Question.
What is the energy in joules, required to shift the electron of the hydrogen atom from the first Bohr orbit to the fifth Bohr orbit and what is the wavelength of the light emitted when the electron returns to the ground state?
The ground state electron energy is $-2.18 \times 10^{-11}$ ergs.
What is the energy in joules, required to shift the electron of the hydrogen atom from the first Bohr orbit to the fifth Bohr orbit and what is the wavelength of the light emitted when the electron returns to the ground state?
The ground state electron energy is $-2.18 \times 10^{-11}$ ergs.
Solution:
Energy $(E)$ of the $\mathrm{n}^{\text {th }}$ Bohr orbit of an atom is given by,
$E_{n}=\frac{-\left(2.18 \times 10^{-18}\right) \mathrm{Z}^{2}}{n^{2}}$
Where,
Z = atomic number of the atom
Ground state energy $=-2.18 \times 10^{-11}$ ergs
$=-2.18 \times 10^{-11} \times 10^{-7} \mathrm{~J}$
$=-2.18 \times 10^{-18} \mathrm{~J}$
Energy required to shift the electron from n = 1 to n = 5 is given as:
$\Delta E=E_{5}-E_{1}$
$=\frac{-\left(2.18 \times 10^{-18}\right)(1)^{2}}{(5)^{2}}-\left(-2.18 \times 10^{-18}\right)$
$=\left(2.18 \times 10^{-18}\right)\left[1-\frac{1}{25}\right]$
$=\left(2.18 \times 10^{-18}\right)\left(\frac{24}{25}\right)=2.0928 \times 10^{-18} \mathrm{~J}$
Wavelength of emitted light $=\frac{h c}{E}$
$=\frac{\left(6.626 \times 10^{-34}\right)\left(3 \times 10^{8}\right)}{\left(2.0928 \times 10^{-18}\right)}$
$=9.498 \times 10^{-8} \mathrm{~m}$
Energy $(E)$ of the $\mathrm{n}^{\text {th }}$ Bohr orbit of an atom is given by,
$E_{n}=\frac{-\left(2.18 \times 10^{-18}\right) \mathrm{Z}^{2}}{n^{2}}$
Where,
Z = atomic number of the atom
Ground state energy $=-2.18 \times 10^{-11}$ ergs
$=-2.18 \times 10^{-11} \times 10^{-7} \mathrm{~J}$
$=-2.18 \times 10^{-18} \mathrm{~J}$
Energy required to shift the electron from n = 1 to n = 5 is given as:
$\Delta E=E_{5}-E_{1}$
$=\frac{-\left(2.18 \times 10^{-18}\right)(1)^{2}}{(5)^{2}}-\left(-2.18 \times 10^{-18}\right)$
$=\left(2.18 \times 10^{-18}\right)\left[1-\frac{1}{25}\right]$
$=\left(2.18 \times 10^{-18}\right)\left(\frac{24}{25}\right)=2.0928 \times 10^{-18} \mathrm{~J}$
Wavelength of emitted light $=\frac{h c}{E}$
$=\frac{\left(6.626 \times 10^{-34}\right)\left(3 \times 10^{8}\right)}{\left(2.0928 \times 10^{-18}\right)}$
$=9.498 \times 10^{-8} \mathrm{~m}$