Derivation of ideal gas law

Question

I looked up on the ideal gas law which our high school textbook derives with the empirical Combined Gas Law. However, the textbook did give a good explanation for this equation $$pV = \frac{N}{3}m\bar{v^2}$$ with which I only need to verify that $$K.E. = \frac{3}{2}k_BT$$ is true. I further looked up this link Average Molecular Kinetic Energy which deduces the result from the Boltzmann distribution $$f(E)=Ae^{-\beta \epsilon}$$ but I could not read any literature deriving $$\beta = \frac{1}{kT}$$ I was wondering if I am in a correct direction and how to derive the thermodynamic beta $\beta$.

Answer 1

The ideal gas law is a combination of many other laws about gases. Some assume the pressure to be constant, others assume the quantity stays constant and others. Now those laws have been set up mostly after experiment and it people working on it noticed that the pressure $P$ according to the small laws seemed to be proportional to the quantity $n$ (in moles), to the temperature $T$ (in kelvins) and inversely proportional to the volume. Know when we find such proportionality, we always need to multiply by a proportionality constant (here they called it R), that could be different than one and that would obviously nkt change the proportions. That proportionality constant, they have found its value by experiment and putting all this together gives the familiar $$P=nRT/V$$.

Hope you understand !

Answer 2

In classical thermodynamics, temperature $T$ is defined through ideal gas equation $$pV = nRT$$ from which we conclude that $$K.E. = \frac{3}{2}k_BT$$ is true for any ideal monatomic gas which cannot exist in real life anyways. Statistical mechanics provides postulates that is broader in context. It redefines the temperature through the second law $$dE = TdS$$ Now from $$p_i = \frac{e^{-\beta \epsilon_i}}{Z}$$ obviously we could obtain $$\beta = \frac{d \ln \Omega}{dE}$$ and by Boltzmann's assumption $$S = k_b \ln \Omega$$ we could have $$\beta = \frac{1}{k_BT}$$ so everything boils down to the definition of absolute temperature.