# Doubt over Kelvin-Planck statement of thermodynamics' second law

I have a doubt over the Kelvin and Planck's statement of thermodynamics' second law, in particular applied to a cycle. Let's take a Carnot cycle as an example, and let's call the first two transformations (the isotherm and the adiabatic) done. Now, isn't it obvious that the machine has to give up heat to go back to the initial state? Isn't it something that follows from the fact that the cycle has to be continuous?

• Can you explicitly write the statement of the second law you are using? – joshphysics Jun 27 '13 at 18:04
• Sure. "It's impossible to realise a process which has as the sole result the transformation in work of the heat given by a reservoir at a uniform temperature" – Gennaro Marco Devincenzis Jun 27 '13 at 18:14
• Right so in the Carnot cycle, the engine takes some heat $|Q_H|$ from a hot reservoir, and transforms some of it into work $W$ and the rest into exhaust heat $|Q_L|$ which is given up to a lower temperature reservoir. The exhaust heat is precisely what causes the law to not be violated. – joshphysics Jun 27 '13 at 18:20
• Thank you. My doubt was just that a general cycle, once has done work, has to close itself in some way (I'm thinking of it in a p-V diagram), and I couldn't think of any plausible way in which a cycle could close without giving up some exhaust heat. The engine would have to "jump" in some manner to go back to the initial state. But as I understood from your comment, that's precisely what the Kelvin and Planck's statement says, right? – Gennaro Marco Devincenzis Jun 27 '13 at 18:39
• Yeah that's basically right. Any cycle is going to have to give up some exhaust heat to satisfy the Kelvin statement. I'm going to make this a bit more precise in an answer. – joshphysics Jun 27 '13 at 18:41

Every thermodynamic system satisfies the first law during a given process; $$\Delta E = Q-W$$ Here $\Delta E$ is the change in its internal energy, $Q$ is the heat transferred to the system, and $W$ is the work done by the system. For a system undergoing a cyclic process, namely one for which it starts and ends in the same thermodynamic state, one has $\Delta E = 0$, and the first law then tells us that $$Q = W$$ Now, suppose that the system is taking some heat $Q_H>0$ from a reservoir, and turning it into some work $W$. Let's define $$Q_\mathrm{exhuast} = Q-Q_H$$ then we can write $$Q_H + Q_\mathrm{exhaust} = W$$ The Kelvin statement tells us that we must have $Q_\mathrm{exhaust}\neq 0$ because otherwise, we would have $Q_H = W$; the sole result of the process would have been to transform the heat it took in into work. The symbol $Q_\mathrm{exhaust}$ is what one commonly calls the exhaust heat.