Heat capacity ratio


GasTemp.
H2−1811.597
H2−761.453
H2201.410
H21001.404
H24001.387
H210001.358
H220001.318
He201.66
Ar−1801.760
Ar201.670
O2−1811.450
O2−761.415
O2201.400
O21001.399
O22001.397
O24001.394
N2−1811.470
Cl2201.340
Ne191.640
Xe191.660
Kr191.680
Hg3601.670
H2O201.330
H2O1001.324
H2O2001.310
CO201.310
CO2201.300
CO21001.281
CO24001.235
CO210001.195
CO201.400
NO201.400
N2O201.310
CH4−1151.410
CH4−741.350
CH4201.320
NH3151.310
SO2151.290
C2H6151.220
C3H8161.130
Dry air-151.404
Dry air01.403
Dry air201.400
Dry air2001.398
Dry air4001.393
Dry air10001.365

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 for an ideal gas or , the isentropic exponent for a real gas. The symbol is used by aerospace and chemical engineers.
where is the heat capacity, the molar heat capacity, and the specific heat capacity 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.

Thought experiment

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. Doing this work, air inside the cylinder will cool to below the target temperature.
To return to the target temperature, 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, or if work is done by the system, which changes its temperature. applies only if, 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, 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 is higher for this constant-pressure case.

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.,, where 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 from the more easily measured value of :
This relation may be used to show the heat capacities may be expressed in terms of the heat capacity ratio and the gas constant :

Relation with degrees of freedom

The classical equipartition theorem predicts that the heat capacity ratio for an ideal gas can be related to the thermally accessible degrees of freedom 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 the noble gases He, Ne, and Ar all have nearly the same value of, 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
For example, terrestrial air is primarily made up of diatomic gases, 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%.
For a linear triatomic molecule such as, there are only 5 degrees of freedom, 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, for which one quantum of vibration is a fairly large amount of energy, than for the bending or stretching vibrations of.
For a non-linear triatomic gas, such as water vapor, which has 3 translational and 3 rotational degrees of freedom, this model predicts

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. Conversely, as the temperature is lowered, rotational degrees of freedom may become unequally partitioned as well. As a result, both and 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, 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.
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 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 can also be calculated by determining from the residual properties expressed as
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, 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.