(Based on McQuarrie 20-20) For an ensemble of A distinguishable, isolated system
ID: 3309140 • Letter: #
Question
(Based on McQuarrie 20-20) For an ensemble of A distinguishable, isolated systems with j states of the same energy, we learned that the number of ways of having ai systems in state 1, a2 systems in state 2,.., and aj systems in state j is given by: A! W(a1,a2,... ,lj)ai!a2!...aj Consider a case in which there are only two states, 1 and 2. Show that W(ai,a2) (and therefore also the entropy)is at its largest when aa2. (The textbook suggests considering In W using Stirling's approximation, and treating a and a2 as continuous variables. You're welcome to do this, but in my opinion, there are other ways that are less mathematically involved.)Explanation / Answer
When there are two states involved; it a classic example of a two-level system. Here, there are two degrees of freedom involved in the system and there are a1 particles in each state. Since entropy is an extensive quantity, it should scales as like volume or number of particles, hence the maximum entropy is proportional to a1 log 2, where two comes because of two states are available in the system.
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