In
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
and
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...
, a degree of freedom is an independent physical parameter in the chosen parameterization of a
physical system
A physical system is a collection of physical objects under study. The collection differs from a set: all the objects must coexist and have some physical relationship.
In other words, it is a portion of the physical universe chosen for analys ...
. More formally, given a parameterization of a physical system, the number of degrees of freedom is the smallest number
of parameters whose values need to be known in order to always be possible to determine the values of ''all'' parameters in the chosen parameterization. In this case, any set of
such parameters are called degrees of freedom.
The location of a
particle
In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from s ...
in
three-dimensional space
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values ('' coordinates'') are required to determine the position of a point. Most commonly, it is the three- ...
requires three
position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three
velocity
Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
components, each in reference to the three dimensions of space. So, if the
time evolution of the system is
deterministic
Determinism is the metaphysical view that all events within the universe (or multiverse) can occur only in one possible way. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping mo ...
(where the state at one instant uniquely determines its past and future position and velocity as a function of time), such a system has six degrees of freedom. If the motion of the particle is constrained to a lower number of dimensions – for example, the particle must move along a wire or on a fixed surface – then the system has fewer than six degrees of freedom. On the other hand, a system with an extended object that can rotate or vibrate can have more than six degrees of freedom.
In
classical mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
, the state of a
point particle
A point particle, ideal particle or point-like particle (often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take ...
at any given time is often described with position and velocity coordinates in the
Lagrangian formalism, or with position and
momentum
In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
coordinates in the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
formalism.
In
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, a degree of freedom is a single
scalar number describing the
microstate
A microstate or ministate is a sovereign state having a very small population or land area, usually both. However, the meanings of "state" and "very small" are not well-defined in international law. Some recent attempts to define microstates ...
of a system.
The specification of all microstates of a system is a point in the system's
phase space
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
.
In the 3D
ideal chain
An ideal chain (or freely-jointed chain) is the simplest model in polymer chemistry to describe polymers, such as nucleic acids and proteins. It assumes that the monomers in a polymer are located at the steps of a hypothetical random walker that ...
model in chemistry, two
angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
s are necessary to describe the orientation of each monomer.
It is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a quadratic function to the energy of the system.
Depending on what one is counting, there are several different ways that degrees of freedom can be defined,
each with a different value.
Thermodynamic degrees of freedom for gases
By the
equipartition theorem
In classical physics, classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energy, energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, ...
, internal energy per mole of gas equals , where is
absolute temperature
Thermodynamic temperature, also known as absolute temperature, is a physical quantity which measures temperature starting from absolute zero, the point at which particles have minimal thermal motion.
Thermodynamic temperature is typically expres ...
and the specific heat at constant volume is c
v = (f)(R/2).
R = 8.314 J/(K mol) is the universal gas constant, and "f" is the number of thermodynamic (quadratic) degrees of freedom,
counting the number of ways in which energy can occur.
Any atom or molecule has three degrees of freedom associated with translational motion (kinetic energy) of the
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weight function, weighted relative position (vector), position of the d ...
with respect to the x, y, and z axes. These are the only degrees of freedom for a monoatomic species, such as
noble gas
The noble gases (historically the inert gases, sometimes referred to as aerogens) are the members of Group (periodic table), group 18 of the periodic table: helium (He), neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), radon (Rn) and, in some ...
atoms.

For a structure consisting of two or more atoms, the whole structure also has rotational kinetic energy, where the whole structure turns about an axis.
A
linear molecule, where all atoms lie along a single axis,
such as any
diatomic molecule
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 mol ...
and some other molecules like
carbon dioxide
Carbon dioxide is a chemical compound with the chemical formula . It is made up of molecules that each have one carbon atom covalent bond, covalently double bonded to two oxygen atoms. It is found in a gas state at room temperature and at norma ...
(CO
2),
has two rotational degrees of freedom, because it can rotate about either of two axes perpendicular to the molecular axis.
A nonlinear molecule, where the atoms do not lie along a single axis, like
water
Water is an inorganic compound with the chemical formula . It is a transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance. It is the main constituent of Earth's hydrosphere and the fluids of all known liv ...
(H
2O), has three rotational degrees of freedom, because it can rotate around any of three perpendicular axes.
In special cases, such as adsorbed large molecules, the rotational degrees of freedom can be limited to only one.
