I have 2 questions regarding the derivation of the formula which calculates the probability of molecules having a particular amount of kinetic energy $E_x$ in a system of $N$ molecules. It states that: $$p(E_x) = \frac{\exp(-E_x/kT)}{\sum_{i=0}^\infty \exp(-E_i/kT)}$$ This formula is initially derived from finding the maximum number of permutations $\Omega$ with $N$ molecules distributed over $n$ energy compartments within a system according to: $$\Omega = \frac{N!}{n_1!n_2!n_3!...n_i!}$$ Question 1. It is explained that one must find the maximum number of permutations by differentiating this permutation function and solving for $0$. However, I can already see from the formula that the maximum number of permutations is achieved if the denominator is equal to 1 (it can't be less), which means that there should be 1 molecule in each energy compartment $n_i$. Why isn’t it possible to reason the maximum that way? I'm aware this would need $N$ number of energy compartments resulting in some having very improbable high energy levels, but can't the difference in energy levels between the compartments be very small?
Question 2. Suppose the molecules only possess translational kinetic energy. I understand that translational kinetic energy is continuous: it does not have discrete energy levels like rotational or vibrational kinetic energy. Since it is continuous, there are no energy compartments with exact energy values. The number of possible energy compartments within such a system should therefore be infinite and one could only derive probability densities. How is it possible nonetheless to calculate a limited number of permutations using a limited number of energy compartments? Does each energy compartment cover a certain range of energy levels?
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