# Particle number

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{{short description|Number of particles in a thermodynamic system}}
{{Conjugate variables (thermodynamics)}}

In [thermodynamics](/source/thermodynamics), the '''particle number''' (symbol {{mvar|N}}) of a [thermodynamic system](/source/thermodynamic_system) is the [number](/source/number_of_entities) of constituent [particle](/source/particle)s in that system.<ref name="HoP">{{cite book|last1=Benenson|first1=Walter|last2=Harris|first2=John|last3=Stöcker|first3=Horst|title=Handbook of Physics |url=https://books.google.com/books?id=c60mCxGRMR8C&q=%22Particle+number%22&pg=PA637|publisher=Springer|isbn=0-387-95269-1|year=2002}}</ref> The particle number is a fundamental [thermodynamic property](/source/List_of_thermodynamic_properties) which is [conjugate](/source/Conjugate_variables_(thermodynamics)) to the [chemical potential](/source/chemical_potential). Unlike most [physical quantities](/source/physical_quantities), the particle number is a [dimensionless quantity](/source/dimensionless_quantity), specifically a [countable quantity](/source/countable_quantity). It is an [extensive property](/source/extensive_property), as it is directly proportional to the size of the system under consideration and thus meaningful only for [closed system](/source/Thermodynamic_system)s.

A '''constituent particle''' is one that cannot be broken into smaller pieces at the scale of [energy](/source/energy) {{mvar|k·T}} involved in the process (where {{mvar|k}} is the [Boltzmann constant](/source/Boltzmann_constant) and {{mvar|T}} is the [temperature](/source/temperature)). For example, in a thermodynamic system consisting of a [piston](/source/piston) containing [water vapour](/source/water_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.

==Determining the particle number==
The concept of particle number plays a major role in [theoretical](/source/Theoretical_physics) considerations. In situations where the actual particle number of a given thermodynamical system needs to be determined, mainly in [chemistry](/source/chemistry), it is not practically possible to measure it directly by [counting](/source/counting) the particles. If the material is homogeneous and has a known ''[amount of substance](/source/amount_of_substance)'' ''n'' expressed in ''[mole](/source/mole_(unit))s'', the particle number ''N'' can be found by the relation :
<math> N = nN_A</math>,
where ''N<sub>A</sub>'' is the [Avogadro constant](/source/Avogadro_constant).<ref name="HoP"/>

==Particle number density==
A related [intensive system parameter](/source/intensive_parameter) is the '''particle [number density](/source/number_density)''' (or [particle number concentration](/source/particle_number_concentration), PNC), a quantity of kind [volumetric number density](/source/volumetric_number_density) obtained by dividing the particle number of a system by its [volume](/source/volume). This parameter is often denoted by the lower-case letter ''n''.

==In quantum mechanics==
In [quantum mechanical](/source/quantum_mechanical) processes, the total number of particles may not be preserved. The concept is therefore generalized to the [particle number operator](/source/particle_number_operator), that is, the [observable](/source/observable) that counts the number of constituent particles.<ref>{{cite book|last1=Schumacher|first1=Benjamin|last2=Westmoreland|first2=Michael|title=Quantum Processes, Systems, and Information |year=2010|publisher=[Cambridge University Press](/source/Cambridge_University_Press)|bibcode=2010qpsi.book.....S }}</ref> In [quantum field theory](/source/quantum_field_theory), the particle number operator (see [Fock state](/source/Fock_state)) is conjugate to the phase of the ''classical'' wave (see [coherent state](/source/coherent_state)).

==In air quality==
One measure of [air pollution](/source/air_pollution) used in air quality standards is the atmospheric concentration of [particulate matter](/source/Atmospheric_particulate_matter). This measure is usually expressed in μg/m<sup>3</sup> ([micrograms](/source/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 as ''PN'', with units [#/km] or [#/kWh]. In this case, PN expresses a quantity of particles per unit distance (or work).

==References==
{{Reflist}}

{{Mole concepts}}
<!--Categories-->
Category:Thermodynamics
Category:Dimensionless numbers of thermodynamics
Category:Countable quantities
Category:State functions

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Adapted from the Wikipedia article [Particle number](https://en.wikipedia.org/wiki/Particle_number) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Particle_number?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.