(i) To find the degree of freedom of mono-atomic gases let us first discuss what degree of freedom is and how it is calculated thereafter we will discuss what is mono-atomic gases
Degree of freedom of any gas is defined as the total number of independent modes (ways) in which a system can possess energy.
So, to calculate the degree of a system we use the following formula
f = 3N − k
Where, f is the degree of freedom
N is the number of particles in the system
And, K is the independent relationship among the system
Now, let us discuss what mono-atomic gas is.
A monatomic gas is defined as the one where the atoms of the gas that do not chemically combine to form molecules of more than one atom. Normally, the only atoms which act this way are the inert gases which are present at the rightmost column of the periodic table. These are Helium, argon, neon etc. Their outer electron shells are naturally entirely filled, so they have no openings to form covalent bonds, and the atoms are electrically neutral so they will not form ionic bonds.
So, here in mono-atomic gases the number of particles in the system is 1 and the independent relationship among the particles is 0.
So, according the formula of degree of freedom
f = 3N − k
for monatomic molecules N = 1 and k = 0
= 3 × 1 − 0
= 3
So, here according to the above solution we can say that the degree of freedom of the monatomic gases is 3.
(ii) Number of degree of freedom
d = 3N − 1
where N is the number of atoms in a molecules
In diatomic molecules,
N = 2
⇒ d = 3(2) − 1 = 5
Hence diatomic molecule has 5 degrees of freedom (3 translational and 2 rotational).
(iii) We have to find the Degree of freedom of a triatomic molecule. A triatomic gas molecule has 3 atoms in it.
We now consider the possible movements of this molecule in the x, y and z axis.
Here this triatomic gas can have a translatory motion along the x, y, and z axis. I.e. triatomic molecules can move along x direction, y direction and z direction.
Hence the translatory degree of freedom of this molecule is 3
Now let us consider the rotational degree of freedom of this molecule.
For that we place two atoms of the molecule on the x axis. Then it can rotate about y axis and z axis. It also has a significant rotation about x axis because here the third atom has a moment of inertia about x axis even if the other two atoms do not have the inertia.
And thus the rotational degree of freedom of this molecule is also three.
Hence the degree of freedom = Translatory degree of freedom + Rotational degree of freedom
= 3 + 3
= 6
Therefore at moderate temperature the degree of freedom of a triatomic gas equals to 6.
