# Biological Physics/Stirling's Approximation

Calculating entropy directly calls for us to calculate factorials. Taking factorials of relatively small numbers is not a problem (at least with a computer). However, once you begin to work in systems that have a very large number of particles and energy packets (on the order of a mole, or ), it becomes clear that a direct calculation for factorials is not feasible. Rather, an approximation for the entropy must be developed. We will look more closely at what is known as **Stirling's Approximation**.

Recall that the multiplicity Ω for ideal solids is

and entropy is

.

.

By the rules of logarithms,

.

All terms are essentially logarithms of factorials, so let's study the general case.

,

which implies by the rules of logarithms , or

.

For very large values (think on the order of magnitude of moles), this can be approximated by

, which has a solution .

And by the fundamental theorem of calculus, . For very large numbers, the *1* becomes negligible, so . So this gives us the following approximations

So .

With some simplification,

Once again, let's make the assumption that these numbers are very large, making 1 negligible. So

Therefore, entropy can be approximated with . This is Stirling's Approximation. If you recall that numbers out in front of a logarithm can be written as the power of the number inside the logarithm, so , or . For calculations, the first equation would be better since it doesn't exponentiate before it takes a logarithm.