- #1

- 73

- 1

## Homework Statement

Hi all,

There is a question from the course book:

## Homework Equations

[tex]S=k_B ln W[/tex]

## The Attempt at a Solution

My solution:

So first of all, for each molecule, there are 2 motions: translational and rotational.

For rotational I get:

[tex]W_1 =\Omega \left ( \theta \right )[/tex]

For translational, I should calculate the free (non forbidden volume) for molecule to move in. Since the overall volume is V and there are N molecules, the volume allowed for moving is:

[tex]W_2 =V-NA \left ( \theta \right )[/tex]

Since molecules are distinguishable, I need to divide by (N!)^(1/N) for each molecule, so:

[tex]W_2 =\frac{V-NA \left ( \theta \right )}{\sqrt[N]{N!}}[/tex]

Therefore, the entropy per molecule is:

[tex]s= k_B ln \left ( \frac{V-NA \left ( \theta \right )}{\sqrt[N]{N!}}\Omega \left ( \theta \right ) \right )[/tex]

And since there are N molecules:

[tex]S= k_B N ln \left ( \frac{V-NA \left ( \theta \right )}{\sqrt[N]{N!}}\Omega \left ( \theta \right ) \right )=k_B ln \left (\frac{1}{N!} \left ( V-NA \left ( \theta \right ) \right )^N \left ( \Omega \left ( \theta \right ) \right )^N \right )[/tex]

But according the book, the answer is:

So I have 2 questions:

1. Do they mixed up between Omega and A? Or it's me?

2. Where do a factor of 2 came from in the expression for the volume?

Thanks!