Question: After Count Rumford (Benjamin Thompson) and James Prescott Joule had shown the equivalence of mechanical energy and heat, it was natural that engineers believed it possible to make a "heat engine" (e.g., a steam engine) that would convert heat completely into mechanical energy. Sadi Carnot considered a hypothetical piston engine that contained n moles of an ideal gas, showing first that it was reversible, and most importantly that?regardless of the specific heat of the gas?it had limited efficiency, defined as e=W/Qh, where W is the net work done by the engine and Qh is the quantity of heat put into the engine at a (high) temperature Th. Furthermore, he showed that the engine must necessarily put an amount of heat Qc back into a heat reservoir at a lower temperature Tc. The cycle associated with a Carnot engine is known as a Carnot cycle. A pV plot of the Carnot cycle is shown in the figure. (Figure 1) The working gas first expands isothermally from state A to state B, absorbing heat Qh from a reservoir at temperature Th. The gas then expands adiabatically until it reaches a temperature Tc, in state C. The gas is compressed isothermally to state D, giving off heat Qc. Finally, the gas is adiabatically compressed to state A, its original state.

B) Find the total work W done by the gas after it completes a single Carnot cycle. Express the work in terms of any or all of the quantities Qh, Th, Qc, and Tc.

C) Suppose there are n moles of the ideal gas, and the volumes of the gas in states A and Bare, respectively, VA and VB. Find Qh, the heat absorbed by the gas as it expands from state A to state B. Express the heat absorbed by the gas in terms of n, VA, VB, the temperature of the hot reservoir, Th, and the gas constant R.

D) The volume of the gas in state C is VC, and its volume in state D is VD. Find Qc, the magnitude of the heat that flows out of the gas as it proceeds from state C to state D. Express your answer in terms of n, VC, VD, Tc (the temperature of the cold reservoir), and R.

E) Now, by considering the adiabatic processes (from B to C and from D to A), find the ratio VC/VD in terms of VA and VB.

F) Using your expressions for Qh and Qc (found in Parts C and D), and your result from Part E, find a simplified expression for Qc/Qh. No volume variables should appear in your expression, nor should any constants (e.g., n or R).