A chemist observes a weak IR absorption at approximately 3400 cm⁻¹ — close to, but slightly below, twice the frequency of the fundamental O-H stretch at 3700 cm⁻¹. What is this peak, and what physical phenomenon causes it?
AA hot band from the v=1→2 transition, caused by thermal population of the v=1 O-H state at room temperature
BThe first overtone of the O-H stretch, appearing at slightly less than 2×3700 cm⁻¹ because anharmonicity causes vibrational energy levels to converge rather than remaining evenly spaced
CA combination band arising from simultaneous excitation of two O-H bending modes near 1700 cm⁻¹ each
DAn instrumental artifact from detector nonlinearity at high absorbance values
This is the first overtone — the v=0→2 transition. In an anharmonic oscillator, the Δv=±1 selection rule relaxes and Δv=±2 becomes weakly allowed. Crucially, anharmonicity also makes energy levels converge: the gap between v=1 and v=2 is slightly smaller than between v=0 and v=1. So the first overtone appears at slightly less than exactly 2×ν, not exactly 2×ν. The hot band (v=1→2) would also appear below the fundamental, but it requires the v=1 state to be populated, and it would shift with temperature — the overtone is present regardless of temperature.
Question 2 Multiple Choice
A spectroscopist measures an IR spectrum of a diatomic gas at room temperature and then at 100 K. A certain weak absorption slightly below the fundamental frequency disappears at 100 K. What type of transition is this, and why does it disappear?
AAn overtone transition — at low temperatures, molecules lack the energy needed to jump two vibrational quanta at once
BA hot band (e.g., v=1→2) — at 100 K, very few molecules occupy the v=1 level, so almost none can make this transition
CA combination band — low temperatures prevent two different modes from being simultaneously active
DThe fundamental transition itself — it shifts to a lower frequency at low temperature, moving away from its usual position
Hot bands arise from thermally populated excited states: molecules already in v=1 absorb to reach v=2. The v=1 population follows the Boltzmann distribution — at room temperature there is a small but measurable population, but at 100 K (much less than the vibrational temperature for most bonds), almost all molecules are in v=0. With no molecules in v=1, the hot band disappears. This temperature dependence is the defining diagnostic for hot bands. Overtones originate from v=0 (which is always the most populated level), so their intensity does not depend on temperature in the same way.
Question 3 True / False
Vibrational overtones occur because the harmonic oscillator selection rule (Δv = ±1) breaks down in real molecules whose potential energy is anharmonic, allowing Δv = ±2, ±3, and higher transitions.
TTrue
FFalse
Answer: True
This is exactly correct. The Δv = ±1 rule is a consequence of the harmonic oscillator wavefunction mathematics — the transition dipole moment integrals vanish for Δv ≠ ±1 in a perfect harmonic potential. Anharmonicity (the deviation of the real potential from a perfect parabola, well-described by a Morse potential) mixes wavefunctions from adjacent levels, giving non-zero transition dipole moments for Δv = ±2, ±3, etc. The overtones are intrinsically weaker than the fundamental because the mixing is small, and higher overtones are weaker still.
Question 4 True / False
A hot band and the first overtone for the same vibrational mode appear at the same frequency in the IR spectrum because they both involve the same energy gap between adjacent vibrational levels.
TTrue
FFalse
Answer: False
They appear at different frequencies. The first overtone is the v=0→2 transition, which spans two level spacings — at an anharmonic frequency slightly less than 2ν. The hot band is the v=1→2 transition, which spans one level spacing — but a smaller one than v=0→1, because anharmonic level spacings decrease with increasing v. So the hot band appears at a frequency slightly LOWER than the fundamental (not at the same frequency as the overtone). The fundamental, hot band, and overtone all appear at distinct positions in the spectrum.
Question 5 Short Answer
What is the key experimental observation that allows you to distinguish a hot band from an overtone in a vibrational spectrum, and why does that observation work?
Think about your answer, then reveal below.
Model answer: Temperature dependence. Hot bands strengthen as temperature increases and weaken as temperature decreases (eventually disappearing at very low temperatures). Overtones show no such intensity change with temperature. This works because hot bands require molecules to already be in an excited vibrational state (e.g., v=1) before they can absorb; the population of that excited state follows the Boltzmann distribution and is therefore temperature-sensitive. Overtones originate from v=0, which is always the dominant population regardless of temperature, so their intensity is relatively temperature-insensitive.
This is the diagnostic: if you cool the sample and a weak band disappears (or heat it and the band grows), it is a hot band. If a weak band shows no intensity change with temperature, it is an overtone or combination band. In practice, variable-temperature spectroscopy is a standard technique for assigning ambiguous features in complex molecular spectra.