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The heat capacity ratio or adiabatic index or ratio of specific heats, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume .






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Heat Capacity Ratio for various gases
Temp. Gas ?   Temp. Gas ?   Temp. Gas ?
-181°C H2 1.597 200°C Dry Air 1.398 20°C NO 1.40
-76°C 1.453 400°C 1.393 20°C N2O 1.31
20°C 1.41 1000°C 1.365 -181°C N2 1.47
100°C 1.404 2000°C 1.088 15°C 1.404
400°C 1.387 0°C CO2 1.310 20°C Cl2 1.34
1000°C 1.358 20°C 1.30 -115°C CH4 1.41
2000°C 1.318 100°C 1.281 -74°C 1.35
20°C He 1.66 400°C 1.235 20°C 1.32
20°C H2O 1.33 1000°C 1.195 15°C NH3 1.310
100°C 1.324 20°C CO 1.40 19°C Ne 1.64
200°C 1.310 -181°C O2 1.45 19°C Xe 1.66
-180°C Ar 1.76 -76°C 1.415 19°C Kr 1.68
20°C 1.67 20°C 1.40 15°C SO2 1.29
0°C Dry Air 1.403 100°C 1.399 360°C Hg 1.67
20°C 1.40 200°C 1.397 15°C C2H6 1.22
100°C 1.401 400°C 1.394 16°C C3H8 1.13


The heat capacity ratio or adiabatic index or ratio of specific heats, 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) or (kappa
Kappa

File:Greek lowercase kappa variant.svgKappa is the 10th letter of the Greek alphabet, used to represent the voiceless velar stop, or "k", sound in Ancient Greek and Modern Greek....
). The latter symbol kappa is primarily used by chemical engineers. Mechanical engineers use the Roman
Latin alphabet

The Latin alphabet, also called the Roman alphabet, is the most widely used alphabetic writing system in the world today. It evolved from the western variety of the Greek alphabet called the Cumae alphabet, and was initially developed by the Ancient Romes to write the Latin....
 letter .

where, is the heat capacity or the specific heat capacity
Specific heat capacity

Specific heat capacity, also known simply as specific heat, is the measure of the energy required to increase the temperature of a of a substance by a certain Celsius#Temperatures_and_intervals....
 of a gas, suffix and refer to constant pressure and constant volume conditions respectively.

To understand this relation, consider the following experiment:

A closed cylinder with a locked piston contains air. The pressure inside is equal to the outside air pressure. This cylinder is heated to a certain target temperature. Since the piston cannot move, the volume is constant, while temperature and pressure rise. When the target temperature is reached, the heating is stopped. The piston is now freed and moves outwards, expanding without exchange of heat (adiabatic expansion
Adiabatic process

In thermodynamics, an adiabatic process or an isocaloric process is a thermodynamic process in which no heat is transferred to or from the working fluid....
). Doing this work
Mechanical work

In physics, mechanical work is the amount of energy transferred by a force acting through a distance. Like energy, it is a scalar quantity, with SI of joules....
 cools the air inside the cylinder to below the target temperature. To return to the target temperature (still with a free piston), the air must be heated. This extra heat amounts to about 40% of 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 (e.g. by moving a piston so as to compress the contents of a cylinder), or if work is done by the system which changes its volume (e.g. heating the gas in a cylinder to cause a piston to move). applies only if - that is, the work done - is zero. 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
Enthalpy

In thermodynamics and chemistry, the enthalpy is a quotient or description of thermodynamic potential of a system, which can be used to calculate the heat transfer during a quasistatic process taking place in a closed system thermodynamic system under constant pressure....
  as and the internal energy
Internal energy

In thermodynamics, the internal energy of a thermodynamic system, or a physical body with well-defined dimension, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules and the potential energy associated with the vibrational and electricity energy of atoms within molecules or crysta...
 as . Thus, it can also be said that the heat capacity ratio is the ratio between the enthalpy to the internal energy:

