What is the work function of the metal if the light of wavelength 4000 Å generates photoelectrons
of velocity 6 × 10⁵ ms–1 form it ?

(Mass of electron = 9 × 10^–31 kg Velocity of
light = 3×10⁸ ms–1 Planck’s constant = 6.626× 10^–34Js Charge of electron = 1.6 × 10^–19 JeV^–1)

To calculate the work function of the metal, we use the photoelectric equation that relates the energy of the incident photon to the work function and the kinetic energy of the emitted photoelectrons:

Total Energy (E) = Work Function (ϕ) + Kinetic Energy (KE)

Rearranging to find the work function:

ϕ = Total Energy − Kinetic Energy

Step 1: Calculate the Total Energy of the Photon

The energy of a photon is calculated using the formula:

Total Energy (E) = (h × c) / λ

• h = 6.626 × 10⁻³⁴ Js (Planck's constant)
• c = 3 × 10⁸ m/s (speed of light)
• λ = 4000 Å = 4000 × 10⁻¹⁰ m

Substituting the values:

E = (6.626 × 10⁻³⁴ Js) × (3 × 10⁸ m/s) / (4000 × 10⁻¹⁰ m)

After calculation:

E ≈ 4.9695 × 10⁻¹⁹ J

Step 2: Calculate the Kinetic Energy of the Emitted Photoelectrons

The kinetic energy (KE) is given by:

KE = (1/2) × m × v²

• m = 9 × 10⁻³¹ kg (mass of the electron)
• v = 6 × 10⁵ m/s (velocity of the photoelectrons)

Substituting the values:

KE = (1/2) × (9 × 10⁻³¹) × (6 × 10⁵)²

After calculation:

KE ≈ 1.62 × 10⁻¹⁹ J

Step 3: Calculate the Work Function

Now, substitute the values for total energy and kinetic energy into the formula for the work function:

ϕ = E − KE

ϕ = (4.9695 × 10⁻¹⁹ J) − (1.62 × 10⁻¹⁹ J)

After calculation:

ϕ ≈ 3.3495 × 10⁻¹⁹ J

Step 4: Convert the Work Function to Electron Volts

To convert the work function from joules to electron volts, use the conversion factor:

1 eV = 1.6 × 10⁻¹⁹ J

ϕ (in eV) = (3.3495 × 10⁻¹⁹ J) / (1.6 × 10⁻¹⁹ J/eV)

After calculation:

ϕ ≈ 2.1 eV

Final Answer

The work function of the metal is approximately:

ϕ ≈ 2.1 eV