22.1.4 Threshold Frequency

• The concept of a threshold frequency is required in order to explain why a low frequency source, such as a filament lamp, was unable to liberate any electrons in the gold leaf experiment
• The threshold frequency is defined as:

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

• The threshold wavelength, related to threshold frequency by the wave equation, is defined as:

The longest wavelength of incident electromagnetic radiation that would remove a photoelectron from the surface of a metal

• Threshold frequency and wavelength are properties of a material, and vary from metal to metal

Threshold frequencies and wavelengths for different metals

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

The threshold energy + the 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 gives the emitted photoelectron the remaining amount as kinetic energy
• This equation is known as the photoelectric equation:

E = hf = Φ + ½mv²_max

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

• This equation demonstrates:
  • If the incident photons do not have a high enough frequency (f) and energy to overcome the work function (Φ), then no electrons will be emitted
  • When hf₀ = Φ, where f₀ = threshold frequency, photoelectric emission only just occurs
  • E_kmax 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 E_kmax

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:

E_kmax = hf - Φ

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

Graph of maximum kinetic energy of photoelectrons against photon frequency

• 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

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

       straight line

E = hf = Φ + ½mv²_max         →    E_kmax = 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