Past exam of the chemistry A course of the University of Cambridge / 2025 / Paper 1 / A3 / d

This is a rovibrational spectrum: the vibrational excitation is accompanied by quantized molecular rotation. The high-wavenumber R branch has $\Delta J=+1$, and the low-wavenumber P branch has $\Delta J=-1$. In the simplest common-$B$ approximation,Consecutive lines within either branch are therefore separated by roughly $2B$, while the gap between P(1) and R(0) is roughly $4B$. There is no Q branch: for this diatomic parallel $\Sigma \to \Sigma$ transition, the electric-dipole rotational selection rule allows $\Delta J=\pm 1$, and the $\Delta J=0$ line strength vanishes.

The intensity envelope comes from rotational populations, approximately $(2J+1)e^{-hcBJ(J+1)/(k_{B}T)}$, together with the rotational line strengths. The paired peaks arise from the two chlorine isotopologues, ${\mathrm{H}}^{35}Cl$ and ${\mathrm{H}}^{37}Cl$: their different reduced masses shift both rotational and vibrational spacings. The more abundant ${}^{35}\mathrm{Cl}$ gives the stronger member. The changing spacing away from the band centre reflects $B_1\ne B_0$ and, at higher $J$, centrifugal distortion; it is not evidence for additional vibrational fundamentals.

Use the two-body reduced mass, rather than the chlorine mass or hydrogen mass alone:The bond force constant givesThe requested “frequency in cm−1” is a spectroscopic wavenumber; the ordinary frequency is the value in s−1 above. This calculation uses the question's force constant in the harmonic model, rather than replacing it with a tabulated spectroscopic constant.

Read the stronger members of the inset doublets at about $2945$, $2926$ and $2906\mathrm{c}{\mathrm{m}}^{-1}$. Their spacings are roughly $19$–$20\mathrm{c}{\mathrm{m}}^{-1}$, so the common-$B$ estimate isUsing the harmonic wavenumber from the preceding calculation and labelling lines by the lower-state $J$,These are graph-based estimates. Finite reading precision and different upper/lower rotational constants limit their accuracy to a few cm−1; extra decimal places would not be justified. The R1 result refers to $J=1\to 2$, while R0 is the line nearest the high-wavenumber side of the band gap.