# Physics with Calculus/Thermodynamics/Entropy

## Definition[edit]

Entropy is a measure of how organised a system is. A system with low entropy is one with a organised, or unlikely configuration. For example: if you have a glass of water with average temperature 20 degrees, a low entropy state would be the top half 30 degrees and bottom half 10 degrees, whereas the highest entropy state is the entire glass 20 degrees.

## Mathematically[edit]

To quantify this, we find the total number of microstates(Ω); all the possible states the system could be in. For a coin: Ω = 2 (heads / tails), 2 coins: Ω = 4 (HH/HT/TH/TT), N coins: Ω = 2^{N}. If you have two systems with Ω_{1} and Ω_{2} microstates, Ω_{total} = Ω_{1}Ω_{2}.

To apply this to a system of particles, use a volume V with N atoms. Classically, atoms can have an infinite number of states, making Ω infinite. But due to results from quantum physics, Ω is in fact finite. Given a total energy E and N atoms with mass m:

c is a constant that depends on geometry.

h is Planck's constant.

A few notes here:

- It is possible to derive this from a pseudo-quantum argument, which says that there are only finite positions in the box equal to the volume divided by h
^{3}and number of possible momenta is the surface area of a hypersphere. But since these are both aphysical and irrelevant, I might as well just state the formula(learn quantum mechanics!). - I put a constant in the formula, because we are mainly interested in how the number of states grow rather than the exact number.
- The N! is only for indistinguishable particles. This is another quantum effect that means nature can't tell two atoms of the same kind apart.
- The numbers that statistical mechanics works with are on the order of N = 10
^{27}, so Ω is generally very, very big.

Entropy is defined as , where k_{b} is Boltzmann's constant. So:

Note: the units are fuddled in the last bit, it won't matter though. Stirling's approximation says for large N: . So:

Since the important part is the change, I put the constants into D(assuming particle conservation). Also, , so:

at constant temperature.

at constant volume.

It is also useful to define temperature in terms of entropy: