Force Constant and Isotope Shift in Diatomic IR Spectra
A bond behaves like a spring. A stiff spring vibrates fast; a heavy mass on it vibrates slowly. The infrared absorption of a diatomic molecule gives the vibrational wavenumber, and from it you can find the force constant k, a direct measure of bond stiffness. Questions then ask you to predict how the band moves when an atom is replaced by an isotope. This page shows the calculation with HCl, DCl and CO.
The formulas
k = 4π2 c2 ν̃2 μ
μ = m1 m2 / (m1 + m2)
Isotope shift: ν̃2 / ν̃1 = √(μ1 / μ2)
ν̃ is the wavenumber (cm−1), c = 2.998 × 1010 cm/s when ν̃ is in cm−1, and μ is the reduced mass in kg (1 u = 1.6605 × 10−27 kg). k is in N/m. For a quick check, 1 N/m = 1 kg s−2. The isotope formula follows from the first one, because the force constant depends on the electrons and not on nuclear mass, so it stays the same when an isotope replaces an atom.
This is the harmonic model. A real band is the fundamental (v = 0 → 1) and is slightly lower than the harmonic frequency because of anharmonicity. Exam questions normally give you the wavenumber and expect you to use it directly as above.
Worked example 1: force constant of H–35Cl
Take the fundamental band of H35Cl as 2886 cm−1. Masses: H = 1.007825 u, 35Cl = 34.968853 u.
μ = (1.007825 × 34.968853) / (1.007825 + 34.968853) = 35.2425 / 35.97668 = 0.97959 u = 0.97959 × 1.66054 × 10−27 = 1.6267 × 10−27 kg.
ν = c ν̃ = 2.99792 × 1010 × 2886 = 8.652 × 1013 Hz.
k = 4π2 ν2 μ = 39.478 × (8.652 × 1013)2 × 1.6267 × 10−27 = 39.478 × 7.486 × 1027 × 1.6267 × 10−27 = 481 N/m.
Worked example 2: where does D–35Cl absorb?
Deuterium has mass 2.014102 u. The force constant of the bond stays 481 N/m.
μ(DCl) = (2.014102 × 34.968853) / (2.014102 + 34.968853) = 70.4308 / 36.98296 = 1.9044 u.
ν̃(DCl) = 2886 × √(0.97959 / 1.9044) = 2886 × √0.51437 = 2886 × 0.71720 = 2070 cm−1.
The band moves down by about 816 cm−1. The ratio 0.717 is close to 1/√2 = 0.707 because replacing H by D roughly doubles the reduced mass, as the heavy Cl atom hardly moves.
Worked example 3: force constant of CO and what it says about the bond
The fundamental of 12C16O is 2143 cm−1. Masses: 12.000 u and 15.9949 u.
μ = (12.000 × 15.9949) / (27.9949) = 191.939 / 27.9949 = 6.8562 u = 1.1385 × 10−26 kg.
ν = 2.99792 × 1010 × 2143 = 6.4245 × 1013 Hz. k = 39.478 × (6.4245 × 1013)2 × 1.1385 × 10−26 = 39.478 × 4.1274 × 1027 × 1.1385 × 10−26 = 1855 N/m.
This is 3.86 times the value for HCl. The C–O bond is much stiffer, which agrees with the triple-bond character of CO in molecular orbital theory. Compare force constants only for similar bond types; different masses change the wavenumber even when stiffness is the same, which is why k and not ν̃ is the better measure of bond strength.
Worked example 4: zero-point energy and isotope effect
The lowest vibrational level lies half a quantum above the potential minimum: E0 = ½ h c ν̃.
For HCl: E0 = 0.5 × 6.626 × 10−34 × 2.998 × 1010 × 2886 × 6.022 × 1023 J/mol = 17.26 kJ/mol. For DCl (2070 cm−1): 12.38 kJ/mol.
The difference is 4.88 kJ/mol. A bond that starts at a lower energy level needs more energy to break; this is the origin of the primary kinetic isotope effect, using the same model.
Common mistakes
- Using the ordinary mass or the sum of masses instead of the reduced mass μ.
- Using grams or atomic mass units in the formula. Convert μ to kg, or k will be wrong by a factor of 1000 or 1027.
- Using c in m/s with cm−1. If ν̃ is in cm−1, take c in cm/s, or convert ν̃ to m−1 by multiplying by 100.
- Changing k for an isotope. k stays the same; only μ changes.
- Taking ν̃ ∝ μ instead of ν̃ ∝ 1/√μ.
- Confusing the fundamental with the harmonic frequency when the question gives both.
Exam relevance
| Result | Value from this page |
|---|---|
| μ for H35Cl | 0.9796 u |
| k for HCl (from 2886 cm−1) | about 481 N/m |
| ν̃ of DCl | about 2070 cm−1 |
| k for CO (from 2143 cm−1) | about 1855 N/m |
| Isotope shift rule | ν̃2 / ν̃1 = √(μ1 / μ2) |
The wavenumbers used here are practice values taken as typical textbook figures. In an exam, use the numbers given in the question, and check the current official notification for the syllabus of your paper.
There is no separate infrared tool, but you can use the Scientific Calculator for the square roots and powers, and the constants list for h, c and NA.
Open the Chemistry Calculator Suite →Turn this concept into marks: solve previous-year CSIR-NET questions and take a timed quiz.
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