Fugacity of Bose-Einstein Condensation

Question

I'm studying Bose Einstein Condensation.

In the book "Huang K Statistical Mechanics 2 edition", page 288, the author gets the following result for the fugacity ($z$) as a function of temperature and specific volume (lambda is the thermal wavelength and small $v$ the specific volume):

I understand how $z$ is equal to 1. I don't understand how can one obtain the last result for the value of $z$ above the critical temperature.

Answer 1

I am assuming you don't know how to get from 12.41 to the second equation of 12.52?

If you start from 12.41:

$$ \frac{1}{v} = \frac{1}{\lambda^3}g_{3/2}(z) + \frac{1}{V}\frac{z}{1-z}, $$

and take the infinite volume limit $V\rightarrow \infty$, so that the $1/V$ term above goes to $0$.

Then, you are left with: $$ \frac{1}{v} = \frac{1}{\lambda^3}g_{3/2}(z), \\ g_{3/2}(z) = \frac{\lambda^3}{v}. $$

Then you solve it graphically by plotting each side of the equation and find their intersection (the "root"):

For finite volumes, the error goes as $\mathcal{O}(1/V)$.