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## Main Question or Discussion Point

Why is there a logarithm in the entropy formula? Why is it S=kln(N) where k is the Boltzmann constant and N is the number of microstates? Why isn't it S=N?

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Why is there a logarithm in the entropy formula? Why is it S=kln(N) where k is the Boltzmann constant and N is the number of microstates? Why isn't it S=N?

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CompuChip

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Suppose that we have two systems, with N

However, to be an extensive quantity, the entropy should scale as

S = S

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Why do we want entropy to be an extensive quantity? Multiplying the microstates to calculate the entropy seems just as easy/useful as adding the entropies.Iknow of, is that we require entropy to be an extensive property.

Suppose that we have two systems, with N_{1}and N_{2}microstates, respectively, and we join them. From basic statistics it follows that the new system has N = N_{1}N_{2}microstates.

However, to be an extensive quantity, the entropy should scale as

S = S_{1}+S_{2}.

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DrDu

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CompuChip

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dE = T dS - p dV + N dμ

and we definitely want

(Note by the way that the quantities occur in combinations of extensive and intensive: two systems with entropy S and temperature T have total entropy 2S but temperature T, two systems with pressure p and volume V have pressure 2V but pressure p, etc)

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But is ln the only function for which f(xy) = f(x)+f(y)?

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morrobay

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delta S for n moles of a gas in isothermal expansion =

integral V_{1} to V_{2} nR dV/V = delta S= nR ln V_{2}/V_{1}

Given that a change in entropy in statistical mechanics from a system with probability of W1

to one of W2 = k ln W2/W1 , it should follow that

delta S = integral w_{1} to w_{2} k = k ln w_{2}/w_{1}

And since w_{2} = all the possible states in phase space and w_{1} = one state

Then S = k ln w

integral V

Given that a change in entropy in statistical mechanics from a system with probability of W1

to one of W2 = k ln W2/W1 , it should follow that

delta S = integral w

And since w

Then S = k ln w

Last edited:

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CompuChip

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I answered that mathematically in your other threadBut is ln the only function for which f(xy) = f(x)+f(y)?

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I have summarized my thoughts on the original question: http://proteinsandwavefunctions.blogspot.com/2012/01/where-does-ln-come-from-in-s-k-lnw.html

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