IR units refresher: What is the mathematical relationship between wavelength and wavenumber used in IR spectroscopy?

Introduction / Context:
IR spectra are commonly plotted versus wavenumber (cm^-1). Understanding how wavenumber relates to wavelength prevents unit errors when converting between frequency-like and length units during spectral analysis.

Given Data / Assumptions:

• Wavenumber (ν̄) is defined as the reciprocal of wavelength expressed in centimeters.
• If wavelength is provided in nanometers, an appropriate conversion is needed.
• We use classical spectroscopic conventions (cm and cm^-1).

Concept / Approach:
By definition, ν̄ = 1/λ (with λ in cm). If λ is in nm, first convert: 1 nm = 1×10^-7 cm, so ν̄ (cm^-1) = 10^7 / λ(nm). The core relationship remains reciprocal with proper units.

Step-by-Step Solution:
State definition: ν̄ (cm^-1) = 1 / λ(cm).Recognize options that either ignore units or misuse dimensional analysis.Select the option that explicitly states the reciprocal in centimeters.

Verification / Alternative check:
Test with λ = 10 μm = 1×10^-3 cm → ν̄ = 1000 cm^-1, a typical mid-IR value.

Why Other Options Are Wrong:

• Wavenumber − wavelength = 1: Dimensionally inconsistent.
• Wavelength × wavenumber = 1 (nm basis): Would only hold with the constant 10^7 for nm.
• “None of the above”: Incorrect because the correct definition is present.

Common Pitfalls:
Mixing nm and cm without using the factor 10^7; always convert units correctly.

Final Answer:
Wavenumber = 1 / wavelength in centimeters.