Degree of freedom

Degree of freedom

Edited By Vishal kumar | Updated on Jul 02, 2025 06:35 PM IST

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.

This Story also Contains
  1. Definition of Degree of Freedom
  2. Value of Degree of Freedom
  3. Solved Examples Based on Degree of Freedom
  4. Summary

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

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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.

Frequently Asked Questions (FAQs)

1. What is meant by degrees of freedom in the kinetic theory of gases?
Degrees of freedom refer to the number of independent ways a gas molecule can store energy. In the kinetic theory of gases, these are typically associated with the molecule's motion and internal energy states. For monatomic gases, there are three translational degrees of freedom (movement in x, y, and z directions). For diatomic and polyatomic gases, additional rotational and vibrational degrees of freedom may be present.
2. Why does the heat capacity of a gas change with temperature?
The heat capacity of a gas changes with temperature because different degrees of freedom become active at different temperatures. As temperature increases, previously "frozen" degrees of freedom (like vibration in diatomic molecules) can become active, allowing the gas to store more energy and increasing its heat capacity.
3. What is the significance of the 3N in the degrees of freedom for a system of N particles?
In a system of N particles, 3N represents the total number of coordinates needed to specify the positions of all particles in three-dimensional space. Each particle has three coordinates (x, y, z), so N particles require 3N coordinates. This is often the starting point for counting degrees of freedom in a system, before considering constraints or additional internal degrees of freedom.
4. How do constraints affect the number of degrees of freedom?
Constraints reduce the number of degrees of freedom in a system. For example, if N particles are confined to move on a surface, one coordinate is constrained, leaving 2N degrees of freedom. In general, the number of degrees of freedom is calculated by subtracting the number of constraints from the total number of coordinates.
5. Why don't we usually consider translational degrees of freedom for gases in a container?
While individual gas molecules have translational degrees of freedom, when considering the gas as a whole in a container, we don't typically include the overall translation of the entire gas. This is because the container constrains the gas's overall motion, and we're more interested in the internal energy and motion of the molecules relative to each other.
6. What is the connection between degrees of freedom and the efficiency of heat engines?
The efficiency of heat engines, such as those operating on the Carnot cycle, depends on the ratio of specific heats (γ) of the working gas. This ratio is directly related to the number of degrees of freedom (f) by γ = (f+2)/f. Gases with fewer degrees of freedom (higher γ) can potentially lead to more efficient heat engines. This is one reason why monatomic gases, with their higher γ, are theoretically more efficient as working fluids in heat engines compared to diatomic or polyatomic gases.
7. Why do monatomic gases have only three degrees of freedom?
Monatomic gases, such as helium or neon, consist of single atoms. These atoms can only move in three spatial dimensions (x, y, and z), which correspond to three translational degrees of freedom. They cannot rotate or vibrate internally, so they lack rotational and vibrational degrees of freedom.
8. How many degrees of freedom does a diatomic gas molecule typically have?
A diatomic gas molecule typically has five degrees of freedom at room temperature: three translational (movement in x, y, and z directions) and two rotational (rotation around two axes perpendicular to the molecular axis). At higher temperatures, vibrational degrees of freedom may also become active.
9. How do degrees of freedom affect the heat capacity of a gas?
The heat capacity of a gas is directly related to its degrees of freedom. More degrees of freedom mean the gas can store more energy as temperature increases, resulting in a higher heat capacity. The molar heat capacity at constant volume (Cv) is given by (f/2)R, where f is the number of degrees of freedom and R is the gas constant.
10. How do degrees of freedom relate to the equipartition theorem?
The equipartition theorem states that in thermal equilibrium, each degree of freedom contributes 1/2 kT of energy on average, where k is Boltzmann's constant and T is the absolute temperature. This means that the total energy of a gas molecule is directly related to its number of degrees of freedom, with each degree contributing equally to the molecule's energy.
11. What determines whether a degree of freedom is "active" or "frozen"?
A degree of freedom is considered "active" if the gas molecule has enough energy to excite that particular mode of motion or energy storage. The activation of degrees of freedom depends on temperature. At low temperatures, some degrees (like vibration) may be "frozen" because there's not enough thermal energy to excite them. As temperature increases, more degrees of freedom become active.
12. How do degrees of freedom affect the speed of sound in a gas?
The speed of sound in a gas is given by the equation v = √(γRT/M), where γ is the ratio of specific heats, R is the gas constant, T is temperature, and M is the molar mass. Since γ depends on the degrees of freedom, gases with fewer degrees of freedom (like monatomic gases) have a higher γ and thus a higher speed of sound compared to gases with more degrees of freedom at the same temperature.
13. What is the relationship between degrees of freedom and the equipartition theorem?
