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**1. A system containing 38 particles has three equally spaced energy levels available. Two population distributions are A (18, 12, 8) and B (17, 14, 7) (populations listed lowest to highest energy).**

a. Show that both distributions have the same energy.

a. Show that both distributions have the same energy.

This part is pretty easy. Since each energy level is listed lowest to highest and they are equally spaced we can assume the energy levels follow E = 1, 2, and 3 respectively. Therefore,

Energy of A = 18 * 1 + 12 * 2 + 8 * 3 = 66

Energy of B = 17 * 1 + 14 * 2 + 7 * 3 = 66

**b. Is either distribution a Boltzmann distribution?**

I think I know the answer for this but I need a good, solid justification. According to the equation for the weight of a system:

W = N!/n0!n1!n2!, The Boltzmann distribution is the distribution with the greatest weight while satisfying the conditions where the total number of molecules and total energy of the system remains constant.

Weight of A = 38!/18!12!8! = 4.23 x 10^15

Weight of B = 38!/17!14!7! = 3.35 x 10^15

I think A is the Boltzmann because the weight of the configuration is greater than that of B or any other valid configurations that meet the constraints that I can think of, but I don't have a good justification.

A way I can prove this is if I come up with an equation for dW = 0, showing that is when the weight is maximized, but I don't know how to do that while keeping the constraints.

If anyone can help me out by coming up with a mathematical justification or if you have anything else to add, I would greatly appreciate it...thanks!