2.28 The Photoelectric Equation

• Since energy is always conserved, the energy of an incident photon is equal to:

The work function + the maximum kinetic energy of the photoelectron

• The energy within a photon is equal to hf
• This energy is transferred to the electron to release it from a material (the work function) and the remaining amount is given as kinetic energy to the emitted photoelectron
• This equation is known as the photoelectric equation:

E = hf = Φ + ½ mv²_max

•  Where:
  • h = Planck's constant (J s)
  • f = the frequency of the incident radiation (Hz)
  • Φ = the work function of the material (J)
  • ½ mv²_max= KE_max = the maximum kinetic energy of the photoelectrons (J)

• This equation demonstrates:
  • If the incident photons do not have a high enough frequency and energy to overcome the work function (Φ), then no electrons will be emitted
  • hf₀ = Φ, where f₀ = threshold frequency, photoelectric emission only just occurs
  • KE_max depends only on the frequency of the incident photon, and not the intensity of the radiation
  • The majority of photoelectrons will have kinetic energies less than KE_max

Work Function

• The work function Φ, or threshold energy, of a material, is defined as:

The minimum energy required to release a photoelectron from the surface of a metal

• Consider the electrons in a metal as trapped inside an ‘energy well’ where the energy between the surface and the top of the well is equal to the work function Φ
• A single electron absorbs one photon
• Therefore, an electron can only escape from the surface of the metal if it absorbs a photon which has an energy equal to Φ or higher

In the photoelectric effect, a single photon may cause a surface electron to be released if it has sufficient energy

Graphical Representation of Work Function

• The photoelectric equation can be rearranged into the straight line equation:

y = mx + c

• Comparing this to the photoelectric equation:

KE_max = hf - Φ

•  A graph of maximum kinetic energy KE_max against frequency f can be obtained

• The key elements of the graph:
  • The work function Φ is the y-intercept
  • The threshold frequency f₀ is the x-intercept
  • The gradient is equal to Planck's constant h
  • There are no electrons emitted below the threshold frequency f₀

Threshold Frequency

• The threshold frequency is defined as:

The minimum frequency of incident electromagnetic radiation required to remove a photoelectron from the surface of a metal

Step 1: Write out the photoelectric equation and rearrange to fit the equation of a straight line

E = hf = Φ + ½ mv²_max         →    KE_max = hf - Φ

y = mx + c

 Step 2: Identify the threshold frequency from the x-axis of the graph

When E_k = 0, f = f₀

Therefore, the threshold frequency is f₀ = 4 × 10¹⁴ Hz

Step 3: Calculate the work function

From the graph at f₀, ½ mv_max² = 0

Φ = hf₀ = (6.63 × 10^-34) × (4 × 10¹⁴) = 2.652 × 10^-19 J

Step 4: Convert the work function into eV

1 eV = 1.6 × 10^-19 J                 J → eV: divide by 1.6 × 10^-19