An isochoric process, also called a constant-volume process, an isovolumetric process, or an isometric process, is a thermodynamic process during which the volume of the closed system undergoing such process remains constant. In nontechnical terms, an isochoric process is exemplified by the heating or the cooling of the contents of a sealed non-deformable 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 deform imposes the constant-volume condition.
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Theory
If the volume stays constant: (ΔV = 0), this implies that the process does no pressure-volume work, since such work is defined by
- ΔW = PΔV,
where P is pressure (no minus sign; this is work done by the system).
By applying the first law of thermodynamics, we can deduce that ΔU the change in the system's internal energy, is
- ΔU = Q
for an isochoric process: all the heat being transferred to the system is added to the system's internal energy, U. If the quantity of gas stays constant, then this increase in energy is proportional to an increase in temperature,
- Q = nCVΔT
where CV is molar specific heat for constant volume.
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 gas stays constant, then the increase in energy is proportional to an increase in temperature and pressure. Take 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.
Applications
Ideal Otto Cycle
In the ideal Otto cycle we found an example of an isochoric process when we assume an instantaneous burning of the gasoline-air mixture in an internal combustion engine car. There is an increase in the temperature and the pressure of the gas inside the cylinder while the volume remains the same.
Derivation
From the First Law of Thermodynamics, for a reversible process (thermodynamics) it is known that
dU = dQ − dW
Replacing work (physics) with a change in volume gives
dU = dQ − PdV
Since the process is isochoric, dV = 0, the previous equation now gives
dU = dQ
Using the definition of specific heat capacity at constant volume
Cv = dU / dT
dQ = nCvdT
Integrating both sides yields
Where C is specific heat capacity at constant volume, a is initial temperature and b is final temperature. We conclude with:
Etymology
The noun isochor and the adjective isochoric derive from the Greek word stems ἴσος (isos) meaning "equal", and χώρος (choros) meaning "space."
See also
References
http://lorien.ncl.ac.uk/ming/webnotes/Therm1/revers/isocho.htm
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