0
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.