Question

To calculate the size of a hydrogen ion using the Bohr's model, we assume that its two electrons move in an orbit such that they are always on diametrically opposite sides of the nucleus. With each electron having the angular momentum $\hbar=h / 2 \pi$ and taking electron interaction into account the radius of the orbit in terms of the Bohr's radius of hydrogen atom

• A. $a_B$
• B. $\frac{4}{3} a_{B}$
• C. $\frac{2}{3} a_{B}$
• D. $\frac{3}{2} a_{B}$

Answer 1

Answer: (B)

Let the velocity of the electron be $v$.
Angular momentum of electron $mvr =\hbar$ (given)
Net electrostatic force acting on the electron
$F _{ e }=\frac{ e ^{2}}{4 \pi \epsilon_{0} r ^{2}}-\frac{ e ^{2}}{4 \pi \epsilon_{0}\left(2 r ^{2}\right)}=\frac{3}{4} \frac{ e ^{2}}{4 \pi \epsilon_{0} r ^{2}}$
Centrifugal force $F _{ c }=\frac{ mv ^{2}}{ r }=\frac{( mvr )^{2}}{ r \times mr ^{2}}=\frac{\hbar^{2}}{ mr ^{3}}$
But $F _{ c }= F _{ e }$
$\therefore \frac{\hbar^{2}}{ mr ^{3}}=\frac{3}{4} \frac{ e ^{2}}{4 \pi \epsilon_{0} r ^{2}}$
$\Longrightarrow r =\frac{4}{3} \frac{4 \pi \epsilon_{0} \hbar^{2}}{ me ^{2}} $
$=\frac{4}{3} a _{ B }$