Question

Calculate the energy in joules required to shift an electron from the first Bohr orbit to the fourth Bohr orbit. Also, find the wavelength of light emitted when the electron returns to the ground state?

Answer 1

Dear student,


The electron is moving from n = 1 to n = 4 orbit

=> $∆E=E_4-E_1$

The energy of electron in ground state is -2.18 x 10^-18 J

The energy of an electron is given by:

$$\begin{gathered} E_n=\frac{-2{\mathrm{\pi }}^{2}m{e}^4{\mathrm{z}}^2}{{\mathrm{n}}^2{\mathrm{h}}^2} \\ {E}_n=\frac{-2.18\times {10}^{-18}\left(z^2\right)}{n^2} \end{gathered}$$

Where, m is mass of electron,
z is atomic number of atom,
e is charge on electron
h is Planck's constant

$$\begin{gathered} Onsubstitutingthesevaluesweget, \\ {E}_n=\frac{-\left(2.18\times {10}^{-18}\right){\left(1\right)}^2}{\left(4^2\right)}-\left(-2.18\times {10}^{-18}\right) \\ =\left(2.18\times {10}^{-18}\right)\left[1-\frac{1}{16}\right] \\ =\left(2.18\times {10}^{-18}\right)\left(\frac{15}{16}\right) \\ =2.04375\times {10}^{-18}J \end{gathered}$$

The wavelength of the emitted light when the electron returns to ground state:

$$\begin{gathered} E=\frac{hc}{\lambda } \\ =>\lambda =\frac{hc}{E} \\ \lambda =\frac{(6.626\times {10}^{-34}J.s)x(3\times {10}^{8}m/s)}{2.04375\times {10}^{-18}J}=9.726\times {10}^{-8}m \end{gathered}$$

Regards