Furthermore, the heat capacities can be expressed in terms of heat capacity ratio ( ) and the gas constant
Gas constant

The gas constant is a physical constant which is featured in a large number of fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation....
 ( ):

It can be rather difficult to find tabulated information for , since is more commonly tabulated. The following relation, can be used to determine :

Relation with degrees of freedom

The heat capacity ratio ( ) for an ideal gas can be related to the degrees of freedom
Degrees of freedom (physics and chemistry)

Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters....
 ( ) of a molecule by: Thus we observe that for a monatomic
Monatomic

In physics and chemistry, monatomic is a combination of the words "mono" and "atomic," and means "single atom." It is usually applied to gases: a monatomic gas is one in which atoms are not bound to each other....
 gas, with three degrees of freedom: , while for a diatomic
Diatomic

Diatomic molecules are molecules made only of two atoms, of either the same or different chemical elements. The prefix di- means two in Greek....
 gas, with five degrees of freedom (at room temperature): .

E.g.: The terrestrial air
AIR

Air is the part of Earth's atmosphere that humans breath and as such Air .Air may also refer to:...
 is primarily made up of diatomic
Diatomic

Diatomic molecules are molecules made only of two atoms, of either the same or different chemical elements. The prefix di- means two in Greek....
 gasses (~78% nitrogen
Nitrogen

Nitrogen is a chemical element that has the symbol N and atomic number 7 and atomic mass 14.00674?. Elemental nitrogen is a colorless, odorless, tasteless and mostly inert diatomic gas at standard conditions, constituting 78% by volume of Earth's atmosphere....
 (N2) and ~21% oxygen
Oxygen

Oxygen no O2 produced; 2) O2 produced, but absorbed in oceans & seabed rock; 3) O2 starts to gas out of the oceans, but is absorbed by land surfaces and formation of ozone layer; 4-5) O2 sinks filled and the gas accumulates]]...
 (O2)) and, at standard conditions 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
.
This is consistent with the measured adiabatic index of approximately 1.403 (listed above in the 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 . For a real gas, and usually increase with increasing temperature and decreases. Some correlations exist to provide values of as a function of the temperature....

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:

Values for are readily available and recorded, but values for 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 values. Values can also be determined through numerical derivatives (peturb T and P (independently!) and calculate and ).

Adiabatic process


This ratio also gives the important relation for an isentropic (quasistatic
Quasistatic process

In thermodynamics, a quasistatic process is a thermodynamic process that happens infinitely slowly. In practice, such processes can be approximated by performing them "very slowly"....
, adiabatic process
Adiabatic process

In thermodynamics, an adiabatic process or an isocaloric process is a thermodynamic process in which no heat is transferred to or from the working fluid....
, reversible) process of a simple compressible calorically perfect ideal gas
Ideal gas

The ideal gas model is a model of matter in which the molecules are treated as non-interacting point particles which are engaged in a random motion that obeys conservation of energy....
:

where, is the pressure and 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
    Specific heat capacity

    Specific heat capacity, also known simply as specific heat, is the measure of the energy required to increase the temperature of a of a substance by a certain Celsius#Temperatures_and_intervals....
  • Speed of sound
    Speed of sound

    Sound is a vibration that travels through an elasticity medium as a wave. The speed of sound describes how much distance such a wave travels in a certain amount of time....
  • Thermodynamic equations
    Thermodynamic equations

    In thermodynamics, there are a large number of equations relating the variousthermodynamic quantities. In chemical thermodynamics, which is a sub-branch of thermodynamics, for example, there are millions of useful equations....
  • Thermodynamics
    Thermodynamics

    In physics, thermodynamics is the study of the conversion of heat energy into different forms of energy ; different energy conversions into heat energy; and its relation to macroscopic variables such as temperature, pressure, and volume....
  • Volumetric heat capacity
    Volumetric heat capacity

    Volumetric heat capacity describes the ability of a given volume of a substance to store internal energy while undergoing a given temperature change, but without undergoing a phase change....