Heat capacity ratio for various gases<ref>{{cite book|last=White|first=Frank M.|title=Fluid Mechanics|date=October 1998|publisher=[[McGraw Hill]]|isbn=978-0-07-228192-7|edition=4th|location=New York}}</ref><ref>{{cite book|last=Lange|first=Norbert A.|title=Lange's Handbook of Chemistry|publisher=[[McGraw Hill]]|year=1967|isbn=978-0-07-036261-1|edition=10th|location=New York|page=1524}}</ref>
In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume ().
It is sometimes also known as the isentropic expansion factor and is denoted by  (gamma) for an ideal gasγ first appeared in an article by the French mathematician, engineer, and physicist Siméon Denis Poisson:   *    On p. 332, Poisson defines γ merely as a small deviation from equilibrium which causes small variations of the equilibrium value of the density ρ.
In Poisson's article of 1823 – *   γ was expressed as a function of density D (p. 8) or of pressure P (p. 9).
<br>  Meanwhile, in 1816 the French mathematician and physicist Pierre-Simon Laplace had found that the speed of sound depends on the ratio of the specific heats.
*   However, he didn't denote the ratio as γ.
<br> In 1825, Laplace stated that the speed of sound is proportional to the square root of the ratio of the specific heats: *    On p. 127, Laplace defines the symbols for the specific heats, and on p. 137 (at the bottom of the page), Laplace presents the equation for the speed of sound in a perfect gas.
In 1851, the Scottish mechanical engineer William Rankine showed that the speed of sound is proportional to the square root of Poisson's γ: *   It follows that Poisson's γ is the ratio of the specific heats — although Rankine didn't state that explicitly.
*  See also:   or  (kappa), the isentropic exponent for a real gas.
The symbol  is used by aerospace and chemical engineers.
\gamma = \frac{C_P}{C_V} = \frac{\bar{C}_P}{\bar{C}_V} = \frac{c_P}{c_V},
where  is the heat capacity, {\bar{C}} the molar heat capacity (heat capacity per mole), and  the specific heat capacity (heat capacity per unit mass) of a gas.
The suffixes  and  refer to constant-pressure and constant-volume conditions respectively.
The heat capacity ratio is important for its applications in thermodynamical reversible processes, especially involving ideal gases; the speed of sound depends on this factor.
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 , with  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 , whereas the total amount of heat added is proportional to .
Therefore, the heat capacity ratio in this example is 1.4.
Another way of understanding the difference between  and  is that  applies if work is done to the system, which causes a change in volume (such as by moving a piston so as to compress the contents of a cylinder), or if work is done by the system, which changes its temperature (such as heating the gas in a cylinder to cause a piston to move).
applies only if P\,\mathrm{d}V = 0, that is, no work is done.
Consider the difference between adding heat to the gas with a locked piston and adding heat with a piston free to move, so that pressure remains constant.
In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere.
The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston.
In the first, constant-volume case (locked piston), there is no external motion, and thus no mechanical work is done on the atmosphere;  is used.
In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case.
Ideal-gas relations
For an ideal gas, the heat capacity is constant with temperature.
Accordingly, we can express the enthalpy  as  and the internal energy as .
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 () and the gas constant ():
C_P = \frac{\gamma n R}{\gamma - 1} \quad \text{and} \quad C_V = \frac{n R}{\gamma - 1},
where  is the amount of substance in moles.
Mayer's relation allows to deduce the value of  from the more commonly tabulated value of :
C_V = C_P - nR.
Relation with degrees of freedom
The heat capacity ratio () for an ideal gas can be related to the degrees of freedom () of a molecule by
\gamma = 1 + \frac{2}{f},\quad \text{or} \quad f = \frac{2}{\gamma - 1}.
Thus we observe that for a monatomic gas, with 3 degrees of freedom:
\gamma = \frac{5}{3} = 1.6666\ldots,
while for a diatomic gas, with 5 degrees of freedom (at room temperature: 3 translational and 2 rotational degrees of freedom; the vibrational degree of freedom is not involved, except at high temperatures):
\gamma = \frac{7}{5} = 1.4.
For example, the terrestrial air is primarily made up of diatomic gases (around 78% nitrogen, N2, and 21% oxygen, O2), and at standard conditions it can be considered to be an ideal gas.
The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above).
For a non-colinear triatomic gas as water vapor having 6 degrees of freedom :
\gamma = \frac{8}{6} = 1.3333\ldots.
For a co-linear triatomic molecule such as , there are only 5 degrees of freedom, assuming vibrational modes are not excited.
In general however, as mass increases and the frequency of vibrational modes decreases, vibrational degrees of freedom start to enter into the equation at far lower temperatures.
For example, it requires a far larger temperature to excite vibrational modes for , for which one quantum of vibration is a much larger energy, than for .
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 .
For a real gas, both  and  increase with increasing temperature, while continuing to differ from each other by a fixed constant (as above, ), which reflects the relatively constant  difference in work done during expansion for constant pressure vs. constant volume conditions.
Thus, the ratio of the two values, , decreases with increasing temperature.
For more information on mechanisms for storing heat in gases, see the gas section of specific heat capacity.
While at 273 K (0 °C), Monatomic gases such as the noble gases He, Ne, and Ar all have the same value of , that being 1.664.
Thermodynamic expressions
Values based on approximations (particularly ) 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  can also be calculated by determining  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)_V^2}{\left(\frac{\partial P}{\partial V}\right)_T}.
Values for  are readily available and recorded, but values for  need to be determined via relations such as these.
See relations between specific heats for the derivation 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  values.
Values can also be determined through finite-difference approximation.
Adiabatic process
This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas:
PV^\gamma is constant
Using the ideal gas law, PV = nRT:
P^{1-\gamma} T^\gamma is constant
TV^{\gamma-1} is constant
where  is pressure in Pa,  is the volume of the gas in m^3 and  is the temperature in K.
In gas dynamics we are interested in the local relations between pressure, density and temperature, rather than considering a fixed quantity of gas.
By considering the density \rho = M/V as the inverse of the volume for a unit mass, we can take \rho = 1/V in these relations.
Since for constant entropy, S, we have P \propto \rho^\gamma, or  \ln P = \gamma \ln \rho + \mathrm{constant}, it follows that
\gamma = \left.
\frac{\partial \ln P}{\partial \ln \rho}\right|_{S}.
For an imperfect or non-ideal gas, Chandrasekhar defined three different adiabatic indices so that the adiabatic relations can be written in the same form as above; these are used in the theory of stellar structure:
\begin{align}                        \Gamma_1 &= \left.
\frac{\partial \ln P}{\partial \ln \rho}\right|_{S}, \\[2pt]   \frac{\Gamma_2 - 1}{\Gamma_2} &= \left.
\frac{\partial \ln T}{\partial \ln P}\right|_{S}, \\[2pt]                    \Gamma_3 - 1 &= \left.
\frac{\partial \ln T}{\partial \ln \rho}\right|_{S}.
\end{align}
All of these are equal to \gamma in the case of a perfect gas.
See also
Relations between heat capacities
Heat capacity
Specific heat capacity
Speed of sound
Thermodynamic equations
Thermodynamics
Volumetric heat capacity
References
Notes
