Light as photons
Light behaves like a wave and also like a stream of tiny energy packets called photons.
- Wave speed: c = fλ (c = 3.00 × 10⁸ m/s, f = frequency in Hz, λ = wavelength in m).
- Photon energy: E = hf = hc/λ, with Planck's constant h = 6.63 × 10⁻³⁴ J s.
- Shorter wavelength → higher frequency → more energy per photon. Ultraviolet photons can damage skin; infrared photons only warm it.
Visible light runs from about 400 nm (violet) to 700 nm (red).
Emission and absorption spectra
Electrons in an atom can only have certain energies, called energy levels. When an electron falls from a higher level to a lower one, it gives out one photon with energy ΔE = hf, equal to the gap.
- Emission spectrum: bright coloured lines on a dark background (from a hot gas).
- Absorption spectrum: dark lines on a rainbow, where the gas has taken in exactly those photons.
The lines are at the same wavelengths in both. Each element has its own pattern, so spectra identify elements, for example sodium's yellow lines in street lamps or helium found first in the Sun's spectrum.
Colour of solutions and choosing the wavelength
A solution looks coloured because it absorbs some wavelengths and transmits the rest. The colour we see is the complementary colour of the one absorbed (opposite on the colour wheel).
- Blue copper(II) sulfate absorbs orange-red light.
- Purple potassium permanganate absorbs green-yellow light (about 525 nm).
For measurements we pick the wavelength the solution absorbs most, called λmax. A coloured filter or a monochromator picks it. Mixing coloured lights adds colours (additive synthesis); paints and filters remove colours (subtractive synthesis).
The spectrophotometer: transmittance and absorbance
A spectrophotometer has a lamp, a wavelength selector, a sample cell called a cuvette (usually 1 cm wide), and a detector.
- I₀ = light going in, I = light coming out.
- Transmittance T = I/I₀ (often given as %).
- Absorbance A = log₁₀(I₀/I) = −log₁₀ T. A has no unit.
A = 0 means all light passes; A = 1 means 10% passes; A = 2 means 1% passes. First we zero the machine with a blank (cuvette with only the solvent).
The Beer-Lambert law and the calibration curve
A = ε l c
- ε = molar absorptivity (L mol⁻¹ cm⁻¹): how strongly that substance absorbs at that wavelength.
- l = path length through the sample (cm).
- c = concentration (mol/L).
Calibration curve: make 4–6 standard solutions of known c, measure A for each at λmax, and plot A against c. The best-fit line passes through zero. Measure the unknown and read c from the line. Give the result with an uncertainty, for example c = 0.60 ± 0.02 mmol/L. The law works best for dilute solutions (A below about 1.5); at high concentration the line bends.
Infrared (IR) spectroscopy in brief
IR photons have too little energy to move electrons, but they make bonds vibrate. Each bond type absorbs at its own wavenumber (cm⁻¹):
- O–H (alcohol): broad band near 3200–3550 cm⁻¹.
- C=O (ketone, acid, ester): strong sharp band near 1700 cm⁻¹.
- C–H: near 2850–3000 cm⁻¹.
So an IR spectrum tells us which functional groups a molecule has.
Key formulas and definitions
- c = fλ
- E = hf = hc/λ
- ΔE = E(high) − E(low) = hf
- T = I/I₀
- A = log₁₀(I₀/I) = −log₁₀ T
- A = ε l c (Beer-Lambert law)
Worked examples
1. Find the energy of a photon of wavelength 500 nm.
E = hc/λ = (6.63 × 10⁻³⁴ × 3.00 × 10⁸) / (500 × 10⁻⁹) = 3.98 × 10⁻¹⁹ J.
2. Light of 100 units enters a sample and 25 units come out. Find T and A.
T = 25/100 = 0.25 (25%). A = log₁₀(100/25) = log₁₀ 4 = 0.60.
3. A solution has A = 0.45 in a 1 cm cuvette. ε = 1500 L mol⁻¹ cm⁻¹. Find c.
c = A/(εl) = 0.45/(1500 × 1) = 3.0 × 10⁻⁴ mol/L.
4. The same solution is placed in a 2 cm cuvette. What is A now?
A is proportional to l, so A doubles: 0.90.
5. Standards give A = 0.10, 0.20, 0.40 for c = 1, 2, 4 mg/L. An unknown has A = 0.30. Find c.
The slope is 0.10 per mg/L. c = 0.30/0.10 = 3 mg/L.
6. An electron falls between levels 3.0 × 10⁻¹⁹ J apart. Find the wavelength of the photon.
λ = hc/ΔE = (6.63 × 10⁻³⁴ × 3.00 × 10⁸)/(3.0 × 10⁻¹⁹) = 6.63 × 10⁻⁷ m = 663 nm (red).
Common mistakes
- Thinking a blue solution absorbs blue light. It transmits blue and absorbs orange.
- Mixing up T and A: high absorbance means LOW transmittance.
- Forgetting to change nm to m (1 nm = 10⁻⁹ m) in E = hc/λ.
- Using the Beer-Lambert law at very high concentrations, where the graph is no longer straight, or forgetting to zero with a blank.