Degrees of freedom (physics and chemistry)

External images
External images
	https://chem.libretexts.org/@api/deki/files/9669/h2ovibrations.gif?revision=1
	https://chem.libretexts.org/@api/deki/files/9668/co2vibrations.gif?revision=1

Last updated November 26, 2024

In physics and chemistry, a degree of freedom is an independent physical parameter in the chosen parameterization of a physical system. More formally, given a parameterization of a physical system, the number of degrees of freedom is the smallest number ${\textstyle n}$ 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 ${\textstyle n}$ such parameters are called degrees of freedom.

The location of a particle in three-dimensional space 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. So, if the time evolution of the system is deterministic (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.^{[ citation needed ]} 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, 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 coordinates in the Hamiltonian formalism.

In statistical mechanics, a degree of freedom is a single scalar number describing the microstate of a system.^[1] The specification of all microstates of a system is a point in the system's phase space.

In the 3D ideal chain model in chemistry, two angles 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.^[2]

Thermodynamic degrees of freedom for gases

By the equipartition theorem, internal energy per mole of gas equals $c v T$ , where $T$ is absolute temperature 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 with respect to the x, y, and z axes. These are the only degrees of freedom for a monoatomic species, such as noble gas 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 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.^[3]

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 mode: the two atoms oscillate back and forth with the chemical bond between them acting as a spring. A molecule with $N$ atoms has more complicated modes of molecular vibration, with $3 N - 5$ vibrational modes for a linear molecule and $3 N - 6$ modes for a nonlinear molecule.^[4] As specific examples, the linear CO₂ molecule has 4 modes of oscillation,^[5] and the nonlinear water molecule has 3 modes of oscillation^[6] Each vibrational mode has two energy terms: the kinetic energy of the moving atoms and the potential energy of the spring-like chemical bond(s). Therefore, the number of vibrational energy terms is $2(3 N - 5)$ modes for a linear molecule and is $2(3 N - 6)$ modes for a nonlinear molecule.

Both the rotational and vibrational modes are quantized, requiring a minimum temperature to be activated.^[7] 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.^[1] The "vibrational temperature" necessary for substantial vibration is between 10³ K and 10⁴ K, 3521 K for N₂ and 2156 K for O₂.^[1] 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 gases keep the troposphere warm by absorbing infrared from the Earth's surface, which excites their vibrational modes.^[8] 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 $γ$ ≈ $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠5/3⁠$ for monatomic gases and $γ$ ≈ $⁠ 7 / 5 ⁠$ for diatomic gases at room temperature.^[1]

Since the air is dominated by diatomic gases (with nitrogen and oxygen contributing about 99%), its molar internal energy is close to $c v T$ = (5/2) $R$ $T$ , determined by the 5 degrees of freedom exhibited by diatomic gases.^{[ citation needed ]}^[11]^{[ circular reference ]} See the graph at right. For 140 K < $T$ < 380 K, c_v differs from (5/2) $R$ _d by less than 1%. Only at temperatures well above temperatures in the troposphere and stratosphere do some molecules have enough energy to activate the vibrational modes of N₂ and O₂. The specific heat at constant volume, c_v, increases slowly toward (7/2) $R$ as temperature increases above T = 400 K, where c_v is 1.3% above (5/2) $R$ _d = 717.5 J/(K kg).

Component degrees of freedom for $N$ atoms (3 $N$ total coordinate motions)
	Monatomic	Linear molecules	Non-linear molecules
Translation ( $x$ , $y$ , and $z$ )	3	3	3
Rotation ( $x$ , $y$ , and $z$ )	0	2	3
Vibration (high temperature)	0	$2 (3 N - 5)$	$2 (3 N - 6)$

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 $(x 1, y 1, z 1)$ and the other has coordinate $(x 2, y 2, z 2)$ with $z 2$ unknown. Application of the formula for distance between two coordinates

d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}

results in one equation with one unknown, in which we can solve for $z 2$ . One of $x 1$ , $x 2$ , $y 1$ , $y 2$ , $z 1$ , or $z 2$ can be unknown.

Contrary to the classical equipartition theorem, at room temperature, the vibrational motion of molecules typically makes negligible contributions to the 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 ( $k B T$ ).^[1]

Independent degrees of freedom

The set of degrees of freedom $X 1, ... , X N$ of a system is independent if the energy associated with the set can be written in the following form:

E=\sum _{i=1}^{N}E_{i}(X_{i}),

where $E i$ is a function of the sole variable $X i$ .

example: if $X 1$ and $X 2$ are two degrees of freedom, and $E$ 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 $X 1$ and $X 2$ is a coupling term that describes an interaction between the two degrees of freedom.

For $i$ from 1 to $N$ , the value of the $i$ th degree of freedom $X i$ is distributed according to the Boltzmann distribution. Its probability density function is the following:

p_{i}(X_{i})={\frac {e^{-{\frac {E_{i}}{k_{B}T}}}}{\int dX_{i}\,e^{-{\frac {E_{i}}{k_{B}T}}}}}

,

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 _{i=1}^{N}\langle E_{i}\rangle .

