Consider a hydrogen-like ionized atom with atomic number with a single electron. In the emission spectrum of this atom, the photon emitted in the 2 to 1 transition has energy 74.8 higher than the photon emitted in the 3 to 2 transition. The ionization energy of the hydrogen atom is 13.6. The value of is __________.

Answer

`3.00`

Explanation

`DeltaE_(2 rarr 1) = 13.6 xx Z^(2) [ 1 - (1)/(4) ] = 13.6 xx Z^(2) [ (3)/(4)]`
`Delta E_(3 rarr 2 ) = 13.6 xx Z^(2) [(1)/(4)- (1)/(9)] = 13.6 xx Z^(2) [ (5)/(36)]`
`Delta E_(2 rarr 1 ) = Delta E_(3-2) + 74 .8 `
`13.6 xx Z^(2) [ (3)/(4) ] = 13.6 xx Z^(2) [(5)/(36)] + 74.8`
`13. 6 xx Z^(2) [ (3)/(4) - (5)/(36) ] = 74. 8 `
`Z^(2) =9`
`Z = +3 `