Heat capacity ratio

Heat capacity ratio for various gases
Temp.	Gas	$γ$	Temp.	Gas	$γ$	Temp.	Gas	$γ$
−181 °C	H₂	1.597	200 °C	Dry air	1.398	20 °C	NO	1.400
−76 °C		1.453	400 °C		1.393	20 °C	N₂O	1.310
20 °C		1.410	1000 °C		1.365	−181 °C	N₂	1.470
100 °C		1.404	2000 °C		1.088	15 °C	N₂	1.404
400 °C		1.387	0 °C	CO₂	1.310	20 °C	Cl₂	1.340
1000 °C		1.358	20 °C		1.300	−115 °C	CH₄	1.410
2000 °C		1.318	100 °C		1.281	−74 °C		1.350
20 °C	He	1.660	400 °C		1.235	20 °C		1.320
20 °C	H₂O	1.330	1000 °C		1.195	15 °C	NH₃	1.310
100 °C		1.324	20 °C	CO	1.400	19 °C	Ne	1.640
200 °C		1.310	−181 °C	O₂	1.450	19 °C	Xe	1.660
−180 °C	Ar	1.760	−76 °C		1.415	19 °C	Kr	1.680
20 °C	Ar	1.670	20 °C		1.400	15 °C	SO₂	1.290
0 °C	Dry air	1.403	100 °C		1.399	360 °C	Hg	1.670
20 °C		1.400	200 °C		1.397	15 °C	C₂H₆	1.220
100 °C		1.401	400 °C		1.394	16 °C	C₃H₈	1.130

In thermal physics and thermodynamics, the heat capacity ratio or adiabatic index or ratio of specific heats or Poisson constant, is the ratio of the heat capacity at constant pressure ( $C P$ ) to heat capacity at constant volume ( $C V$ ). It is sometimes also known as the isentropic expansion factor and is denoted by $γ$ (gamma) for an ideal gas or $κ$ (kappa), the isentropic exponent for a real gas. The symbol gamma is used by aerospace and chemical engineers.

where $C$ is the heat capacity and $c$ the specific heat capacity (heat capacity per unit mass) of a gas. The suffixes $P$ and $V$ refer to constant pressure and constant volume conditions, respectively.

To understand this relation, consider the following thought experiment. A closed pneumatic cylinder contains air. The piston is locked. The pressure inside is equal to atmospheric pressure. This cylinder is heated to a certain target temperature. Since the piston cannot move, the volume is constant. The temperature and pressure will rise. When the target temperature is reached, the heating is stopped. The amount of energy added equals $C V Δ T$ , with $Δ T$ representing the change in temperature. The piston is now freed and moves outwards, stopping as the pressure inside the chamber reaches atmospheric pressure. We assume the expansion occurs without exchange of heat (adiabatic expansion). Doing this work, air inside the cylinder will cool to below the target temperature. To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume since the piston is free to move as the gas is reheated. This extra heat amounts to about 40% more than the previous amount added. In this example, the amount of heat added with a locked piston is proportional to $C V$ , whereas the total amount of heat added is proportional to $C P$ . Therefore, the heat capacity ratio in this example is 1.4.

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