Hong Life CalculatorsUnderstanding the adiabatic and isothermal process calculator — concept and principle
When a gas is compressed or expanded, the result depends on what happens to the heat. In an adiabatic process no heat is exchanged, so the gas heats up when compressed; in an isothermal process heat flows out so the temperature stays constant. Enter the initial pressure, the initial and final volumes, the starting temperature and the heat-capacity ratio γ, and the calculator gives the final pressure, temperature and work for both cases side by side.
How it is calculated
- Adiabatic pressure: P₂ = P₁ (V₁/V₂)^γ
- Adiabatic temperature: T₂ = T₁ (V₁/V₂)^(γ−1)
- Adiabatic work on the gas: W = (P₂V₂ − P₁V₁) / (γ − 1)
- Isothermal pressure: P₂ = P₁V₁ / V₂
- Isothermal work on the gas: W = P₁V₁ ln(V₁/V₂), which equals nRT ln(V₁/V₂)
Pressures are in kPa and volumes in litres; since kPa × L = J, the work is divided by 1000 to give kJ. Positive work means work is done on the gas (compression).
The principle
The first law says ΔU = Q + W. With Q = 0, all work done on the gas goes into internal energy, so its temperature rises. For an ideal gas this leads to PV^γ = constant, where γ = c_p/c_v is about 1.4 for air and other diatomic gases near room temperature and 5/3 for monatomic gases. In the isothermal case ΔU = 0 for an ideal gas, so every joule of work leaves as heat, and PV stays constant. The adiabatic curve is steeper, which is why it needs more work for the same compression.
Worked example
- Inputs: P₁ = 100 kPa, V₁ = 10 L, V₂ = 2 L, T₁ = 300 K, γ = 1.4. Compression ratio V₁/V₂ = 5.
- Adiabatic P₂ = 100 × 5^1.4 = 100 × 9.51827 = 951.8269694 kPa.
- Adiabatic T₂ = 300 × 5^0.4 = 300 × 1.90365 = 571.0961816 K.
- Adiabatic work = (951.827 × 2 − 100 × 10) / 0.4 = 903.654 / 0.4 = 2259.13 J = 2.259134847 kJ.
- Isothermal P₂ = 100 × 10 / 2 = 500 kPa.
- Isothermal work = 100 × 10 × ln 5 = 1609.44 J = 1.609437912 kJ.
| γ (air 1.4) | Adiabatic P₂ (kPa) |
|---|---|
| 0.35 | 175.6465005 |
| 0.7 | 308.5169314 |
| 1.05 | 541.8991934 |
| 1.4 | 951.8269694 |
| 1.75 | 1,671.850762 |
| 2.1 | 2,936.547358 |
| 2.8 | 9,059.745796 |
How to read the result
Compressing air five-fold without cooling almost doubles its absolute temperature, here from 27 °C to about 298 °C, and needs about 40% more work than the cooled isothermal case. Real compressors fall between the two, which is why multi-stage compressors use intercoolers. For expansion (V₂ larger than V₁) the work becomes negative: the gas does work on its surroundings and, adiabatically, cools down.
Common mistakes
- Using °C for T₁; the formula needs kelvin.
- Using gauge pressure instead of absolute pressure.
- Confusing the sign convention: some textbooks report work done by the gas, which has the opposite sign.
Frequently asked questions
Why does a bicycle pump get hot?
Rapid compression is close to adiabatic, so the work you do raises the air temperature, and friction adds more.
What γ should I use for other gases?
About 1.67 for helium and argon, about 1.4 for air, nitrogen and oxygen, and around 1.3 for carbon dioxide or steam.
Sources: Y. A. Çengel and M. A. Boles, Thermodynamics: An Engineering Approach, McGraw-Hill · P. Atkins and J. de Paula, Atkins' Physical Chemistry, Oxford University Press