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Example: Calculating entropy by counting microstates

Page history last edited by Joe Redish 6 years, 1 month ago

Working Content II > Example: Entropy and Heat Flow

 

Prerequisites:

 

Let's do an example of calculating entropy by counting. Will take the a highly simplified example so that we can see all the details.

 

Consider an example of an isolated box of volume 2V divided into two equal compartments. An ideal gas occupies half of the container and the other half is empty. When the partition separating the two halves of the box is removed and the system reaches equilibrium again, how does the new entropy of the gas compare to the energy of the original system?  

 

   ➡︎  

 

One way to do this is to imagine breaking up the left side of the box into M small volumes. (Small compared to the size of the box, but large compared to the size of an atom, so we don't have to worry about atoms "filling up one of the small volume".) 

 

We can put one molecule into the left side of the box into the box in M ways. We can do this for each molecule, so the total number of ways we can put the molecules into the bins is MxMxM...xM (N times) so

W1 = MN

Now this isn't quite right, since if we put more than one atom into a box it doesn't matter in which order we put it in. We should really divide by the number of permutations of the atoms in each box. But if we have a dilute gas, there is a lot of space, and very few of our boxes will have more than one atom in it. We can fix this, but it's not really worth it. We'll ignore this for now.

 

We can now calculate the entropy of the initial state:

 

S1 = kB log(MN) = kBN log(M)

 

Now pull out the partition. The N molecules can now be put into 2M different small volumes.

 

We get the new number of microstates and the entropy to be

 

W2 = (2M)N = 2NMN = 2NW1

S2 = kB log((2M)N) = kBN log(2M) =  kBN {log(2) + log(M)} = kBN log(2) + S1

 

We get the cool result that the change in the entropy, ΔS = S2 - S1 kBN log(2) is proportional to N (so extensive) and is independent of the size of M, so the change doesn't depend on the size of the box.

 

Follow ons:

 

 

Joe Redish 2/15/18 

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