Heat capacity ratio
Encyclopedia
Heat Capacity Ratio for various gases
Temp. Gas γ   Temp. Gas γ   Temp. Gas γ
−181°C H2 1.597 200°C Dry Air 1.398 20°C NO 1.400
−76°C 1.453 400°C 1.393 20°C N2O 1.310
20°C 1.410 1000°C 1.365 −181°C N2 1.470
100°C 1.404 2000°C 1.088 15°C 1.404
400°C 1.387 0°C CO2 1.310 20°C Cl2 1.340
1000°C 1.358 20°C 1.300 −115°C CH4 1.410
2000°C 1.318 100°C 1.281 −74°C 1.350
20°C He 1.660 400°C 1.235 20°C 1.320
20°C H2O 1.330 1000°C 1.195 15°C NH3 1.310
100°C 1.324 20°C CO 1.400 19°C Ne 1.640
200°C 1.310 −181°C O2 1.450 19°C Xe 1.660
−180°C Ar 1.760 −76°C 1.415 19°C Kr 1.680
20°C 1.670 20°C 1.400 15°C SO2 1.290
0°C Dry Air 1.403 100°C 1.399 360°C Hg 1.670
20°C 1.400 200°C 1.397 15°C C2H6 1.220
100°C 1.401 400°C 1.394 16°C C3H8 1.130


The heat capacity ratio or adiabatic index or ratio of specific heats, is the ratio of the heat capacity
Heat capacity
Heat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...

 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
Kappa is the 10th letter of the Greek alphabet, used to represent the voiceless velar stop, or "k", sound in Ancient and Modern Greek. In the system of Greek numerals it has a value of 20. It was derived from the Phoenician letter Kaph...

). 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 recognized alphabet used in the world today. It evolved from a western variety of the Greek alphabet called the Cumaean alphabet, which was adopted and modified by the Etruscans who ruled early Rome...

 letter .

where, is the heat capacity and the specific heat capacity (heat capacity per unit mass) 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
Piston
A piston is a component of reciprocating engines, reciprocating pumps, gas compressors and pneumatic cylinders, among other similar mechanisms. It is the moving component that is contained by a cylinder and is made gas-tight by piston rings. In an engine, its purpose is to transfer force from...

 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 the net heat transfer to or from the working fluid is zero. Such a process can occur if the container of the system has thermally-insulated walls or the process happens in an extremely short time,...

). Doing this work
Mechanical work
In physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work...

 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% 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 (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 temperature (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
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...

  as and the internal energy
Internal energy
In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...

 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 many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

 ( ):

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

Where n is the amount of substance
Amount of substance
Amount of substance is a standards-defined quantity that measures the size of an ensemble of elementary entities, such as atoms, molecules, electrons, and other particles. It is sometimes referred to as chemical amount. The International System of Units defines the amount of substance to be...

  in moles.

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)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

 ( ) of a molecule by:
Thus we observe that for a monatomic gas, with three degrees of freedom:,
while for a diatomic
Diatomic
Diatomic molecules are molecules composed only of two atoms, of either the same or different chemical elements. The prefix di- means two in Greek. Common diatomic molecules are hydrogen , nitrogen , oxygen , and carbon monoxide . Seven elements exist in the diatomic state in the liquid and solid...

 gas, with five degrees of freedom (at room temperature):.

E.g.: The terrestrial air is primarily made up of diatomic
Diatomic
Diatomic molecules are molecules composed only of two atoms, of either the same or different chemical elements. The prefix di- means two in Greek. Common diatomic molecules are hydrogen , nitrogen , oxygen , and carbon monoxide . Seven elements exist in the diatomic state in the liquid and solid...

 gases (~78% nitrogen
Nitrogen
Nitrogen is a chemical element that has the symbol N, atomic number of 7 and atomic mass 14.00674 u. Elemental nitrogen is a colorless, odorless, tasteless, and mostly inert diatomic gas at standard conditions, constituting 78.08% by volume of Earth's atmosphere...

 (N2) and ~21% oxygen
Oxygen
Oxygen is the element with atomic number 8 and represented by the symbol O. Its name derives from the Greek roots ὀξύς and -γενής , because at the time of naming, it was mistakenly thought that all acids required oxygen in their composition...

 (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, both and increase with increasing temperature, while continuing to differ from each other by a fixed constant (as above, = ) which reflects the relatively constant P*V 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.

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 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
Finite difference method
In mathematics, finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.- Derivation from Taylor's polynomial :...

.

Adiabatic process

This ratio gives the important relation for an isentropic (quasistatic
Quasistatic process
In thermodynamics, a quasistatic process is a thermodynamic process that happens infinitely slowly. However, it is very important of note that no real process is quasistatic...

, reversible, adiabatic process
Adiabatic process
In thermodynamics, an adiabatic process or an isocaloric process is a thermodynamic process in which the net heat transfer to or from the working fluid is zero. Such a process can occur if the container of the system has thermally-insulated walls or the process happens in an extremely short time,...

) process of a simple compressible calorically perfect ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

:


where is the pressure and is the volume.

See also

  • Heat capacity
    Heat capacity
    Heat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...

  • Specific heat capacity
  • Speed of sound
    Speed of sound
    The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

  • Thermodynamic equations
    Thermodynamic equations
    Thermodynamics is expressed by a mathematical framework of thermodynamic equations which relate various thermodynamic quantities and physical properties measured in a laboratory or production process...

  • Thermodynamics
    Thermodynamics
    Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

  • Volumetric heat capacity
    Volumetric heat capacity
    Volumetric heat capacity , also termed volume-specific 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...

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