A structure consisting of two or more atoms also has vibrational energy, where the individual atoms move with respect to one another. A diatomic molecule has one
molecular vibration
A molecular vibration is a Periodic function, periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The infrared spectroscopy correlation table, typical vibrational fre ...
mode: the two atoms oscillate back and forth with the chemical bond between them acting as a spring. A molecule with atoms has more complicated modes of
molecular vibration
A molecular vibration is a Periodic function, periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The infrared spectroscopy correlation table, typical vibrational fre ...
, with vibrational modes for a linear molecule and modes for a nonlinear molecule.
As specific examples, the linear CO
2 molecule has 4 modes of oscillation, and the nonlinear water molecule has 3 modes of oscillation
Each vibrational mode has two energy terms: the
kinetic energy
In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion.
In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
of the moving atoms and the
potential energy
In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
of the spring-like chemical bond(s).
Therefore, the number of vibrational energy terms is modes for a linear molecule and is modes for a nonlinear molecule.
Both the rotational and vibrational modes are quantized, requiring a minimum temperature to be activated. The "
rotational temperature" to activate the rotational degrees of freedom is less than 100 K for many gases. For N
2 and O
2, it is less than 3 K.
The "
vibrational temperature" necessary for substantial vibration is between 10
3 K and 10
4 K, 3521 K for N
2 and 2156 K for O
2.
Typical atmospheric temperatures are not high enough to activate vibration in N
2 and O
2, which comprise most of the atmosphere. (See the next figure.) However, the much less abundant
greenhouse gas
Greenhouse gases (GHGs) are the gases in the atmosphere that raise the surface temperature of planets such as the Earth. Unlike other gases, greenhouse gases absorb the radiations that a planet emits, resulting in the greenhouse effect. T ...
es keep the
troposphere
The troposphere is the lowest layer of the atmosphere of Earth. It contains 80% of the total mass of the Atmosphere, planetary atmosphere and 99% of the total mass of water vapor and aerosols, and is where most weather phenomena occur. From the ...
warm by absorbing
infrared
Infrared (IR; sometimes called infrared light) is electromagnetic radiation (EMR) with wavelengths longer than that of visible light but shorter than microwaves. The infrared spectral band begins with the waves that are just longer than those ...
from the Earth's surface, which excites their vibrational modes.
Much of this energy is reradiated back to the surface in the infrared through the "
greenhouse effect
The greenhouse effect occurs when greenhouse gases in a planet's atmosphere insulate the planet from losing heat to space, raising its surface temperature. Surface heating can happen from an internal heat source (as in the case of Jupiter) or ...
."
Because room temperature (≈298 K) is over the typical rotational temperature but lower than the typical vibrational temperature, only the translational and rotational degrees of freedom contribute, in equal amounts, to the
heat capacity ratio
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 vol ...
. This is why ≈ for
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 ...
gases and ≈ for
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 mol ...
gases at room temperature.

Since the
air
An atmosphere () is a layer of gases that envelop an astronomical object, held in place by the gravity of the object. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A stellar atmosph ...
is dominated by diatomic gases (with
nitrogen
Nitrogen is a chemical element; it has Symbol (chemistry), symbol N and atomic number 7. Nitrogen is a Nonmetal (chemistry), nonmetal and the lightest member of pnictogen, group 15 of the periodic table, often called the Pnictogen, pnictogens. ...
and
oxygen
Oxygen is a chemical element; it has chemical symbol, symbol O and atomic number 8. It is a member of the chalcogen group (periodic table), group in the periodic table, a highly reactivity (chemistry), reactive nonmetal (chemistry), non ...
contributing about 99%), its molar internal energy is close to = (5/2), determined by the 5 degrees of freedom exhibited by diatomic gases.
[ Equipartition theorem#Diatomic gases]
See the graph at right. For 140 K < < 380 K, c
v differs from (5/2)
d by less than 1%.
Only at temperatures well above temperatures in the
troposphere
The troposphere is the lowest layer of the atmosphere of Earth. It contains 80% of the total mass of the Atmosphere, planetary atmosphere and 99% of the total mass of water vapor and aerosols, and is where most weather phenomena occur. From the ...
and
stratosphere
The stratosphere () is the second-lowest layer of the atmosphere of Earth, located above the troposphere and below the mesosphere. The stratosphere is composed of stratified temperature zones, with the warmer layers of air located higher ...
do some molecules have enough energy to activate the vibrational modes of N
2 and O
2. The specific heat at constant volume, c
v, increases slowly toward (7/2) as temperature increases above T = 400 K, where c
v is 1.3% above (5/2)
d = 717.5 J/(K kg).
