Numford
Photoelectric

Work function of metals

The work function is the least energy needed to free an electron from a metal surface. It runs from 2.14 eV for caesium to 5.65 eV for platinum; aluminium is 4.28 eV, so 250 nm light frees electrons with K max = 0.69 eV.

Photoelectric effect

Pick a metal and a colour of light. Each photon carries hν; it frees an electron only if hν is more than the work function φ. Brighter light means more photons, not bigger ones. Values use h = 6.63 × 10⁻³⁴ J s and e = 1.6 × 10⁻¹⁹ C.

Metal
Light
Photon energy4.97 eV7.96 × 10⁻¹⁹ J
Threshold1.03 × 10¹⁵ Hzλ₀ = 290 nm
K max0.69 eVfastest at 4.93 × 10⁵ m/s
Stopping potential0.69 Vstops even the fastest
Photons per second2.5 × 10¹⁵at 2 mW

Al plate in light of 250 nm: electrons leave with up to 0.69 eV.

51015-2.52.5ν (10¹⁴ Hz)K max (eV)ν₀ = 10.33-φ = -4.28 eVslope h
K max against ν
-10123V (volt)Current2 mW1 mW-V₀
Current against voltage
Show the working
  1. ν = c / λ = 3 × 10⁸ / (250 × 10⁻⁹) = 1.2 × 10¹⁵ Hz
  2. E = hν = 6.63 × 10⁻³⁴ × 1.2 × 10¹⁵ = 7.96 × 10⁻¹⁹ J = 7.96 × 10⁻¹⁹ / 1.6 × 10⁻¹⁹ = 4.97 eV
  3. ν₀ = φ / h = 4.28 × 1.6 × 10⁻¹⁹ / 6.63 × 10⁻³⁴ = 1.03 × 10¹⁵ Hz, so λ₀ = c / ν₀ = 290 nm
  4. K max = hν - φ = 4.972 - 4.28 = 0.69 eV
  5. eV₀ = K max, so V₀ = 0.69 V
  6. v max = √(2K / m) = √(2 × 1.11 × 10⁻¹⁹ / 9.11 × 10⁻³¹) = 4.93 × 10⁵ m/s
  7. Photons per second = P / E = 2.0 × 10⁻³ W / 7.96 × 10⁻¹⁹ J = 2.5 × 10¹⁵
Working

Step by step

  1. ν = c / λ = 3 × 10⁸ / (250 × 10⁻⁹) = 1.2 × 10¹⁵ Hz
  2. E = hν = 6.63 × 10⁻³⁴ × 1.2 × 10¹⁵ = 7.96 × 10⁻¹⁹ J = 7.96 × 10⁻¹⁹ / 1.6 × 10⁻¹⁹ = 4.97 eV
  3. ν₀ = φ / h = 4.28 × 1.6 × 10⁻¹⁹ / 6.63 × 10⁻³⁴ = 1.03 × 10¹⁵ Hz, so λ₀ = c / ν₀ = 290 nm
  4. K max = hν - φ = 4.972 - 4.28 = 0.69 eV
  5. eV₀ = K max, so V₀ = 0.69 V
  6. v max = √(2K / m) = √(2 × 1.11 × 10⁻¹⁹ / 9.11 × 10⁻³¹) = 4.93 × 10⁵ m/s
  7. Photons per second = P / E = 2.0 × 10⁻³ W / 7.96 × 10⁻¹⁹ J = 2.5 × 10¹⁵
NCERT Table 11.1

Work functions of ten metals

Metalφλ₀Green light?
Caesium (Cs)2.14 eV581 nmEmits
Potassium (K)2.3 eV540 nmEmits
Sodium (Na)2.75 eV452 nmNo
Calcium (Ca)3.2 eV388 nmNo
Molybdenum (Mo)4.17 eV298 nmNo
Aluminium (Al)4.28 eV290 nmNo
Copper (Cu)4.65 eV267 nmNo
Silver (Ag)4.7 eV264 nmNo
Nickel (Ni)5.15 eV241 nmNo
Platinum (Pt)5.65 eV220 nmNo
Derivation

Why it works

  1. Light arrives in packets, photons, each of energy E = hν.
  2. One photon is absorbed by one electron, all at once, so there is no delay however dim the light.
  3. The electron must spend at least the work function φ to leave the metal.
  4. The fastest electrons keep the rest: K max = hν - φ. This is Einstein's photoelectric equation.
  5. A retarding voltage V₀ stops even these, so eV₀ = K max. When hν = φ, K max = 0: that frequency is the threshold ν₀ = φ / h.
Questions

Work function of metals: quick answers

What is the work function?

The minimum energy an electron needs to escape from the surface of a metal. It is written φ and usually given in electron volts; 1 eV = 1.6 × 10⁻¹⁹ J.

Which metal has the lowest work function?

Of the NCERT table, caesium, at 2.14 eV. That is why caesium and potassium are used in photocells that work with visible light.

Does the work function depend on the light?

No. It is a property of the metal and its surface. The light decides hν; the metal decides φ; the difference is K max.

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