PHYSICS 313
Problem set 5 2002
Given Oct 18
Due Oct 28

Problem 1: It is found that for a certain substance the internal energy $U$ depends on the volume $V$, the number of particles $N$ and the entropy $S$ according to

$$U=const.\; N (\frac{N}{V})^{0.4}\exp(\frac{0.4\;S}{Nk_B})$$

a: Show that the substance satisfies the ideal gas law.
b:Find the coefficient $\gamma$ in the adiabatic equation of state.

$$PV^\gamma=const.$$

Problem 2: A certain device undergoes a thermodynamic cycle where the working substance can be taken to be an ideal gas with $\gamma=C_P/C_V=1.4$. The system starts with the pressure $P_A=1$ bar, temperature $T_A$. It is compressed adiabatically until the pressure is $P_B=10$ bar. Next it is heated and expands at constant pressure to temperature $T_C$. It next expands adiabatically until the pressure returns to $P_A$. Finally it is cooled and compressed at constant pressure until the temperature returns to its original value $T_A$.
a: Will the device work as a heat engine or a heat pump?
b: Show that the coefficient of performance is independent of $T_A$ and $T_C$ and find the COP.
Problem 3: One mol of an ideal gas is mixed with 2 mols of another ideal gas. Initially the pressure of both gases is 1 bar and the temperature is $T=300$ K. In the final state the pressure and temperature is unchanged. Find the entropy of mixing.