Structure of simple molecules has been determined by spectroscopy, where the interaction between the — Physical Chemistry — Kinetics Chemistry Question
Internuclear distance of a hetero-nuclear diatomic molecule
Structure of simple molecules has been determined by spectroscopy, where the interaction between the radiation and the molecules is observed as a function of wavelength. The rotational spectrum of molecules appears in the far infrared or microwave region. Since microwave frequencies can be measured very precisely, internuclear distance of a diatomic molecule with a permanent dipole moment can be determined with high accuracy. A spectrum for the H35Cl molecule is shown in Fig. 1. The rotational lines are separated by ν = 6.26 × 10^11 s–1.
According to the simple model of a rotating diatomic molecule, the rotational energy, EJ, is discrete, which can be written as
EJ = h^2 / (8π^2µRe^2) J(J+1) J = 0,1,2,...
where µ and Re are reduced mass and internuclear distance, respectively. The rotational energy depends on the quantum number J. Under irradiation of microwave, transitions between rotational levels from the rotational state J” to the rotational state J’ are allowed, if J' - J" = ±1.
Calculate internuclear distance, Re, of H35Cl.
Fig. 1
Note:
Reduced mass, µ, is the effective inertial mass appearing in the two-body problem. For two bodies, one with mass, m1, and the other with mass m2, it is given by
1/µ = 1/m1 + 1/m2.
Model Answer
According to the simple model (a rigid rotator model), the allowed frequencies for absorption (J' = J" + 1) are
∆E = EJ' - EJ" = h^2 / (8π^2µRe^2) [(J"+1)(J"+2) - J"(J"+1)] = h^2 / (4π^2µRe^2) (J"+1) = hν.
Hence, frequency of microwave resonant to the J’ = 1 ← J” = 0 transition is 6.26×10^11 s–1. As reduced mass of HCl35 is 1.61 × 10–27 kg,
Re = sqrt(h / (4π^2µν)) = sqrt(6.63 × 10^-34 / (4π^2 × 1.61 × 10^-27 × 6.26 × 10^11)) = 1.29 × 10^-10