# Partition function of a gas of $N$ identical classical particles

Partition function of a gas of $N$ identical classical particles is given by

$$Z~=~\frac {1}{N! h^{3N}} \int \exp[-\beta H(p_1.......p_n, x_1....x_n)]d^3p_1...d^3p_n,d^3x_1...d^3x_n$$

in this above equation we use $N!$ as the total number of sub-systems of a system of identical particles. and $h^{3N}$ to make the partition function dimensionless. I am not clear how $h^{3N}$ is used to make it dimensionless.

The easiest way to think about it is that $\exp(\dots)$ is just a number and doesn't affect the dimension. However, you still have $3N$ factors of the momenta and the position lying around that will give you dimensions of [Length x Momentum]${}^{3N}$. Planck's constant has the units of Length x Momentum, so the $3N$ factors of $h$ cancel the $3N$ factors coming from the integral.
Notice that the $e^{-\beta H}$ is dimensionless, while each factor of $dp$ contributes one factor with the dimensions of momentum while each $dx$ contributes one factor with the dimensions of length. Therefore each factor $dp dx$ contributes a factor with dimensions of angular momentum. Since there are $3N$ of these factors (N particles and 3 dimensions) in the integration measure, the integral has a total dimension of angular momentum to the power of $3N$. On the other hand, $h$ has dimensions of angular momentum, so dividing by $h^{3N}$ makes the full expression dimensionless.