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## Homework Statement

Hello everyone. My problem is as follows: In a spontaneous process where two bodies at different temperatures [itex]T_{1}[/itex] and [itex]T_{2}[/itex], where [itex]T_{1}>T_{2}[/itex], are put together until they reach thermal equilibrium. The number of atoms or molecules of the first is [itex]N_{1}[/itex] and [itex]N_{2}[/itex] for the second one, with [itex]N_{1} \neq N_{2}[/itex], and they have heat capacities equal to [itex]C_{V_{1}}=aN_{1}k[/itex] and [itex]C_{V_{2}}=aN_{2}k[/itex], with [itex]a[/itex] given with the appropriate units. Past some sufficiently large time, the system reaches a temperature [itex]T[/itex], provided that [itex]T_{1}>T>T_{2}[/itex], which is in function of the initial temperatures and the number of atoms or molecules of the two bodies. The problem is that i can't demonstate that the change of the entropy of the system as a whole is positive, i.e. [tex]\bigtriangleup S>0[/tex]

## Homework Equations

When i compute the change of the entropy for the i-th body, i get

[tex]\bigtriangleup S_{i}=\int_{T_{i}}^T \! \frac{1}{T} \, \mathrm{d} Q=\int_{T_{i}}^T \! \frac{aN_{i}k}{T} \, \mathrm{d} T=aN_{i}k\int_{T_{i}}^T \! \frac{1}{T} \, \mathrm{d} T=aN_{i}k\ln{\frac{T}{T_{i}}}[/tex]

With the hypothesis that the entropy is an extensive property, then [tex]\bigtriangleup S=\bigtriangleup S_{1}+\bigtriangleup S_{2}=aN_{1}k\ln{\frac{T}{T_{1}}}+aN_{2}k\ln{\frac{T}{T_{2}}}[/tex]

So i just have to prove that [tex]N_{1}\ln{\frac{T}{T_{1}}}+N_{2}\ln{\frac{T}{T_{2}}} > 0[/tex]

## The Attempt at a Solution

I think that i have to use the two cases ([itex]N_{1}>N_{2}[/itex] and [itex]N_{1}<N_{2}[/itex]), and using the fact that [itex]T_{1}>T>T_{2}[/itex], to prove the inequality, but i have tried to do it in very different ways, and i get nothing, so i think there is some trick to demonstrating that, but i'm still a bit of an amateur in proving tricky inequalities.