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Question: A double charged lithium atom is equivalent to hydrogen whose atomic number is 3. The wavelength of ...

A double charged lithium atom is equivalent to hydrogen whose atomic number is 3. The wavelength of required radiation for emitting electron from first to third Bohr orbit in Li++Li^{+ +} will be (Ionisation energy of hydrogen atom is 13.6 eV)

A

182.51 Å

B

177.17 Å

C

142.25 Å

D

113.74 Å

Answer

113.74 Å

Explanation

Solution

Energy of a electron in nth orbit of a hydrogen like atom is given by En=−13.6Z2n2eVE_{n} = - 13.6\frac{Z^{2}}{n^{2}}eV, and Z = 3 for Li

Required energy for said transition

ΔE=E3−E1=13.6Z2(112−132)=13.6×32[89]\Delta E = E_{3} - E_{1} = 13.6Z^{2}\left( \frac{1}{1^{2}} - \frac{1}{3^{2}} \right) = 13.6 \times 3^{2}\left\lbrack \frac{8}{9} \right\rbrack

=108.8eV=108.8×1.6×10−19J= 108.8eV = 108.8 \times 1.6 \times 10^{- 19}J

Now using ΔE=hcλ\Delta E = \frac{hc}{\lambda} ⇒ λ=hcΔE\lambda = \frac{hc}{\Delta E}

⇒ λ=6.6×10−34×3×108108.8×1.6×10−19=0.11374×10−7m\lambda = \frac{6.6 \times 10^{- 34} \times 3 \times 10^{8}}{108.8 \times 1.6 \times 10^{- 19}} = 0.11374 \times 10^{- 7}m

⇒ λ=113.74A˚\lambda = 113.74Å