User:Milton Beychok/Sandbox

Heat Capacity Ratio for 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:

$C$ = the heat capacity or specific heat of a gas
$p$ = the suffix referring to constant pressure conditions
$v$ = the suffix referring 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:

Failed to parse (unknown function "\k"): {\displaystyle \k = \frac{H}{U}}

Furthermore, the heat capacities can be expressed in terms of the heat capacity ratio ( Failed to parse (unknown function "\k"): {\displaystyle \k} ) and the gas constant ( $R$ ):

Failed to parse (unknown function "\k"): {\displaystyle C_p = \frac{k R}{\k - 1} \qquad \mbox{and} \qquad C_v = \frac{R}{\k - 1}}

It can be rather difficult to find tabulated information for $C_{v}$. When needed, given $C_{p}$, the following equation can be used to determine $C_{v}$:

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

Relation with degrees of freedom

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 (~78% nitrogen (N₂) and ~21% oxygen (O₂)) and, at 0 °C and an absolute pressure of 101.325 kPa, 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:

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

This is consistent with the value of approximately 1.403 listed for oxygen in the above table.

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.

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