In the Carnot cycle, a gas is heated at constant temperature TH and cooled at constant temperature TC. Furthermore, no other heat transfer occurs, and all other steps of the cycle are reversible. The laws of thermodynamics state that any such cycle must have efficiency η=W/Qin=1−(TC/TH). Below we will explore two other heat engines, which recover this efficiency in certain limits.
a. Consider the following heat engine involving one mole of ideal monatomic gas. The gas begins at temperature T0, pressure P0, and volume V0, and undergoes four reversible steps.
[Figure: P-V diagram showing cycle with points at (V₀, P₀), labeled temperatures T₀ and (1+ε)T₀, and pressure P₀/α]
1. The gas is expanded at constant pressure until its temperature rises to (1+ϵ)T0.
2. The gas is expanded at constant temperature until its pressure falls to P0/α.
3. The gas is contracted at constant pressure until its temperature falls back to T0.
4. The gas is contracted at constant temperature until its pressure rises back to P0.
i. Which steps require heat to be transferred to the gas? For each such step, give the total heat input in terms of P0, V0, α, and ϵ.
ii. Under what conditions on α and ϵ would we expect the efficiency of this heat engine to approach that of a Carnot cycle working between the same maximum and minimum temperatures?
iii. Find the efficiency of this heat engine for general α and ϵ.
b. Now consider a heat engine built around the freezing and melting of water, which occurs at a pressure-dependent temperature Tc(P). Initially, a volume of V of water is squeezed underneath a piston, so that it experiences a total pressure P1, and the water is on the edge of freezing, with temperature Tc(P1). The engine then undergoes four reversible steps.
[Figure: P-T diagram showing rectangular cycle with corners at (T_c(P₁), P₁), (T_c(P₁), P₂), (T_c(P₂), P₂), (T_c(P₂), P₁)]
1. A mass is slowly placed on the piston, raising the total pressure to P2.
2. The water is cooled to temperature Tc(P2) and frozen.
3. The mass is slowly removed from the piston, lowering the pressure back to P1.
4. The ice is heated back to temperature Tc(P1) and melted.
Assume that water and ice are incompressible, with fixed densities ρw and ρi.
i. What is the net work done by this engine, in terms of P1, P2, V, and the densities?
ii. Assume the latent heat per unit mass L to melt ice is large, so that freezing and melting account for essentially all of the heat transfer in the cycle. What is the efficiency of the engine, in terms of P1, P2, L