An unknown homonuclear diatomic molecule is investigated using three spectroscopic…

CHEMISTRY · Multi-part · IB Diploma Programme · Chemistry · Functional groups: Classification of organic compounds · Functional groups: Classification of organic compounds

Problem

An unknown homonuclear diatomic molecule

\mathrm{X}_2

is investigated using three spectroscopic techniques. • Microwave spectroscopy shows successive rotational lines separated by

2.88\,\text{cm}^{-1}

. • Infrared spectroscopy reveals a fundamental vibrational absorption at

\tilde{\nu} = 1580\,\text{cm}^{-1}

. • UV–visible spectroscopy records an electronic absorption maximum at

\lambda = 250\,\text{nm}

. Useful constants:

h = 6.626\times10^{-34}\,\text{J}\,\text{s}

c = 3.00\times10^{8}\,\text{m}\,\text{s}^{-1}

N_A = 6.022\times10^{23}\,\text{mol}^{-1}

. (1) Part 1: (a) Assuming the rigid-rotor model and that the rotational line spacing equals

2B

, calculate the bond length of

\mathrm{X}_2

. Express your answer in picometres. (2) Part 2: (b) Treating the molecule as a harmonic oscillator, use the given vibrational wavenumber to estimate the force constant

k

of the X–X bond. Report

k

\text{N}\,\text{m}^{-1}

. (3) Part 3: (c) Compare quantitatively the energies (in

\text{kJ}\,\text{mol}^{-1}

) associated with the rotational transition (

J=0\rightarrow1

), the vibrational transition (

v=0\rightarrow1

), and the electronic transition responsible for the 250 nm absorption. Discuss the trend you observe. (4) Part 4: (d) Considering the wavelength of the electronic absorption, predict whether gaseous

\mathrm{X}_2

would appear coloured to the human eye under ordinary white light illumination. Provide a justification based on the relationship between absorbed and transmitted wavelengths.

Related theory

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