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VITEEE 2010 Paper

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Question : 31 of 120

Marks: +1, -0

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

$\frac{1}{n}, \frac{1}{n^{2}}, n^{2}$
$\frac{1}{n}, n^{2}, \frac{1}{n^{2}}$
$n^{2}, \frac{1}{n^{2}}, n^{2}$
$n, \frac{1}{n^{2}}, \frac{1}{n^{2}}$

Solution:

According to Bohr's theory of hydrogen atom,

(i) The speed of the electron in the nth orbit

V_{n} = \frac{ \frac{1}{n} e^{2} }{ 4\pi \varepsilon_{0} \left( \frac{h}{2\pi} \right) }

or

v_{n} \propto \frac{1}{n}

(ii) The energy of the electron in the nth orbit

E_{n} = \frac{ m e^{4} }{ 8 n^{2} \varepsilon_{0}^{2} h^{2} } = -\frac{13.6}{ n^{2} } \,\mathrm{eV}

or

E_{n} \propto \frac{1}{n^{2}}

(iii) The radius of the electron in the nth orbit

r_{n} = \frac{ n^{2} h^{2} \varepsilon_{0} }{ \pi m e^{2} } = n^{2} a_{0}

or

r_{n} \propto n^{2}

where

a_{0} = \frac{ h^{2} \varepsilon_{0} }{ \pi m e } = 5.29 \times 10^{-11}

m

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