The energy states A, B and C having energy `E_(A)ltE_(B)ltE_(C)`. If the transition `CtoB,BtoA` and `CtoA` have the corresponding wavelength `lamda_(1),lamda_(2)` and `lamda_(3)` espectively, then which of following relationship is obtained ?

Options

A. `lamda_(1)+lamda_(2)+lamda_(3)=0`
B. `lamda_(3)^(2)=lamda_(1)^(2)+lamda_(2)^(2)`
C. `lamda_(3)=(lamda_(1)lamda_(2))/(lamda_(1)+lamda_(2))`
D. `lamda_(3)=lamda_(1)+lamda_(2)`

Answer

C

Explanation

`E_(A)ltE_(B)ltE_(C)`

For transition `CtoB`
`E_(C)-E_(B)=(hc)/(lamda_(1))" ".......(1)`
For transition `BtoA`
`E_(B)-E_(A)=(hc)/(lamda_(2))" ".......(2)`
Adding equation (1) and (2),
`E_(C)-E_(A)=(hc)/(lamda_(1))-(hc)/(lamda_(2))" ".......(3)`
Now for transition `CtoA`.
`E_(C)-E_(A)=(hc)/(lamda_(3))" ".......(4)`
`:.` From equation (3) and (4),
`(hc)/(lamda_(3))=(hc)/(lamda_(1))+(hc)/(lamda_(2))`
`:.(1)/(lamda_(3))=(1)/(lamda_(1))+(1)/(lamda_(2))`
`:.lamda_(3)=(lamda_(1)lamda_(2))/(lamda_(1)+lamda_(2))`