In A STUDENT'S GUIDE TO ENTROPY by Don S. Lemons, the author derives how entropy depends on energy and temperature as follows;

Due to the additivity, "the entropy increment of the composite system is the sum of the entropy increments of its parts: $f(T,Q)=2f(T, \frac{Q}{2})$" where $Q$ is the heat absorbed by the system and $T$ is the temperature. "If instead of dividing the heart reservoir into two identical parts we divide it into $n$ identical parts, then


The solution can be obtained by "taking the partial derivative of this with respect to $n$, and solving the resulting partial differential equation", and we have,


Taking a partial derivative with respect to $n$, I got $nf(T,\frac{Q}{n})=Q\frac{\partial{f}}{\partial{n}}(T,\frac{Q}{n})$. I tried to see if we can separate the variables by $f(T,\frac{Q}{n})=g(T)h(\frac{Q}{n})$, but this got me nowhere.

Any help would be appreciated as to how to solve the above partial differential equation.


You don't need a differential equation. The equation $$ f(T,Q)=n f(T,Q/n) $$ holds for any $n$, not just whole numbers. So just take $n=aQ$ (where the constant $a$ has dimensions
Joules$^{-1}$ so the units match) to get $$ f(T,Q)= aQf(T,1/a) $$ Then set $af(T,1/a)=g(T)$. Thus $$ f(T,Q)= g(T)Q. $$

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