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Q.
A hydrogen atom initially in the ground level absorbs a photon, which excites it to the $n=4$ level. Determine the wavelength and frequency of photon.
Atoms
Solution:
For ground state $n_1 = 1$ to $n_2 = 4$
Energy absorbed by photon : $E = E_2 − E_1$
$= + 13.6 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \times 1.6 \times 10^{-16} J$
$= 13.6 \left(\frac{1}{1} -\frac{1}{4^2}\right) \times 1.6 \times 10^{-19}$
$= 13.6 \times 1.6 \times 10^{-19}\left(\frac{15}{16}\right)$
$= 20.4\times 10^{-19}$
or $E = h \nu = 20.4\times 10^{-19}$
Frequency $\nu = \frac{20.4 \times 10^{-19}}{h} = \frac{20.4\times 10^{-19}}{6.63\times 10^{-34}}$
$= 3.076\times 10^{15}$
$ = 3.1 \times 10^{15}\,Hz$
Wavelength of photon $\lambda = \frac{c}{\nu}$
$= \frac{3\times 10^{8}}{3.076\times 10^{15}}$
$ = 9.74\times 10^{-8}\,m$
Thus, the wavelength is $9.7\times 10^{-8}\,m$ and frequency is $3.1\times 10^{15} Hz$