Quadratic degrees of freedom

A degree of freedom $X i$ 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 $Y$ is a linear combination of other quadratic degrees of freedom.

example: if $X 1$ and $X 2$ are two degrees of freedom, and $E$ 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, the dynamics of a system of quadratic degrees of freedom are controlled by a set of homogeneous linear differential equations with constant coefficients.

Quadratic and independent degree of freedom

$X 1, ... , X N$ 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 _{i=1}^{N}\alpha _{i}X_{i}^{2}

Equipartition theorem

In the classical limit of statistical mechanics, at thermodynamic equilibrium, the internal energy of a system of $N$ quadratic and independent degrees of freedom is:

U=\langle E\rangle =N\,{\frac {k_{B}T}{2}}

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}(X_{i})={\frac {\int dX_{i}\,\,\alpha _{i}X_{i}^{2}\,\,e^{-{\frac {\alpha _{i}X_{i}^{2}}{k_{B}T}}}}{\int dX_{i}\,\,e^{-{\frac {\alpha _{i}X_{i}^{2}}{k_{B}T}}}}}

\langle E_{i}\rangle ={\frac {k_{B}T}{2}}{\frac {\int dx\,\,x^{2}\,\,e^{-{\frac {x^{2}}{2}}}}{\int dx\,\,e^{-{\frac {x^{2}}{2}}}}}={\frac {k_{B}T}{2}}

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 in its phase space, although mathematically convenient, is thought to be fundamentally inaccurate. In quantum mechanics, the motion degrees of freedom are superseded with the concept of wave function, and operators 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, and individual degrees of freedom can be distinguished.

Related Research Articles

<span class="mw-page-title-main">Diatomic molecule</span> Molecule composed of any two atoms

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. Otherwise, if a diatomic molecule consists of two different atoms, such as carbon monoxide or nitric oxide, the molecule is said to be heteronuclear. The bond in a homonuclear diatomic molecule is non-polar.

In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's energy spectrum or its set of energy eigenvalues, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

In physics, the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.

Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.

<span class="mw-page-title-main">Quantum harmonic oscillator</span> Important, well-understood quantum mechanical model

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stated by Benoît Paul Émile Clapeyron in 1834 as a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. The ideal gas law is often written in an empirical form:

<span class="mw-page-title-main">Kinetic theory of gases</span> Understanding of gas properties in terms of molecular motion

The kinetic theory of gases is a simple classical model of the thermodynamic behavior of gases. Its introduction allowed many principal concepts of thermodynamics to be established. It treats a gas as composed of numerous particles, too small to be seen with a microscope, in constant, random motion.These particles are now known to be the atoms or molecules of the gas. The kinetic theory of gases uses their collisions with each other and with the walls of their container to explain the relationship between the macroscopic properties of gases, such as volume, pressure, and temperature, as well as transport properties such as viscosity, thermal conductivity and mass diffusivity.

A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle in physics, a phonon is an excited state in the quantum mechanical quantization of the modes of vibrations for elastic structures of interacting particles. Phonons can be thought of as quantized sound waves, similar to photons as quantized light waves.

<span class="mw-page-title-main">Equipartition theorem</span> Theorem in classical statistical mechanics

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. The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among all of its various forms; for example, the average kinetic energy per degree of freedom in translational motion of a molecule should equal that in rotational motion.

<span class="mw-page-title-main">Stark effect</span> Spectral line splitting in electrical field

The Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to the presence of an external electric field. It is the electric-field analogue of the Zeeman effect, where a spectral line is split into several components due to the presence of the magnetic field. Although initially coined for the static case, it is also used in the wider context to describe the effect of time-dependent electric fields. In particular, the Stark effect is responsible for the pressure broadening of spectral lines by charged particles in plasmas. For most spectral lines, the Stark effect is either linear or quadratic with a high accuracy.

In quantum mechanics, a rotational transition is an abrupt change in angular momentum. Like all other properties of a quantum particle, angular momentum is quantized, meaning it can only equal certain discrete values, which correspond to different rotational energy states. When a particle loses angular momentum, it is said to have transitioned to a lower rotational energy state. Likewise, when a particle gains angular momentum, a positive rotational transition is said to have occurred.

In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters, the traces of the matrices representing group elements of the column's class in the given row's group representation. In chemistry, crystallography, and spectroscopy, character tables of point groups are used to classify e.g. molecular vibrations according to their symmetry, and to predict whether a transition between two states is forbidden for symmetry reasons. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry devote a chapter to the use of symmetry group character tables.

In statistical mechanics, the grand canonical ensemble is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium with a reservoir. The system is said to be open in the sense that the system can exchange energy and particles with a reservoir, so that various possible states of the system can differ in both their total energy and total number of particles. The system's volume, shape, and other external coordinates are kept the same in all possible states of the system.

In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat bath, so that the states of the system will differ in total energy.

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 sample of the substance divided by the amount of substance of the sample; or also the specific heat capacity of the substance times its molar mass. The SI unit of molar heat capacity is joule per kelvin per mole, J⋅K⁻¹⋅mol⁻¹.

