Question

An electron in a hydrogen atom makes a transition from the ground state, n=1, to the excited state, n=5. Calculate the energy in Joules, frequency in Hertz ( $ \dfrac{1}{s} $ ), and wavelength in nm of the photon?

Answer 1

Hint: To solve the given problem we will use the Rydberg’s equation to find the wave number. From the wavenumber we can find the wavelength. The frequency and wavelength can be related by using the equation $ v = \dfrac{c}{\lambda } $ . The energy in joules can be found out by using the equation $ E = hv $ .

Complete answer:
We are given an electron which excites from ground state to n=5(fourth excited state). The Rydberg’s equation for finding the wavelength can be given as:
 $ \dfrac{1}{\lambda } = R{Z^2}\left( {\dfrac{1}{{n_2^2}} - \dfrac{1}{{n_1^2}}} \right) $
In here, R = Rydberg’s constant which has the value of $ 1.097 \times {10^7}/m $
Z = atomic number
 $ {n_1} $ = Initial energy level
 $ {n_2} $ = Final energy level
The information given to us are: Z = 1 (hydrogen atom), $ {n_1} = $ 1 (ground state) and $ {n_2} = $ 5 (excited state)
Substituting the values in the equation we get, $ \dfrac{1}{\lambda } = (1.097 \times {10^7}){(1)^2}\left( {\dfrac{1}{{{5^2}}} - \dfrac{1}{{{1^2}}}} \right) $
 $ \dfrac{1}{\lambda } = 1.097 \times {10^7} \times \left( {\dfrac{1}{{25}} - 1} \right) $
 $ \dfrac{1}{\lambda } = 1.097 \times {10^7} \times \dfrac{{1 - 25}}{{25}} = - 1.097 \times {10^7} \times \dfrac{{24}}{{25}} $
 $ \dfrac{1}{\lambda } = - 1.053 \times {10^7}{m^{ - 1}} $
The negative charge indicates the energy being absorbed. The wavelength will be the inverse of the answer found.
Therefore, $ \lambda = \dfrac{1}{{1.053 \times {{10}^7}}} = 9.496 \times {10^{ - 8}}m = 94.96nm $
The frequency can be found out by using the formula $ v = \dfrac{c}{\lambda } $
Where, $ \lambda = $ wavelength of light and c is the velocity of light. Substituting the values to find the frequency.
 $ v = \dfrac{{3 \times {{10}^8}m/s}}{{94.96 \times {{10}^{ - 9.}}m}} = 3.157 \times {10^{15}}{s^{ - 1}} $
Hence this is the value of Frequency. The energy can be found out by using the equation $ E = hv $ . Where h = Planck's Constant.
Substituting the values we get, $ E = 6.626 \times {10^{ - 34}}J.s \times 3.157 \times {10^{15}}{s^{ - 1}} = 2.092 \times {10^{ - 19}}J $
Hence the required answer is, wavelength $ \lambda = 94.96nm $ , $ v = 3.157 \times {10^{15}}{s^{ - 1}} $ and the energy $ E = 2.092 \times {10^{ - 19}}J $

Note:
This problem can be easily solved using the Planck's equation and simple wavelength to frequency conversion formula. The energy is to be found in electron volt (eV), the energy in Joules can be divided by $ 1.6 \times {10^{ - 19}} $