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
$a_n=\frac{v_n^2}{r_n}=\frac{\frac{C_1^2}{C_2}.\left(\frac{Z^2}{n^2}\right)}{\left(\frac{n^2}{Z}\right)}$
$\Rightarrow a_n\alpha \frac{Z^3}{n^4}$
So, ratio of magnitude of acceleration of electron in nth orbit of hydrogen atom and that of singly ionised helium atom is given by
$\frac{a_H}{a_{\text{He}}}=\frac{\left(\frac{Z_H^3}{n^4}\right)}{\left(\frac{Z_{He}^3}{n^4}\right)}=\frac{Z_H^3}{Z_{He}^3}$
where, $Z$ = atomic number
$\Rightarrow Z_H=1$ and $Z_{\text{He}}=2$
$\Rightarrow \frac{a_H}{a_{\text{He}}}={\left(\frac{1}{2}\right)}^3=\frac{1}{8}$