The equipartition theorem states that in thermal equilibrium, each degree of freedom contributes 1/2 kT of energy on average. This means that the total energy of a gas molecule is directly proportional to its number of degrees of freedom. The theorem allows us to calculate the average energy of a molecule by multiplying 1/2 kT by the number of degrees of freedom.
14. What is the significance of the Dulong-Petit law in relation to degrees of freedom?
The Dulong-Petit law states that the molar heat capacity of many solid elements is approximately 3R, where R is the gas constant. This can be explained using the concept of degrees of freedom. In a solid, each atom typically has 3 degrees of freedom (vibration in x, y, and z directions). By the equipartition theorem, each contributes 1/2 kT of energy, leading to a total of 3 × 1/2 R = 3/2 R per mole for heat capacity at constant volume, or 3R for heat capacity at constant pressure.
15. How do molecular collisions affect the degrees of freedom in a gas?
Molecular collisions don't directly change the number of degrees of freedom a molecule has, but they play a crucial role in energy distribution among these degrees of freedom. Collisions allow energy to be transferred between molecules and between different degrees of freedom within a molecule. This process helps maintain the equipartition of energy across all available degrees of freedom, keeping the gas in thermal equilibrium.
16. Why do some gases deviate from the expected behavior based on degrees of freedom at very high temperatures?
At very high temperatures, gases may deviate from expected behavior because additional degrees of freedom can become active. For example, electronic excitation can occur, adding new ways for the gas to store energy. Additionally, at extremely high temperatures, molecules may dissociate or ionize, fundamentally changing the particle nature and thus the degrees of freedom in the system. These effects can lead to unexpected changes in properties like heat capacity.
17. How does the activation of vibrational degrees of freedom affect the behavior of diatomic gases at high temperatures?
At high temperatures, vibrational degrees of freedom in diatomic gases become "unfrozen" or active. This leads to several effects:
18. How does the concept of degrees of freedom relate to the ideal gas law?
The ideal gas law (PV = nRT) doesn't explicitly include degrees of freedom, but it's closely related. The temperature T in the equation is directly linked to the average kinetic energy of the gas molecules, which depends on their degrees of freedom. The more degrees of freedom a gas has, the more energy it can store at a given temperature, affecting its pressure and volume relationships.
19. What is the relationship between degrees of freedom and the gamma (γ) in the adiabatic process equation?
The ratio of specific heats, gamma (γ), is related to the degrees of freedom (f) by the equation γ = (f+2)/f. This comes from the relationship between heat capacities at constant pressure (Cp) and constant volume (Cv), which depend on the degrees of freedom. For monatomic gases with 3 degrees of freedom, γ = 5/3, while for diatomic gases with 5 degrees of freedom at room temperature, γ ≈ 7/5.
20. Why do some degrees of freedom "freeze out" at low temperatures?
Degrees of freedom "freeze out" at low temperatures because there isn't enough thermal energy to excite certain modes of motion or energy storage. For example, vibrational modes in diatomic molecules require more energy to excite than rotational or translational modes. At low temperatures, molecules may not have enough energy to vibrate, effectively "freezing" this degree of freedom.
21. How does quantum mechanics affect our understanding of degrees of freedom in gases?
Quantum mechanics introduces the concept of energy quantization, which affects how we understand degrees of freedom, especially at very low temperatures. In the quantum view, energy states are discrete, not continuous. This means that some degrees of freedom may not be accessible at low energies, leading to phenomena like the "freezing out" of rotational and vibrational states in molecules at low temperatures.
22. What is the difference between external and internal degrees of freedom in molecules?
External degrees of freedom refer to the motion of the molecule as a whole, typically translational motion in three dimensions. Internal degrees of freedom relate to the motion or energy storage within the molecule itself, such as rotation around its axes or vibration of its bonds. Monatomic gases only have external degrees of freedom, while polyatomic molecules have both external and internal degrees of freedom.
23. How do degrees of freedom affect the Maxwell-Boltzmann distribution?
The Maxwell-Boltzmann distribution describes the distribution of molecular speeds in a gas. While the shape of the distribution doesn't directly depend on the degrees of freedom, the average energy per molecule does. More degrees of freedom mean more ways for a molecule to store energy, which affects the total energy distribution among the molecules and can influence properties derived from the Maxwell-Boltzmann distribution.
24. Why is the vibrational degree of freedom often ignored at room temperature for diatomic gases?
The vibrational degree of freedom is often ignored for diatomic gases at room temperature because the energy required to excite molecular vibrations is typically much higher than the average thermal energy available at room temperature. Most molecules remain in their vibrational ground state, effectively "freezing" this degree of freedom until higher temperatures are reached.
25. How does molecular structure affect the degrees of freedom in a gas?
Molecular structure greatly influences the degrees of freedom. Monatomic gases have only 3 translational degrees. Diatomic gases add 2 rotational degrees at room temperature. More complex molecules like H2O or CH4 have 3 rotational degrees. Additionally, polyatomic molecules have more vibrational modes. The more complex the molecule, the more potential degrees of freedom it has.
