The logarithm in the entropy formula

[gsingh2011]

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?

 

[CompuChip]

The reason that I know of, is that we require entropy to be an extensive property.
Suppose that we have two systems, with N₁ and N₂ microstates, respectively, and we join them. From basic statistics it follows that the new system has N = N₁N₂ microstates.

However, to be an extensive quantity, the entropy should scale as
S = S₁ + S₂.

 

[gsingh2011]

The reason that I know of, is that we require entropy to be an extensive property.
Suppose that we have two systems, with N₁ and N₂ microstates, respectively, and we join them. From basic statistics it follows that the new system has N = N₁N₂ microstates.

However, to be an extensive quantity, the entropy should scale as
S = S₁ + S₂.

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.

 

[DrDu]

Because Entropy was defined as an extensive quantity long before people knew about statistical mechanics.

 

[CompuChip]

Well, again there is a lot I'm omitting, but one good reason is that entropy contributes to the energy of the system as
dE = T dS - p dV + N dμ
and we definitely want that to be extensive, don't we?
(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)

 

[jhjensen]

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

 

[morrobay]

delta S for n moles of a gas in isothermal expansion =
integral V₁ to V₂ nR dV/V = delta S= nR ln V₂/V₁
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 w₁ to w₂ k = k ln w₂/w₁
And since w₂ = all the possible states in phase space and w₁ = one state
Then S = k ln w

 

[CompuChip]

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

I answered that mathematically in your other thread

 

[jhjensen]

Thank you!

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