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 volume. It is sometimes also known as the isentropic expansion factor and is denoted by $γ$ (gamma) for an ideal gas or $κ$ (kappa), the isentropic exponent for a real gas. The symbol $γ$ is used by aerospace and chemical engineers. $where C is the heat capacity, the molar heat capacity, and c the specific heat capacity of a gas. The suffixes P and V refer to constant-pressure and constant-volume conditions respectively.$

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 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm⁻¹ and wavelengths of approximately 30 to 3 μm.

In chemistry, the rotational partition function relates the rotational degrees of freedom to the rotational part of the energy.

Heat transfer physics describes the kinetics of energy storage, transport, and energy transformation by principal energy carriers: phonons, electrons, fluid particles, and photons. Heat is thermal energy stored in temperature-dependent motion of particles including electrons, atomic nuclei, individual atoms, and molecules. Heat is transferred to and from matter by the principal energy carriers. The state of energy stored within matter, or transported by the carriers, is described by a combination of classical and quantum statistical mechanics. The energy is different made (converted) among various carriers. The heat transfer processes are governed by the rates at which various related physical phenomena occur, such as the rate of particle collisions in classical mechanics. These various states and kinetics determine the heat transfer, i.e., the net rate of energy storage or transport. Governing these process from the atomic level to macroscale are the laws of thermodynamics, including conservation of energy.

Molecular symmetry in physics and chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in the application of Quantum Mechanics in physics and chemistry, for example it can be used to predict or explain many of a molecule's properties, such as its dipole moment and its allowed spectroscopic transitions, without doing the exact rigorous calculations. To do this it is necessary to classify the states of the molecule using the irreducible representations from the character table of the symmetry group of the molecule. Among all the molecular symmetries, diatomic molecules show some distinct features and they are relatively easier to analyze.

References

1 2 3 4 5 Reif, F. (2009). Fundamentals of Statistical and Thermal Physics. Long Grove, IL: Waveland Press, Inc. p. 51. ISBN 978-1-57766-612-7.
↑ "Physical chemistry - Does a diatomic gas have one or two vibrational degrees of freedom?".
↑ Waldmann, Thomas; Klein, Jens; Hoster, Harry E.; Behm, R. Jürgen (2013). "Stabilization of Large Adsorbates by Rotational Entropy: A Time-Resolved Variable-Temperature STM Study". ChemPhysChem. 14 (1): 162–9. doi:10.1002/cphc.201200531. PMID 23047526. S2CID 36848079.
↑ Molecular vibration ^{[ user-generated source ]}
↑ For drawings, see http://www.colby.edu/chemistry/PChem/notes/NormalModesText.pdf
↑ For drawings, see https://sites.cns.utexas.edu/jones_ch431/normal-modes-vibration
↑ Section 12-7 (pp. 376-379) of Sears and Salinger, 1975: Thermodynamics, Kinetic Theory, and Statistical Thermodynamics. Third edition. Addison-Wesley Publishing Co.
↑ "Molecules Vibrate". UCAR Center for Science Education. Archived from the original on 2014-10-10. Retrieved 2021-01-19.
↑ "Air - Specific Heat vs. Temperature at Constant Pressure".
↑ Gatley, D. P., S. Herrmann, H.-J. Kretzshmar, 2008: A twenty-first century molar mass for dry air. HVAC&R Research, vol. 14, pp. 655-662.
↑ Equipartition theorem#Diatomic gases

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[:0-1] 1 2 3 4 5 Reif, F. (2009). Fundamentals of Statistical and Thermal Physics. Long Grove, IL: Waveland Press, Inc. p. 51. ISBN 978-1-57766-612-7.

[2] "Physical chemistry - Does a diatomic gas have one or two vibrational degrees of freedom?".

[3] Waldmann, Thomas; Klein, Jens; Hoster, Harry E.; Behm, R. Jürgen (2013). "Stabilization of Large Adsorbates by Rotational Entropy: A Time-Resolved Variable-Temperature STM Study". ChemPhysChem. 14 (1): 162–9. doi:10.1002/cphc.201200531. PMID 23047526. S2CID 36848079.

[4] Molecular vibration ^{[ user-generated source ]}

[5] For drawings, see http://www.colby.edu/chemistry/PChem/notes/NormalModesText.pdf

[6] For drawings, see https://sites.cns.utexas.edu/jones_ch431/normal-modes-vibration

[7] Section 12-7 (pp. 376-379) of Sears and Salinger, 1975: Thermodynamics, Kinetic Theory, and Statistical Thermodynamics. Third edition. Addison-Wesley Publishing Co.

[8] "Molecules Vibrate". UCAR Center for Science Education. Archived from the original on 2014-10-10. Retrieved 2021-01-19.

[9] "Air - Specific Heat vs. Temperature at Constant Pressure".

[10] Gatley, D. P., S. Herrmann, H.-J. Kretzshmar, 2008: A twenty-first century molar mass for dry air. HVAC&R Research, vol. 14, pp. 655-662.

[11] Equipartition theorem#Diatomic gases

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[11]