Modern Physics 1 Question 50

46. A hydrogen-like atom (described by the Bohrs model) is observed to emit six wavelengths, originating from all possible transitions between a group of levels. These levels have energies between 0.85eV and 0.544eV (including both these values).

(2002,5 M)

(a) Find the atomic number of the atom.

(b) Calculate the smallest wavelength emitted in these transitions.

(Take hc=1240eVnm, ground state energy of hydrogen atom =13.6eV)

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Solution:

  1. (a) Total 6 lines are emitted. Therefore,

n(n1)2=6 or n=4

So, transition is taking place between mth  energy state and (m+3)th  energy state.

Em=0.85eV or 13.6z2m2=0.85 or zm=0.25

Similarly, Em+3=0.544eV

or 13.6z2(m+3)2=0.544

or z(m+3)=0.2

Solving Eqs. (i) and (ii) for z and m, we get

m=12 and z=3

(b) Smallest wavelength corresponds to maximum difference of energies which is obviously Em+3Em

λmin=hcΔEmax=12400.306=4052.3nm.

ΔEmax=0.544(0.85)=0.306eV



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