Question

Compare the energy of recoil of an atom when it emits an X-ray photon (λ = 1 Å) to that when it emits a photon of visible light of wavelength 5000 Å
1. 1 : 5000
2. 5000 : 1
3. \(\sqrt{5000} : 1\)
4. (5000)² : 1

Answer 1

Correct Answer - Option 2 : 5000 : 1

Concept:

Recoil effect:

• Whenever light energy particle is released from the body at rest, releasing body pushed backwards called recoil effect.
• This is true for the absorption of high energy particle also.
• Recoil happens to conserve the linear momentum.

When an atom or molecule emits a quantum of energy E, the momentum will be E/c,

P = mV_R = - E/c 

where c is the speed of light in m/s2

V_R = recoil velocity and the negative sign shows its direction is opposite

Recoil energy: 

• In a transition of a nucleus from a higher to a lower energy state with emission of gamma rays, the nucleus to recoil.
• This takes energy from the emitted gamma rays.

Recoil energy is given by 

\(E\ = \ \frac{hc}{λ}\ = \ \frac{p^2}{2m} \)

Calculation:

Given that,

Wavelength of X-ray photon λ₁ = 1 Å

Wavelength of visible light λ₂  = 5000 Å

So, using the relation \(E\ = \ \frac{hc}{λ}\ \)

\(E_1\ = \ \frac{hc}{1}\ \ = hc\)         ......(1)

\(E_2\ = \ \frac{hc}{5000}\ \)           .......(2)

Dividing the equation (1) and (2)

\(\Rightarrow \frac{E_1}{E_2} \ = \ \frac{5000}{1}\)

Hence, recoil energy will be in the ratio of  5000 :1.