Specific Heat Ratio Calculator

The specific heat ratio (adiabatic index) γ = Cp/Cv determines how a gas behaves in adiabatic processes, sets the speed of sound, and governs isentropic compression.

🔥 Thermodynamics📐 γ = Cℙ/Cᵥ💨 Gas Dynamics
Cp (J/kg·K)
Cv (J/kg·K)

Common values: Air: Cp=1005, Cv=718  |  Argon: Cp=520, Cv=312  |  CO₂: Cp=844, Cv=655

Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Specific Heat Ratioγγ = Cp/Cv—
Speed of Soundvsvs = √(γRT/M)m/s
Monatomic Gasγ5/3 = 1.667 (He, Ar, Ne)—
Diatomic Gasγ7/5 = 1.400 (N₂, O₂, air)—
Polyatomic Gasγ~1.3 (CO₂, H₂O vapor)—

Step-by-Step Examples

Example 1
Air at Standard Conditions

Air: Cp = 1005 J/kg·K, Cv = 718 J/kg·K.

  • γ = Cp/Cv = 1005/718
  • γ = 1.400
  • R = Cp - Cv = 287 J/kg·K
✓ gamma = 1.4 — standard value for air
Example 2
Monatomic Argon

Argon: Cp = 520 J/kg·K, Cv = 312 J/kg·K.

  • γ = 520/312 = 1.667
  • Theoretical: 5/3 = 1.667
✓ gamma = 5/3 — as predicted by kinetic theory
Example 3
CO2 Gas

CO₂: Cp = 844 J/kg·K, Cv = 655 J/kg·K.

  • γ = 844/655 = 1.29
  • Lower gamma due to additional vibrational modes
✓ gamma = 1.29 — polyatomic molecule

Real-World Applications

💨
Gas Dynamics
Gamma determines shock wave behavior, nozzle flow patterns, and explosion propagation.
🌴
Engine Design
Diesel cycle efficiency uses gamma: eta = 1 - (1/r^(gamma-1)). Higher gamma improves efficiency.
🔊
Speed of Sound
v_s = sqrt(gamma*RT/M). Gamma is why sound in helium is faster than in air.
🚀
Rocket Nozzles
Isentropic flow equations use gamma to relate pressure ratios to Mach number in nozzles.

Common Mistakes to Avoid

⚠️
Confusing Cp and Cv

Cp includes work done expanding against atmosphere; Cv is heat added at constant volume. Always Cp > Cv.

⚠️
Forgetting gamma depends on temperature

At high temperatures, more vibrational modes activate and gamma decreases. Air gamma drops from 1.4 at 20°C to ~1.2 at 2000°C.

⚠️
Applying to non-ideal gases

Real gases deviate from ideal behavior at high pressure/low temperature. Gamma is defined for ideal gases.

Frequently Asked Questions

Why is gamma = 5/3 for monatomic gases? ▾
Kinetic theory: monatomic gases have 3 translational degrees of freedom. Cp = (5/2)R, Cv = (3/2)R per mole, so gamma = 5/3.
Why does diatomic air have gamma = 1.4? ▾
Diatomic molecules (N2, O2) have 3 translational + 2 rotational = 5 active degrees at room temperature. Cp = (7/2)R, Cv = (5/2)R, gamma = 7/5 = 1.4.
What does gamma = 1 mean? ▾
Isothermal process (infinite heat exchange). No real gas achieves this, but processes with very slow compression approach it.
How does gamma affect speed of sound? ▾
v_s = sqrt(gamma*RT/M). Helium (gamma=5/3, M=4 g/mol) gives much higher sound speed than air (gamma=1.4, M=29 g/mol).
What is isentropic process? ▾
Reversible adiabatic process where entropy is constant. Governed by P*V^gamma = constant. Used to model fast compression/expansion.
How does gamma relate to entropy? ▾
For an isentropic process: T*V^(gamma-1) = constant and P*V^gamma = constant. Gamma determines how steeply temperature rises with compression.
Why does sound travel faster in helium? ▾
Helium: gamma=5/3, M=4 g/mol. Air: gamma=7/5, M=29 g/mol. Ratio sqrt((5/3 * 29)/(7/5 * 4)) = 2.9. Sound in helium is ~2.9x faster.
Can gamma be less than 1? ▾
No, by thermodynamic stability, Cp > Cv always, so gamma > 1. For gamma = 1 + 2/f where f = degrees of freedom, minimum is 1.

Related Physics Calculators

💨 Ideal Gas Law⚙️ Carnot Efficiency💨 Gas Pressure🔊 Speed of Sound🔥 Adiabatic Process

Formula Explorer connections

Interpretation: This formula tracks heat, temperature, work, entropy or transport in a thermodynamic system. Assumption: Use absolute temperature where required and consistent energy units. Constant properties, equilibrium, ideal gases or negligible losses may be assumed.

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