We now turn to verifying that the entropy  is a signpost of spontaneous change, in the sense that the total entropy must increase for any spontaneous change.

We consider a system at temperature T, in thermal and mechanical contact with its surroundings, which are also at temperature T. (Note that the system and surroundings are not necessarily in mechanical equilibrium - eg they may be at different pressures). Any change of state is accompanied by changes in the entropy of the system, dS_sys , and the entropy of the surroundings dS_sur .

From the Second Law, we write:

; or equivalently

The entropy change is equal to zero if the process occurs reversibly. If it occurs irreversibly, then there is, by definition,  some heat loss from the system and an increase in entropy.

We have already encountered the relation dS_sur  =  dq_sur / T.  Clearly, since the heat transfer occurs between the system and the surroundings, it must be the case that dq_sur = -dq_sys , and thus by implication dS_sur = -dq_sys / T . It then follows, from the second expression above, that:

which is the Clausius inequality (or Clausius relation). Consider the situation where the system is isolated from its surroundings. In this case, dq_sys = 0 , and the Clausius relation implies

This tells us that the entropy of an isolated system cannot decrease in the course of a spontaneous change.

Since the universe is itself an isolated system, this result shows that we can use entropy as the signpost of spontaneous change. Processes are only spontaneous if they cause an increase in the total entropy of the universe.