Number density

The number density $n$ is the particle number per volume . It is a central intensive descriptor of a material’s concentration.

For a homogeneous system with particle number $N$ and volume $V$, the number density is

Equivalently, $N = nV$ for a homogeneous equilibrium state.

In spatially varying situations (e.g., local equilibrium), one often uses a field $n(\mathbf{r})$ such that integrating over the volume gives the total particle number.

Physical interpretation

Number density tells you how tightly packed particles are in space. Holding $n$ fixed while taking the thermodynamic limit keeps the “bulk” state of matter comparable as the system size grows.

Because it is a ratio of two extensive variables , $n$ is a typical intensive variable and is an example of a specific quantity (per volume).

Key relations

• Equation of state viewpoint: Many equations of state express pressure as a function of temperature and number density, $p=p(T,n)$. For an ideal gas this becomes $p = n k_B T$, where $k_B$ is the Boltzmann constant .

• Link to chemical potential: In many systems the chemical potential can be viewed (holding temperature fixed) as encoding how the free energy changes with density, so $n$ is often the natural “composition” variable in equilibrium thermodynamics.