It is found experimentally that \(13.6\mathit{eV}\) 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 \(-13.6\mathit{eV}= -13.6\times 1.6\times 10^{-19}J=-2.2\times 10^{-18}\mathit{J.}\)
Thus, we have
\(\frac{-e^2}{8\pi \varepsilon _0r}=-2.2\times 10^{-18}J\)
This gives the orbital radius
\(r=\frac{-e^2}{8\pi \varepsilon _0E}=\frac{-(9\times 10^9Nm^2/C^2)(1.6\times 10^{-19}C)^2}{(2)(-2.2\times 10^{-18}J)}\)
\(=5.3\times 10^{-11}m\)
The velocity of the revolving electron can be computed with \(m=9.1\times 10^{-31}\mathit{kg}\)
\(v=\frac e{\sqrt{4\pi \varepsilon _0\mathit{mv}}}=2.2\times 10^6m\text /s\)
Thus, we have
\(\frac{-e^2}{8\pi \varepsilon _0r}=-2.2\times 10^{-18}J\)
This gives the orbital radius
\(r=\frac{-e^2}{8\pi \varepsilon _0E}=\frac{-(9\times 10^9Nm^2/C^2)(1.6\times 10^{-19}C)^2}{(2)(-2.2\times 10^{-18}J)}\)
\(=5.3\times 10^{-11}m\)
The velocity of the revolving electron can be computed with \(m=9.1\times 10^{-31}\mathit{kg}\)
\(v=\frac e{\sqrt{4\pi \varepsilon _0\mathit{mv}}}=2.2\times 10^6m\text /s\)
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