Spectroscopy Mock Tests

In electron spin resonance (ESR) spectroscopy, the methyl radical (·CH₃) has a planar structure with three equivalent hydrogen atoms. How many lines are observed in the ESR spectrum of the methyl radical, and what is their intensity ratio? (Consider hyperfine coupling between the unpaired electron and the three equivalent ¹H nuclei, each with nuclear spin I = 1/2.)

The rotational constant B for the ¹H³⁵Cl molecule is 10.593 cm⁻¹. The wavenumber of the rotational transition from J = 0 to J = 1 is:

A particle of mass m is confined in a one-dimensional infinite potential well of length L. The wavefunction for the n=3 state is given by ψ₃(x) = √(2/L) sin(3πx/L). What is the probability of finding the particle in the region L/4 ≤ x ≤ 3L/4?

The fundamental vibrational wavenumber of the ¹²C¹⁶O molecule is 2143 cm⁻¹. Assuming the C−O bond can be modelled as a harmonic oscillator, what is the force constant k of the C−O bond? (Use the relation: ν̃ = (1/(2πc)) × √(k/μ), where μ is the reduced mass and c = 3 × 10¹⁰ cm/s. The atomic masses are: ¹²C = 12.000 amu, ¹⁶O = 15.995 amu, and 1 amu = 1.6605 × 10⁻²⁷ kg.)

The number of IR-active vibrational modes of CO₂ in the linear geometry is:

In the ¹H NMR spectrum of ethanol (CH₃CH₂OH) recorded at room temperature in a non-polar solvent, the signal for the OH proton appears as:

The ¹H NMR spectrum of ethanol (CH₃CH₂OH) recorded at room temperature in a pure sample (without D₂O exchange) shows how many signals and what splitting pattern for the CH₂ protons, respectively?

In the rotational spectrum of a rigid diatomic molecule such as ¹H³⁵Cl, the transitions follow the selection rule ΔJ = +1 (absorption). If the rotational constant B is defined as B = h/(8π²Ic) in wavenumber units (cm⁻¹), where I is the moment of inertia, what is the separation between consecutive lines in the rotational spectrum? (The rotational energy levels are given by E_J = B·J(J+1) in cm⁻¹.)