# Thermodynamic limit for an ideal gas

In S. Salinas, Introduction to Statistical Physics (Springer, 2001), the author states (p. 68):"...the thermodynamical limit is essential to allow the connection between the average values of statistical mechanics and the macroscopic values of the thermodynamic quantities". However, on p. 80 the author states: "As a matter of fact, it is not even necessary to invoke the thermodynamic limit to obtain the equations of state of the ideal gas" and goes on to show: $$S(E,V,N;\delta E)=k_B\ln\Omega(E,V,N;\delta E)=\frac{3}{2}k_B\ln E+k_B\ln V+f(N;\delta E),$$ where $$f(N;\delta E)$$ is a function of $$N$$ and $$\delta E$$. Therefore, $$\frac{1}{T}=\frac{\partial S}{\partial E}=\frac{3k_B}{2E}$$ and $$\frac{p}{T}=\frac{\partial S}{\partial V}=\frac{k_B}{V}.$$ Is there a contradiction here? How is the temperature (a macroscopic, averaged quantity) related to the derivative of the entropy without taking the thermodynamic limit?

The existence of exceptional cases like the one you are reporting is not a contradiction. Actually, in your example, the critical observation is that the term containing the dependence on the ensemble is the function $$f(N,\delta E)$$. More precisely, the ensemble dependence is the dependence of the entropy on $$\delta E$$. It is generally possible to show that dependence gives a vanishing contribution to the entropy per particle (or the entropy per unit volume) at the thermodynamic limit if $$\delta E$$ does not scale with the size. However, for a perfect gas, the $$\delta E$$ dependence is confined to an additive term, which exactly disappears when a partial derivative of entropy is taken with respect to volume or energy.
The relations $$\frac{1}{T} = \frac{\partial S}{\partial E}$$ and $$\frac{P}{T} = \frac{\partial S}{\partial V}$$ are pure thermodynamic relations valid for every possible thermodynamic system.