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The heat capacity ratio or adiabatic index or ratio of specific heats, 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 $\gamma$ (gamma) or $\kappa$ (kappa). The latter symbol kappa is primarily used by chemical engineers. Mechanical engineers use the more common Roman letter $k$.

$\gamma ={\frac {C_{P}}{C_{V}}}$

where, $C$ is the heat capacity or the specific heat capacity of a gas, suffix $P$ and $V$ refer to constant pressure and constant volume conditions respectively.

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 is the ratio between the enthalpy to the internal energy:

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

Furthermore, the heat capacities can be expressed in terms of heat capacity ratio ( $\gamma$ ) and the gas constant ( $R$ ):

$C_{P}={\frac {\gamma R}{\gamma -1}}\qquad {\mbox{and}}\qquad C_{V}={\frac {R}{\gamma -1}}$

It can be rather difficult to find tabulated information for $C_{V}$, since $C_{P}$ is more commonly tabulated. The following relation, can be used to determine $C_{V}$:

$C_{V}=C_{P}-R$

Relation with degrees of freedom

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

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

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

$\gamma \ ={\frac {5}{3}}=1.67$,

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

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

E.g.: The terrestrial air is primarily made up of diatomic gasses (~78% nitrogen (N₂) and ~21% oxygen (O₂)) and, at standard conditions it 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

$\gamma ={\frac {5+2}{5}}={\frac {7}{5}}=1.4$.

This is consistent with the measured adiabatic index of approximately 1.403 (listed above in the table).

Real gas relations

Template:Expand-section 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.

Thermodynamic Expressions

Values based on approximations (particularly $C_{p}-C_{v}=R$) are in many cases not sufficiently accurate for practical engineering calculations such as flow rates through pipes and valves. An experimental value should be used rather than one based on this approximation, where possible. A rigorous value for the ratio ${\frac {C_{p}}{C_{v}}}$ can also be calculated by determining $C_{v}$ from the residual properties expressed as:

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

Values for $C_{p}$ are readily available and recorded, but values for $C_{v}$ need to be determined via relations such as these. See here for the derviation of the thermodynamic relations between the heat capacities.

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. Values can also be determined through numerical derivatives (peturb T and P (independently!) and calculate ${\frac {\Delta V}{\Delta T}}$ and ${\frac {\Delta V}{\Delta P}}$).

Adiabatic process

This ratio also gives the important relation for an isentropic (quasistatic, adiabatic process, reversible) process of a simple compressible calorically perfect ideal gas:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle p_{1}{V_{1}}^{\gamma }=p_{2}{V_{2}}^{\gamma }={\emph {constant}}}

where, $p$ is the 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.

See also

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