Heat Capacity Ratio Calculator

The heat capacity ratio gamma, also called the adiabatic index or isentropic exponent, is Cp divided by Cv. It is always greater than one because heating a gas at constant pressure must also pay for the expansion work it does on its surroundings, so Cp exceeds Cv by the Mayer relation Cp - Cv = R = 8.314 J/(mol*K) for an ideal gas. Gamma is what sets the exponent in the reversible adiabatic relation P*V^gamma = constant and it enters the speed of sound as c = sqrt(gamma*R*T/M), which is why it can be measured acoustically. The kinetic-theory route comes from equipartition: each active quadratic degree of freedom contributes R/2 to molar Cv, so Cv = (f/2)*R and gamma = (f + 2)/f. Monatomic gases have only three translational degrees of freedom, giving gamma = 5/3 = 1.667, and helium and argon measure 1.66 to 1.67. Diatomic molecules add two rotational modes, so f = 5 and gamma = 7/5 = 1.400; nitrogen is about 1.404 and oxygen about 1.395 near 298 K. Carbon dioxide, with low-lying bending modes already partly excited, sits near 1.289 at 300 K, and steam is around 1.33. The value 5/3 is therefore an upper bound for an ideal gas. The main caveat is temperature: as vibrational modes unfreeze, f rises and gamma falls, so nitrogen drops toward 1.30 near 2000 K. Near the critical point Cp diverges and the ideal-gas treatment collapses entirely, so tabulated real-fluid data should be used there.

The practical importance of gamma was recognized before anyone understood heat capacities. Isaac Newton's 1687 calculation of the speed of sound assumed isothermal compression and came out roughly 15 percent too low, near 290 m/s against a measured value close to 340 m/s. Pierre-Simon Laplace resolved the discrepancy in 1816 by arguing that the compressions in a sound wave are too rapid for heat exchange and are therefore adiabatic, which inserts the factor gamma into the expression. Simeon Denis Poisson formalised the adiabatic relations in 1823, and Julius Robert von Mayer stated the Cp minus Cv relation in 1842 as part of his argument for the mechanical equivalent of heat. Measuring gamma directly became possible with the adiabatic expansion method of Nicolas Clement and Charles Desormes, published in 1819, in which a large vessel is briefly opened to the atmosphere and the pressure recovery is recorded. Eduard Ruchhardt introduced a neat oscillating-ball variant in 1929. The theoretical picture came from the kinetic theory of James Clerk Maxwell and Ludwig Boltzmann, but classical equipartition predicted values that experiments contradicted, since vibrational modes that should have counted appeared to be missing. Maxwell called this the greatest difficulty facing the molecular theory. Quantum mechanics supplied the answer: vibrational levels are frozen out until thermal energy approaches their spacing.