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Isochoric process

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From Wikipedia, the free encyclopedia

Thermodynamic process of a closed system in which volume remains constant

Thermodynamics

The classicalCarnot heat engine

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Equilibrium / Non-equilibrium

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State
Equation of state Ideal gas Real gas State of matter Phase (matter) Equilibrium Control volume Instruments
Processes
Isobaric Isochoric Isothermal Adiabatic Isentropic Isenthalpic Quasistatic Polytropic Free expansion Reversibility Irreversibility Endoreversibility
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Heat engines Heat pumps Thermal efficiency

System properties

Note:Conjugate variables initalics

Process functions
Property diagrams Intensive and extensive properties
Work Heat
Functions of state
Temperature / Entropy (introduction) Pressure / Volume Chemical potential / Particle number Vapor quality Reduced properties

Material properties

Property databases

Specific heat capacity

c=

$T {\displaystyle T}$	$\partial S$
$N {\displaystyle N}$	$\partial T$

Compressibility

\beta =-

$1 {\displaystyle 1}$	$\partial V$
$V {\displaystyle V}$	$\partial p$

Thermal expansion

\alpha =

$1 {\displaystyle 1}$	$\partial V$
$V {\displaystyle V}$	$\partial T$

Equations

Potentials

Internal energy
$U(S,V)$
Enthalpy
$H(S,p)=U+pV$
Helmholtz free energy
$A(T,V)=U-TS$
Gibbs free energy
$G(T,p)=H-TS$

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Inthermodynamics, anisochoric process, also called aconstant-volume process, anisovolumetric process, or anisometric process, is athermodynamic process during which thevolume of theclosed system undergoing such a process remains constant. An isochoric process is exemplified by the heating or the cooling of the contents of a sealed,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 todeform imposes the constant-volume condition.

Formalism

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An isochoric thermodynamicquasi-static process is characterized by constantvolume, i.e.,ΔV = 0.^[1]The process does nopressure-volumework, since such work is defined by $W=P\Delta V,$ whereP is pressure.^{[citation needed]} The sign convention is such that positive work is performed by the system on the environment.^{[citation needed]}

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

For areversible process, thefirst law of thermodynamics gives the change in the system'sinternal energy: $dU=dQ-dW$

Replacingwork with a change in volume gives $dU=dQ-P\,dV$

Since the process is isochoric,dV = 0, the previous equation now gives $dU=dQ$

Using the definition ofspecific heat capacity at constant volume,c_v = (dQ/dT)/m, wherem is the mass of the gas, we get $dQ=mc_{\mathrm {v} }\,dT$

Integrating both sides yields $\Delta Q\ =m\int _{T_{1}}^{T_{2}}\!c_{\mathrm {v} }\,dT,$ wherec_v is the specific heat capacity at constant volume,T₁ is the initialtemperature andT₂ is the finaltemperature. We conclude with: $\Delta Q\ =mc_{\mathrm {v} }\Delta T$

Isochoric process in thepressure volume diagram. In this diagram, pressure increases, but volume remains constant.

On apressure volume diagram, an isochoric process appears as a straight vertical line. Its thermodynamic conjugate, anisobaric process would appear as a straight horizontal line.

Ideal gas

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If anideal gas is used in an isochoric process, and the quantity ofgas stays constant, then the increase inenergy is proportional to an increase intemperature 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

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The idealOtto cycle is an example of an isochoric process when it is assumed that the burning of thegasoline-air mixture in aninternal 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.