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Inthermodynamics, theparticle number (symbolN) of athermodynamic system is thenumber of constituentparticles in that system.[1] The particle number is a fundamentalthermodynamic property which isconjugate to thechemical potential. Unlike mostphysical quantities, the particle number is adimensionless quantity, specifically acountable quantity. It is anextensive property, as it is directly proportional to the size of the system under consideration and thus meaningful only forclosed systems.
Aconstituent particle is one that cannot be broken into smaller pieces at the scale ofenergyk·T involved in the process (wherek is theBoltzmann constant andT is thetemperature). For example, in a thermodynamic system consisting of apiston containingwater vapour, the particle number is the number of water molecules in the system. The meaning of constituent particles, and thereby of particle numbers, is thus temperature-dependent.
The concept of particle number plays a major role intheoretical considerations. In situations where the actual particle number of a given thermodynamical system needs to be determined, mainly inchemistry, it is not practically possible to measure it directly bycounting the particles. If the material is homogeneous and has a knownamount of substancen expressed inmoles, the particle numberN can be found by the relation :,whereNA is theAvogadro constant.[1]
A relatedintensive system parameter is theparticlenumber density （orparticle number concentration PNC), a quantity of kindvolumetric number density obtained by dividing the particle number of a system by itsvolume. This parameter is often denoted by the lower-case lettern.
Inquantum mechanical processes, the total number of particles may not be preserved. The concept is therefore generalized to theparticle number operator, that is, theobservable that counts the number of constituent particles.[2] Inquantum field theory, the particle number operator (seeFock state) is conjugate to the phase of theclassical wave (seecoherent state).
One measure ofair pollution used in air quality standards is the atmospheric concentration ofparticulate matter. This measure is usually expressed in μg/m3 (micrograms per cubic metre). In the current EU emission norms for cars, vans, and trucks and in the upcoming EU emission norm for non-road mobile machinery, particle number measurements and limits are defined, commonly referred to asPN, with units [#/km] or [#/kWh]. In this case, PN expresses a quantity of particles per unit distance (or work).