Question

The magnitude of acceleration of the electron in the nth orbit of hydrogen atom is $a _H$ and that of singly ionised helium atom is $a_ {He}$ . The ratio of $a_ H: a_ {He}$ is

• A. 1 : 8
• B. 1 : 4
• C. 1 : 2
• D. dependent on n

Answer 1

Answer: (A)

In a hydrogen ion, velocity and radius of nth orbit are
$v_{n}=C_{1}.\left(\frac{Z}{n}\right)$ and $r_{n}=C_{2}.\left(\frac{n^{2}}{Z}\right)$
where, $C_1$ and $C_2$ are constants.
So, acceleration of electron in nth orbit is
$a_{n}=\frac{v^{2}_{n}}{r_{n}}=\frac{\frac{C^{2}_{1}}{C_{2}}.\left(\frac{Z^{2}}{n^{2}}\right)}{\left(\frac{n^{2}}{Z}\right)}$
$\Rightarrow a_{n }\alpha \frac{Z^{3}}{n^{4}}$
So, ratio of magnitude of acceleration of electron in nth orbit of hydrogen atom and that of singly ionised helium atom is given by
$\frac{a_{H}}{a_{\text{He}}}=\frac{\left(\frac{Z^{3}_{H}}{n^{4}}\right)}{\left(\frac{Z^{3}_{He}}{n^{4}}\right)}=\frac{Z^{3}_{H}}{Z^{3}_{He}} $
where, $Z$ = atomic number
$\Rightarrow Z_{H}=1$ and $Z_{\text{He}}=2$
$\Rightarrow \frac{a_{H}}{a_{\text{He}}}=\left(\frac{1}{2}\right)^{3}=\frac{1}{8}$