Derivation of Boltzwann equation for n moles of gas

[Cameron Sasmor 1G]

I don't understand how k_B is changed into nR in the Boltzwann equation for n moles of gas. Can someone explain this please?

[Neil DSilva 1L]

All of this assumes two states.
So we have as our equation.
For 1 mole (N_A),

We use the log rule to get
, so we get
This is for one mole, so if we have n moles, we just multiple that by n to get


If you have n moles, that's , and plugging that into the equations above and following all the same rules will get you

Let's assume
We have
Then, we use the log rule to get:
This simplifies to , where , so we have

[Chem_Mod]

I assume you are probably more comfortable with the gas constant R rather than the Boltzmann constant k. These two are really just the same constant, but on different scales.

R has units of J/mol*K and relates to the entropy of 1 mole of particles. R is "macroscopic"
k has units of J/K and relates to the entropy of just 1 particle, so we often say k is "microscopic"

S = k*ln(W) gives the entropy per particle of a system with W states. So Nk*ln(W) is the total entropy of N particles. Dividing N by avogadro's number gives n, the number of moles. Multiplying k by avogadro's number gives R. Since we divided and multiplied by the same number, nothing has changed, and Nk = nR.

[Justin Le 2I]

I would recommend remembering the log rule where the exponent can be turned into a coefficient. If the equation is left as it is, your calculator might overflow like with one of the homework problems.