Fugacity of Bose-Einstein Condensation

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.

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 $$\mathcal{O}(1/V)$$.

• Thank you very much, it was "the root of" the source of my confusion. Commented Jan 1, 2021 at 19:07