Question

The magnitude of acceleration of the electron in the ${{n}^{th}}$ orbit of hydrogen atom is ${{a}_{H}}$ and that of single ionized helium atom is ${{a}_{He}}$. The ratio ${{a}_{H}}:{{a}_{He}}$ is?
$\begin{aligned} & A)1:8 \\\ & B)1:4 \\\ & C)1:2 \\\ & D)\text{Dependent on n} \\\ \end{aligned}$

Answer 1

Hint: The formula for velocity of an electron in the ${{n}^{th}}$ orbit of an atom and the formula for the radius of the electron in the ${{n}^{th}}$ orbit of the atom will give a relation between the acceleration of the electron and the atomic number of the particular atom under consideration. The proportionality condition can be directly used to determine the ratio of the acceleration of the electron in the ${{n}^{th}}$ orbit of the given atoms.

Complete step by step solution:
Let us consider that ${{v}_{n}}$ is the velocity of the electron in the ${{n}^{th}}$ orbit of hydrogen and ${{r}_{n}}$ is the radius of the ${{n}^{th}}$ orbit. Now if $Z$ is the atomic number of a particular atom taken into consideration, then we know that,
${{v}_{n}}\alpha \dfrac{Z}{n}$ and also, ${{r}_{n}}\alpha \dfrac{{{n}^{2}}}{Z}$
So, now if we consider that ${{a}_{n}}$ is the acceleration of the electron in the ${{n}^{th}}$ orbit of hydrogen, then we have,
$\begin{aligned} & {{a}_{n}}=\dfrac{v_{n}^{2}}{r} \\\ & \Rightarrow {{a}_{n}}\alpha \dfrac{{{Z}^{3}}}{{{n}^{4}}} \\\ \end{aligned}$
Therefore,
$\begin{aligned} & \dfrac{{{a}_{H}}}{{{a}_{He}}}=\dfrac{Z_{H}^{3}}{{{n}^{4}}}\times \dfrac{{{n}^{4}}}{Z_{He}^{3}} \\\ & \Rightarrow \dfrac{{{a}_{H}}}{{{a}_{He}}}=\dfrac{Z_{H}^{3}}{Z_{He}^{3}} \\\ & \Rightarrow \dfrac{{{a}_{H}}}{{{a}_{He}}}=\dfrac{{{1}^{3}}}{{{2}^{3}}} \\\ & \Rightarrow \dfrac{{{a}_{H}}}{{{a}_{He}}}=\dfrac{1}{8} \\\ \end{aligned}$
So, the answer to the given question is option $A)1:8$.
Additional information: Students can also solve this question by elaborately stating the formula for the velocity and radius of an electron in a Bohr orbit, which might be quite time taking. However, in order to solve these types of questions in a very short period of time, it is always better to remember the relation between the quantities mentioned in the question.

Note: In the question, the electron is in the ${{n}^{th}}$ orbit of both the atoms and so we could cancel out $n$ during our calculations, but if the electron would have been in different orbits for both the cases then, the value of $n$ would have been different for both cases.