Temperature Dependence of Thermal Efficiency in Carnot Cycle

Question

Suppose a Carnot Cycle operates between a hot reservoir at temp $T_h$ and a cold reservoir at temp $T_c$. Suppose also that the system is initially at point $(V_1,p_1)$ in the pV-plane and that the subsequent states of the system in the pV-plane are $(V_2,p_2)$, $(V_3,p_3)$ and $(V_4,p_4)$, respectively. Finally, suppose also that the heat absorbed by the system during isothermal expansion is $Q_{12}$ and that the heat surrendered to the cold reservoir during isothermal compression is $Q_{43}$. Then, the thermal efficiency of the cycle is given by

\begin{equation} \eta = 1 - \frac{Q_{43}}{Q_{12}} = 1 - \frac{T_c}{T_h} \quad (1) \end{equation}

However, by the Ideal Gas Law, the above can be written as

$$ \eta = 1 - \frac{T_c\cdot \ln(V_3/V_4)}{T_h\cdot \ln(V_2/V_1)} $$

Now, for the second equality in (1) to hold, it must be the case that $V_3/V_4 = V_2/V_1$. How do I know apriori that this is true? There's no obvious reason I can discern that the above statement would necessarily be true.

Answer 1

Write your result as $$ \eta = 1 - \frac{T_c \Delta S_{34}}{T_h\Delta S_{21}} $$ and recall that the Carnot cycle is reversible, so that $\Delta S_{12}+\Delta S_{34}=0$, which gives the correct efficiency.

This answers your question indirectly (and not only for ideal gas): Since your equation is correct, you ought to be able to prove it. Fix enough information to calculate the pressure and volume in all states, then show the relationship between the volumes. Or follow the hint in the comment by @naturallyInconsistent.

Answer 2

You just have to equate the volumes through their laws in the cicle, two isothermal (1-2 & 3-4) and two adiabatic curves (2-3 & 4-1). In the first you have (i) $p_1V_1 = p_2V_2$ (iii) $p_3 V_3 = p_4 V_4$ Then for the adiabatic curves, (ii) $p_2 V_2^\alpha = p_4 V_4^\alpha$ (iv) $p_2 V_2^\alpha = p_4 V_4^\alpha$ in which $\alpha$ depends on the gas.

Now, working with eqs. (i) to (iv) you can demonstrate that your (1) is the same as your second equation, that is $V_3/V_4=V_2/V_1$.