In
thermal physics
Thermal physics is the combined study of thermodynamics, statistical mechanics, and kinetic theory of gases. This umbrella-subject is typically designed for physics students and functions to provide a general introduction to each of three core hea ...
and
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
, 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
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity i ...
at constant pressure () to heat capacity at constant volume (). It is sometimes also known as the ''
isentropic
In thermodynamics, an isentropic process is an idealized thermodynamic process that is both adiabatic and reversible. The work transfers of the system are frictionless, and there is no net transfer of heat or matter. Such an idealized process ...
expansion factor'' and is denoted by (
gamma
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...
) for an ideal gas
[γ first appeared in an article by the French mathematician, engineer, and physicist ]Siméon Denis Poisson
Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electri ...
:
* On p. 332, Poisson defines γ merely as a small deviation from equilibrium which causes small variations of the equilibrium value of the density ρ.
In Poisson's article of 1823 –
*
γ was expressed as a function of density D (p. 8) or of pressure P (p. 9).
Meanwhile, in 1816 the French mathematician and physicist Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
had found that the speed of sound depends on the ratio of the specific heats.
*
However, he didn't denote the ratio as γ.
In 1825, Laplace stated that the speed of sound is proportional to the square root of the ratio of the specific heats:
* On p. 127, Laplace defines the symbols for the specific heats, and on p. 137 (at the bottom of the page), Laplace presents the equation for the speed of sound in a perfect gas.
In 1851, the Scottish mechanical engineer William Rankine
William John Macquorn Rankine (; 5 July 1820 – 24 December 1872) was a Scottish mechanical engineer who also contributed to civil engineering, physics and mathematics. He was a founding contributor, with Rudolf Clausius and William Thomson ( ...
showed that the speed of sound is proportional to the square root of Poisson's γ:
*
It follows that Poisson's γ is the ratio of the specific heats — although Rankine didn't state that explicitly.
* See also: or (
kappa
Kappa (uppercase Κ, lowercase κ or cursive ; el, κάππα, ''káppa'') is the 10th letter of the Greek alphabet, representing the voiceless velar plosive sound in Ancient and Modern Greek. In the system of Greek numerals, has a value o ...
), 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
The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. Alternatively, it is the heat capacity of a ...
(heat capacity per mole), and the
specific heat capacity
In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
(heat capacity per unit mass) 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
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
; the
speed of sound
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as w ...
depends on this factor.
To understand this relation, consider the following
thought experiment
A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences.
History
The ancient Greek ''deiknymi'' (), or thought experiment, "was the most anc ...
. A closed
pneumatic cylinder
Pneumatic cylinders (sometimes known as air cylinders) are mechanical devices which use the power of compressed gas to produce a force in a reciprocating linear motion.
Like hydraulic cylinders, something forces a piston to move in the desir ...
contains air. The
piston
A piston is a component of reciprocating engines, reciprocating pumps, gas compressors, hydraulic cylinders and pneumatic cylinders, among other similar mechanisms. It is the moving component that is contained by a cylinder and is made gas-tig ...
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 (
adiabatic expansion
In thermodynamics, an adiabatic process (Greek: ''adiábatos'', "impassable") is a type of thermodynamic process that occurs without transferring heat or mass between the thermodynamic system and its environment. Unlike an isothermal process ...
). Doing this
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal t ...
, 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. 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 (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). 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 (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 molar heat capacity is at most a function of temperature, since the
internal energy
The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
is solely a function of temperature for a
closed system
A closed system is a natural physical system that does not allow transfer of matter in or out of the system, although — in contexts such as physics, chemistry or engineering — the transfer of energy (''e.g.'' as work or heat) is allowed.
In ...
, i.e.,
, where is the
amount of substance
In chemistry, the amount of substance ''n'' in a given sample of matter is defined as the quantity or number of discrete atomic-scale particles in it divided by the Avogadro constant ''N''A. The particles or entities may be molecules, atoms, ions, ...
in moles. In thermodynamic terms, this is a consequence of the fact that the
internal pressure
Internal pressure is a measure of how the internal energy of a system changes when it expands or contracts at constant temperature. It has the same dimensions as pressure, the SI unit of which is the pascal.
Internal pressure is usually given th ...
of an ideal gas vanishes.
