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Atoms and Nuclei question

2021 · 27 Aug · Shift 2 · Q60
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Atoms and Nuclei question

2021 · 27 Aug · Shift 2 · Q60

JEE MainPhysicsAtoms and NucleiNumerical+4 / −1
X different wavelengths may be observed in the spectrum from a hydrogen sample if the atoms are exited to states with principal quantum number n = 6 ? The value of X is ‾\underline{\hspace{2cm}}​.
Numerical answer
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Correct answer: 15

  1. In a hydrogen atom, if electrons are excited up to the level with principal quantum number n=6n=6n=6, then spectral lines are produced by all possible downward transitions between the levels 1,2,3,4,5,6.1,2,3,4,5,6.1,2,3,4,5,6.

  2. Each distinct transition between two energy levels gives one distinct wavelength.

  3. So we need the number of possible pairs (ni,nf)(n_i,n_f)(ni​,nf​) such that ni>nf,n_i > n_f,ni​>nf​, with ni,nf∈{1,2,3,4,5,6}n_i,n_f \in \{1,2,3,4,5,6\}ni​,nf​∈{1,2,3,4,5,6}.

  4. The number of such transitions is the number of ways to choose 2 levels out of 6: X=(62)=6⋅52=15.X = {6 \choose 2} = \frac{6\cdot 5}{2} = 15.X=(26​)=26⋅5​=15.

  5. Therefore, the total number of different wavelengths is 15.\boxed{15}.15​.

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