§ C-17 Optical Spectrophotometry & the Beer–Lambert Law (A = ε·l·C) (Grade 10–11 · Beer–Lambert Law)
Monochromatic light attenuation through a colored solution in a quartz cuvette, optical absorbance A = ε(λ)lC, and transmittance T = 10⁻ᴬ. Each successive layer of equal thickness in an absorbing solution absorbs an equal fraction of the monochromatic light beam passing through it.
Theory Summary & Scientific Mechanism
Spectrophotometry determines solute concentrations by measuring the attenuation of a monochromatic light beam (λ) passing through a sample cuvette. Photons whose energy hν matches an electronic transition of the dissolved chromophore (such as charge-transfer in purple MnO₄⁻ at λₘax = 525 nm) are selectively absorbed.
According to the Beer–Lambert Law, transmitted light intensity decays exponentially with optical path length l and molar concentration C: I = I₀ · 10⁻ε(λ) l C. Defining optical absorbance as A = lg (I₀ / I) = -lg T yields the fundamental linear law of analytical spectroscopy: A = ε(λ)· l· C, where ε(λ) is the molar absorptivity.
Because absorbance A scales linearly with concentration C while transmittance T = 10⁻A varies exponentially, calibration curves plot A versus C at λₘax to achieve maximum analytical sensitivity.
Everyday Analogy
Looking at a light through a glass of tea or purple KMnO₄, the darker the brew (concentration C) or wider the glass (path length l), the dimmer the transmitted beam. A spectrophotometer uses A = ε l C to measure concentration from light absorption!
Interactive Question & Derivation
How do optical absorbance A and light transmittance T = I/I₀ change if the chromophore molar concentration C or cuvette path length l is doubled?
- A: Optical absorbance A = ε l C doubles linearly, whereas transmitted light intensity T = 10⁻A decreases exponentially
- B: Absorbance A is halved while transmittance T doubles
- C: Both absorbance A and transmittance T vary linearly with concentration
Study Hint: By the Beer–Lambert law, optical absorbance A = ε l C increases linearly with concentration C or path length l, while transmittance T = I/I₀ = 10⁻A decreases exponentially.
Proof & Explanation: According to the Beer–Lambert law, optical absorbance A = lg (I₀/I) = ε(λ)· l· C is directly proportional to molar absorptivity ε(λ), optical path length l, and concentration C. Because A is logarithmic in intensity, transmittance T = I/I₀ = 10⁻A decays exponentially.
Key Equations & 3D Simulation Parameters
A = ε(λ) l C, T = 10⁻A· 100%
- Chromophore molar concentration C: 0.05 … 1 mmol/L (default: 0.4 mmol/L)
- Cuvette optical path length l: 0.5 … 3 cm (default: 1.5 cm)
- Monochromator wavelength λ: 400 … 700 nm (default: 525 nm)
- Solute (1=KMnO₄ 525 nm, 2=CuSO₄ 650 nm, 3=K₂Cr₂O₇ 450 nm): 1 … 3 (default: 1)