Question

Using einstiens photoelectric equation to find the expressions for work function in terms of l,n and mass m of electron

Answer 1

In photoelectric effect, energy from incident photon is absorbed by an electron in the valency (conduction) band. It excites from energy state n to n+1.

Work function = h f0 = energy gap
      f0 = threshold frequency of the electron in the atom.

Kinetic energy KE of electron emitted = KE = h (f - f0)
      f = frequency of the photon incident on the atom
h f0 = energy gap between energy states n and n+1

h = Planck's constant
K = 1/(4πε) = 9 * 10⁹ units,  Z = atomic number, e = electron charge
Radius of electron in n th energy state = R = n² h² / (4π² m K e² Z)
Velocity of electron in Hydrogen like atom = v
Angular momentum = m v R = n (h/2π) 
     => v = n h /(2π m R) = 2π K e² Z / (n h)

(potential energy of electron is twice the kinetic energy and is negative)
Total energy of the electron in n th orbit = - 1/2 m v² 
      = - 2π² m K² Z² e⁴ / (n² h²)

Work function equals the total energy of the electron.
W = 2π² m K² Z² e⁴ / (n² h²)