# 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?

## Answers and Replies

Science Advisor
Homework Helper
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.

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

Science Advisor
Homework Helper
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)?

Gold Member
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

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

I answered that mathematically in your other thread