Clausius inequality
In an isolated system the entropy cannot decrease when a spontaneous change occurs as a consequence of the second principle. Any change in entropy of the system dS is accompianed by a change in entropy of the surroundings dSsurr
dS + dSsurr ≥0
ovvero
dS ≥ − dSsurr
We have seen that the variation of entropy of the surroundings is dSsurr = −δq/Tsurr, where dq is the heat absorbed by the system during the process (that is dqsurr = −δq), it follows that for a general transformation
dS + dSsurr ≥ 0
dS ≥ δq/Tsurr
This is inequality is known as Clausius inequality.
Spontaneous cooling. Consider the transfer of energy as heat from one system—the hot source—at a temperature Th to another system—the cold sink—at a temperature Tc When |δq| leaves the hot source (so δqh ≤ 0), the Clausius inequality implies that dS ≥ δqh/Th. When |dq| enters the cold sink the Clausius inequality implies that dS ≥ δqc /Tc (with δqc > 0). Overall, therefore
dS ≥ δqh/Th + δqh/Tc
However, δqh = −δqc, so
dS ≥ −δqc/Th + δqc/Tc = δqc (1/Tc − 1/Th)
which is positive (because dqc > 0 and Th ≥ Tc). Hence, cooling (the transfer of heat from hot to cold) is spontaneous, as we know from experience.