Verified Solution • 2025–26 Session moderate

QJEEM18Ph72fb: JEE Main 2018 Physics - Class 12 JEEM18Ph72fb Solution: An electron from various excited states of hydrogen atom emit…

An electron from various excited states of hydrogen atom emit radiation to come to the ground state. Let λn{\lambda _n} , λg{\lambda _g} be the de Broglie wavelength of the electron in the nth state and the ground state respectively. Let Λn{\Lambda _n} be the wavelength of the emitted photon in the transition from the nth state to the ground state. For large n, (A, B are constants)

Multiple Choice Options

  • A Λn≈A+Bλn2{\Lambda _n} \approx A + {B \over {\lambda _n^2}}
  • B Λn≈A+Bλn{\Lambda _n} \approx A + B{\lambda _n}
  • C Λn2≈A+Bλn2\Lambda _n^2 \approx A + B\lambda _n^2
  • D Λn2≈λ\Lambda _n^2 \approx \lambda

Step-by-Step Verified Solution

We know,

Wavelength of emitted photon from n2 state to n1 state is

1λ{1 \over \lambda }   =  RZ2 (1n12−1n22)\left( {{1 \over {n_1^2}} - {1 \over {n_2^2}}} \right)

Here electron comes from nth state to ground state (n = 1),

then the wavelength of photon is ,

1Λn{1 \over {{\Lambda _n}}}  =  RZ2 (112−1n2)\left( {{1 \over {{1^2}}} - {1 \over {{n^2}}}} \right)

⇒\Rightarrow    \,\,\, Λ\Lambda n   =  1RZ2(1−1n2)−1{1 \over {R{Z^2}}}{\left( {1 - {1 \over {{n^2}}}} \right)^{ - 1}}

As n is very large, so using binomial theorem

Λ\Lambda n  =  1RZ2(1+1n2){1 \over {R{Z^2}}}\left( {1 + {1 \over {{n^2}}}} \right)

⇒\Rightarrow    \,\,\, Λ\Lambda n =  1RZ2+1RZ2(1n2){1 \over {R{Z^2}}} + {1 \over {R{Z^2}}}\left( {{1 \over {{n^2}}}} \right)

We know,

λ\lambda n =  2πrn{{2\pi r} \over n}

=  2 π\pi (n2h24π2mZC2)×1n{\left( {{{{n^2}{h^2}} \over {4{\pi ^2}mZ{C^2}}}} \right)\times{{1 \over n}}}

∴   \therefore\,\,\, λ\lambda n ∝\propto n

⇒\Rightarrow    \,\,\, n  =  K λ\lambda n

∴   \therefore\,\,\, Λ\Lambda n  =  1RZ2+1RZ2(1(K λn)2){1 \over {R{Z^2}}} + {1 \over {R{Z^2}}}\left( {{1 \over {{{\left( {K\,{\lambda _n}} \right)}^2}}}} \right)

Let A  =   1RZ2{1 \over {R{Z^2}}}  and   B  =  1K2RZ2{1 \over {{K^2}R{Z^2}}}

⇒\Rightarrow    \,\,\, Λ\Lambda n  =  A + Bλn2{B \over {\lambda _n^2}}

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