# What ds>dQ/T mean?

I read the derivation on page 216 over here.

First, it considers an irreversible process between states 1 and 2, followed by a reversible process between states 2 and 1. From my interpretation, equation 8.31 means that the entropy change of a reversible process is greater than that of an irreversible one between the two states. From equation 8.32, did they generalise and change the RHS to a reversible integral?

(We know that $$\delta S = \frac{\delta Q}{T}$$ in the case of equality, hence the inequality must be the case of an irreversible process. Is that how to interpret it?)

Then I read in my class notes that the inequality can be removed by adding an entropy generation term: $$\delta S = \frac{\delta Q}{T} + c$$ where c is the entropy generation term.

For an irreversible process, $$c$$ is positive. Where does entropy generated go, the system or surroundings?

• There is no objective identification which part of the system is the system and which part is the environment. It's the total entropy that grows more than the bound. This extra growth may take place both in the system and in the environment.If the environment has blue and red ink which gets mixed to a purple ink, the entropy of the environment goes up. If the same inks appear within the system, the entropy of the system goes up, even without any heat transfer. May 1 '16 at 8:09

There is something they forgot to mention in your notes (either from ignorance, or out of omission). The temperature within the system is spatially non-uniform during an irreversible process. So what value of the temperature are you supposed to use in the integral of dq/T? The Clausius inequality calls for the use of the temperature at the boundary interface with the surroundings (where the actual heat transfer is occurring). This requirement is rarely emphasized in textbooks or online sources. So the Clausius inequality should really read: $$\Delta S\geq \left(\int_1^2\frac{\delta q}{T}\right)_{interface}\tag{1}$$The right hand side of this equation is sometimes interpreted as the entropy entering the system from the surroundings. That, combined with the entropy generated within the system by irreveribilities within the system gives rise to the total entropy change of the system.