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) \[n,\frac{1}{{{n}^{2}}},\frac{1}{{{n}^{2}}}\]
Correct Answer: A
Solution :
According to Bohrs 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 \[{{v}_{n}}\alpha \frac{1}{n}\] (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 \[{{E}_{n}}\,\,\alpha \,\,\frac{1}{{{n}^{2}}}\] (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 \[{{a}_{0}}=\frac{{{h}^{2}}{{\varepsilon }_{0}}}{\pi me}=5.29\times {{10}^{-11}}m,\], is called Bohrs radius, or \[{{r}_{n}}\,\,\alpha \,\,\,{{n}^{2}}\]
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