Where did the mathematical description of entropy come from? I understand the qualitative idea of disorder and microstates and all of that but why does dS = dQ/T? This equation is derived from the first law of thermodynamics for an ideal gas heated isothermically. Why is this important? The equation at the end of the derivation was dV/V = dQ/nRT = dS/nR where n and R are the number of moles in the gas and the gas law constant. Why should dQ/T be entropy and not dQ/nRT? I can't figure it out! Can anybody else?

The exact description of entropy is not simple. It's possible to show that

~~d~~Q

_{R}/T is a function of state, then it's possible to show that this is equal to the statistical definition of entropy S = klnW but all the demonstration are quite complex.

~~d~~ = not-exact differential

Q

_{R} = reversibly exchanged heat

Simplistically we could say: we know that entropy increases in a dissipative process, that is when heat flows between to bodies which temperature difference is finite (and not infinitesimal), so dS have to be proportional to this heat; furthermore, since heat flows from a body with higher T to another with lower T, the increase in entropy of the colder body have to be greater than that of the hotter, so that the total variation of entropy is positive; so dS of a body have to be inversely (proportional) to the body's temperature: dS = dQ/T. I wrote proportional in brackets because with this reasoning we don't have proved the inverse proportionality, but only that one increases when the other decreases.