# Why entropy is logarithmic

#### Prerequisites

In Entropy -- implications of the 2nd law of thermodynamicse, we defined the entropy, $S$ of a particular macrostate of a system as equal to $k_B \ln W$, where $W$ is the number of possible arrangements of the system (microstates) corresponding to that macrostate.  But why? Why not just say that entropy is the number of arrangements? Let's think through why it has to be defined this way.

We want to define entropy to be an extensive property. This means that if I have two systems A and B, the total entropy should be the entropy of A plus the entropy of B. This is like mass (2 kg + 2 kg = 4 kg), and not an intensive property like temperature. (If you combine two systems that are each at 300 K, you have a system at 300 K, not at 600 K!)

What happens to the number of possible arrangements when you combine two systems? If system A can be in 3 different arrangements and system B can be in 5 different arrangements, then there are $3 \times 5 = 15$ possible combinations. They multiply! This '80s music video explains why.

So we can't just define entropy as the number of possible arrangements, because we need the entropy to add, not multiply, when we combine two systems.

How do you turn multiplication into addition?  Just take the logarithm: $3 \times 5 = 15$, but $\ln 3 + \ln 5 = \ln 15$.

So that's why entropy is defined as a constant times ln $W$.  $W \times$ (the number of arrangements) is a dimensionless number, so $\ln W$ is too.

The constant out in front could be any constant, but we use Boltzmann's constant, $k_B = 1.38 \times 10^{-23} \mathrm{ J/K}$.  When we get to Gibbs free energy, we'll see that this constant has the right units, we see that it's very convenient for entropy to be in units of energy/temperature.

Ben Dreyfus 1/9/12
Article 599