A stationary hydrogen atom of mass M emits a photon corresponding to the first line of Lyman series. If R is the Rydberg's constant, the speed of recoil of the atom after the emission of the photon is

Question

A stationary hydrogen atom of mass M emits a photon corresponding to the first line of Lyman series. If R is the Rydberg's constant, the speed of recoil of the atom after the emission of the photon is (A) (3 R h/4 M) (B) (R h/4 M) (C) (R h/2 M) (D) (R h/M)

Tardigrade Admin · Accepted Answer

From conservation of momentum overset arrow pH+ overset arrow pphoton=0 |( overset arrow p)H|=|( overset arrow p)photon|=(h/λ )=hR((1/12) - (1/22))=(3 R h/4) vH=(3 R h/4 M)

Q. A stationary hydrogen atom of mass $M$ emits a photon corresponding to the first line of Lyman series. If $R$ is the Rydberg's constant, the speed of recoil of the atom after the emission of the photon is

$\frac{3 R h}{4 M}$

$\frac{R h}{4 M}$

$\frac{R h}{2 M}$

$\frac{R h}{M}$

From conservation of momentum $p \to_{H} + p \to_{p h o t o n} = 0$
$∣ ∣ (p \to)_{H} ∣ ∣ = ∣ ∣ (p \to)_{p h o t o n} ∣ ∣ = \frac{h}{λ} = h R (\frac{1}{1 ^{2}} - \frac{1}{2 ^{2}}) = \frac{3 R h}{4}$
$v_{H} = \frac{3 R h}{4 M}$

Q. A stationary hydrogen atom of mass M emits a photon corresponding to the first line of Lyman series. If R is the Rydberg's constant, the speed of recoil of the atom after the emission of the photon is

4M3Rh​

4MRh​

2MRh​

MRh​