So I have learned that entropy is the measure of disorder of a system. For the IPhO this was of course not enough as we need to be able to calculate entropy changes of ideal gases. Those equations were derived from another definition of entropy, the integral definition dS = dE / T. This relates temperature with entropy. Now I never understood how this is equivalent to the statistical definition S = k * ln Π, with Π the multiplicity of the system. What is the proof? I couldn't google it. Also, I always thought temperature was a measure for the average internal energy, but how does that relate to the entropy definition? I do see it implies that adding ∆E will affect S more if T is low than when T is high. I interpret this as the more E a system has, the more ways there are to distribute the energy over the particles, thus a higher Π. Could you clarify these fundamental concepts and definitions?