# Where does the factor of $1/V^N$ come from in the classical non-ideal partition function?

I'm reading about classical non-ideal gases from this source. On page 5, they say that the non-ideal partition function is a product of the ideal partition function and the non-ideal partition function, which makes perfect sense.

They write the interaction part of the non-ideal partition function in equation (3.1.4) as

$$\frac{1}{V^N} \int \mathrm{e}^{-\left(\sum_{i

with no explanation as to where the factor $$1/V^N$$ comes from. I can't seem to figure it out in any sort of detail. Any ideas?

It is just a convention choice, not every textbook uses this. It's been a while, but I think the choice to have that factor is due to the fact that the positions are uniformly distributed over the volume $$V$$, so we choose to perform the integral with this factor to represent this probability. So,
$$Q = \int \prod_i \left(\frac{d^3q_i}{V}\right) U(q_1, ... q_n)$$
It doesn't change the physics, since $$V$$ is a constant in these integrals anyway.
The total volume for one particle would be $$V = \int \mathrm{d}^3q$$, so that for $$N$$ particles is $$(\int \mathrm{d}^3q)^N = V^N.$$