Question

According to the Bohr's theory of hydrogen atom, the speed of the electron, energy and the radius of its orbit vary with the principal quantum number $n$, respectively, as

• A. $\frac{1}{n}, \frac{1}{n^{2}}, n^{2}$
• B. $\frac{1}{n}, n^{2}, \frac{1}{n^{2}}$
• C. $n^{2}, \frac{1}{n^{2}}, n^{2}$
• D. $n, \frac{1}{n^{2}}, \frac{1}{n^{2}}$

Answer 1

Answer: (A)

According to Bohr's theory of hydrogen atom,
(i) The speed of the electron in the $nth$ orbit is
$v_{n}=\frac{1}{n} \frac{e^{2}}{4 \pi \varepsilon_{0}(h / 2 \pi)}$
or $v_{n} \propto \frac{1}{n}$
(ii) The energy of the electron in the nth orbit is
$E_{n}=-\frac{m e^{4}}{8 n^{2} \varepsilon_{0}^{2} h^{2}}=\frac{-13.6}{n^{2}} eV$
or $E_{n} \propto \frac{1}{n^{2}}$
(iii) The radius of the electron in the nth orbit is
$r_{n}=\frac{n^{2} h^{2} \varepsilon_{0}}{\pi m e^{2}}=n^{2} a_{0}$
where $a_{0}=\frac{h^{2} \varepsilon_{0}}{\text { \pi me }}=5.29 \times 10^{-11} m$, is called Bohr's radius,
or $r_{n} \propto n^{2}$