Question

A vessel contains a non-linear triatomic gas. If $50\%$ of gas dissociate into individual atom, then find new value of degree of freedom by ignoring the vibrational mode and any further dissociation:
(A) $2.15$
(B) $3.75$
(C) $5.25$
(D) $6.35$

Answer 1

A non-linear triatomic gas means the gas molecule which contains three different kinds of atoms. In the triatomic gas, it is given that $50\%$ gets dissociated to individual atoms, then the remaining $50\%$ of gas is triatomic. The degree of freedom can be determined by multiplying the number of moles in remaining gas and the degree of freedom of nonlinear triatomic gas. Some examples of the triatomic gases are carbon dioxide.

Complete step by step solution:
Given that,
A vessel contains a non-linear triatomic gas, the number of atoms is $3$.
$50\%$ of gas dissociates into individual atoms.
For the remaining $50\%$ of triatomic gas,
The percentage is converted to moles, then
The remaining triatomic gas is $0.5\,mole$.
Degree of freedom for $0.5\,mole$ of triatomic gas, which is undissociated is,
$\Rightarrow 0.5 \times 6 = 3$ (Here $6$ is the degree of freedom for nonlinear triatomic gas).
The degree of freedom for $0.5\,mole$ of dissociated gas,
$\Rightarrow 0.5 \times 1.5 = 7.5$ (Here $1.5$ is the number of atoms dissociated)
Then, the total degree of freedom is,
$\Rightarrow 3 + 0.75 = 3.75$

Hence, the option (B) is the correct answer.

Note: Assume that, we have $1\,mole$ of triatomic gas, so it has $3$ atoms in it. First $50\%$ is dissociated, so the number of moles will be $0.5\,mole$, so it has $1.5$ atoms in it. This is the reason for substituting this value in the step to find the degree of freedom for $0.5\,mole$ of dissociated gas.