# Is Clausius' inequality a statement of the second law of thermodynamics?

From the Kelvin and Clausius statements of the 2nd law we can prove Carnot's theorem (no engine is more efficient than a heat engine), and this therefore becomes an additional statement of the second law. Combining all 3 together gives us the Clausius inequality, and from this we can derive that $$dS \geq 0\tag{1}$$ for a thermally isolated process, and this is presented as another statement of the second law.

However, in what I've read the Clausius inequality $$\oint \frac{dQ}{T} \leq 0 \tag{2},$$ is not explicitly presented as a statement of the second law. Is there any non-trivial reason for this or am I correct in understanding that this is indeed an equivalent second law statement?

First of all, it should be pointed out that, in the equation you have written for the Clausius inequality, the T should be evaluated at the boundary temperature $$T_b$$between the system and the surroundings through which the increment of heat dQ flows during the process (not at the average temperature of the system). (See Thermodynamics by Fermi, and Fundamentals of Engineering Thermodynamics by Moran et al). I would also express the inequality in a little different form as $$\int_A^B{\frac{dQ}{T_b}}\leq \Delta S$$where A is the initial thermodynamic equilibrium state of the system and B is the final thermodynamic equilibrium state. This seems to me to be the simplest statement of the 2nd law of thermodynamics, and the one that I personally relate best to (since it is a straightforward mathematical relationship). The other statements of the 2nd law, in my judgment, are important only because of their historical significance.
• In the theoretical (idealistic) case where there is a discontinuity of temperature at the boundary, $T_{\rm{sys}}≠T_{\rm{sur}}$, which one should be used in the integral? Commented May 13, 2023 at 19:21