after having done all the derivatives, the remaining equation is still quite complicated.
Oh wait... there's no variable "T" inside it. Maybe I should get T as $\frac{1}{\frac{\partial S }{\partial U}}$? Then I consider all variables but "V" as constants? And then graph that function?Edit: This cannot be, I get $T=\frac{2U}{3NR}$ that way. Absolutely no dependence between T and V...

 

[NateD]

For 2): Isothermal processes are horizontal lines in a P-V diagram, while adiabatic processes are oblique lines. As such, the four processes in the cycle are two isothermals and two adiabatics.For 3) A refrigerator operates by taking heat from a cold space, transferring it to a hot space, and then rejecting the heat to the surrounding environment. In Carnot cycle, the heat is taken from a low temperature reservoir (cold space), and rejected to a high temperature reservoir (hot space).For 4) A heat pump takes heat from a low temperature reservoir and transfers it to a higher temperature reservoir. In a Carnot cycle, heat is taken from a low temperature reservoir (cold space) and transfered to a higher temperature reservoir (hot space).