Question

The energy of a silver atom with a vacancy in K shell is 25.31 keV, in L shell is 3.56 keV and in M shell is 0.530 keV higher than the energy of the atom with no vacancy. Find the frequency of K_α, K_β and L_α X-rays of silver.

(Use Planck constant h = 6.63 × 10^-34 Js= 4.14 × 10^-15 eVs, speed of light c = 3 × 10⁸ m/s.)

Answer 1

Given:-
Energy of electron in the K shell, E_k = 25.31 keV
Energy of electron in the L shell, E_L = 3.56 keV
Energy of electron in the M shell, E_M = 0.530 keV

Let f be the frequency of Kα X-ray and f₀ be the frequency of K_β X-ray.
Let f₁ be the frequency of L_α X-rays of silver.

∴ K_α = E_K − E_L = hf
Here, h = Planck constant
         f = frequency of `Kα` X-ray

`f = (E_K - E_L)/h`

`f = ((25.31 xx 3.56))/(6.63 xx 10^-34) xx 1.6 xx 10^-19 xx 10^3`

`f = (21.75 xx 10^3 xx 10^15)/6.67`

`f = 5.25 xx 10^18  "Hz"`

`K_β = E_K - E_M = hf_0`

`⇒ f_0 = (E_K - E_M)/h`

`⇒ f_0 = ((25.31 - 0.53))/(6.67 xx 10^-34) xx 10^3 xx 1.6 xx 10^-19`

`⇒ f_0 = 5.985 xx 10^18  "Hz"`

`K_L = E_L - E_M = hf_1`

`f_1 = (E_L - E_M)/h`

`f_1 = (3.56 - 0.530)/(6.63 xx 10^-34) xx 10^3 xx 1.6 xx 10^-19`

`f_1 = 7.32 xx 10^17  "Hz"`