The ratio for the speed of the electron in the 3rd orbit of \( \mathrm{He^{+}} \) to the speed of the electron in the 3rd orbit of the hydrogen atom will be
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 2 : 1
In the Bohr model of a hydrogen-like ion of atomic number \( Z \), the speed of the electron in the \( n \)th orbit is
\( \displaystyle v_n = \frac{Ze^{2}}{2\varepsilon_0 h n} \;\Rightarrow\; v_n \propto \frac{Z}{n} \)
Both electrons are in the same orbit, \( n = 3 \), so the ratio depends only on \( Z \): \( Z = 2 \) for \( \mathrm{He^{+}} \) and \( Z = 1 \) for hydrogen.
\( \displaystyle \frac{v_{\mathrm{He^{+}}}}{v_{\mathrm{H}}} = \frac{Z_{\mathrm{He^{+}}}}{Z_{\mathrm{H}}} = \frac{2}{1} = 2 : 1 \)