Question
It is found experimentally that $13.6 eV$ energy is required to separate a hydrogen atom into a proton and an electron. Compute the orbital radius and the velocity of the electron in a hydrogen atom.
✓
Answer
Total energy of the electron in hydrogen atom is $-13.6 eV =$ $-13.6 \times 1.6 \times 10^{-19} J =-2.2 \times 10^{-18} J$. Thus from Eq. (12.4), we have
$
E=-\frac{e^2}{8 \pi \varepsilon_0 r}=-2.2 \times 10^{-18} J
$
This gives the orbital radius
$
\begin{aligned}
r & =-\frac{e^2}{8 \pi \varepsilon_0 E}=-\frac{\left(9 \times 10^9 N m ^2 / C ^2\right)\left(1.6 \times 10^{-19} C \right)^2}{(2)\left(-2.2 \times 10^{-18} J \right)} \\
& =5.3 \times 10^{-11} m .
\end{aligned}
$
The velocity of the revolving electron can be computed from Eq. (12.3) with $m=9.1 \times 10^{-31} kg$,
$
v=\frac{e}{\sqrt{4 \pi \varepsilon_0 m r}}=2.2 \times 10^6 m / s
$
$
E=-\frac{e^2}{8 \pi \varepsilon_0 r}=-2.2 \times 10^{-18} J
$
This gives the orbital radius
$
\begin{aligned}
r & =-\frac{e^2}{8 \pi \varepsilon_0 E}=-\frac{\left(9 \times 10^9 N m ^2 / C ^2\right)\left(1.6 \times 10^{-19} C \right)^2}{(2)\left(-2.2 \times 10^{-18} J \right)} \\
& =5.3 \times 10^{-11} m .
\end{aligned}
$
The velocity of the revolving electron can be computed from Eq. (12.3) with $m=9.1 \times 10^{-31} kg$,
$
v=\frac{e}{\sqrt{4 \pi \varepsilon_0 m r}}=2.2 \times 10^6 m / s
$
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