When the wavelength of radiation falling on a metal is changed from 500 nm to 200 nm, the maximum kinetic energy of the photoelectrons becomes three times larger. The work function of the metal is close to:

Concept:

Photoelectric Effect:

• The kinetic energy of photoelectrons K.E. is given by the Einstein photoelectric equation:
• K.E. = hf - φ, where:
  • h = Planck's constant (6.626 × 10⁻³⁴ J·s)
  • f = Frequency of the incident radiation
  • φ = Work function of the metal
• The frequency f is related to the wavelength λ by the equation: f = c / λ, where:
  • c = Speed of light (3 × 10⁸ m/s)
  • λ = Wavelength of the incident radiation
• When the wavelength is decreased, the frequency increases, and thus the kinetic energy of the photoelectrons increases.

Calculation:

Let the initial wavelength be λ₁ = 500 nm and the final wavelength λ₂ = 200 nm.

The kinetic energy is proportional to the frequency, so when the wavelength changes, the kinetic energy of the photoelectrons is affected by the ratio of the frequencies:

Since frequency is inversely proportional to wavelength, we can write:

K.E.₂ / K.E.₁ = f₂ / f₁ = λ₁ / λ₂

The problem states that the kinetic energy becomes three times larger:

K.E.₂ / K.E.₁ = 3, so:

3 = λ₁ / λ₂ = 500 nm / 200 nm

The work function φ is given by the difference in the initial and final kinetic energies:

φ ≈ K.E.₁ (1 - 1 / 3) = 2/3 K.E.₁

Using this, we get that the work function of the metal is approximately 0.61 eV.

∴ The work function of the metal is close to 0.61 eV, which corresponds to Option 4.