Degrees of freedom (physics and chemistry) Totally Explained

Everything about Degrees Of Freedom Physics And Chemistry totally explained

ť For information on degrees of freedom in other sciences, see degrees of freedom. For other uses of degree, see Degree

Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. In mathematical terms, the degrees of freedom are the dimensions of a phase space.

Degrees of freedom in mechanics (physics)

In mechanics, for each particle belonging to a system, and for each independent direction in which movement is possible, two degrees of freedom are defined, one describing the particle's momentum in that direction, the other describing the particle's position along an axis defined by that direction.
Note that "degrees of freedom" has a different meaning in the context of engineering and machines.

A more general definition

In statistical mechanics, a degree of freedom is a single scalar number describing the classical micro-state of a system. The micro-state of a system is completely described by the set of all values of all its degrees of freedom.
   If the system studied can be described as a set of mechanical particles, then degrees of freedom are defined in the same manner as above. Thus, a micro-state of the system is a point in the system's phase space.
   It must be noted that for a system, a micro-state defined by using degrees of freedom is intrinsically a classical state. This is because for a quantum micro-state, defining a precise value of both the position and momentum of a particle violates the Heisenberg uncertainty principle. The description of a system through a set of degrees of freedom is thus only valid in the classical (or high temperature) limit of statistical mechanics.
   In some cases, when the system isn't appropriately described as a set of mechanical particles, other types of degrees of freedom have to be defined. For example, in the 3D ideal chain model, two angles are necessary to describe each monomer's orientation. The value of each of these angles can each be a degree of freedom.

Example: classical ideal diatomic gas

In 3D, there are 6 degrees of freedom associated to the movement of a mechanical particle, 3 for its position, and 3 for its momentum.
   There are 6 degrees of freedom in total. Another way to justify this figure is to consider that the movement of the molecule will be described by the movement of the two mechanical particles representing its two atoms, and 6 degrees of freedom are attached to each particle, as above. With this alternative breakdown, it appears that different sets of degrees of freedom can be defined to describe the movement of the molecule. In fact a set of degrees of freedom for a mechanical system is a set of independent axes in the phase space of the system, and that allows the generation of the whole phase space. For a multidimensional space like phase space, there's more than one possible set of axes.
   It is notable that not all degrees of freedom of the hydrogen molecule participate in the above expression of its energy. For example, those degrees of freedom associated to the position of the center of mass of the particle don't weigh in the energy.
   In the table below the degrees which are disregarded are like this because of their low effect on total energy, unless they're at very very high temperatures or energies. The diatomic rotation is disregarded due to rotation about the molecules axis. Monatomic rotation is disregarded for the same reason as diatomic, but this effect continues into the other 2 directions.

	Monatomic	Linear molecules	Non-Linear molecules
Position (x, y and z)	3	3	3
Rotation (x, y and z)	0	2	3
Vibration	0	3N - 5	3N - 6
Total	3	3N	3N

Independent degrees of freedom

Definition

The set of degrees of freedom

X_1, ldots, X_N

of a system is independent if the energy associated with the set can be written in the following form: ť

E = sum_

Since the degrees of freedom are independent, the internal energy of the system is equal to the sum of the mean energy associated to each degree of freedom, which demonstrates the result.

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