Light and Spectra (c = λν, E = hν)电磁波与光谱（c = λν、E = hν）

Big Idea · Structure and Properties (SAP)AP Science Practice 5 · Mathematical RoutinesAP link · Unit 3~35 min

Learning goals

  • Use c = λν to convert between wavelength and frequency, and list the regions of the electromagnetic spectrum in order
  • Use E = hν to calculate the energy of a photon, and convert it to the energy of a mole of photons
  • Distinguish continuous spectra from line spectra, and explain how a flame test works

Before you start

  • Development of the atomic model (2.2.1)

Why are fireworks different colors?

Fireworks get their colors from metal salts: sodium salts glow yellow, strontium salts red, and copper salts green. Each element gives off its own characteristic colors of light, like a fingerprint. To understand why atoms emit particular colors, you first need to understand light itself.

Light is an electromagnetic wave

Light is an electromagnetic wave电磁波. Two quantities describe a wave:

  • Wavelength, λ: the distance from one crest to the next, usually in meters or nanometers (1 nm = 10⁻⁹ m).
  • Frequency, ν: the number of waves passing a point each second, in hertz (1 Hz = 1 s⁻¹).

All electromagnetic waves travel through a vacuum at the same speed, the speed of light, c = 3.00 × 10⁸ m/s. Wavelength, frequency, and the speed of light are related by:

c=λνc = \lambda \nu

The shorter the wavelength, the higher the frequency. From shortest to longest wavelength, the electromagnetic spectrum runs: gamma rays, X-rays, ultraviolet, visible light, infrared, microwaves, radio waves. Human eyes see only a narrow slice, about 380 to 750 nm, from violet to red.

Photons: energy in packets

In 1900, Max Planck proposed that energy is given off in small packets, and in 1905 Albert Einstein went further: light itself is made of packets called photons光子. A photon’s energy is proportional to the light’s frequency:

E=hν=hcλE = h\nu = \frac{hc}{\lambda}

Here h = 6.626 × 10⁻³⁴ J·s is Planck’s constant. The J (joule) is a unit of energy, and 1 kJ = 1000 J; lesson 2.10.1 covers energy units in detail. The higher the frequency (the shorter the wavelength), the more energy each photon carries. That’s why ultraviolet light causes sunburn and X-rays pass through the body, while infrared only feels warm.

One key piece of evidence for photons is the photoelectric effect: shining even very bright red light on certain metals knocks out no electrons, yet very dim ultraviolet light knocks them out immediately. Whether electrons are ejected depends on the energy of each photon, not on how bright the light is.

🔧 Math tool card · Scientific NotationSkip it if you already know this

To multiply numbers in scientific notation, multiply the coefficients and add the exponents; to divide, divide the coefficients and subtract the exponents. Then adjust so the coefficient is between 1 and 10. For example:

(6.626×10−34)×(3.00×108)=19.878×10−26=1.99×10−25(6.626 \times 10^{-34}) \times (3.00 \times 10^{8}) = 19.878 \times 10^{-26} = 1.99 \times 10^{-25}

What is (3.0 × 10⁸) ÷ (6.0 × 10⁻⁷)?

One photon carries very little energy, so chemists usually give the energy of a mole of photons instead. The mole (mol) is chemistry’s counting unit, like a dozen for eggs: 1 mol is 6.022 × 10²³ particles. That number is Avogadro’s number, NA = 6.022 × 10²³ mol⁻¹, and lesson 2.6.1 covers it in detail. Multiplying the energy of one photon by NA gives the energy of a mole of photons in kJ/mol (kilojoules per mole). That makes it easy to compare with the energy of chemical bonds, a few hundred kJ/mol.

Drag the slider, or click a familiar kind of light, to see its frequency and photon energy:

Try it: drag the slider to change the wavelength and watch the frequency and photon energy change
← short wavelength, high energylong wavelength, low energy →

Continuous and line spectra

White light passed through a prism spreads into an unbroken band of colors from red to violet: a continuous spectrum. But when an element’s gas is energized by electricity or heat, its light passes through a prism as just a few separate bright lines: a line spectrum (or emission spectrum). Every element’s line spectrum is different, so it can be used to identify the element.

A flame test uses exactly this: some metals and their compounds give a flame a characteristic color.

ElementSodiumPotassiumLithiumCalciumStrontiumCopper
Flame colorYellow-orangeLilac (violet)CrimsonOrange-redRedGreen

Potassium’s faint lilac is usually viewed through blue cobalt glass, which blocks the yellow glow from any sodium impurity. Why does each element emit only certain wavelengths? Lesson 2.2.3 answers that with the Bohr model.

Example 1: A sodium lamp

A sodium lamp gives off yellow light with a wavelength of 589 nm. Find its frequency and the energy of one photon.

