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Inphysics andchemistry, adegree of freedom is an independent physical parameter in the chosen parameterization of aphysical system. More formally, given a parameterization of a physical system, thenumber 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 ofall parameters in the chosen parameterization. In this case, any set of such parameters are calleddegrees of freedom.
The location of aparticle inthree-dimensional space requires threeposition coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of threevelocity components, each in reference to the three dimensions of space. So, if thetime evolution of the system isdeterministic (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.
Inclassical mechanics, the state of apoint particle at any given time is often described with position and velocity coordinates in theLagrangian formalism, or with position andmomentum coordinates in theHamiltonian formalism.[citation needed]
Instatistical mechanics, a degree of freedom is a singlescalar number describing themicrostate of a system.[1] The specification of all microstates of a system is a point in the system'sphase space.
In the 3Dideal chain model in chemistry, twoangles 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]
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By theequipartition theorem, internal energy per mole of gas equalscvT, whereT isabsolute temperature 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 thecenter of mass with respect to the x, y, and z axes. These are the only degrees of freedom for a monoatomic species, such asnoble 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.Alinear molecule, where all atoms lie along a single axis,such as anydiatomic molecule and some other molecules likecarbon 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, likewater (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 onemolecular vibration mode: the two atoms oscillate back and forth with the chemical bond between them acting as a spring. A molecule withN atoms has more complicated modes ofmolecular vibration, with3N − 5 vibrational modes for a linear molecule and3N − 6 modes for a nonlinear molecule.As specific examples, the linear CO2 molecule has 4 modes of oscillation,[4] and the nonlinear water molecule has 3 modes of oscillation[5]Each vibrational mode has two energy terms: thekinetic energy of the moving atoms and thepotential energy of the spring-like chemical bond(s).Therefore, the number of vibrational energy terms is2(3N − 5) modes for a linear molecule and is2(3N − 6) modes for a nonlinear molecule.
Both the rotational and vibrational modes are quantized, requiring a minimum temperature to be activated.[6] 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 abundantgreenhouse gases keep thetroposphere warm by absorbinginfrared from the Earth's surface, which excites their vibrational modes.[7]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 theheat capacity ratio. This is whyγ≈5/3 formonatomic gases andγ≈7/5 fordiatomic gases at room temperature.[1]
Since theair is dominated by diatomic gases (withnitrogen andoxygen contributing about 99%), its molar internal energy is close tocv T = (5/2)RT, determined by the 5 degrees of freedom exhibited by diatomic gases.[citation needed]
See the graph at right. For140 K <T < 380 K, cv differs from (5/2)Rd by less than 1%.Only at temperatures well above temperatures in thetroposphere andstratosphere 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)R as temperature increases above T = 400 K, where cv is 1.3% above (5/2)Rd = 717.5 J/(K kg).
| Monatomic | Linear molecules | Non-linear molecules | |
|---|---|---|---|
| Translation (x,y, andz) | 3 | 3 | 3 |
| Rotation (x,y, andz) | 0 | 2 | 3 |
| Vibration (high temperature) | 0 | 2 (3N − 5) | 2 (3N − 6) |
One can also count degrees of freedom using the minimum number of coordinates required to specify a position. This is done as follows:
Let's say one particle in this body has coordinate(x1, y1, z1) and the other has coordinate(x2, y2, z2) with z2 unknown. Application of the formula for distance between two coordinatesresults in one equation with one unknown, in which we can solve forz2.One ofx1,x2,y1,y2,z1, orz2 can be unknown.
Contrary to the classicalequipartition theorem, at room temperature, the vibrational motion of molecules typically makes negligible contributions to theheat capacity. This is because these degrees of freedom arefrozen because the spacing between the energyeigenvalues exceeds the energy corresponding to ambienttemperatures (kBT).[1]
The set of degrees of freedomX1, ... , XN of a system is independent if the energy associated with the set can be written in the following form:
whereEi is a function of the sole variableXi.
example: ifX1 andX2 are two degrees of freedom, andE is the associated energy:
Fori from 1 toN, the value of theith degree of freedomXi is distributed according to theBoltzmann distribution. Itsprobability density function is the following:
In this section, and throughout the article the brackets denote themean of the quantity they enclose.
Theinternal energy of the system is the sum of the average energies associated with each of the degrees of freedom:
A degree of freedomXi is quadratic if the energy terms associated with this degree of freedom can be written as
whereY is alinear combination of other quadratic degrees of freedom.
example: ifX1 andX2 are two degrees of freedom, andE is the associated energy:
For example, inNewtonian mechanics, thedynamics of a system of quadratic degrees of freedom are controlled by a set of homogeneouslinear differential equations withconstant coefficients.
X1, ... , XN are quadratic and independent degrees of freedom if the energy associated with a microstate of the system they represent can be written as:
In the classical limit ofstatistical mechanics, atthermodynamic equilibrium, theinternal energy of a system ofN quadratic and independent degrees of freedom is:
Here, themean energy associated with a degree of freedom is:
Since the degrees of freedom are independent, theinternal energy of the system is equal to the sum of themean energy associated with each degree of freedom, which demonstrates the result.
The description of a system's state as apoint in its phase space, although mathematically convenient, is thought to be fundamentally inaccurate. Inquantum mechanics, the motion degrees of freedom are superseded with the concept ofwave function, andoperators which correspond to other degrees of freedom havediscrete spectra. For example,intrinsic angular momentum operator (which corresponds to the rotational freedom) for anelectron orphoton has only twoeigenvalues. This discreteness becomes apparent whenaction has anorder of magnitude of thePlanck constant, and individual degrees of freedom can be distinguished.
