Calculate adiabatic compression or expansion temperature, pressure, volume ratios, work, and energy from initial gas conditions and γ.

Adiabatic Compression Temperature Calculator

Use absolute pressure and temperature values.

Compression Temp
Expansion Temp
Advanced Solver
Work & Energy
Compression input
Expansion input
Final state input
Show Calculation Steps ▼

Adiabatic Compression Temperature Formula

The calculator uses reversible adiabatic ideal-gas relationships. Use absolute temperature and absolute pressure values. Temperatures entered in °C or °F are converted to kelvin for the calculation, then converted back for display.

T_2 = T_1\left(\frac{P_2}{P_1}\right)^{(\gamma-1)/\gamma}
T_2 = T_1\left(\frac{V_1}{V_2}\right)^{\gamma-1}
\frac{P_2}{P_1}=\left(\frac{V_1}{V_2}\right)^\gamma
P_1V_1^\gamma=P_2V_2^\gamma
T_1V_1^{\gamma-1}=T_2V_2^{\gamma-1}
Q=0
C_v=\frac{R}{\gamma-1}
\Delta U=nC_v(T_2-T_1)
W_{by\ gas}=\frac{P_1V_1-P_2V_2}{\gamma-1}
  • T₁ = initial absolute temperature
  • T₂ = final absolute temperature
  • P₁ = initial absolute pressure
  • P₂ = final absolute pressure
  • V₁ = initial volume
  • V₂ = final volume
  • γ = heat capacity ratio, equal to Cₚ/Cᵥ
  • R = ideal gas constant, 8.314462618 J/(mol·K)
  • n = amount of gas in moles
  • Q = heat transfer, zero for an adiabatic process
  • ΔU = internal energy change
  • W by gas = work done by the gas

Compression Temp: calculates the final temperature after adiabatic compression from an outlet pressure, pressure ratio, or volume compression ratio.

Expansion Temp: uses the same temperature relation, but the pressure ratio is usually less than 1, so the gas cools during expansion.

Advanced Solver: fills missing state values using the pressure, volume, and temperature adiabatic relations when enough state data is supplied.

Work & Energy: solves the final state and calculates Cᵥ, Cₚ, internal energy change, work done by the gas, and heat transfer.

Common Gamma Values and Result Interpretation

Use a custom γ value if you have a gas-specific value at the temperature range of your process.

Gas Typical γ Notes
Air, nitrogen, oxygen 1.400 Common default for room-temperature diatomic gases
Helium, argon 1.667 Monatomic gases, larger temperature change for the same pressure ratio
Carbon dioxide 1.289 Lower γ, smaller temperature change for the same pressure ratio
Steam 1.330 Approximate ideal-gas value, not suitable near saturation

Condition Pressure ratio P₂/P₁ Temperature result Energy result
Compression Greater than 1 T₂ is greater than T₁ ΔU is positive, work by gas is negative
Expansion Less than 1 T₂ is less than T₁ ΔU is negative, work by gas is positive
No pressure change Equal to 1 T₂ equals T₁ No ideal adiabatic work or internal energy change

Examples

Example 1: Air compressed from 1 atm to 10 atm

You have air with γ = 1.4, T₁ = 300 K, P₁ = 1 atm, and P₂ = 10 atm.

T_2=300\left(\frac{10}{1}\right)^{(1.4-1)/1.4}

The exponent is 0.285714, so:

T₂ = 579.21 K

The compression ratio is:

\frac{V_1}{V_2}=10^{1/1.4}=5.1795

Example 2: Air expanding from 10 bar to 1 bar

You have air with γ = 1.4, T₁ = 600 K, P₁ = 10 bar, and P₂ = 1 bar.

T_2=600\left(\frac{1}{10}\right)^{(1.4-1)/1.4}

T₂ = 310.73 K

The expansion ratio is:

\frac{V_2}{V_1}=\left(\frac{10}{1}\right)^{1/1.4}=5.1795

FAQ

Do you need absolute pressure or gauge pressure?

Use absolute pressure. If you have gauge pressure, add atmospheric pressure before entering it. For example, 100 psi gauge is about 114.7 psi absolute at normal atmospheric pressure.

Why does temperature rise during adiabatic compression?

In adiabatic compression, no heat is transferred out of the gas. Work is done on the gas, increasing its internal energy. For an ideal gas, higher internal energy means higher temperature.

When is this calculation not accurate?

It becomes less accurate when the gas is not close to ideal, when heat transfer is significant, when friction and losses are important, or when γ changes strongly with temperature. Real compressors and turbines often need efficiency corrections in addition to the ideal adiabatic result.