JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Modern Physics

A hydrogen atom changes its state from

n = 3

n = 2

. Due to recoil, the percentage change in the wave length of emitted light is approximately

1 \times 1 0^{- n}

. The value of

n

is ____ .
[Given Rhc

= 13.6 eV

, hc

= 1242 eV nm

h = 6.6 \times 1 0^{- 34} J

s, mass of the hydrogen atom

= 1.6 \times 1 0^{- 27} kg

]

Answer: 7

Solution

Photon energy for

n = 3 \to 2

Δ E = 13.6 (\frac{1}{2 ^{2}} - \frac{1}{3 ^{2}}) = 13.6 \times \frac{5}{36} \approx 1.9 eV .

Because the atom recoils, the emitted wavelength shifts by the fractional amount

\frac{Δ λ}{λ} \approx \frac{Δ E}{2 M c ^{2}} = \frac{1.9 \times 1.6 \times 1 0 ^{- 19}}{2 ( 1.6 \times 1 0 ^{- 27} ) ( 9 \times 1 0 ^{16} )} \approx 1 \times 1 0^{- 9} .

As a percentage this is

\approx 1 \times 1 0^{- 7} %

, i.e.

1 \times 1 0^{- n}

with

n = 7

.
Answer: 7.

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Integer answer