The ratio of speed of electrons in the first excited state of hydrogen atom to the speed of light in vacuum…

The ratio of speed of electrons in the first excited state of hydrogen atom to the speed of light in vacuum is [Given, Planck's constant $=6.625 \times 10^{-34} \mathrm{Js}$ and permittivity of free space is $8.85 \times 10^{-12} \mathrm{Fm}^{-1}$ ]
  1. $5 \times 10^{-3}$
  2. $7.3 \times 10^{-3}$
  3. $3.6 \times 10^{-3}$
  4. $36.5 \times 10^{-3}$

Solution

Velocity of electron in the nth orbit of an atom is given by $v=\frac{2 \pi k e^2}{h} \times \frac{Z}{n}$
where, $\frac{2 \pi k e^2}{h}$ is a constant quantity and $\mathrm{Z}$ is atomic number.
Hence, speed of electron in first excited state
(n = 2) is
$\begin{aligned}
v & =2.188 \times 10^6 \times \frac{1}{2} \quad(Z=1, \text { for hydrogen atom }) \\
& =1.094 \times 10^6 \mathrm{~ms}^{-1}
\end{aligned}$
Ratio of speed of electron in lst excited state to
$\begin{aligned}
\text {the speed of light in vacuum } & =\frac{1.094 \times 10^6}{3 \times 10^8} \\
& =0.364 \times 10^{-2} \\
& =3.6 \times 10^{-3}
\end{aligned}$

Asked in: AP EAMCET 2021 (25 Aug Shift 2)

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