# Partial Differential Equation and Additivity of Entropy

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

$$f(T,Q)=nf(T,\frac{Q}{n})$$"

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,

$$f(T,Q)=g(T)Q$$

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 $$f(T,Q)= g(T)Q.$$