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A vessel contains a mixture of one mole of oxygen and two moles of nitrogen at 300 K. The ratio of the average rotational kinetic energy per O₂ molecule to that per N₂ molecule is
  • a)
    1 : 1
  • b)
    1 : 2
  • c)
    2 : 1
  • d)
    Depends on the moments of inertia of the two molecules
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
A vessel contains a mixture of one mole of oxygen and two moles of nit...
The ratio of the rotational kinetic energy of both the gases is 1:1.
Explanation:
Kinetic energy per degree of freedom = 1/2 kT
Since both Oxygen and nitrogen are diatomic gases and diatomic gas possess two degree of freedom for rotational motion therefore ,
rotational   K.E. = 2(1/2kT) = kT
In the problem both gases (oxygen and nitrogen) are diatomic and have same temperature (300 K) therefore ratio of average rotational kinetic energy will be equal to one.
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Most Upvoted Answer
A vessel contains a mixture of one mole of oxygen and two moles of nit...
Given:
- The vessel contains a mixture of one mole of oxygen and two moles of nitrogen.
- The temperature of the mixture is 300 K.

To find:
The ratio of the average rotational kinetic energy per oxygen molecule to that per nitrogen molecule.

Explanation:
The average rotational kinetic energy of a molecule can be given by the formula:

E_rot = (1/2) * I * ω^2

Where:
- E_rot is the rotational kinetic energy
- I is the moment of inertia of the molecule
- ω is the angular velocity of the molecule

Step 1: Calculate the moment of inertia:
The moment of inertia of a molecule depends on its mass and shape. For diatomic molecules like oxygen (O2) and nitrogen (N2), the moment of inertia can be approximated as:

I = μ * R^2

Where:
- μ is the reduced mass of the molecule
- R is the bond length of the molecule

The reduced mass can be calculated using the formula:

μ = (m1 * m2) / (m1 + m2)

Where:
- m1 and m2 are the masses of the atoms in the molecule

Step 2: Calculate the angular velocity:
The angular velocity of a molecule can be calculated using the formula:

ω = √(8 * R * T) / (π * I)

Where:
- R is the gas constant (8.314 J/mol·K)
- T is the temperature in Kelvin

Step 3: Calculate the rotational kinetic energy:
Using the calculated moment of inertia and angular velocity, we can now calculate the rotational kinetic energy for oxygen and nitrogen molecules.

E_rot_O2 = (1/2) * I_O2 * ω_O2^2
E_rot_N2 = (1/2) * I_N2 * ω_N2^2

Step 4: Calculate the ratio:
Finally, we can calculate the ratio of the average rotational kinetic energy per oxygen molecule to that per nitrogen molecule:

Ratio = (E_rot_O2 / 1) / (E_rot_N2 / 2)
= 2 * (E_rot_O2 / E_rot_N2)
= 2 * [(1/2) * I_O2 * ω_O2^2] / [(1/2) * I_N2 * ω_N2^2]
= (I_O2 * ω_O2^2) / (I_N2 * ω_N2^2)

Conclusion:
Since the moment of inertia and angular velocity are constants for a given molecule at a given temperature, the ratio of the average rotational kinetic energy per oxygen molecule to that per nitrogen molecule is independent of the moments of inertia of the two molecules. Thus, the correct answer is option 'A' - 1 : 1.
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A vessel contains a mixture of one mole of oxygen and two moles of nitrogen at 300 K. The ratio of the average rotational kinetic energy per O₂ molecule to that per N₂ molecule isa)1 : 1b)1 : 2c)2 : 1d)Depends on the moments of inertia of the two moleculesCorrect answer is option 'A'. Can you explain this answer?
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