Library / Thermodynamics / THE-18
Thermodynamics
Boundary work in a polytropic process
Work done by the gas in a quasi-equilibrium polytropic process: positive for expansion and negative for compression.
- Ref
- THE-18
- Rev
- v1
- Field
- Thermodynamics
- Result
- Wout [J]
For a polytropic process with n ≠ 1 the boundary work done by the gas is W = (p₂V₂ − p₁V₁)/(1 − n), which comes from integrating p dV along pVn = constant. For n = 1 it is W = p₁V₁ ln(V₂/V₁). A positive W is work done by the gas (expansion); a negative W means work is done on the gas (compression). Pressure in kPa and volume in litres give joules.
Work done by the gas in a quasi-equilibrium polytropic process: positive for expansion and negative for compression. For n = 1 the limiting logarithmic form is used.
Calculator
- Work put into the gas (positive for compression), −W288.7 J
- Final pressure1,493 kPa
- Isothermal work for the same volumes, p₁V₁ ln(V₂/V₁)−207.9 J
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 needs 288.7 J. The same compression done isothermally would need 207.9 J, so cooling during compression saves 28 % of the work.
- The work is the area under the p–V curve; a steeper curve (larger n) encloses more area, which is why multistage compression with intercooling is more efficient than a single uncooled stage.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (J) | Local step | Change in result |
|---|---|---|---|---|---|---|
| V1 Initial volume | 0.1 to 10 L | −9,078 | 2.158 | 9,080 | +10% | +16.9 % |
| p1 Initial absolute pressure | 10 to 1,000 kPa | −2,887 | −28.87 | 2,858 | +10% | +10 % |
| V2 Final volume | 0.05 to 10 L | −485.5 | 166.3 | 651.8 | +10% | −6.07 % |
| n Polytropic exponent | 0.5 to 1.67 | −451.9 | −129.3 | 322.6 | +10% | +16.4 % |
Sweep of Final volume
| Final volume (L) | Work done by the gas (J) |
|---|---|
| 0.05 | −485.5 |
| 1.294 | 24.78 |
| 2.537 | 81.24 |
| 3.781 | 109.7 |
| 5.025 | 128 |
| 6.269 | 141.1 |
| 7.512 | 151.3 |
| 8.756 | 159.5 |
| 10 | 166.3 |
Sweep one variable
Variables and how each one contributes
Work done by the gas
What the equation gives you. Everything else feeds it.
Initial absolute pressure
The initial absolute pressure, in kPa. Makes the work larger in size (more work needed to compress, or more produced on expansion), in direct proportion. Raising it by 10 kPa (to 110 kPa) moves the result by +10 %, from −288.7 to −317.6 J.
Initial volume
The initial volume, in litres. Makes the work larger in size, because there is more gas and a higher compression ratio. Raising it by 0.1 L (to 1.1 L) moves the result by +16.9 %, from −288.7 to −337.4 J.
Final volume
The final volume, in litres. Makes the work smaller in size as the final volume approaches the initial volume, because less compression is done. Raising it by 0.025 L (to 0.15 L) moves the result by −11.5 %, from −288.7 to −255.6 J.
Polytropic exponent
The polytropic exponent. The formula switches form at n = 1. Makes the work larger in size on compression: a larger exponent means the pressure rises faster, so more work is needed. Raising it by 0.05 (to 1.35) moves the result by +5.95 %, from −288.7 to −305.9 J.
Worked example with the values above
- Final pressure: p₂ = 100 × (8)1.3 = 1,493 kPa.
- Products: p₂V₂ = 186.6 J and p₁V₁ = 100 J.
- W = (186.6 − 100) / (1 − 1.3) = −288.7 J (negative: work is done on the gas).
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.
- Using the n ≠ 1 formula at n = 1, which divides by zero.
Statistics and uncertainty
Work depends on both the final pressure and n: a small error in n changes p₂ and the denominator 1 − n together. Near n = 1 the denominator approaches zero and the formula needs the logarithmic form, which this entry applies when n is within 10−9 of 1; values of n very slightly different from 1 are numerically delicate.
Where it is used
- Work needed per cycle in a compressor or gas spring, and work output of an expanding gas.
- Comparing compression options: isothermal, polytropic and adiabatic.
Maths you need
- Ideal gas law: pV = mRT
- Polytropic process: pVn = constant, with n set by the heat exchange