Question:
The ratio of the degrees of freedom of a monoatomic gas and a nonlinear polyatomic gas having 2 vibrational modes is:
The ratio of the degrees of freedom of a monoatomic gas and a nonlinear polyatomic gas having 2 vibrational modes is:
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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 $10$.
$E_{\text{vib}} = \frac{1}{2}m v^2 + \frac{1}{2}k x^2$.
Non-vibrational polyatomic baseline is $6$ ($3$ trans + $3$ rot). Adding $2 \times 2 = 4$ gives $10$.
Updated On: Jul 22, 2026
- $3:10$
- $3:7$
- $3:5$
- $3:11$
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
The question asks for the ratio of the degrees of freedom of a monoatomic gas to those of a non-linear polyatomic gas that contains exactly 2 vibrational modes.
Step 2: Key Formula and Approach:
The total degrees of freedom ($f$) of a gas molecule is the sum of its translational, rotational, and vibrational degrees of freedom:
\[ f = f_{\text{trans}} + f_{\text{rot}} + f_{\text{vib}} \] We will calculate $f$ for both the monoatomic gas and the specified non-linear polyatomic gas.
Step 3: Detailed Explanation:
• Degrees of freedom of a monoatomic gas ($f_{\text{mono}}$):
A monoatomic gas molecule (e.g., Helium, Neon) is represented as a point mass.
It can move in three independent spatial directions, so it has only translational motion.
\[ f_{\text{mono}} = 3_{\text{trans}} = 3 \]
• Degrees of freedom of a non-linear polyatomic gas ($f_{\text{poly}}$):
A non-linear polyatomic molecule (e.g., Water, Ammonia) has:
- $3$ translational degrees of freedom ($f_{\text{trans}} = 3$).
- $3$ rotational degrees of freedom ($f_{\text{rot}} = 3$).
Each active vibrational mode contributes $2$ degrees of freedom (one for kinetic energy and one for potential energy of vibration).
Since it has $2$ vibrational modes:
\[ f_{\text{vib}} = 2 \times 2 = 4 \] Total degrees of freedom:
\[ f_{\text{poly}} = 3_{\text{trans}} + 3_{\text{rot}} + 4_{\text{vib}} = 10 \]
• Calculate the ratio:
\[ \text{Ratio} = \frac{f_{\text{mono}}}{f_{\text{poly}}} = \frac{3}{10} \]
Step 4: Final Answer:
The ratio of their degrees of freedom is $3:10$, which corresponds to Option (A).
The question asks for the ratio of the degrees of freedom of a monoatomic gas to those of a non-linear polyatomic gas that contains exactly 2 vibrational modes.
Step 2: Key Formula and Approach:
The total degrees of freedom ($f$) of a gas molecule is the sum of its translational, rotational, and vibrational degrees of freedom:
\[ f = f_{\text{trans}} + f_{\text{rot}} + f_{\text{vib}} \] We will calculate $f$ for both the monoatomic gas and the specified non-linear polyatomic gas.
Step 3: Detailed Explanation:
• Degrees of freedom of a monoatomic gas ($f_{\text{mono}}$):
A monoatomic gas molecule (e.g., Helium, Neon) is represented as a point mass.
It can move in three independent spatial directions, so it has only translational motion.
\[ f_{\text{mono}} = 3_{\text{trans}} = 3 \]
• Degrees of freedom of a non-linear polyatomic gas ($f_{\text{poly}}$):
A non-linear polyatomic molecule (e.g., Water, Ammonia) has:
- $3$ translational degrees of freedom ($f_{\text{trans}} = 3$).
- $3$ rotational degrees of freedom ($f_{\text{rot}} = 3$).
Each active vibrational mode contributes $2$ degrees of freedom (one for kinetic energy and one for potential energy of vibration).
Since it has $2$ vibrational modes:
\[ f_{\text{vib}} = 2 \times 2 = 4 \] Total degrees of freedom:
\[ f_{\text{poly}} = 3_{\text{trans}} + 3_{\text{rot}} + 4_{\text{vib}} = 10 \]
• Calculate the ratio:
\[ \text{Ratio} = \frac{f_{\text{mono}}}{f_{\text{poly}}} = \frac{3}{10} \]
Step 4: Final Answer:
The ratio of their degrees of freedom is $3:10$, which corresponds to Option (A).
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