AP Chemistry · Lesson 4

Light, photons and Beer's law

Know Know it

Light travels at c=λνc = \lambda \nu, where λ is the wavelength and ν is the frequency. Each photon carries energy E=hν=hcλE = h\nu = \frac{hc}{\lambda}, with h=6.626×10−34h = 6.626 \times 10^{-34} J·s and c=2.998×108c = 2.998 \times 10^{8} m/s. Shorter wavelength means higher energy.

Each region of the spectrum does a different job. Microwaves change how a molecule rotates. Infrared light changes how its bonds vibrate. Visible and ultraviolet light move electrons to higher energy levels.

In a spectrophotometer, light of one wavelength passes through a sample, and the absorbance A is measured. The Beer-Lambert law is A=εbcA = \varepsilon b c: ε is the molar absorptivity (fixed for a substance at one wavelength), b is the path length and c is the concentration.

To find an unknown concentration, measure standards of known concentration, plot A against c (a straight line through the origin), and read the unknown's concentration from its absorbance. Choose the wavelength where the substance absorbs most.

Worked example Easy

What is the energy of one photon of green light with a wavelength of 500. nm? (h=6.626×10−34h = 6.626 \times 10^{-34} J·s, c=2.998×108c = 2.998 \times 10^{8} m/s)

  1. 3.97×10−283.97 \times 10^{-28} J
  2. 1.11×10−481.11 \times 10^{-48} J
  3. 3.97×10−193.97 \times 10^{-19} JAnswer
  4. 2.39×1052.39 \times 10^{5} J

How to solve it

  1. Convert: 500. nm = 5.00×10−75.00 \times 10^{-7} m.
  2. E=hcλ=(6.626×10−34)(2.998×108)5.00×10−7E = \frac{hc}{\lambda} = \frac{(6.626 \times 10^{-34})(2.998 \times 10^{8})}{5.00 \times 10^{-7}}.
  3. E=3.97×10−19E = 3.97 \times 10^{-19} J.

Why each choice is right or wrong

  • A. Incorrect. 3.97×10−283.97 \times 10^{-28} J leaves the wavelength in nm; 500. nm must be written as 5.00×10−75.00 \times 10^{-7} m.
  • B. Incorrect. 1.11×10−481.11 \times 10^{-48} J multiplies h by the wavelength and divides by c, which inverts the formula.
  • C. Correct. E=hcλ=(6.626×10−34)(2.998×108)5.00×10−7=3.97×10−19E = \frac{hc}{\lambda} = \frac{(6.626 \times 10^{-34})(2.998 \times 10^{8})}{5.00 \times 10^{-7}} = 3.97 \times 10^{-19} J.
  • D. Incorrect. 2.39×1052.39 \times 10^{5} J is the energy of a whole mole of these photons, not of one photon.

Trap Absorbance is not transmittance

A darker solution absorbs more light and lets less through. Beer's law is linear in absorbance, not in the percent of light transmitted. If a question gives percent transmittance, the more concentrated solution has the lower transmittance and the higher absorbance.

Explain Explain it

Task verbs: predict, explain

Lab errors in Beer's law work through A = εbc. If something makes the measured absorbance too high, the calculated concentration is too high; if it makes the absorbance too low, the concentration is too low.

For example, measuring at a wavelength away from the peak lowers ε, so A is smaller for each concentration. For any other error, ask whether it changes the light reaching the detector or the amount of the substance in the light path.

Worked example Medium

Calibration standards for a fictional blue dye, measured at one wavelength in a 1.00 cm cell

Concentration (mM)0.1000.2000.3000.400
Absorbance0.1200.2400.3600.480

A sample of the dye in the same cell has an absorbance of 0.300. What is its concentration?

  1. 0.360 mM
  2. 0.300 mM
  3. 0.250 mMAnswer
  4. 0.400 mM

How to solve it

  1. The standards lie on a line through the origin with slope 0.1200.100=1.20\frac{0.120}{0.100} = 1.20 per mM.
  2. c=A1.20=0.3001.20=0.250c = \frac{A}{1.20} = \frac{0.300}{1.20} = 0.250 mM, between the 0.200 and 0.300 mM standards.

Why each choice is right or wrong

  • A. Incorrect. 0.360 mM multiplies the absorbance by the slope, 0.300 × 1.20; the concentration is the absorbance divided by the slope.
  • B. Incorrect. 0.300 mM assumes absorbance equals concentration, but the slope of the line is 1.20 per mM, not 1.
  • C. Correct. The slope is 0.120 ÷ 0.100 = 1.20 per mM, so the concentration is 0.300 ÷ 1.20 = 0.250 mM.
  • D. Incorrect. 0.400 mM is the concentration of the standard with absorbance 0.480, which is more than the sample's 0.300.

Predict Your turn

A student measures a set of standards correctly and makes a Beer's law calibration line. Before measuring an unknown, the student rinses the cuvette with water but does not rinse it with the unknown before filling it, so a few drops of water remain inside. Predict whether the concentration the student calculates for the unknown will be too high, too low or unaffected, and explain why.

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Compare with a model answer

A full answer does this

  • Predicts that the calculated concentration will be too low
  • Explains that the leftover water dilutes the sample, so the measured absorbance is lower, and by A = εbc the concentration calculated from it is lower

Model answer

The calculated concentration will be too low. The drops of water left in the cuvette dilute the sample, so the solution in the light path is less concentrated than the original. It absorbs less light, so the measured absorbance is lower, and since A = εbc, the concentration calculated from that absorbance is lower than the true value.

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