It is sometimes also known as the isentropic expansion factor and is denoted by γ (gamma) for an ideal gas[note 1] or κ (kappa), the isentropic exponent for a real gas.
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.
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.
In this example, the amount of heat added with a locked piston is proportional to CV, whereas the total amount of heat added is proportional to CP.
Another way of understanding the difference between CP and CV is that CP 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).
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.
In the first, constant-volume case (locked piston), there is no external motion, and thus no mechanical work is done on the atmosphere; CV 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.
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.,
In thermodynamic terms, this is a consequence of the fact that the internal pressure of an ideal gas vanishes.
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
Thus we observe that for a monatomic gas, with 3 translational degrees of freedom per atom:
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.
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 a non-linear triatomic gas, such as water vapor, which has 3 translational and 3 rotational degrees of freedom, this model predicts
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.
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.
Values based on approximations (particularly CP − CV = 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.
A rigorous value for the ratio CP/CV can also be calculated by determining CV from the residual properties expressed as
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 CV values.
This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: Using the ideal gas law,
For an imperfect or non-ideal gas, Chandrasekhar[3] 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: