Suppose two amounts of water of the same mass, but with different temperature, are mixed. Then the entropy of the hot water decreases, but the entropy of the cold water increases due to heat transfer.
But how is the entropy of whole system increased? And how is this irreversible?
(Or am I getting the concept of entropy wrong?)
Best Answer
The bottom line is that hot water loses heat at high temperature, giving a small negative entropy change while the cold water gain heat at low temperature resulting in a high entropy change. The net entropy change is positive. We can explicitly see this:
At any instant, the infinitesimal change in the entropy of the system is $$dS=\frac{dQ_H}{T_H}+\frac{dQ_C}{T_C},$$ where $dQ_H<0$ and $dQ_C>0$ are the heat exchanged by the hot and cold water respectively. The corresponding temperatures are $T_H$ and $T_C$. Since $$|dQ_H|=|dQ_C|\equiv dQ>0,$$ we can write $$dS=dQ\left(\frac{1}{T_C}-\frac{1}{T_H}\right)=dQ\left(\frac{T_H-T_C}{T_HT_C}\right)>0.$$ At any instant the temperature of the hot water is greater than the temperature of the cold water. So the $dS$ above is always positive and the process is irreversible at any intermediate state.