Question

Calculate the wavelength in angstrom of the photon that is emitted when an electron in Bohr orbit n=2 returns to the orbit n=1 in the hydrogen atom. The ionization potential of the ground state of the hydrogen atom is \[2.17\times {{10}^{-11}}ergato{{m}^{-1}}\].

Answer 1

Hint: Bohr produces a set of postulates to explain the hydrogen atom. We know the energy difference between two energy levels is equal to the energy of a photon and its equation is given as $\Delta E=h\nu $ , where $\Delta E$ is the energy difference between two energy levels, h is the Planck’s constant and $\nu $ is the frequency.


Complete step by step solution:
In the question, we have to find the difference between two energy levels, which give the wavelength of the photon that is emitted. This is explained by using a formula called Rydberg formula.
\[E=R(\dfrac{1}{n_{f}^{2}}-\dfrac{1}{n_{i}^{2}})\] Here, \[n_{f}^{{}}\] is the final energy level, that is\[n_{f}^{{}}=2\]. And \[n_{i}^{{}}\]is the initial energy level, that is \[n_{i}^{{}}=1\] and E is the energy level and R is the Rydberg constant. So, For 2 energy levels${{E}_{2}}$ and ${{E}_{2}}$ .
\[{{E}_{2}}-{{E}_{1}}=-2.17\times {{10}^{-11}}(\dfrac{1}{4}-\dfrac{1}{1})\]=\[2.17\times {{10}^{-11}}\dfrac{3}{4}\]
Here, \[n_{f}^{{}}\]is the final energy level, that is \[n_{f}^{{}}=2\]. And \[n_{i}^{{}}\]is the initial energy level, that is \[n_{i}^{{}}=1\]
\[{{E}_{2}}-{{E}_{1}} = \Delta E=hv=\dfrac{c}{\lambda }h\]
We can write the above equation in another manner too based on wavelength.
\[\lambda = \dfrac{hc}{\Delta E}\]
Where, \[\lambda \]is the wavelength, h is the Planck’s constant, c is the velocity of light and \[\Delta E\] difference in energy levels. We know, \[h=6.62\times {{10}^{-27}}erg\], \[c=3\times {{10}^{10}}cm\] and \[\Delta E\] is given as\[2.17\times {{10}^{-11}}\dfrac{3}{4}\].
\[\lambda =\dfrac{6.62\times {{10}^{-27}}erg\times 3\times {{10}^{10}}cm}{2.17\times {{10}^{-11}}\dfrac{3}{4}} = \dfrac{2.648}{2.17}{{10}^{-5}}cm\]
$=1.22\times { 10 }^{ -5 }cm = 1220\mathring { A } $

Additional Information:
Niels Bohr produces the atomic Hydrogen model. In it he describes, a positively charged nucleus consists of protons and neutrons and is surrounded by negatively charged electron clouds. The atom is held together by electrostatic forces between the positively charged nucleus and negatively charged surrounding, that is electrons. The structure of hydrogen in Bohr’s model has energy levels.

Note: In the question, wavelength in angstrom is asked. We are calculating wavelength in centimetres. We should always see to it that it is converted to angstrom units. 1 Armstrong =${{10}^{-8}}$ centimetres.