Monotherm cycle, negative $Q_{net}$

Question

In the demonstration of the Clausius theorem in classical thermodynamics my book uses the fact that $Q_{net}\le0$ is negative in a cyclic monotherm (meaning with one heat reservoir) transformation. This fact, however, left me a bit uneasy. Taking, as usual, a Carnot cycle as an example, and calling the adiabatic and the isotherm expansions done, couldn't I just do an adiabatic compression to bring it back to the initial state? The second law wouldn't be violated: the heat absorbed wouldn't entirely be converted into work. The expression of the first principle, though, is indeed a bit strange: since $\Delta U$ is $0$ in any cycle, then $Q = W$, and my reasoning breaks down. I can't really see what's going on here, would it be theoretically possible a process similar to the one I described? If not, why so?

Answer 1

Let's assume that the thermodynamic state of the system can be represented in two dimensions (like on a pressure-volume diagram for an ideal gas). Let's say further that the cycle starts at point $a$, undergoes an isothermal expansion to point $b$, then undergoes an adiabatic expansion to point $c$.

As far as I understand, you now want to take the system from point $c$ to point $a$ by way of an adiabatic compression. This is not possible because there is only one adiabatic curve passing through point $c$; it is precisely the curve along which the system underwent the adiabatic expansion in the first place. There is no adiabatic curve connecting $a$ and $c$.