# Inversion temperature

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{{Short description|Concept in thermodynamics}}
The '''inversion temperature''' in [thermodynamics](/source/thermodynamics) and [cryogenics](/source/cryogenics) is the critical temperature below which a gas that is expanding at constant [enthalpy](/source/enthalpy) will experience a temperature decrease, and above which will experience a temperature increase. This temperature change is known as the [Joule–Thomson effect](/source/Joule%E2%80%93Thomson_effect), and is exploited in the [liquefaction of gases](/source/liquefaction_of_gases). Inversion temperature depends on the nature of the gas. The temperature of an [ideal gas](/source/ideal_gas) remains constant during expansion.

For a [van der Waals gas](/source/Van_der_Waals_equation) we can calculate the [enthalpy](/source/enthalpy) <math>H</math> using [statistical mechanics](/source/statistical_mechanics) as
: <math>H = \frac{5}{2} N k_\mathrm B T + \frac{N^2}{V} (b k_\mathrm B T - 2a)</math>
where <math>N</math> is the number of molecules, <math>V</math> is volume, <math>T</math> is temperature (in the [Kelvin scale](/source/Kelvin_scale)), <math>k_\mathrm B</math> is the [Boltzmann constant](/source/Boltzmann_constant), and <math>a</math> and <math>b</math> are constants depending on intermolecular forces and molecular volume, respectively.

From this equation, if enthalpy is kept constant and there is an increase of volume, temperature must change depending on the sign of <math>b k_\mathrm B T - 2a</math>. Therefore, our inversion temperature is given where the sign flips at zero, or
: <math> T_\text{inv} = \frac{2a}{b k_\mathrm B} = \frac{27}{4} T_\mathrm c </math>,
where <math>T_\mathrm c</math> is the [critical temperature](/source/critical_temperature) of the substance. So for <math>T > T_\text{inv}</math>, an expansion at constant enthalpy increases temperature as the [work](/source/work_(thermodynamics)) done by the repulsive interactions of the gas is dominant, and so the change in kinetic [energy](/source/energy) is positive. But for <math>T < T_\text{inv}</math>, expansion causes temperature to decrease because the work of attractive intermolecular forces dominates, giving a negative change in average molecular speed, and therefore kinetic energy.<ref>{{cite book|author=Charles Kittel and Herbert Kroemer|title=Thermal Physics|edition=2nd|publisher=W.H. Freeman|year=1980|isbn=0-7167-1088-9}}</ref>

== See also ==
* [Critical point (thermodynamics)](/source/Critical_point_(thermodynamics))
* [Phase transition](/source/Phase_transition)
* [Joule–Thomson effect](/source/Joule%E2%80%93Thomson_effect)

== References ==
{{reflist}}

== External links ==
* [http://stp.clarku.edu/notes/chap2.pdf Thermodynamic Concepts and Processes (Chapter 2)] (part of the Statistical and Thermal Physics (STP) Curriculum Development Project at [Clark University](/source/Clark_University))

Category:Temperature
Category:Thermodynamic properties
Category:Engineering thermodynamics
Category:Industrial gases
Category:Gases

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