### Derivation of Boltzwann equation for n moles of gas

Posted:

**Sat Jan 17, 2015 8:12 pm**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?Created by Dr. Laurence Lavelle

https://lavelle.chem.ucla.edu/forum/

https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=132&t=5030

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Posted: **Sat Jan 17, 2015 8:12 pm**

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?

Posted: **Sun Jan 18, 2015 3:24 pm**

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

So we have as our equation.

For 1 mole (N

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

Posted: **Tue Jan 20, 2015 10:16 pm**

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.

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.

Posted: **Tue Jan 27, 2015 11:56 pm**

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.