Question

(a) Using Bohr's postulates, obtain the expression for total energy of the electron in the $n^{th}$ orbit of hydrogen atom.
(b) What is the significance of negative sign in the expression for the energy?
(c) Draw the energy level diagram showing how the line spectra corresponding to Paschen series occur due to transition between energy levels.

Answer 1

(a) According to Bohr's postulates, in a hydrogen atom, a single electron revolves around a nucleus of charge$+e$. For an electron moving with a uniform speed in a circular orbit on a given radius, the centripetal force is provided by the coulomb force of attraction between the electron and the nucleus.

Therefore,

$\frac{m{v}^2}{r}=\frac{1\left(Ze\right)\left(e\right)}{4\pi {ϵ}_0{r}^2}$

$⟹$ $m{v}^2=\frac{1}{4\pi {ϵ}_0}\frac{Z{e}^2}{r}$ ....(1)

So,

Kinetic Energy, $K.E=\frac{1}{2}m{v}^2$

$K.E=\frac{1}{4\pi {ϵ}_0}\frac{Z{e}^2}{r}$

Potential Energy is given by, $P.E=\frac{1}{4\pi {ϵ}_0}\frac{\left(Ze\right)(-e)}{r}$

Therefore total energy is given by, $E=K.E+P.E$

$E=\frac{1}{4\pi {ϵ}_0}\frac{Z{e}^2}{2r}+(-\frac{1}{4\pi {ϵ}_0}\frac{Z{e}^2}{r})$

$E=-\frac{1}{4\pi {ϵ}_0}\frac{Z{e}^2}{2r}$

For, $n^{th}$ orbit, $E$ can be written as $E_n$,

$E_n=-\frac{1}{4\pi {ϵ}_0}\frac{Z{e}^2}{2r_n}$ ......(2)

Now, using Bohr's postulate for quantization of angular momentum, we have

$mvr=\frac{nh}{2\pi }$

$⟹$ $v=\frac{nh}{2\pi mr}$

Putting the values of $v$ in equation (1), we get

$\frac{m}{r}$ $(\frac{nh}{2\pi mr}{)}^2=\frac{1}{4\pi {ϵ}_0}\frac{Z{e}^2}{r^2}$

$=-\frac{m{Z}^2{e}^4}{8{ϵ}_0{h}^2{n}^2}$

$⟹$ $\frac{{ϵ}_0{h}^2{n}^2}{\pi mZ{e}^2}$

Now putting a]value of $r_n$ in equation (2), we get

$E_n=-\frac{1}{4\pi {ϵ}_0}\frac{Z{e}^2}{2\left(\frac{{ϵ}_0{h}^2{n}^2}{\pi m}\right)}$

$=-\frac{m{Z}^2{e}^4}{8{ϵ}_0{h}^2{n}^2}$

$⟹$ $E_n=-\frac{Z^{2}Rhc}{n^2}$, where $R=\frac{m{e}^4}{8{ϵ}_0^{2}c{h}^3}$

where R is the Rydberg constant.

(b) Negative sign shows that electron remain bound with the nucleus.

(c) If electron jumps from $n_i=4,5,6$ to $n_f=3$, the energy of the line spectra

$△E=\frac{m{e}^4}{8{ϵ}_0^2{h}^2}(\frac{1}{n_f^2}-\frac{1}{n_i^2})$