What is the heat capacity ratio (γ)?

The heat capacity ratio, γ (gamma) or κ (kappa), is the ratio of the heat capacity at constant pressure (Cp) to the heat capacity at constant volume (Cv). For ideal gases, γ = Cp / Cv. It governs adiabatic processes, the speed of sound, and isentropic flow. Typical values: monatomic ideal gas (He, Ar) γ = 5/3 ≈ 1.667; diatomic (N₂, O₂, air) γ ≈ 1.4; polyatomic (CO₂, NH₃) γ < 1.4.

How does γ relate to degrees of freedom?

For an ideal gas with f degrees of freedom, Cv = (f/2)R and Cp = ((f+2)/2)R, so γ = (f+2)/f. Monatomic gases have f = 3 (translational only), giving γ = 5/3. Diatomic gases at room temperature have f = 5 (3 translational + 2 rotational), giving γ = 7/5 = 1.4. Polyatomic gases have more modes and therefore lower γ.

How is the speed of sound related to γ?

The speed of sound in an ideal gas is c = √(γRT/M), where R is the universal gas constant (8.314 J/mol·K), T is temperature in Kelvin, and M is molar mass in kg/mol. A higher γ or higher temperature both increase the speed of sound. In dry air at 20 °C, this gives approximately 343 m/s.

What is an adiabatic process?

An adiabatic process is one in which no heat is exchanged with the surroundings (Q = 0). For a reversible adiabatic process of an ideal gas, the relation PVγ = constant holds. Using this, you can relate pressure and volume: P₁V₁γ = P₂V₂γ, and temperature and volume: T₁V₁^(γ−1) = T₂V₂^(γ−1).

What is the difference between Cp and Cv?

Cv is the heat capacity at constant volume — energy added goes entirely into raising the internal energy. Cp is the heat capacity at constant pressure — extra energy is also needed to do work against the surroundings (PdV work). For ideal gases, Cp − Cv = R (the universal gas constant, 8.314 J/mol·K), a result known as Mayer's relation.

Why does γ decrease for more complex molecules?

More complex molecules have more vibrational and rotational degrees of freedom that can absorb energy. This raises Cv faster than Cp, bringing γ = Cp/Cv closer to 1. Monatomic gases have only 3 translational modes, so γ stays at 5/3. Polyatomic molecules can have 6 or more modes active at room temperature, pushing γ below 1.3.

Heat Capacity Ratio Calculator

Name: Heat Capacity Ratio Calculator
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Author: Nham Vu

Calculate the heat capacity ratio (γ = Cp/Cv) for ideal gases and derive adiabatic properties including speed of sound and adiabatic temperature change.

Gas Parameters

Gas Type Molar Mass (g/mol) Temperature (K)

Adiabatic Process (optional)

Volume ratio V₁/V₂

Pressure ratio P₁/P₂

Enter one ratio to compute the adiabatic temperature change (T₁ → T₂).

Fill in the parameters and click Calculate to see results.

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Summary

Calculate the heat capacity ratio (γ = Cp/Cv) for ideal gases and derive adiabatic properties including speed of sound and adiabatic temperature change.

How it works

Select a gas type preset (monatomic, diatomic, linear polyatomic, non-linear polyatomic) or choose "Custom" to enter Cp and Cv manually.
Enter the molar mass of the gas (g/mol) for the speed-of-sound calculation.
Enter the temperature (K) for the speed-of-sound and adiabatic calculations.
The tool computes γ = Cp / Cv and displays it alongside Cp − Cv (which equals R for ideal gases).
Optionally enter a volume ratio (V₁/V₂) or pressure ratio (P₁/P₂) to compute adiabatic temperature changes.
Click Calculate to see all results with unit labels and brief explanations.

Use cases

Determine the adiabatic index for a specific gas in thermodynamics coursework.
Calculate the speed of sound in air, argon, hydrogen, or other gases at a given temperature.
Find the temperature change during adiabatic compression or expansion in a piston cycle.
Verify ideal-gas heat capacity relationships (Cp − Cv = R) in physical chemistry problems.
Estimate isentropic process properties for compressor or turbine design.
Compare γ values across monatomic, diatomic, and polyatomic gases.
Solve AP Chemistry or university thermodynamics exam problems involving adiabatic processes.
Convert between pressure and volume ratios for adiabatic expansions using γ.

Frequently Asked Questions

Last updated: 2026-06-18 · Reviewed by Nham Vu