Question

The ratio of the degrees of freedom of a monoatomic gas and a nonlinear polyatomic gas having 2 vibrational modes is:

• A. $3:10$
• B. $3:7$
• C. $3:5$
• D. $3:11$

Hint:

Remember that each vibrational mode contributes $2$ to the total degrees of freedom because of the potential energy term in the harmonic oscillator description:
$E_{\text{vib}} = \frac{1}{2}m v^2 + \frac{1}{2}k x^2$.
Non-vibrational polyatomic baseline is $6$ ($3$ trans + $3$ rot). Adding $2 \times 2 = 4$ gives $10$.

Answer 1

The Correct Option is A

Step 1: Count each contribution for the monatomic gas.
A single atom can only move along three independent directions, so it has purely translational motion, $f_{mono}=3$.
Step 2: Count each contribution for the nonlinear polyatomic gas separately.
It has $3$ translational degrees of freedom (moving as a whole) plus $3$ rotational degrees of freedom (spinning about three independent axes, since it is nonlinear), plus each vibrational mode contributes $2$ degrees of freedom (one for kinetic energy, one for potential energy of the vibration). With $2$ such modes, that gives $2\times2=4$.
Step 3: Add these up. \[ f_{poly} = 3+3+4 = 10 \]
Step 4: Form the ratio. \[ \boxed{f_{mono}:f_{poly} = 3:10} \]