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In
physics Physics is the natural science that studies matter, its 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 which relat ...
and
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, properties, ...
, a degree of freedom is an independent physical
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
in the formal description of the state of a
physical system A physical system is a collection of physical objects. In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...
. The set of all states of a system is known as the system's phase space, and the degrees of freedom of the system are the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
s of the phase space. The location of a particle in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
requires three position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. If the time evolution of the system is
deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
(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 physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mec ...
, the state of a point particle 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 (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. If is an object's mass and ...
coordinates in the 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. It does not assume or postulate any natural laws, but explains the macroscopic b ...
, 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 very small land area, usually both. However, the meanings of "state" and "very small" are not well-defined in international law.Warrington, E. (1994). "Lilliputs ...
of a system. The specification of all microstates of a system is a point in the system's phase space. In the 3D
ideal chain Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
model in chemistry, two
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
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 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. ...
, internal energy per mole of gas equals , where is
absolute 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 thermodynami ...
and the specific heat at constant volume is cv = (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 balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
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 also the inert gases; sometimes referred to as aerogens) make up a class of chemical elements with similar properties; under standard conditions, they are all odorless, colorless, monatomic gases with very low che ...
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 ...
and some other molecules like carbon dioxide (CO2), 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 (H2O), 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 motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 1013 Hz to approximately 1014 ...
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 motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 1013 Hz to approximately 1014 ...
, with vibrational modes for a linear molecule and modes for a nonlinear molecule. As specific examples, the linear CO2 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 energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accel ...
of the moving atoms and the
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potenti ...
of the spring-like chemical bond(s). Therefore, the number of vibrational energy terms for a linear molecule and 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 N2 and O2, it is less than 3 K. The " vibrational temperature" necessary for substantial vibration is between 103 K and 104 K, 3521 K for N2 and 2156 K for O2. Typical atmospheric temperatures are not high enough to activate vibration in N2 and O2, which comprise most of the atmosphere. (See the next figure.) However, the much less abundant
greenhouse gas A greenhouse gas (GHG or GhG) is a gas that absorbs and emits radiant energy within the thermal infrared range, causing the greenhouse effect. The primary greenhouse gases in Earth's atmosphere are water vapor (), carbon dioxide (), methane ...
es keep the
troposphere The troposphere is the first and lowest layer of the atmosphere of the Earth, and contains 75% of the total mass of the planetary atmosphere, 99% of the total mass of water vapour and aerosols, and is where most weather phenomena occur. From ...
warm by absorbing
infrared Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from around ...
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." 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. This is why ≈ for monatomic 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. Ot ...
gases at room temperature. Since the
air The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing ...
is dominated by diatomic gases (with nitrogen and
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 well a ...
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, cv differs from (5/2) d by less than 1%. Only at temperatures well above temperatures in the
troposphere The troposphere is the first and lowest layer of the atmosphere of the Earth, and contains 75% of the total mass of the planetary atmosphere, 99% of the total mass of water vapour and aerosols, and is where most weather phenomena occur. From ...
and
stratosphere The stratosphere () is the second layer of the Earth's atmosphere, atmosphere of the Earth, located above the troposphere and below the mesosphere. The stratosphere is an atmospheric layer composed of Atmospheric stratification, stratified tem ...
do some molecules have enough energy to activate the vibrational modes of N2 and O2. The specific heat at constant volume, cv, increases slowly toward (7/2) as temperature increases above T = 400 K, where cv 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 :$d=\sqrt$ 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 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. ...
, 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 ...
. This is because these degrees of freedom are ''frozen'' because the spacing between the energy eigenvalues exceeds the energy corresponding to ambient temperatures ().

# 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: :$E = \sum_^N E_i\left(X_i\right),$ where is a function of the sole variable . example: if and are two degrees of freedom, and is the associated energy: * If $E = X_1^4 + X_2^4$, then the two degrees of freedom are independent. * If $E = X_1^4 + X_1 X_2 + X_2^4$, 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 ...
. Its probability density function is the following: : $p_i\left(X_i\right) = \frac$, In this section, and throughout the article the brackets $\langle \rangle$ denote the mean of the quantity they enclose. The internal energy of the system is the sum of the average energies associated with each of the degrees of freedom: :$\langle E \rangle = \sum_^N \langle E_i \rangle.$

A degree of freedom is quadratic if the energy terms associated with this degree of freedom can be written as :$E = \alpha_i\,\,X_i^2 + \beta_i \,\, X_i Y$, where is a linear combination of other quadratic degrees of freedom. example: if and are two degrees of freedom, and is the associated energy: * If $E = X_1^4 + X_1^3 X_2 + X_2^4$, then the two degrees of freedom are not independent and non-quadratic. * If $E = X_1^4 + X_2^4$, then the two degrees of freedom are independent and non-quadratic. * If $E = X_1^2 + X_1 X_2 + 2X_2^2$, then the two degrees of freedom are not independent but are quadratic. * If $E = X_1^2 + 2X_2^2$, then the two degrees of freedom are independent and quadratic. For example, in
Newtonian mechanics Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motio ...
, 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 defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = ...
s with constant coefficients.

## 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: :$E = \sum_^N \alpha_i X_i^2$

## 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. It does not assume or postulate any natural laws, but explains the macroscopic b ...
, at
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In therm ...
, the internal energy of a system of quadratic and independent degrees of freedom is: : $U = \langle E \rangle = N\,\frac$ Here, the mean energy associated with a degree of freedom is: :$\langle E_i \rangle = \int dX_i\,\,\alpha_i X_i^2\,\, p_i\left(X_i\right) = \frac$ :$\langle E_i \rangle = \frac\frac = \frac$ Since the degrees of freedom are independent, the internal energy of the system is equal to the sum of the mean energy associated with each degree of freedom, which demonstrates the result.

# Generalizations

The description of a system's state as a
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Poin ...
in its phase space, although mathematically convenient, is thought to be fundamentally inaccurate. In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
, the motion degrees of freedom are superseded with the concept of
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ma ...
, and
operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...
which correspond to other degrees of freedom have discrete spectra. For example, intrinsic angular momentum operator (which corresponds to the rotational freedom) for an electron or photon has only two eigenvalues. This discreteness becomes apparent when action has an order of magnitude of the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, and individual degrees of freedom can be distinguished.

# References

{{Authority control Physical quantities Dimension