First slide

Hydrogen -like ion

Question

Consider 3rd orbit of ${He}^{+}$ (Helium), using non-relativistic approach, the speed of electron in this orbit will be $[given K = 9 x 10^{9}$ constant, Z = 2 and h (Planck's Constant) $= 6.6 \times 10^{- 34} J s]$

Moderate

A

$0.73 \times 10^{6} m / s$
B

$3.0 \times 10^{8} m / s$
C

$2.92 \times 10^{6} m / s$
D

$1.46 \times 10^{6} m / s$

Solution

Energy of electron in $H e^{+}$ 3rd orbit
$E_{3} = - 13.6 \times \frac{Z^{2}}{n^{2}} eV = - 13.6 \times \frac{4}{9} eV$
$= - 13.6 \times \frac{4}{9} \times 1.6 \times 10^{- 19} J ≃ 9.7 \times 10^{- 19} J$
As per Bohr's model,
Kinetic energy of electron in the 3rd orbit = - $E_{3}$
$∴ 9.7 \times 10^{- 19} = \frac{1}{2} m_{e} v^{2}$
$v = \sqrt{\frac{2 \times 9.7 \times 10^{- 19}}{9.1 \times 10^{- 31}}} = 1.46 \times 10^{6} m s^{- 1}$

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