The adiabatic index of a gas, is the ratio of its specific heat capacity at constant pressure (c_P) to its specific heat capacity at constant volume (c_V). It is also known as the isentropic expansion factor and is denoted by the greek letter κ (kappa) or the greek letter γ (gamma). The adiabatic index is dependent on the number f, the degree of freedom of the gas particles.

$\backslash kappa\; =\; \backslash frac\{c\_P\}\{c\_V\}\; =\; \backslash frac\{f\; +\; 2\}\{f\}$

To understand this definition consider the following experiment: A closed cylinder with a locked piston contains air. The pressure inside is equal to the outside air pressure. This cylinder is heated. Since the piston cannot move the volume is constant. Temperature and pressure rise. Heating is stopped and the energy added to the system, which is proportional to c_V, is noted.

The piston is now freed and moves outwards, expanding without exchange of energy (adiabatic expansion). Doing this work (proportional to c_P) cools the air to below its starting temperature. To return to the starting temperature (still with a free piston) the air must be heated. This extra heat amounts to about 40% of the previous amount.

Because the composition of terrestrial air is 99% diatomic gasses (78% N₂ and 21% O₂) and, at standard conditions, is virtually ideal, the degrees of freedon of these 99% of the gas particles is 5 (3 translational and 2 rotational degrees of freedom) resulting in κ = (5+2)/5 = 1.4. This is consistent with the measured adiabatic index of approximately 1.403 (International Critical Tables, 1929, 5-81), a result often used in aerodynamics.

Thermodynamics

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