Specific heat ratio/Citable Version

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

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

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

where:

$C$ = the specific heat of a gas

$p$ = refers to constant pressure conditions

$v$ = refers to constant volume conditions

Ideal gas relations

In thermodynamic terminology, $C_{p}$ and $C_{V}$ may be expressed as:

$C_{p}=\left({\frac {\partial H}{\partial T}}\right)_{p}$     and     $C_{V}=\left({\frac {\partial U}{\partial T}}\right)_{V}$

where $T$ stands for temperature, $H$ for the enthalpy and $U$ for the internal energy. 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 specific heat ratio of an ideal gas is the ratio of the enthalpy to the internal energy:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa = \frac{H}{U}}

The specific heats at constant pressure, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p} , of various gases are relatively easy to find in the technical literature. However, it can be difficult to find values of the specific heats at constant volume, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v} . When needed, given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p} , the following equation can be used to determine Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v}  :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p = \frac{\kappa R}{\kappa - 1} \qquad \mbox{and} \qquad C_v = \frac{R}{\kappa - 1}}

Relation with degrees of freedom

The specific heat ratio ( Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} ) for an ideal gas can be related to the degrees of freedom ( Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} ) of a molecule by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa = \frac{f+2}{f}}

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

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

and for a diatomic gas, with five degrees of freedom (three translational and two rotational degrees of freedom, the vibrational degree of freedom is not involved except at high temperatures):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa = \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. This results in a value of:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa = \frac{5 + 2}{5} = \frac{7}{5} = 1.4}

The specific heats of real gases (as differentiated from ideal gases) are not constant with temperature. As temperature increases, higher energy rotational and vibrational states become accessible to molecular gases, thus increasing the number of degrees of freedom and lowering Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa} .

For a real gas, $C_{p}$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v} usually increase with increasing temperature and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa} decreases. Some correlations exist to provide values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa} as a function of the temperature.

Isentropic compression or expansion of ideal gases

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

(1)    $p_{1}{V_{1}}^{\kappa }=p_{2}{V_{2}}^{\kappa }$

where, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} is the absolute pressure and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle pV = nRT} , equation (1) can be re-arranged to:

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

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is the absolute temperature. Re-arranging further:

(3)    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{T_2}{T_1}\right)^\kappa = \left(\frac{p_1}{p_2}\right)\left(\frac{p_2}{p_1}\right)^\kappa = \left(\frac{p_2}{p_1}\right)^{\kappa-1}}

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

(4)     Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{T_2}{T_1} = \left(\frac{p_2}{p_1}\right)^{(\kappa-1)/\kappa}}

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

Determination of  $C_{v}$  values

Values for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p} are readily available, but values for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v} are not as available and often need to be determined. Values based on the ideal gas relation of  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p - C_v = R} are in many cases not sufficiently accurate for practical engineering calculations. If at all possible, an experimental value should be used.

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v = C_p + T \frac{{\; \left( {\frac{\part p}{\part T}} \right) }^2_V} {\left( \frac{\part p}{\part V} \right)_T} }

Equations of state (EOS) (such as the Peng-Robinson equation of state) can be used to solve this relation and to provide values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v} that match experimental values very closely.