The lowest three rotational absorption lines of gaseous H Cl are measured at GHz ( ),…
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Central potentials and canonical three-dimensional systems
Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The Schrödinger equation in three dimensions and central potentials
Problem
The lowest three rotational absorption lines of gaseous 1H35Cl are measured at ν1=625.9 GHz (l=0→1), ν2=1249.3 GHz (l=1→2) and ν3=1870.2 GHz (l=2→3). Model the molecule as a rigid rotor: two point masses held at a fixed separation d, free to rotate about their centre of mass, so that H^=L^2/2I with I=mrd2; the reduced mass is quoted as mr=0.9796 u, and a photon changes l by one. Take ℏ=1.055×10−34 J s and 1 u =1.661×10−27 kg.
((a)) Using ν1 alone, find the moment of inertia I the rigid model implies, and the bond length d, in nm.
((b)) The rigid model predicts every line spaced by exactly ν1. Predict ν2 and ν3 on this basis, in GHz.
((c)) Compute the fractional shortfall (νpredicted−νmeasured)/νpredicted for the second and third lines, to two significant figures.
((d)) The shortfall grows from the second line to the third. Decide, and justify in a sentence or two, whether this pattern is more consistent with (i) a single mismeasurement of ν1, or (ii) a systematic effect the rigid model leaves out, by considering how each possibility would make the fractional shortfall behave across the three lines.
((e)) The bond is known to stretch slightly as the molecule spins faster, an effect this rigid model ignores by construction. Explain in one sentence why a stretching bond makes the measured lines fall below, rather than above, the rigid-rotor prediction.