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When an object is placed in 3D space, the coordinates can be used to describe the object’s movement. For example, if a block is placed on a table, it can move along the surface of the table, so it is said to move along the X and Y axes based on the coordinate system, so the degree of freedom is 1+1 = 2. Similarly, the degree of freedom for current flow is 1. So we can determine the degrees of freedom of any object. Here is the definition. “The total number of coordinates that define the position or configuration of a system is called the degrees of freedom. Although Degrees of freedom can be defined for any body, but the discussion is limited to gas atoms/molecules only. It is designated by the letter “f”.
Although there is a Degrees of Freedom Formula, which is discussed later, it can be determined logically as well. The next few paragraphs are discussed along with some use of Degrees of Freedom to get a complete overview of this topic.
Definition
As it is already mentioned, it is the number of independent coordinates. Still, it can also be defined as the number of independent ways by which a system can exchange its energy. As motion is related to energy only, both terms are analogous here.
Types of DOF
There are 3 types of energies that can be associated with a gaseous molecule. So a total of 3 DOF is possible in gaseous atoms.
Translational DOF
Due to the translational kinetic energy of a molecule, it can exhibit motion along the X, Y and Z-axis. A maximum of 3 translational DOF are possible.
Rotational DOF
Due to the rotational kinetic energy, a molecule exhibits rotation about different axes(depending on the number of atoms and the structure of the molecule). In this case, the maximum number depends upon the number of different axes of rotation present.
Vibrational DOF
Vibrational motion originates due to the interatomic forces acting between every atom. The overall motion of atoms due to these forces can be imagined as every atom being connected via springs with its neighbouring atoms. Each atom can vibrate along the line joining two atoms. Here, both potential and kinetic energy are taken into account. Also, the maximum number depends on how many axes a molecule is allowed to vibrate. But for every axis here, we have to take 2 (potential + kinetic).
DOF for different gases: Degrees of Freedom Example
Monoatomic gas:
Any monatomic gas is free to move in any direction in space. So, it has 3 translational degrees of freedom.
Here, only one axis of rotation is present, which passes through the centre of the molecule. But we neglect this motion, as the radius of an atom is so small that the moment of inertia is nearly equal to zero. So for a monatomic gas, f=3.
Ex: He.
Diatomic gas:
In the case of diatomic gas molecules, it can also move freely in space, so translational degrees of freedom is 3.
A diatomic molecule can be considered two atoms connected with a bond, so it has two axes of rotation. One axis of rotation passes through the centre of any one atom (we don’t consider the axis to be present in each atom because then both the cases will be the same), and the other passes through the middle of the bond. But in this case, we will not consider the axis that passes along the bond through the centre of the two atoms, as in that case, the rotation radius will be much less. So for a diatomic gas, f= 3 (translation) + 2 (rotation) = 5.
Ex: O2.
If temperature increases, then we cannot neglect the vibration motion. So at higher temperature f = 5+2 (vibration) = 7.
Polyatomic gas(3 or more atoms)
Here two cases are possible according to the structure of the molecule.
Linear polyatomic gas
In the case of linear molecules, three (3) translational DOF are there. The axis of rotation is the same here as it is in a diatomic molecule. So f =5.
Ex: CO2.
If temperature increases, then we cannot neglect the vibration motion. So at higher temperature, f = 5+2 (vibration) = 7.
Non-linear polyatomic gases
In the case of non-linear molecules, the value for translational energy remains the same, but the rotational energy changes as per the shape changes. Two of the axes will be the same as in the case of a linear-shaped molecule, but one axis will be here, which is present along the axis from one atom to another. So in this case f= 3(translation) + 3(rotation)= 6.
Ex: NH3.
If temperature increases, then we cannot neglect the vibration motion. So at higher temperature, f = 6+2(vibration) = 8.
General expression for degrees of freedom: Degrees of Freedom Formula
f = 3A − B
where A = Number of independent particles,
B = Number of independent restriction
The quantity ‘independent restriction’ solely depends on the structure of the molecules. For example, in the case of a monoatomic gas, B=0, as it can move freely. A diatomic gas is not allowed to move along the bond.
Degree of freedom for different gases:
Atomicity of gas | Degrees of Freedom Example | A | B | f = 3A – B |
Monoatomic | He, Ne, Ar | 1 | 0 | f = 3 |
Diatomic | O2 | 2 | 1 | f = 5 |
Triatomic non-linear | NH3 | 3 | 3 | f = 6 |
Triatomic linear | CO2 | 3 | 2 | f = 7 |
Degrees of freedom in solid
Solids don’t have dof at room temperature, because all molecules strongly attract each other with intramolecular forces. But at very high temperatures, considerable amounts of vibrational motion is observed. They have three axes along which they vibrate, so f= 2×3=6. (2 is multiplied because potential+kinetic is considered).
Law of equipartition of energy
The equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of 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 translation motion of a molecule should equal that in rotational motion.
Let us consider one mole of a monatomic gas in thermal equilibrium at temperature T. The degrees of freedom for each molecule due to its translational motion is 3. According to the kinetic theory of gases, the average kinetic energy of a molecule is 3/2 kT.
Since molecules move randomly, the average kinetic energy corresponding to each degree of freedom is the same.
Thus, the mean kinetic energy per molecule per degree of freedom is ½ kT.
Conclusion
This is an important topic for understanding the physical as well as thermodynamic behaviour of a gas molecule. The use of Degrees of Freedom is wide in the field of physics and chemistry. This concept is applied to determine specific heats of different gases. In most cases, translational and rotational kinetic energies are considered, and values are determined only with that help.
