What hydrogen-like ion has the wavelength difference the first lines of Balmer and Lyman series equal to `59.3 nm`? `R_(H) = 109.678 cm^(-1)`

Explanation

We have `(1)/(lambda) = R_(H)Z^(2) [(1)/(n_(1)^(2)) - (1)/(n_(2)^(2))]`For first line of Balmer series
`(1)/(lambda_(B)) =R_(H).Z^(2) [(1)/(2^(2)) - (1)/(3^(2))] = (5)/(36) xx R_(H) xx Z^(2)`
or `lambda_(B) = (36)/(5.R_(H).Z^(2))`.......(i)
For first line of Lyman series
`(1)/(lambda_(L)) =R_(H).Z^(2) [(1)/(1^(2)) - (1)/(2^(2))] = (3)/(4) xx R_(H) xx Z^(2)`
or `lambda_(L) = (4)/(3R_(H).Z^(2))`.....(ii)
Given `lambda_(B) - lambda_(L) = 59.3 xx 10^(-7) cm `
or `(36)/(5R_(H).Z^(2)) -(4)/(3R_(H).Z^(2))= 59.8 xx 10^(-7)`
`(1)/(5R_(H).Z^(2))[7.2 - 1.333] = 59.3 xx 10^(-7)`
or `Z^(2) = (5.867)/(R_(H) xx 59.3 xx 10^(-7))`
`= (5.867)/(109678 xx 59.8 xx 10^(-7))`
`:. Z = 3`
`:. "H-like atom is" Li^(2+)`