An ideal gas of n mol is initially at pressure P1, volume V1, and temperature Th. It expands isothermally until its pressure and volume are P2 and V2. It then expands adiabatically until its temperature is Tc and its pressure and volume are P3 and V3. It is then compressed isothermally until it is at a pressure P4 and a volume V4, which is related to its initial volume V1 by TcV4y–1 = ThV1y–1. The gas is then compressed adiabatically until it is back in its original state.

(a) Assuming that each process is quasi-static, plot this cycle on a PV diagram. (This cycle is known as the Carnot cycle for an ideal gas.)

(b) Show that the heat Qh absorbed during the isothermal expansion at Th is Qh = nRTh ln(V2/V1).

(c) Show that the heat Qc given off by the gas during the isothermal compression at Tc is c = nRTc ln(V3/V4).

(d) Using the result that TVy-1 is constant for an adiabatic expansion, show that V2 / V1 = V3/V4.

(e) The efficiency of a Carnot cycle is defined to be the net work done divided by the heat absorbed Qh. Using the first law of thermodynamics, show that the efficiency is 1–Qc/Qh.

(f) Using your results from the previous parts of this problem, show that Qc/Qh = Tc/Th.