Counting the minimum number of co-ordinates to specify a position
One can also count degrees of freedom using the minimum number of coordinates required to specify a position. This is done as follows:
# For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space. Thus its degree of freedom in a 3-D space is 3.
# For a body consisting of 2 particles (ex. a diatomic molecule) in a 3-D space with constant distance between them (let's say d) we can show (below) its degrees of freedom to be 5.
Let's say one particle in this body has coordinate and the other has coordinate with unknown. Application of the formula for distance between two coordinates
results in one equation with one unknown, in which we can solve for .
One of , , , , , or can be unknown.
Contrary to the classical
equipartition theorem
In classical physics, classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energy, energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, ...
, at room temperature, the vibrational motion of molecules typically makes negligible contributions to 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 is a ...
. This is because these degrees of freedom are ''frozen'' because the spacing between the energy
eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s exceeds the energy corresponding to ambient
temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
s ().
Independent degrees of freedom
The set of degrees of freedom of a system is independent if the energy associated with the set can be written in the following form:
where is a function of the sole variable .
example: if and are two degrees of freedom, and is the associated energy:
* If
, then the two degrees of freedom are independent.
* If
, then the two degrees of freedom are ''not'' independent. The term involving the product of and is a coupling term that describes an interaction between the two degrees of freedom.
For from 1 to , the value of the th degree of freedom is distributed according to the
Boltzmann distribution
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability tha ...
. Its
probability density function
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
is the following:
In this section, and throughout the article the brackets
denote the
mean
A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
of the quantity they enclose.
The
internal energy
The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accoun ...
of the system is the sum of the average energies associated with each of the degrees of freedom:
Quadratic degrees of freedom
A degree of freedom is quadratic if the energy terms associated with this degree of freedom can be written as
where is a
linear combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of other quadratic degrees of freedom.
example: if and are two degrees of freedom, and is the associated energy:
* If
, then the two degrees of freedom are not independent and non-quadratic.
* If
, then the two degrees of freedom are independent and non-quadratic.
* If
, then the two degrees of freedom are not independent but are quadratic.
* If
, then the two degrees of freedom are independent and quadratic.
For example, in
Newtonian mechanics
Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows:
# A body r ...
, the
dynamics of a system of quadratic degrees of freedom are controlled by a set of homogeneous
linear differential equation
In mathematics, a linear differential equation is a differential equation that is linear equation, linear in the unknown function and its derivatives, so it can be written in the form
a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x)
wher ...
s with
constant coefficients
In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form
a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x)
where and are arbi ...
.
Quadratic and independent degree of freedom
are quadratic and independent degrees of freedom if the energy associated with a microstate of the system they represent can be written as:
Equipartition theorem
In the classical limit of
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, at
thermodynamic equilibrium
Thermodynamic equilibrium is a notion of thermodynamics with axiomatic status referring to an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable ...
, the
internal energy
The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accoun ...
of a system of quadratic and independent degrees of freedom is:
Here, the
mean
A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
energy associated with a degree of freedom is:
Since the degrees of freedom are independent, the
internal energy
The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accoun ...
of the system is equal to the sum of the
mean
A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
energy associated with each degree of freedom, which demonstrates the result.
Generalizations
The description of a system's state as a
point in its phase space, although mathematically convenient, is thought to be fundamentally inaccurate. In
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, the motion degrees of freedom are superseded with the concept of
wave function
In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
, and
operators which correspond to other degrees of freedom have
discrete spectra. For example,
intrinsic angular momentum
Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin is quantized, and accurate models for the interaction with spin require relativistic ...
operator (which corresponds to the rotational freedom) for an
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
or
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
has only two
eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s. This discreteness becomes apparent when
action has an
order of magnitude
In a ratio scale based on powers of ten, the order of magnitude is a measure of the nearness of two figures. Two numbers are "within an order of magnitude" of each other if their ratio is between 1/10 and 10. In other words, the two numbers are ...
of the
Planck constant
The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
, and individual degrees of freedom can be distinguished.
References
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Physical quantities
Dimension