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Heat Capacity Ratio of various gases
Gas	°C	k		Gas	°C	k
H2	−181	1.597		Dry Air	20	1.40
	−76	1.453			100	1.401
	20	1.41			200	1.398
	100	1.404			400	1.393
	400	1.387		CO2	0	1.310
	1000	1.358			20	1.30
	2000	1.318			100	1.281
He	20	1.66			400	1.235
N2	−181	1.47		NH3	15	1.310
	15	1.404		CO	20	1.40
Cl2	20	1.34		O2	−181	1.45
Ar	−180	1.76			−76	1.415
	20	1.67			20	1.40
CH4	−115	1.41			100	1.399
	−74	1.35			200	1.397
	20	1.32			400	1.394

The heat capacity ratio of a gas is the ratio of the heat capacity at constant pressure, $C_{p}$, to the heat capacity at constant volume, $C_{v}$. It is also often referred to as the adiabatic index or the ratio of specific heats or the isentropic expansion factor.

Either $k$ (Roman letter k), $\gamma$ (gamma) or $\kappa$ (kappa) may be used to denote the heat capacity ratio:

$k=\gamma =\kappa ={\frac {C_{p}}{C_{v}}}$

where:

$C$ = heat capacity or specific heat of a gas

$p$ = refers to constant pressure conditions

$v$ = refers to constant volume conditions

Ideal gas relations

For an ideal gas, the heat capacity is constant with temperature. Accordingly we can express the enthalpy as $H=C_{p}T$ and the internal energy as $U=C_{v}T$. Thus, it can also be said that the heat capacity ratio of an ideal gas is the ratio between the enthalpy to the internal energy:

$k={\frac {H}{U}}$

The heat capacities at constant pressure, $C_{p}$, of various gases are relatively easy to find in the technical literature. However, it can be difficult to find values of the heat capacities at constant volume, $C_{v}$. When needed, given $C_{p}$, the following equation can be used to determine $C_{v}$ :

$C_{v}=C_{p}-R$

where $R$ is the molar gas constant (also known as the Universal gas constant). This equation can be re-arranged to obtain:

$C_{p}={\frac {kR}{k-1}}\qquad {\mbox{and}}\qquad C_{v}={\frac {R}{k-1}}$

Relation with degrees of freedom for ideal gases

The heat capacity ratio ( $k$ ) for an ideal gas can be related to the degrees of freedom ( $f$ ) of a molecule by:

$k={\frac {f+2}{f}}$

Thus for a monatomic gas, with three degrees of freedom:

$k={\frac {5}{3}}=1.67$

and for a diatomic gas, with five degrees of freedom (at room temperature):

$k={\frac {7}{5}}=1.4$.

Earth's atmospheric air is primarily made up of diatomic gases with a composition of ~78% nitrogen (N₂) and ~21% oxygen (O₂). At 20 °C and an absolute pressure of 101.325 kPa, the atmospheric air can be considered to be an ideal gas. A diatomic molecule has five degrees of freedom (three translational and two rotational degrees of freedom, the vibrational degree of freedom is not involved except at high temperatures). This results in a value of:

$k={\frac {5+2}{5}}={\frac {7}{5}}=1.4$

which is consistent with the value of 1.40 listed for oxygen in the above table.

Isentropic compression or expansion of ideal gases

The heat capacity ratio plays an important part in the isentropic process of an ideal gas (i.e., a process that occurs at constant entropy):

(1)    $p_{1}{V_{1}}^{k}=p_{2}{V_{2}}^{k}$

where, $p$ is the absolute pressure and $V$ is the volume. The subscripts 1 and 2 refer to conditions before and after the process, or at any time during that process.

Using the ideal gas law, $pV=nRT$, equation (1) can be re-arranged to:

(2)    ${\frac {p_{1}}{p_{2}}}=\left({\frac {V_{2}}{V_{1}}}\right)^{k}=\left({\frac {T_{2}}{T_{1}}}\right)^{k}\left({\frac {P_{1}}{P_{2}}}\right)^{k}$

where $T$ is the absolute temperature. Re-arranging further:

(3)    $\left({\frac {T_{2}}{T_{1}}}\right)^{k}=\left({\frac {P_{1}}{P_{2}}}\right)\left({\frac {P_{2}}{P_{1}}}\right)^{k}=\left({\frac {P_{2}}{P_{1}}}\right)^{k-1}$

we obtain the the equation for the temperature change that occurs when an ideal gas is isentropically compressed or expanded:

(4)     ${\frac {T_{2}}{T_{1}}}=\left({\frac {P_{2}}{P_{1}}}\right)^{(k-1)/k}$

Equation (4) is widely used to model ideal gas compression or expansion processes in internal combustion engines, gas compressors and gas turbines.

Real gas relations

As temperature increases, higher energy rotational and vibrational states become accessible to molecular gases, thus increasing the number of degrees of freedom and lowering $\gamma$. For a real gas, $C_{P}$ and $C_{V}$ usually increase with increasing temperature and $\gamma$ decreases. Some correlations exist to provide values of $\gamma$ as a function of the temperature.

Determination of  $C_{v}$  values

Values for $C_{p}$ are readily available, but values for $C_{v}$ are not as available and often need to be determined. Values based on the ideal gas relation of  $C_{p}-C_{v}=R$ are in many cases not sufficiently accurate for practical engineering calculations. If available, an experimental value should be used rather than one based on an approximation.

A rigorous value can be calculated by determining $C_{v}$ from the residual property functions (also referred to as departure functions):

$C_{p}-C_{v}\ =\ -T{\frac {\left({\frac {\partial V}{\partial T}}\right)_{P}^{2}}{\left({\frac {\partial V}{\partial P}}\right)_{T}}}\ =\ -T{\frac {{\left({\frac {\partial P}{\partial T}}\right)}^{2}}{\frac {\partial P}{\partial V}}}$

The above definition is the approach used to develop rigorous expressions from equations of state (such as Peng-Robinson), which match experimental values so closely that there is little need to develop a database of ratios or $C_{v}$ values.

See also

• Heat capacity
• Specific heat capacity
• Speed of sound
• Thermodynamic equations
• Thermodynamics
• Volumetric heat capacity