Transition between three energy energy levels in a particular atom give rise to three Spectral line of wavelength , in increasing magnitudes. `lambda_(1), lambda_(2)` and `lambda_(3)`. Which one of the following equations correctly relates `lambda_(1), lambda_(2)` and `lambda_(3)`?

Options

A. `lambda_(1) = lambda_(2) - lambda_(3)`
B. `lambda_(1) = lambda_(3) - lambda_(2)`
C. `(1)/(lambda_(1)) = (1)/(lambda_(2)) + (1)/( lambda_(3))`
D. `(1)/(lambda_(1)) = (1)/(lambda_(3)) + (1)/( lambda_(2))`

Answer

C

Explanation

Let the three energy levels be `E_(1), E_(2), and E_(3)`. The wavelength `lambda_(1), lambda_(2), and lambda_(3)` of the spectral lines corresponding to the three energy transition are depicted as shown in figure.

`E = h f = (h)/(lambda) or E prop (1)/(lambda)`
Thus, for the three wavelength , we have
`{(E_(3) - E_(2) = (h)/(lambda_(3)) ...(i)) , (E_(2) - E_(1) = (h)/(lambda_(2)) ...(ii)) , (E_(3) - E_(1) = (h)/(lambda_(1)) ...(iii)):}`
Now, `E_(3) - E_(1) = (E_(3) - E_(2)) + (E_(2) - E_(1))`
`implies (h)/(lambda_(1)) = (h)/(lambda_(3)) + (h)/(lambda_(2)) implies (1)/(lambda_(1)) = (1)/(lambda_(2)) + (1)/(lambda_(3))`