Two blocks of the same metal having the same mass and at temperature T1 and T2, respectively, are brought in contact with each other and allowed to attain thermal equilibrium at constant pressure. The change in entropy, ΔS, for this process is:

When two blocks are kept in contact with each other, the final temperature will be given as: Tf=(T1+T2)/2
Now, we have ΔSsys = ∫dqrev/T = nCp∫dT/T
For the first block of metal the entropy will be given as:
ΔS1 = nCp∫(Tf to T1)dT/T = nCpln(Tf/T1)
Similarly, ΔS2 = nCp∫(Tf to T2)dT/T = nCpln(Tf/T2)
Now, total change in entropy ΔS1 + ΔS2 = nCp[ln(Tf/T1) + ln(Tf/T2)] = nCp ln[(Tf/T1)(Tf/T2)] = nCp ln[(Tf)²/T1T2]
But , Tf = (T1+T2)/2
∴Final entropy will be: nCp ln[((T1+T2)/2)²/T1T2] = nCp ln[(T1+T2)²/4T1T2]