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Thermodynamics
Pressure after a polytropic process
Pressure of a fixed amount of ideal gas after a process that follows pV^n = constant.
- Ref
- THE-16
- Rev
- v1
- Field
- Thermodynamics
- Result
- p2 [kPa]
A polytropic process follows pVn = constant, where the exponent n describes how heat is exchanged: n = 0 is constant pressure, n = 1 is isothermal (for an ideal gas), n = γ is reversible adiabatic, and n → ∞ is constant volume. Between two states p₂ = p₁(V₁/V₂)n. Pressures must be absolute and the volumes in the same unit.
Pressure of a fixed amount of ideal gas after a process that follows pV^n = constant. An exponent of 1 is isothermal, and the ratio of specific heats gives a reversible adiabatic process.
Calculator
- Compression ratio, V₁/V₂8
- Pressure ratio, p₂/p₁14.93
- Final pressure in bar14.93 bar
- Final pressure if isothermal (n = 1)800 kPa
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Compressing the gas from 1 L to 0.125 L with n = 1.3 raises the pressure from 100 to 1,493 kPa, a ratio of 14.9.
- For the same volumes an isothermal compression (n = 1) would reach 800 kPa and an adiabatic one (n = 1.4) 1,838 kPa: cooling during compression keeps the pressure lower and the work smaller.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| V1 Initial volume | 0.1 to 10 L | 74.82 | 29,790 | 29,710 | +10% | +13.2 % |
| p1 Initial absolute pressure | 10 to 1,000 kPa | 149.3 | 14,930 | 14,780 | +10% | +10 % |
| V2 Final volume | 0.05 to 10 L | 5.012 | 4,913 | 4,908 | +10% | −11.7 % |
| n Polytropic exponent | 0.5 to 1.67 | 282.8 | 3,222 | 2,939 | +10% | +31 % |
Sweep of Final volume
| Final volume (L) | Final pressure (kPa) |
|---|---|
| 0.05 | 4,913 |
| 1.294 | 71.55 |
| 2.537 | 29.8 |
| 3.781 | 17.74 |
| 5.025 | 12.26 |
| 6.269 | 9.197 |
| 7.512 | 7.269 |
| 8.756 | 5.956 |
| 10 | 5.012 |
Sweep one variable
Variables and how each one contributes
Final pressure
What the equation gives you. Everything else feeds it.
Initial absolute pressure
The initial absolute pressure, in kPa. Raises the final pressure in direct proportion. Raising it by 10 kPa (to 110 kPa) moves the result by +10 %, from 1,493 to 1,642 kPa.
Initial volume
The initial volume, in litres. Raises the final pressure, because the volume ratio V₁/V₂ is larger. Raising it by 0.1 L (to 1.1 L) moves the result by +13.2 %, from 1,493 to 1,690 kPa.
Final volume
The final volume, in litres. Lowers the final pressure, because the gas is compressed less. Raising it by 0.025 L (to 0.15 L) moves the result by −21.1 %, from 1,493 to 1,178 kPa.
Polytropic exponent
The polytropic exponent: 1 for isothermal, about 1.4 for adiabatic air, typically 1.2 to 1.35 for a cooled compressor. Raises the final pressure when the gas is compressed: a larger exponent means less heat is lost and so the pressure rises more. Raising it by 0.05 (to 1.35) moves the result by +11 %, from 1,493 to 1,656 kPa.
Worked example with the values above
- Volume ratio: V₁/V₂ = 1 / 0.125 = 8.
- Raise to the power n: 81.3 = 14.93.
- p₂ = 100 × 14.93 = 1,493 kPa.
Assumptions and limits
- An ideal gas following pVn = constant, with absolute pressure and temperature.
- A closed system with a fixed amount of gas, and a reversible, quasi-equilibrium process.
Common mistakes
- Using gauge pressure or Celsius temperature: both must be absolute.
- Using γ for n when the process loses heat: n is lower than γ for a cooled compressor.
Statistics and uncertainty
The exponent n is usually fitted to pressure–volume measurements and carries an uncertainty of a few hundredths. With a volume ratio of 8, an error of 0.05 in n changes the final pressure by about 11 % (80.05 = 1.11), so n matters for compressor sizing.
Where it is used
- Pressure after compression or expansion in compressors, engine cylinders and gas springs.
- Fitting a measured pressure–volume record to find the effective exponent.
Maths you need
- Ideal gas law: pV = mRT
- Polytropic process: pVn = constant, with n set by the heat exchange