Spectroscopy Mock Tests

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.)

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:

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.)

In the infrared (IR) spectrum of acetone (CH₃COCH₃), which of the following vibrational modes would give the strongest, most characteristic absorption band, and at approximately what wavenumber? (Consider the selection rule for IR activity: a vibration is IR-active if it produces a change in the molecular dipole moment. The C=O stretch is particularly intense due to the large dipole moment change.)

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

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 UV-Vis absorption spectrum of the permanganate ion (MnO₄⁻, tetrahedral, T_d symmetry) shows an intense purple colour with a molar absorptivity ε ≈ 2000 L/(mol·cm) at approximately 525 nm. This absorption band is due to which type of electronic transition? (Mn is in the +7 oxidation state with d⁰ configuration. The T_d point group has no centre of inversion. The valence electrons reside in oxygen-based ligand orbitals, and the lowest energy transitions involve charge transfer from oxygen p orbitals to empty manganese d orbitals.)