question_answer

Derive the expression for the radius of the \[\mathbf{nth}\] orbit of hydrogen atom using Bohr's postulates. Show graphically the (nature of) variation of the radius of orbit with the principal quantum number, n.

Answer:

                For a circular orbit of the electron, \[\frac{m{{\upsilon }^{2}}}{r}=\frac{kZe.e}{{{r}^{2}}}=\frac{kZ{{e}^{2}}}{{{r}^{2}}}\]                 or            \[m{{\upsilon }^{2}}=\frac{kZ{{e}^{2}}}{r}\]                 or            \[r=\frac{kZ{{e}^{2}}}{m{{\upsilon }^{2}}}\]                                         ? (i) Using Bohr?s quantisation condition for angular momentum, \[L=m\upsilon r=\frac{nh}{2\pi }\] or                            \[r=\frac{nh}{2\pi m\upsilon }\]                                ?(ii) From (i) and (ii), \[\frac{kZ{{e}^{2}}}{m{{\upsilon }^{2}}}=\frac{nh}{2\pi m\upsilon }\] or            \[\upsilon =\frac{2\pi kZ{{e}^{2}}}{nh}\] \[\therefore \]  \[r=\frac{nh}{2\pi m}.\frac{nh}{2\pi kZ{{e}^{2}}}\] \[=\frac{{{n}^{2}}{{h}^{2}}}{4{{\pi }^{2}}mkZ{{e}^{2}}}\] As \[r\propto {{n}^{2}},\]the graph of r versus n is a parabola as shown in Fig.