Degree of freedom

The phrase ‘degree of freedom’ used in physics denotes the different modes by which a system can move its particles, e.g., energy can be stored as kinetic inertial translational electronic etc., Thus, for example, a three-dimensional space may be occupied by a gas molecule or else rotate or vibrate making them represent many degrees of freedom. An appreciation for degrees of freedom forms the bedrock upon which an examination of energy distribution within any given system is premised, consequently contributing towards the computation of such compound qualities as specific heat and towards projection concerning activities of various molecules. There is nothing more basic than this. In this article, we will cover the concept of the 'degree of freedom’ and provide examples for better understanding.

Definition of Degree of Freedom

The degree of freedom of systems is defined as the possible independent motions, systems can have.

Or

The degree of freedom of systems is defined as the number of independent coordinates required to describe the system completely.

The independent motions can be translational, rotational vibrational or any combination of these.

So the degree of freedom is of three types :
(i) Translational degree of freedom
(ii) Rotational degree of freedom
(iii) Vibrational degree of freedom

$\text { The degree of freedom is denoted by } f$

It is given by

$
f=3 N-R
$

Where
$N=$ no. of particle
$R=$ no. of relation

Now, let's study about the value of the degree of freedom for different gases

Value of Degree of Freedom

• Monoatomic gas

A monoatomic gas can only have a translational degree of freedom.

l.e $f=3$

Example- $\mathrm{He}, \mathrm{Ne}, \mathrm{Ar}$

• Diatomic gas

A diatomic gas can have three translational degrees of freedom and two rotational degrees of freedom.

I.e $f=5$

Example- $\mathrm{H}_2, \mathrm{O}_2, \mathrm{~N}_2$

• Triatomic gas

A triatomic gas can have three translational degrees of freedom and three rotational degrees of freedom.

I.e $f=6$

Example- $\mathrm{H}_2 \mathrm{O}$

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Solved Examples Based on Degree of Freedom

Example 1: For monoatomic gas, the incorrect statement is:

(i) It has all translational degrees of freedom

(ii) examples He, H₂, Ne

(iii) It can have a maximum of 6 degrees of freedom 3 translational and 3 rotational

1) Only (ii)

2) only (i)

3) (ii) and (iii)

4) (i) and (iii)

Solution:

Value of degree of freedom for monoatomic gas

f = 3

wherein

A monoatomic gas can only have a translational degree of freedom.

H₂ is an example of a diatomic gas

monoatomic gas can have a maximum of three degrees of freedom and is all translational.

Hence, the answer is the option of 3.

Example 2: Which statement is true for the term degree of freedom?

(i) Monatomic gas has all translational degrees of freedom

(ii) There can be a maximum of two rotational degrees of freedom

(iii) Total degree of freedom is of three types

1) (i), (ii), (iii)

2) (i) and (ii)

3) (i) and (iii)

4) (ii) and (iii)

Solution:

Degree of freedom

$\begin{aligned}
& f=3 N-R \\
& N=\text { no. of particle } \\
& R=\text { no.of relation }
\end{aligned}$

monatomic gas has only three degrees of freedom and is all translational.

The total degree of freedom is of three types

(i) Translational

(ii) Rotational

(iii) Vibrational

Hence, the answer is the option (3).

Example 3: A gas has n degrees of freedom. The ratio of the specific heat of the gas at constant volume to the specific heat of the gas at constant pressure will be :

1) $\frac{n}{n+2}$
2) $\frac{n+2}{n}$
3) $\frac{n}{2 n+2}$
4) $\frac{n}{n-2}$

Solution:

$
\frac{c_{\mathrm{v}}}{c_{\mathrm{p}}}=\frac{1}{\gamma}=\frac{\mathrm{f}}{2+\mathrm{f}}
$

It is given that, Degree of freedom $=\mathrm{f}=\mathrm{n}$
$
\therefore \frac{c_v}{c_p}=\frac{n}{2+n}
$

Hence, the answer is the option (1).

Example 4: For a triatomic ( non-linear) gas, the total no. of degrees of freedom is:

1) 2

2) 4

3) 6

4) 8

Solution:

Value of degree of freedom for triatomic gas -

f = 6

A triatomic gas can have three translational degrees of freedom and three rotational degrees of freedom.

3(translational) + 3(rotational) = 6

Hence, the answer is the option (3).

Summary

The above degrees of freedom are shown at room temperature. Further at high temperatures, in the case of diatomic or polyatomic molecules, the atoms within the molecule may also vibrate with respect to each other. In such cases, the molecule will have 2 additional degrees of freedom, due to vibrational motion. I.e One for the potential energy and one for the kinetic energy of vibration. So A diatomic molecule that is free to vibrate (in addition to translation and rotation) will have 7 degrees of freedom.