Step-by-step solution

  1. Convert units. The speed of light is in m/s, so the wavelength must be in meters: λ = 589 nm = 589 × 10⁻⁹ m = 5.89 × 10⁻⁷ m.

  2. Frequency. From c = λν, ν = c/λ:

    ν=3.00×108 m/s5.89×10−7 m=5.09×1014 s−1\begin{aligned} \nu &= \frac{3.00 \times 10^{8}\ \text{m/s}}{5.89 \times 10^{-7}\ \text{m}} \\ &= \boxed{5.09 \times 10^{14}\ \text{s}^{-1}} \end{aligned}

  3. Energy of one photon. E = hν; J·s times s⁻¹ gives J:

    E=(6.626×10−34)(5.09×1014)=3.37×10−19 J\begin{aligned} E &= (6.626 \times 10^{-34})(5.09 \times 10^{14}) \\ &= \boxed{3.37 \times 10^{-19}\ \text{J}} \end{aligned}

  4. Check. Visible photons carry energies of order 10⁻¹⁹ J, so this is reasonable. A common mistake is forgetting to convert nm to m, which puts the answer off by 10⁹.

Example 2: Can a microwave oven break chemical bonds?

A microwave oven uses microwaves with a frequency of 2.45 GHz (2.45 × 10⁹ Hz). Find their wavelength and the energy of a mole of these photons, and compare it with a typical bond energy (about 400 kJ/mol).

Step-by-step solution

  1. Wavelength. λ = c/ν:

    λ=3.00×108 m/s2.45×109 s−1=0.122 m=12.2 cm\begin{aligned} \lambda &= \frac{3.00 \times 10^{8}\ \text{m/s}}{2.45 \times 10^{9}\ \text{s}^{-1}} \\ &= 0.122\ \text{m} \\ &= \boxed{12.2\ \text{cm}} \end{aligned}

  2. Energy of one photon.

    E=hν=(6.626×10−34)(2.45×109)=1.623×10−24 J\begin{aligned} E &= h\nu \\ &= (6.626 \times 10^{-34})(2.45 \times 10^{9}) \\ &= 1.623 \times 10^{-24}\ \text{J} \end{aligned}

  3. Energy of a mole of photons. Multiply by Avogadro’s number:

    (1.623×10−24)(6.022×1023)=0.978 J/mol\begin{aligned} & (1.623 \times 10^{-24})(6.022 \times 10^{23}) \\ &= \boxed{0.978\ \text{J/mol}} \end{aligned}

  4. Compare. A bond is about 400 kJ/mol = 4 × 10⁵ J/mol, and 0.978 J/mol is less than one four-hundred-thousandth of that, so microwaves can’t break chemical bonds. They heat food by making water molecules rotate faster.

  5. Watch the units. Convert kJ to J before comparing, or you’ll be off by a factor of 1000.

Practice: which has the higher-energy photons?
  1. Ultraviolet or infrared
  2. Red light or violet light
  3. X-rays or visible light
  4. Radio waves or microwaves
  5. Light at 400 nm or light at 700 nm
  6. Light at 1.0 × 10¹⁵ Hz or light at 5.0 × 10¹⁴ Hz
  7. Green light or blue light
  8. Gamma rays or ultraviolet
  9. Infrared or microwaves

Common mistakes

Mistake 1: “Longer wavelength means more energy.” It’s the opposite: E = hc/λ, so the longer the wavelength, the less energy each photon carries.

Mistake 2: “Brighter light means more energy per photon.” Brighter light just means more photons; each photon’s energy depends only on its frequency. That’s why even very bright red light can’t produce the photoelectric effect.

Mistake 3: “Different colors of light travel at different speeds in a vacuum.” Every electromagnetic wave travels through a vacuum at c = 3.00 × 10⁸ m/s; only the wavelength and frequency differ.

Quick check

What is the frequency of light with a wavelength of 500 nm?

Which of these has the highest-energy photons?

Summary

  • Light is an electromagnetic wave: c = λν, with c = 3.00 × 10⁸ m/s in a vacuum. From short to long wavelength: gamma rays, X-rays, ultraviolet, visible (about 380 to 750 nm), infrared, microwaves, radio waves.
  • Light is made of photons with E = hν = hc/λ, where h = 6.626 × 10⁻³⁴ J·s; multiplying by Avogadro’s number gives the energy per mole of photons.
  • White light gives a continuous spectrum; an element’s gas gives a line spectrum, which is why each element has its own flame-test color.
On to AP (for later)

c = λν and E = hν are both on the AP equation sheet, which lists the speed of light as 2.998 × 10⁸ m/s; use whatever value a problem gives you. AP topic 3.11 adds that microwaves make molecules rotate, infrared makes bonds vibrate, and ultraviolet and visible light move electrons between energy levels; topic 3.12 covers the photoelectric effect.

Practice this chapter: 2.2 · Atomic Structure · Basic, Medium, and Hard levels

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