Apparent contradiction concerning the net work of a Carnot cycle

Question

The net work of a Carnot cycle, $W$, is $$\tag{1}W=Q_{h}-Q_c.$$

This work also satisfies $$\tag{2}W=Q_{h}\bigg ( 1-\frac{T_c}{T_h} \bigg )$$

Let us fix $Q_{h}$, the amount of heat initially extracted from the heat bath with temperature $T_h$.

$Q_c$ is exactly equal to the amount of work $W_3$ we put in during the isothermal compression, i.e., $Q_c = W_3$. Therefore, $W$ in eq. $(1)$ can be made bigger by reducing $W_3$, in other words $W$ depends on us.

Now, eq. $(2)$ says a completely different story. Given $Q_h$, which we already fixed, we no longer have any control over $W$. It is a constant that is determined by the temperatures of the heat baths and the initial amount of heat exctracted $Q_h$.

Where does this apparent contradiction come from?

P.S. We assume that we are working with a perfect gas.

Answer 1

The quantities $Q_h$, $Q_c$, and $W$ all depend on how "big" the Carnot cycle is, i.e. its size on a PV diagram. Once you fix any one of them, you fix all of them.

A Carnot cycle has four steps: (1) isothermal expansion, (2) adiabatic expansion, (3) isothermal contraction, and (4) adiabatic contraction. If you have fixed $Q_h$, then you have already fixed step (1). Then step (2) is fixed by the fact that the final temperature must be $T_c$. Naively you have freedom in step (3) to contract the gas as much as you want, but if you don't contract it just the right amount, you won't return to the initial state in step (4), so you won't have a cycle at all.

Answer 2

𝑄𝑐 is exactly equal to the amount of work 𝑊3 we put in during the isothermal compression, i.e., 𝑄𝑐=𝑊3. Therefore, 𝑊 in eq. (1) can be made bigger by reducing 𝑊3, in other words 𝑊 depends on us.

Keep in mind that in order to reduce the isothermal compression work (lower $Q_c$) you need to lower the temperature of the low temperature reservoir, $T_c$. That increases the Carnot efficiency ζ in equation (2)

$$ζ=1-\frac{T_c}{T_h}$$

Now, eq. (2) says a completely different story. Given 𝑄ℎ, which we already fixed, we no longer have any control over 𝑊. It is a constant that is determined by the temperatures of the heat baths and the initial amount of heat exctracted 𝑄ℎ. Where does this apparent contradiction come from?

There is no contradiction. Equations (1) and (2) are not independent of one another. As i indicated above, in order to lower $Q_c$ in equation (1) you need to lower $T_c$ in equation (2). Lowering $T_c$ while keeping $T_h$ fixed in equation (2) increases the efficiency and therefore increases the net work done.

"in order to reduce the isothermal compression work (lower ) you need to lower the temperature of the low temperature reservoir, " Would you mind explaining why this is the case? I don't really see why the amount of work that we put in depends on the temperature of the thermal reservoir.

To help visualize why this is the case, see the Pv and Ts diagrams for the Carnot Cycle.

The area enclosed in the cycle represents the net work done in the cycle. Note that in order to increase this area, you need to raise the isothermal expansion path and/or lower the isothermal compression path. To do this you need to increase the work done in the isothermal expansion (path 1-2), which means increasing $T_h$ and/or decrease the work done in the isothermal compression (path 4-1), which means lowering $T_c$. Either approach increases the Carnot efficiency and thus the net work done.

Hope this helps.