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.

Explanation

For atom, `DeltaE = hv = (hc)/lambda` …(i)
The energy difference between
n = 1 and n = 4 is
`Delta E = E_4 - E_1 = - 13.6 (1/4^2 - 1/1^2) eV`
` = - 13.6 (1/16 - 1) eV`
` = -13.6 (- 15/16) eV`
`Delta E = 12.75 eV`
` = 12.75 xx 1.6 xx10^(-19)` ...(ii)
From Eqs. (i) and (ii), we get
`1.6 xx 10^(-19) xx 12.75 = (hc)/lambda`
Wavelength, `lambda = (hc)/(12.75 xx 1.6 xx 10^(-19))`
` = (6.6 xx 10^(-34) xx 3 xx 10^8)/(12.75 xx 1.6 xx 10^(-19))`
` = 0.97 xx 10^(-7) m`
As we know, frequency of a proton,
`v = c/lambda = (3xx10^8)/(0.97 xx 10^(-7))`
`= 3.09 xx 10^15 Hz`