26. What is the significance of the "3/2 kT" term often seen in kinetic theory equations?
The term "3/2 kT" represents the average kinetic energy of a monatomic gas molecule at temperature T, where k is Boltzmann's constant. This comes from the equipartition theorem, which states that each degree of freedom contributes 1/2 kT of energy. Monatomic gases have 3 translational degrees of freedom, hence 3/2 kT. This term is fundamental in many kinetic theory equations.
27. How do degrees of freedom relate to the concept of entropy in gases?
Degrees of freedom are closely related to entropy in gases. More degrees of freedom mean more ways for energy to be distributed among molecules, which corresponds to higher entropy. When degrees of freedom "freeze out" at low temperatures, it reduces the number of possible energy states, lowering the entropy. The statistical definition of entropy is directly related to the number of possible microstates, which increases with more active degrees of freedom.
28. Why doesn't the translational kinetic energy of gas molecules depend on their mass?
The average translational kinetic energy of gas molecules doesn't depend on their mass because of the equipartition theorem. Each translational degree of freedom contributes 1/2 kT of energy, regardless of the molecule's mass. Heavier molecules will move more slowly, but they have the same average kinetic energy as lighter molecules at the same temperature.
29. How do degrees of freedom affect the specific heat ratio (γ) of a gas?
The specific heat ratio (γ) is the ratio of heat capacity at constant pressure (Cp) to heat capacity at constant volume (Cv). It's related to degrees of freedom (f) by γ = (f+2)/f. More degrees of freedom lead to a lower γ. For example, monatomic gases (f=3) have γ=5/3, while diatomic gases at room temperature (f=5) have γ≈7/5. This ratio is important in thermodynamic processes and affects phenomena like sound propagation in gases.
30. How do degrees of freedom affect the internal energy of a gas?
The internal energy of a gas is directly related to its degrees of freedom. According to the equipartition theorem, each degree of freedom contributes 1/2 kT of energy. Therefore, the total internal energy per mole of gas is (f/2)RT, where f is the number of degrees of freedom, R is the gas constant, and T is the temperature. More degrees of freedom mean higher internal energy at a given temperature.
31. Why do polyatomic gases generally have higher heat capacities than monatomic or diatomic gases?
Polyatomic gases generally have higher heat capacities because they have more degrees of freedom. In addition to the 3 translational degrees, they typically have 3 rotational degrees and multiple vibrational modes. Each degree of freedom can store energy, so more degrees of freedom allow the gas to absorb more energy for a given temperature increase, resulting in a higher heat capacity.
32. How does the concept of degrees of freedom apply to solids and liquids?
In solids, atoms are typically confined to vibrate around fixed positions, so their degrees of freedom are primarily vibrational. A solid with N atoms generally has 3N vibrational degrees of freedom. In liquids, particles have more freedom to move, but not as freely as in gases. They have both vibrational and translational degrees of freedom, but rotational degrees are often restricted. The concept is more complex in these phases compared to gases.
33. How does the concept of degrees of freedom relate to the zeroth law of thermodynamics?
The zeroth law of thermodynamics establishes the concept of thermal equilibrium. When two systems are in thermal equilibrium, energy is evenly distributed among all available degrees of freedom in both systems, according to the equipartition theorem. This even distribution of energy across degrees of freedom is what allows us to define a common temperature for systems in equilibrium, linking the microscopic concept of degrees of freedom to the macroscopic concept of temperature.
34. What is the relationship between degrees of freedom and the mean free path of gas molecules?
While degrees of freedom don't directly determine the mean free path, they influence it indirectly. The mean free path depends on the size and speed of molecules. The speed distribution of molecules is related to their kinetic energy, which in turn depends on the degrees of freedom through the equipartition theorem. Gases with more degrees of freedom store more energy in non-translational modes, potentially affecting the average translational speed and thus the mean free path.
35. How do degrees of freedom affect the speed of chemical reactions in gases?
Degrees of freedom play a role in chemical reaction rates through their effect on molecular collisions and energy distribution. More degrees of freedom mean more ways for a molecule to store energy, which can affect the likelihood of a collision having enough energy to overcome the activation energy barrier. Additionally, certain degrees of freedom (like vibration) may be more important for specific reactions, influencing the reaction rate and mechanism.
36. Why is the concept of degrees of freedom important in statistical mechanics?
In statistical mechanics, degrees of freedom are crucial for calculating the number of possible microstates of a system, which is fundamental to determining its entropy and other thermodynamic properties. The partition function, a key concept in statistical mechanics, is directly related to the degrees of freedom of the particles in the system. Understanding degrees of freedom allows us to bridge microscopic properties of particles to macroscopic thermodynamic behavior.
37. How do degrees of freedom relate to the concept of enthalpy in gases?
Enthalpy (H) is defined as the sum of internal energy (U) and the product of pressure and volume (PV). The internal energy is directly related to the degrees of freedom through U = (f/2)nRT, where f is the number of degrees of freedom
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