Heat capacity ratio

Ideal-gas relations

For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., ${\displaystyle U=U(n,T)}$, where n is the amount of substance in moles. In thermodynamic terms, this is a consequence of the fact that the internal pressure of an ideal gas vanishes.

Mayer's relation allows us to deduce the value of C_V (molar heat capacity at constant volume) from the more easily measured (and more commonly tabulated) value of C_P (molar heat capacity at constant pressure): $${\displaystyle C_V=C_P-R.}$$

This relation may be used to show the heat capacities may be expressed in terms of the heat capacity ratio (γ) and the gas constant (R): $${\displaystyle \begin{array}{cccc}& C_P=\frac{\gamma R}{\gamma -1},& & C_V=\frac{R}{\gamma -1}\\ & \gamma =\frac{C_P}{C_P-R},& & \gamma =1+\frac{R}{C_V}\end{array}}$$

Relation with degrees of freedom

The classical equipartition theorem predicts that the heat capacity ratio (γ) for an ideal gas can be related to the thermally accessible degrees of freedom (f) of a molecule by $${\displaystyle \gamma =1+\frac{2}{f},\text{or}f=\frac{2}{\gamma -1}.}$$

Thus we observe that for a monatomic gas, with 3 translational degrees of freedom per atom: $${\displaystyle \gamma =\frac{5}{3}=1.6666\dots ,}$$

As an example of this behavior, at 273 K (0 °C) the noble gases He, Ne, and Ar all have nearly the same value of γ, equal to 1.664.

For a diatomic gas, often 5 degrees of freedom are assumed to contribute at room temperature since each molecule has 3 translational and 2 rotational degrees of freedom, and the single vibrational degree of freedom is often not included since vibrations are often not thermally active except at high temperatures, as predicted by quantum statistical mechanics. Thus we have $${\displaystyle \gamma =\frac{7}{5}=\mathrm{1.4.}}$$

For example, 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 linear triatomic molecule such as CO₂, there are only 5 degrees of freedom (3 translations and 2 rotations), assuming vibrational modes are not excited. 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 than is typically the case for diatomic molecules. For example, it requires a far larger temperature to excite the single vibrational mode for H₂, for which one quantum of vibration is a fairly large amount of energy, than for the bending or stretching vibrations of CO₂.

For a non-linear triatomic gas, such as water vapor, which has 3 translational and 3 rotational degrees of freedom, this model predicts $${\displaystyle \gamma =\frac{8}{6}=1.3333\dots .}$$

Real-gas relations

As noted above, as temperature increases, higher-energy vibrational states become accessible to molecular gases, thus increasing the number of degrees of freedom and lowering γ. Conversely, as the temperature is lowered, rotational degrees of freedom may become unequally partitioned as well. As a result, both C_P and C_V increase with increasing temperature.

Despite this, if the density is fairly low and intermolecular forces are negligible, the two heat capacities may still continue to differ from each other by a fixed constant (as above, C_P = C_V + nR), which reflects the relatively constant PV difference in work done during expansion for constant pressure vs. constant volume conditions. Thus, the ratio of the two values, γ, decreases with increasing temperature.

However, when the gas density is sufficiently high and intermolecular forces are important, thermodynamic expressions may sometimes be used to accurately describe the relationship between the two heat capacities, as explained below. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate.

Thermodynamic expressions

Values based on approximations (particularly C_P − C_V = nR) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. An experimental value should be used rather than one based on this approximation, where possible. A rigorous value for the ratio ⁠C_P/C_V⁠ can also be calculated by determining C_V from the residual properties expressed as $${\displaystyle C_P-C_V=-T\frac{{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }T}\right)}_P^2}{{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }P}\right)}_T}=-T\frac{{\left(\frac{\mathrm{\partial }P}{\mathrm{\partial }T}\right)}_V^2}{{\left(\frac{\mathrm{\partial }P}{\mathrm{\partial }V}\right)}_T}.}$$

Values for C_P are readily available and recorded, but values for C_V 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 C_V 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:

$${\displaystyle P{V}^{\gamma }=\text{const.}}$$

Using the ideal gas law, ${\displaystyle PV=nRT}$: $${\displaystyle P^{1-\gamma }{T}^{\gamma }=\text{const.}}$$ $${\displaystyle T{V}^{\gamma -1}=\text{const.}}$$

where P is the pressure of the gas, V is the volume, and T is the thermodynamic temperature.

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 ${\displaystyle \rho =M/V}$ as the inverse of the volume for a unit mass, we can take ${\displaystyle \rho =1/V}$ in these relations. Since for constant entropy, ${\displaystyle S}$, we have ${\displaystyle P\propto {\rho }^{\gamma }}$, or ${\displaystyle \ln P=\gamma \ln \rho +\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}$, it follows that $${\displaystyle \gamma ={\left(\frac{\mathrm{\partial }\ln P}{\mathrm{\partial }\ln \rho }\right)}_S=\frac{\rho }{P}{\left(\frac{\mathrm{\partial }P}{\mathrm{\partial }\rho }\right)}_S.}$$

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: $${\displaystyle \begin{array}{cc}{\mathrm{\Gamma }}_1& ={\left(\frac{\mathrm{\partial }\ln P}{\mathrm{\partial }\ln \rho }\right)}_S=\frac{\rho }{P}{\left(\frac{\mathrm{\partial }P}{\mathrm{\partial }\rho }\right)}_S,\\ 1-{\mathrm{\Gamma }}_2^{-1}& ={\left(\frac{\mathrm{\partial }\ln T}{\mathrm{\partial }\ln P}\right)}_S=\frac{P}{T}{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }P}\right)}_S,\\ {\mathrm{\Gamma }}_3-1& ={\left(\frac{\mathrm{\partial }\ln T}{\mathrm{\partial }\ln \rho }\right)}_S=\frac{\rho }{T}{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }\rho }\right)}_S.\end{array}}$$

All of these are equal to ${\displaystyle \gamma }$ in the case of an ideal gas.