Mayer's relation In the 19th century, German chemist and physicist Julius von Mayer derived a relation between specific heat at constant pressure and the specific heat at constant volume for an ideal gas. Mayer's relation states that
:C_ - C_ = R,
where is the mo ...
allows us to deduce the value of from the more easily measured (and more commonly tabulated) 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
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
():
:
Relation with degrees of freedom
The classical
equipartition theorem
In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. T ...
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
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 a gas in which atoms are not bound to each other. Examples at standard conditions ...
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 , equal to 1.664.
For a
diatomic
Diatomic molecules () are molecules composed of only two atoms, of the same or different chemical elements. If a diatomic molecule consists of two atoms of the same element, such as hydrogen () or oxygen (), then it is said to be homonuclear. Ot ...
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
Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is ...
. Thus we have
:
For example, terrestrial air is primarily made up of
diatomic
Diatomic molecules () are molecules composed of only two atoms, of the same or different chemical elements. If a diatomic molecule consists of two atoms of the same element, such as hydrogen () or oxygen (), then it is said to be homonuclear. Ot ...
gases (around 78%
nitrogen
Nitrogen is the chemical element with the symbol N and atomic number 7. Nitrogen is a nonmetal and the lightest member of group 15 of the periodic table, often called the pnictogens. It is a common element in the universe, estimated at se ...
, N
2, and 21%
oxygen
Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as wel ...
, O
2), 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 , 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 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
An intermolecular force (IMF) (or secondary force) is the force that mediates interaction between molecules, including the electromagnetic forces of attraction
or repulsion which act between atoms and other types of neighbouring particles, e.g. a ...
are negligible, the two heat capacities may still continue to differ from each other by a fixed constant (as above, ), 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
A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaking ...
, in which case thermodynamic expressions arising from simple
equations of state may not be adequate.
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 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
In thermodynamics, the heat capacity at constant volume, C_, and the heat capacity at constant pressure, C_, are extensive properties that have the magnitude of energy divided by temperature.
Relations
The laws of thermodynamics imply the follow ...
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
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are ...
.
Adiabatic process
This ratio gives the important relation for an
isentropic
In thermodynamics, an isentropic process is an idealized thermodynamic process that is both adiabatic and reversible. The work transfers of the system are frictionless, and there is no net transfer of heat or matter. Such an idealized process ...
(
quasistatic,
reversible,
adiabatic process
In thermodynamics, an adiabatic process (Greek: ''adiábatos'', "impassable") is a type of thermodynamic process that occurs without transferring heat or mass between the thermodynamic system and its environment. Unlike an isothermal process, ...
) process of a simple compressible calorically-perfect
ideal gas
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
:
:
is constant
Using the ideal gas law,
:
:
is constant
:
is constant
where is the pressure of the gas, is the volume, and is the
thermodynamic temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
.
In gas dynamics we are interested in the local relations between pressure, density and temperature, rather than considering a fixed quantity of gas. By considering the density
as the inverse of the volume for a unit mass, we can take
in these relations.
Since for constant entropy,
, we have
, or
, it follows that
:
For an imperfect or non-ideal gas,
Chandrasekhar Chandrasekhar, Chandrashekhar or Chandra Shekhar is an Indian name and may refer to a number of individuals. The name comes from the name of an incarnation of the Hindu god Shiva. In this form he married the goddess Parvati. Etymologically, the nam ...
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
Stellar structure models describe the internal structure of a star in detail and make predictions about the luminosity, the color and the future evolution of the star. Different classes and ages of stars have different internal structures, reflec ...
:
:
All of these are equal to
in the case of an ideal gas.
See also
*
Relations between heat capacities
In thermodynamics, the heat capacity at constant volume, C_, and the heat capacity at constant pressure, C_, are extensive properties that have the magnitude of energy divided by temperature.
Relations
The laws of thermodynamics imply the follow ...
*
Heat capacity
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity i ...
*
Specific heat capacity
In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
*
Speed of sound
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as w ...
*
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 is based on a fundamental ...
*
Thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
*
Volumetric heat capacity
The volumetric heat capacity of a material is the heat capacity of a sample of the substance divided by the volume of the sample. It is the amount of energy that must be added, in the form of heat, to one unit of volume of the material in order ...
References
Notes
{{DEFAULTSORT:Heat Capacity Ratio
Thermodynamic properties
Physical quantities
Ratios
Thought experiments in physics