Contents
Some of the most important Physics Topics include energy, motion, and force.
What are the Degrees of Freedom for a Monatomic Gas?
Definition: The minimum number of independent coordinates necessary to specify the instantaneous position of a moving body is called the degree of freedom of the body.
Example:
i) Let us consider the motion of a particle falling freely under gravity We take the initial point as the origin and the downward line as the z -axis. Then the position of the particle at any instant is specified by the z-coordinate only. So, the number of degrees of freedom of the particle is 1. Essentially, the degree of freedom of every one-dimensional motion is 1, e.g., motion of a car along a road, an ant moving on a stationary rope, etc.
ii) Motions of projectiles under gravity, orbital motion of planets around the sun, circular motion, motion of an ant on the floor of a room, etc., are examples of two dimensional motions. In each case, any one point on the plane, is chosen as the origin and two perpendicular axes x and y are considered. Then the position of the object at any instant is specified by the x- and y coordinates. So, the number of degrees of freedom in two-dimensional motion is 2.
iii) According to the kinetic theory, all gas molecules of an ideal gas are point masses and they are in a completely random to and fro motion. At least three coordinates (say x, y, z) are necessary to specify the position of a molecule at any instant. So, the number of degrees of freedom of an ideal gas molecule is 3. Essentially, every three-dimensional motion has 3 degrees of freedom, e.g., Brownian motion, motion of a fly in a room, etc.
In the above examples, the particle, the object and the gas molecule all are considered to be as a point mass, which cannot undergo rotation. If the body is rigid and has a finite size, it can undergo rotation also, about any axis. So, a rigid body will have degrees of freedom both due to its translatory motion and rotatory motion. Like translatory motion, the rotatory can also be resolved into three mutually perpendicular components. Thus a rigid body have six degrees of freedom, 3 for translatory motion and 3 for rotatory motion.
Now, let us consider a system of two particles or two point masses. Each particle has three degrees of freedom, so the system has six degrees of freedom. If the two particles remain at a fixed distance from each other, then there is one definite relationship between them. These definite relationships are known as constraints. As a result, the number of independent coordinates required to describe the configuration of the system reduces by one. Hence, the system has (6 – 1) = 5 degrees of freedom.
These 5 degrees of freedom may be interpreted in another way: degrees of freedom for the translatory motion of the centre of mass = 3 and degrees of freedom for the rotatory motion of the two particles around the centre of mass = 2.
In a system consisting of N particles, if the particles possess k independent relations i.e., constraints among them, then the number of degrees of freedom of the system is given by, f = 3N – k.
Degrees of freedom of different types of gases:
i) Monatomic gas: The molecule of a monatomic gas (e.g., neon, helium, argon etc.) consists of a single atom (a point mass). At least three coordinates (say x y, z) are necessary to specify the postion of the molecule at any instant in the three dimensional space. So, the number of degrees of freedom of a monatomic gas molecule,
f = 3 × 1 – 0 = 3.
ii) Diatomic gas: The molecule of a diatomic gas like hydrogen, oxygen, nitrogen etc. has two atoms in it. Two point atoms have a fixed distance between them (neglecting the vibration of the atoms in the molecules)
i.e., number of constraint is 1. Here, N = 2 and k = 1
∴ f = 3 × 2 – 1 = 5
iii) Triatomic gas: Triatomic gas molecules are of two types:
(a) In a linear molecule such as CO2, CS2, HCN etc. the three atoms are arranged in a straight line [Fig.(a)]. The number of independent relations between them is only two.
∴ f = 3 × 3 – 2 = 7 i.e., such a molecule has seven degrees of freedom.
(b) In a non-linear molecule like H2O, SO2, etc. the three atoms are located at the three vertices of a triangle [Fig.(b)].
Hence, there are three fixed distances among the three atoms.
∴ f = 3 × 3 – 3 = 6
Therefore, a non-linear triatomic molecule has six degrees of freedom.
Degrees of freedom in different cases:
