Heat capacity ratio - AbsoluteAstronomy.com

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	N₂	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	Ar	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...

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...

. 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

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

) 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.

Ideal gas relations

Relation with degrees of freedom

Real gas relations

Thermodynamic expressions

Adiabatic process

See also