The frequency of a microwave absorbed by a molecule when changing from the J = 3 to J = 4 rotation energy state is 4.61 × 10¹¹ Hz. The moment of inertia of the molecule (in kg ⋅ m²) is A) 5.70 × 10−⁴⁵ B) 1.45 × 10−⁴⁶ C) 1.12 × 10−⁴⁴ D) 2.91 × 10−⁴⁶ E) 9.11 × 10−⁴⁶

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Quizplus · Accepted Answer

The energy difference between rotational levels in a molecule can be given by the formula ΔE = hν, where h is Planck's constant (6.626 × 10^-34 J·s) and ν is the frequency of the radiation absorbed or emitted when the molecule transitions between these levels. The rotational energy levels of a rigid rotor (a model for a rotating molecule) are given by E_J = J(J+1)h^2/(8π^2I), where J is the rotational quantum number and I is the moment of inertia of the molecule. The difference in energy between the J = 3 and J = 4 levels can be calculated using the difference in their E_J values, which simplifies to ΔE = 2J + 1 times h^2/(8π^2I) for a transition from J to J+1.Given that the frequency ν = 4.61 × 10^11 Hz, we can find ΔE using ΔE = hν. Then, knowing ΔE and the value of J for the transition (from 3 to 4, so J = 3), we can solve for I.ΔE = hν = (6.626 × 10^-34 J·s)(4.61 × 10^11 Hz) = 3.054 × 10^-22 J.For the transition J = 3 to J = 4, ΔE = 2J + 1 times h^2/(8π^2I) simplifies to ΔE = 7h^2/(8π^2I) because 2J + 1 = 7 for J = 3.Rearranging for I gives I = 7h^2/(8π^2ΔE).Substituting the values for h and ΔE gives I ≈ 1.45 × 10^-46 kg·m^2, which matches choice B.

The Frequency of a Microwave Absorbed by a Molecule When