Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 1 / A3 / b

A vibrational fundamental is the transition $v=0\to 1$. A first vibrational overtone is $v=0\to 2$. A hot band starts from a thermally populated vibrationally excited level, such as $v=1\to 2$.

For a strictly harmonic potential and a dipole linear in displacement, the allowed transitions have $\Delta v=\pm 1$. A fundamental and a $1\to 2$ hot band then share the same frequency, while the overtone has zero intensity. Anharmonicity separates the adjacent gaps and can give overtone intensity; nonlinear displacement dependence of the dipole can also contribute. A hot band is defined by its lower state, not merely by being weak.

The strong band is $0\to 1$, while the weak band is the first vibrational overtone $0\to 2$. Let $w={\tilde{\omega }}_e$ and $q={\tilde{\omega }}_e{x}_e$. Subtracting the Morse oscillator levels givesTwice the fundamental is $881.2\mathrm{c}{\mathrm{m}}^{-1}$, slightly above the overtone because the higher vibrational gap is smaller. Eliminate $w$:soThe implied dimensionless anharmonicity constant is $x_e\simeq 0.004164$. The overtone's weakness follows from its being forbidden in the leading harmonic/linear-dipole model, rather than a lack of population in its lower state.

The relevant allowed absorption is $1\to 2$. Under the stipulated population-only intensity model, the vibrational levels are nondegenerate andSet this ratio to $0.10$ and solve:The energy in the exponential is the measured $0\to 1$ gap, not the harmonic constant $444.3\mathrm{c}{\mathrm{m}}^{-1}$. At this temperature, $N_2/N_0\simeq 0.0102$, so considering the first thermally excited manifold is sensible. Actual band intensity ratios also involve transition moments and rotational structure; the calculation follows the question's simplifying assumption.

Anharmonicity places the $1\to 2$ hot band atonly $3.7\mathrm{c}{\mathrm{m}}^{-1}$ below the fundamental. It can be detected with adequate spectral resolution and signal-to-noise; it is not guaranteed to appear as a separate peak in an unresolved band contour. The predicted ten-percent population-based intensity is appreciable, but overlapping rotational branches and linewidth can hide it beneath the stronger fundamental. No instrumental resolution or linewidth is supplied, so an unconditional assertion of visibility or invisibility would go beyond the information given.