Adiabatic exponent

The adiabatic exponent, γ (gamma), is the ratio of a gas's heat capacity at constant pressure to that at constant volume, γ = C_p/C_v. It controls how steeply a gas's pressure falls when it expands without exchanging heat: along a reversible adiabat PVᵞ = const, and the temperature obeys TVᵞ⁻¹ = const. Because γ always exceeds 1, the adiabat is steeper than the isotherm through the same point.

Its value follows from how many ways a molecule can store energy — its active degrees of freedom f, through γ = (f + 2)/f. A monatomic gas (three translational degrees) has γ = 5/3; a diatomic gas at room temperature (three translational plus two rotational) has γ = 7/5. The deeper reason, the equipartition of energy among degrees of freedom, is treated separately.

The exponent shows up wherever fast or insulated compression and expansion occur: in the speed of sound (which depends on γ because sound waves compress air adiabatically), in diesel ignition, and in the cooling of rising atmospheric air. Measuring γ is one of the cleanest experimental windows on molecular structure.

Entered physics through Poisson's 1823 adiabatic relations and the nineteenth-century discovery that the speed of sound requires the adiabatic exponent; its values were later explained by the equipartition theorem and, for their temperature dependence, by quantum theory.