Fractionation of stable isotopes - Geochemistry: An Introduction - page 50

The positive constant is to be determined and the minus sign ensures that f i remains finite.

From the condition that the fraction proportions f i

β

sum to unity, we get the constant equal

= i =0 e − βE i is known as the partition function. This can be recast as: to Q −1 ,where Q

e − βE i
Q

f i = (3.14)

In order to demonstrate that RT , we need to pay a quick visit to the

concept of entropy. Boltzmann formulated a statistical definition of the entropy S of a system
as R ln

β is actually equal to 1 /

is the number of indistinguishable microscopic configurations (how atoms
distribute themselves among the possible states). The second principle then states that the
entropy of an isolated system increases when it spontaneously evolves towards the state
with the largest number of accessible configurations. The number

,where

can be found using a
simple argument: the total number of permutations of N atoms is N !, where the exclamation
mark stands for the function factorial (e.g. 3! 1). Among the N ! permutations, n 1 !

correspond to indistinguishable occupations of energy level E 1 with a similar outcome for
other energy levels. The number of distinguishable configurations is therefore:

= 3 × 2 ×

N !
n 1 ! n 2 !

= (3.15)

...

This expression would be a nightmare if it was not for the handy Stirling approximation ln N ! ≈

N ln N N , valid for large N , and which gives a new expression for the entropy (remember

that the logarithm of a product of numbers is the sum of the logarithms of these numbers):
S
R =+

−

N ln N − n 1 ln n 1 − n 2 ln n 2 −···− N + n 1 + n 2 +··· (3.16)

= i n i : the second part of the right-hand side therefore vanishes and

We now recall that N

we get:

R
i

S =− f i ln f i (3.17)

Taking the differential of the entropy S at constant T :

R
i

R
i

df i
f i

d S =− d f i ln f i − f i (3.18)

or, using (3.14)

RT ln Q
i

T d S = RT β E i d f i − df i (3.19)

i

At constant volume, i E i d f i =

d U = T d S , which requires that RT β = 1. This ends the

demonstration of one of the most important equations in science.