Adiabatic index

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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 ratio of specific heats, and is denoted by the greek letter ɣ (gamma) or the greek letter κ (kappa). The adiabatic index is dependent on the number f, the degree of freedom of the gas particles.

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.

In the preceding, it may not be obvious how c_P is involved because during the expansion and subsequent heating, the pressure does not remain constant. Another way of understanding the difference between c_P and c_V is to consider the difference between adding heat to the gas with a locked piston, and adding heat with a piston free to move, so that pressure remains constant. In that case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. The heat that is added to the gas goes only partly into heating the gas; the rest is transformed into the mechanical work performed by the piston. In the constant-volume case (locked piston) there is no external motion, and thus no mechanical work is done on the atmosphere. Thus the amount of heat required to raise the gas temperature (the heat capacity) is higher in the constant pressure case.

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 freedom of these 99 % of the gas particles is 5 (3 translational and 2 rotational degrees of freedom) resulting in ɣ = (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.