Particle number

Definition and physical interpretation

The particle number $N$ is an extensive state variable that quantifies the amount of matter in a system by counting particles (or, more generally, counting moles up to a fixed conversion). In equilibrium thermodynamics, $N$ is treated as a macroscopic coordinate alongside volume and entropy for a single-component simple system.

Physically, changes in $N$ represent matter exchange with the environment across the boundary . Accordingly:

• In a closed system , $N$ is fixed.
• In an open system , $N$ can change via particle flow.
• In an isolated system , $N$ is fixed and there is no exchange of matter or energy.

Appearance in the fundamental differential

For a simple compressible single-component system, the energy representation $U=U(S,V,N)$ implies

The coefficient of $dN$ defines the chemical potential $\mu$, showing that $N$ is the extensive variable conjugate to $\mu$.

In this sense, variations in $N$ contribute an energy change even when mechanical work ($p\,dV$) and heat flow ($T\,dS$) are absent; this is the thermodynamic “cost” of adding/removing matter.

Additivity, mixtures, and conservation remarks

• Additivity: Under the additivity postulate , $N$ is additive across subsystems that interact weakly: combining two subsystems gives $N=N_1+N_2$.

• Multiple species: For mixtures, one uses a set $\{N_i\}$ and corresponding chemical potentials $\{\mu_i\}$; chemical exchange and chemical equilibrium then require matching the appropriate $\mu_i$ across phases or subsystems.

• Conservation: In many settings $N$ is conserved by the allowed processes, but thermodynamics itself allows $N$ to vary whenever particle exchange is permitted. In statistical mechanics this distinction is reflected by choosing, for example, the canonical ensemble (fixed $N$) versus the grand canonical ensemble (fluctuating $N$ at fixed $\mu$).

Densities and the thermodynamic limit

In the thermodynamic limit , it is often natural to keep intensive densities finite, especially the number density $n=N/V$. Many macroscopic constitutive relations are more naturally expressed in terms of $n$ rather than $N$ and $V$ separately.