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For a quasi-static reversible adiabatic state change of an ideal gas, pVγ and TVγ−1 remain constant.
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Formula
p in Pa; V in m3; T in K; γ=cp/cv dimensionless
| Symbol | Meaning | Unit |
|---|---|---|
| Absolute gas pressure | Pa | |
| Gas volume | m3 | |
| Absolute temperature | K | |
| Heat-capacity ratio cp/cv | dimensionless |
pV^\gamma=\mathrm{const.},\quad TV^{\gamma-1}=\mathrm{const.}The Poisson relation applies to a reversible quasi-static adiabat of an ideal gas with approximately constant γ. Adiabatic alone is insufficient: an irreversible real compression does not follow this simple path relation. Temperature-dependent heat capacities make γ variable and require an adjusted model. Use absolute pressure and absolute temperature.
The Poisson relation applies to a reversible quasi-static adiabat of an ideal gas with approximately constant γ. Adiabatic alone is insufficient: an irreversible real compression does not follow this simple path relation. Temperature-dependent heat capacities make γ variable and require an adjusted model. Use absolute pressure and absolute temperature.
For a quasi-static reversible adiabatic state change of an ideal gas, pVγ and TVγ−1 remain constant.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Isentropic state change of an ideal gas” and check its unit and magnitude.
For γ=1.4 and volume halved, p2/p1=(V1/V2)^γ=2^1.4≈2.64.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: p in Pa; V in m3; T in K; γ=cp/cv dimensionless.
Formula cluster
For a quasi-static reversible adiabatic state change of an ideal gas, pVγ and TVγ−1 remain constant.
For γ=1.4 and volume halved, p2/p1=(V1/V2)^γ=2^1.4≈2.64.
Analyze idealized compression and expansion in piston machines, nozzles and thermodynamic cycles.
Here is how to work through a typical Isentropic state change of an ideal gas (pVγ = const.; TVγ−1 = const.) task step by step:
Task
Work through the numerical example for “Isentropic state change of an ideal gas” and check its unit and magnitude.
Solution path
For γ=1.4 and volume halved, p2/p1=(V1/V2)^γ=2^1.4≈2.64.
Isentropic means constant entropy. A reversible adiabatic process is isentropic, but a real irreversible adiabatic process generates entropy and is not isentropic. Thus pVγ=constant cannot be inferred from “no heat transfer” alone. Quasi-static reversibility is part of the assumptions. The relation describes a reversible adiabatic ideal-gas path with constant heat-capacity ratio γ. It is a path relation, not an equation of state. Use absolute pressure and kelvin temperatures; irreversible compression or variable cp changes the model. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
γ is the heat-capacity ratio cp/cv for the gas. For an ideal gas with approximately constant heat capacities, it remains constant during the process. Its value depends on gas species and temperature range; do not assume 1.4 for every gas. Variable γ makes the simple power law approximate. The relation describes a reversible adiabatic ideal-gas path with constant heat-capacity ratio γ. It is a path relation, not an equation of state. Use absolute pressure and kelvin temperatures; irreversible compression or variable cp changes the model. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
The relation follows from the ideal-gas model, whose state equation uses absolute pressure. Gauge pressure relative to ambient must not be substituted directly. Convert to absolute pressure before taking powers. Likewise, TVγ−1 uses kelvin, not Celsius. The relation describes a reversible adiabatic ideal-gas path with constant heat-capacity ratio γ. It is a path relation, not an equation of state. Use absolute pressure and kelvin temperatures; irreversible compression or variable cp changes the model. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
Write the quantity and unit from the variable list next to every value: p in Pa; V in m3; T in K; γ=cp/cv dimensionless. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example For γ=1.4 and volume halved, p2/p1=(V1/V2)^γ=2^1.4≈2.64. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. The relation describes a reversible adiabatic ideal-gas path with constant heat-capacity ratio γ. It is a path relation, not an equation of state. Use absolute pressure and kelvin temperatures; irreversible compression or variable cp changes the model. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
The background states this starting condition: The Poisson relation applies to a reversible quasi-static adiabat of an ideal gas with approximately constant γ. Adiabatic. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. The relation describes a reversible adiabatic ideal-gas path with constant heat-capacity ratio γ. It is a path relation, not an equation of state. Use absolute pressure and kelvin temperatures; irreversible compression or variable cp changes the model. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
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Which formula describes Isentropic state change of an ideal gas?
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How do you rearrange Isentropic state change of an ideal gas for Base equation for the target quantity?
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Which common mistake happens with Isentropic state change of an ideal gas?
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