# Carnot engine efficiency

## Homework Statement

A possible ideal gas cycle operates as follows:

(i) From an initial state $$(p_1,V_1)$$, the gas is cooled at constant pressure to $$(p_1,V_2)$$;

(ii) the gas is heated at constant volume to $$(p_2,V_2)$$;

(iii) the gas expands adiabatically back to $$(p_1,V_1)$$.

Assuming constant heat capacities, show that the thermal efficiency is

$$1 - \gamma\frac{(V_1/V_2)-1}{(p_2/p_1) - 1}$$

where $$\gamma = c_p/c_v$$

## Homework Equations

Carnot efficiency: $$\nu = \frac{W}{Q_H} = 1 - \frac{T_l}{T_h}$$

In an adiabatic process, $$pV^{\gamma},TV^{\gamma-1},p^{1-\gamma}T^{\gamma}$$ are all constant.

## The Attempt at a Solution

I've been spinning my wheels a lot with this one, and I think the issue may be algebraic. My first thought was that I'd calculate the work done in the cycle. For part (i), it's just $$p(V_1-V_2)$$. For part (ii), it's zero, since dV = 0. For part (iii), I used the fact that since it's adiabatic, $$pV^{\gamma}$$ is constant, which I'll call k. Then the work becomes

$$W_3 = \frac{k}{\gamma-1}(V_2^{1-\gamma} - V_1^{1-\gamma})$$

You can set $$k = p_1V_1^{\gamma} = p_2V_2^{\gamma}$$, but there doesn't seem to be a preferable way. Adding these together gives you

$$W = p(V_1-V_2) + \frac{p_iV_i^{\gamma}}{\gamma-1}(V_2^{1-\gamma} - V_1^{1-\gamma})$$

Since none of these processes are isothermal, I can't figure out a meaningful expression for $$Q_H$$, so I'm stuck here. Any help is very much appreciated!

Andrew Mason