Increase of entropy

Here we imagine a cycle in which we go from point 1 to point 2 by an irreversible process and back by a reversible process. (See here for an explanation of the picture.)

The entropy change for a reversible process is given by

$$S_2-S_1=\int_1^2 {{}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q^{\rm rev}\over T }$$

What can we say about an irreversible process between the same endpoints? Clearly the entropy change is the same (that's what we mean by saying entropy is a function of state.) But if we consider a cycle involving an irreversible process from 1 to 2 and a reversible process to return to 1, we have a cycle for which Clausius's theorem holds:

$$\begin{eqnarray*} \oint {{}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q\over T }... ...mbol{'040}}\llap{d}Q^{\rm irrev}\over T }\!\!\!&<&\!\!\!S_2-S_1. \end{eqnarray*}$$

Hence in general,

$${}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q\le T{\rm d}S,$$

and so for an isolated system, where ${}\raise0.44ex\hbox{\bf\symbol{'040}}\llap{d}Q=0$,

$\mbox{\large\colorbox{yellow}{\rule[-3mm]{0mm}{10mm} \ $\displaystyle {\rm d}S\ge 0. $ }}$