A free hydrogen atom after absorbing a photon of wavelength λa gets excited from the state n = 1 to the state n = 4. Immediately after that the electron jumps to n = m state by emitting a photon of wavelength λe. Let the change in momentum of atom due to the absorption and the emission are pa and pe, respectively. If , Which of the option(s) is/are correct ?[Use hc = 1242 eV nm; 1 nm = 10–9 m, h and c are Planck's constant and speed of light, respectively]

Engineering

Physics

Bohrs Model of Atom

Question

A free hydrogen atom after absorbing a photon of wavelength λ_a gets excited from the state n = 1 to the state n = 4. Immediately after that the electron jumps to n = m state by emitting a photon of wavelength λ_e. Let the change in momentum of atom due to the absorption and the emission are $∆$ p_a and $∆$ p_e, respectively. If $λ_{a} / λ_{e} = \frac{1}{5}$ , Which of the option(s) is/are correct ?

[Use h_c = 1242 eV nm; 1 nm = 10^–9 m, h and c are Planck's constant and speed of light, respectively]

λ_e = 418nm

${Δp}_{a} / {Δp}_{e} = \frac{1}{2}$

m = 2

The ratio of kinetic energy of the electron in the state n = m to the state n = 1 is $\frac{1}{4}$

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$\frac{hc}{λ_{a}} = 13 .6 [\frac{1}{2} - \frac{1}{4^{2}}]$ .........................(i)

$\frac{hc}{λ_{e}} = 13 .6 [\frac{1}{m^{2}} - \frac{1}{4^{2}}]$ ................................(ii)

(ii) and (i), we get

$\frac{λ_{a}}{λ_{e}} = \frac{[\frac{1}{m^{2}} - \frac{1}{16}]}{[1 - \frac{1}{16}]} = \frac{1}{5}$

$\Rightarrow \frac{1}{m^{2}} - \frac{1}{16} = \frac{15}{16} \times \frac{1}{5} \Rightarrow \frac{1}{m^{2}} - \frac{1}{16} = \frac{3}{16}$

$\Rightarrow \frac{1}{m^{2}} = \frac{3}{16} + \frac{1}{16} \Rightarrow m = 2$

from (ii)

$\frac{hc}{λ_{e}} = 13 .6 [\frac{1}{2^{2}} - \frac{1}{4^{2}}] = 13 .6 \times \frac{3}{16} eV \Rightarrow λ_{e} = \frac{12400 \times 16}{13 .6 \times 3} Å$

⇒ λ_e ≈ 4862Å

we have
${KE}_{n} \propto \frac{z^{2}}{n^{2}}$

$\Rightarrow \frac{{KE}_{2}}{{KE}_{1}} = \frac{1}{4}$

${ΔP}_{a} = \frac{h}{λ_{a}}$

${ΔP}_{e} = \frac{h}{λ_{e}}$