Editing Isochoric process

{{Short description|Thermodynamic process of a closed system in which volume remains constant}}
{{Thermodynamics|cTopic=[[Thermodynamic system|Systems]]}}

In [[thermodynamics]], an '''isochoric process''', also called a '''constant-volume process''', an '''isovolumetric process''', or an '''isometric process''', is a [[thermodynamic process]] during which the [[volume (thermodynamics)|volume]] of the [[closed system]] undergoing such a process remains constant. An isochoric process is exemplified by the heating or the cooling of the contents of a sealed, [[Elasticity (physics)|inelastic]] container: The thermodynamic process is the addition or removal of heat; the isolation of the contents of the container establishes the closed system; and the inability of the container to [[Deformation (physics)|deform]] imposes the constant-volume condition.
==Formalism==
An isochoric thermodynamic [[Quasistatic process|quasi-static]] process is characterized by constant [[Volume (thermodynamics)|volume]], i.e., {{math|1=Δ''V'' = 0}}.<ref>Ansermet, J.-P., Brechet, S.D. (2019). ''Principles of Thermodynamics'', Cambridge University Press, Cambridge UK, p. 113.</ref>
The process does no [[pressure]]-volume [[Work (thermodynamics)|work]], since such work is defined by
<math display="block"> W = P \Delta V ,</math>
where {{mvar|P}} is pressure.{{Cn|date=April 2025}} The sign convention is such that positive work is performed by the system on the environment.{{Cn|date=April 2025}}

If the process is not quasi-static, the work can perhaps be done in a volume constant thermodynamic process.

For a [[reversible process (thermodynamics)|reversible process]], the [[first law of thermodynamics]] gives the change in the system's [[internal energy]]:
<math display="block">dU = dQ - dW</math>

Replacing [[work (physics)|work]] with a change in volume gives
<math display="block">dU = dQ - P \, dV</math>

Since the process is isochoric, {{math|1=''dV'' = 0}}, the previous equation now gives
<math display="block">dU = dQ</math>

Using the definition of [[specific heat capacity]] at constant volume, {{math|1=''c''<sub>v</sub> = (''dQ''/''dT'')/''m''}}, where {{mvar|m}} is the mass of the gas, we get
<math display="block">dQ = m c_\mathrm{v} \, dT</math>

Integrating both sides yields
<math display="block">\Delta Q\ = m \int_{T_1}^{T_2} \! c_\mathrm{v} \, dT,</math>
where {{math|''c''<sub>v</sub>}} is the specific heat capacity at constant volume, {{math|''T''<sub>1</sub>}} is the initial [[temperature]] and {{math|''T''<sub>2</sub>}} is the final [[temperature]]. We conclude with:
<math display="block">\Delta Q\ = m c_\mathrm{v} \Delta T </math>
[[File:isochoric process SVG.svg|thumb|right|250px|Isochoric process in the [[pressure volume diagram]]. In this diagram, pressure increases, but volume remains constant.]]

On a [[pressure volume diagram]], an isochoric process appears as a straight vertical line.  Its thermodynamic conjugate, an [[isobaric process]] would appear as a straight horizontal line.

===Ideal gas===
If an [[ideal gas]] is used in an isochoric process, and the quantity of [[Ideal gas|gas]] stays constant, then the increase in [[energy]] is proportional to an increase in [[temperature]] and pressure. For example a gas heated in a rigid container: the pressure and temperature of the gas will increase, but the volume will remain the same.

==Ideal Otto cycle==
The ideal [[Otto cycle]] is an example of an isochoric process when it is assumed that the burning of the [[gasoline]]-air mixture in an [[internal combustion engine]] car is instantaneous. There is an increase in the temperature and the pressure of the gas inside the cylinder while the volume remains the same.

==Etymology==
The noun "isochor" and the adjective "isochoric" are derived from the [[Ancient Greek|Greek]] words ἴσος (''isos'') meaning "equal", and χῶρος (''khôros'') meaning "space."

==See also==
* [[Isobaric process]]
* [[Adiabatic process]]
* [[Cyclic process]]
* [[Incompressible flow]]
* [[Isothermal process]]
* [[Polytropic process]]

==References==
{{reflist}}



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[[Category:Thermodynamic processes]]