Ratio of speed of electrons in the ground state of hydrogen to speed of light in vac

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12.Atoms

normal

The ratio of the speed of the electrons in the ground state of hydrogen to the speed of light in vacuum is

$1 / 2$

$2 / 137$

$1 / 137$

$1 / 237$

Solution

Speed of electron in $\mathrm{n}^{\text {th }}$ orbit of hydrogen atom

$\mathrm{v}=\frac{\mathrm{e}^{2}}{2 \varepsilon_{0} \mathrm{nh}}$

In ground state $\mathrm{n}=1 \Rightarrow \frac{\mathrm{e}^{2}}{2 \varepsilon_{0} \mathrm{h}}$

$\Rightarrow \frac{\mathrm{v}}{\mathrm{c}}=\frac{\mathrm{e}^{2}}{2 \varepsilon_{0} \mathrm{ch}}=\frac{\left(1.6 \times 10^{-19}\right)^{2}}{2 \times 8.85 \times 10^{-12} \times 3 \times 10^{8} \times 6.6 \times 10^{-34}}$

$=\frac{1}{137}$

Standard 12

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