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Derive an expression for the radius of n^(th) Bohr's orbit of hydrogen atom hence write the expression for the radius of first orbit of hydrogen atom. |
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Answer» Solution :Consider a ATOM with effective nuclear charge + Ze. LET an electron revolves around the nucleus with speed v in the orbit of radius r as shown in figure. The necessary centripetal force on electron is provided by the electrostatic force between the electron and the nucleus. Therefore we have, centripetal force= electrostatic force `(mv^(2))/(r )= (1)/(4PI epsi_(0))(Z e.e)/(r^(2))` `mv^(2)r= (Ze^(2))/(4pi epsi_(0))` ....(1) From Bohr.s angular momentum quantization rule, `mvr= (nh)/(2pi)` `m^(2)v^(2)r^(2)= (n^(2)h^(2))/(4pi^(2))` ....(2) Dividing equation (2) by equation (1) we have, `(m^(2)v^(2)r^(2))/(mv^(2)r)= (n^(2)h^(2))/(4pi^(2)) XX (4pi epsi_(0))/(Ze^(2))` `mr= (n^(2)h^(2) epsi_(0))/(pi Ze^(2))rArr r= (n^(2)h^(2) epsi_(0))/(pi m Ze^(2))` For `n^(th)` orbit, `r_(n)= (n^(2)h^(2) epsi_(0))/(pi mZe^(2))` For H atom Z=1 and for `n^(th)` orbit, `r_(n)= (n^(2)h^(2)epsi_(0))/(pi m e^(2))` 
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