The ionization energy of a hydrogen-like Bohr atom is `4` rydbergs.
a. What is the wavelength of radiation emitted when the electron jumps from the first excited state to the ground state ?
b. What is the radius of the first orbit for this atom ? Given that bohr radius of hydrogen atom `= 5 xx 10^(-11) m` and `1` rydberg`= 2.2 xx 10^(-18 J`.

Explanation

Energy of first orbit of H- like atom `= 4 R_(h) = 4 xx 2.18 xx 10^(-19) J`
`E_(1)` for = `- 2.18 xx 10^(-19) J`
`:' E("H-like atom") = E_(IH) xx Z^(2)`
`-4 xx 2.18 xx 10^(-19)= - 2.18 xx Z^(2)`
`:. Z = 2`
i.e. atomic number of H-like atom is `2` or it is `He^(o+)`
a For de-excitation of e in `He^(o+)` from `n_(2) = 2` to `n_(1) = 1`
`E_(2) - E_(1) =(hc)/(lambda)`
Now `E_(1) = - 54R_(n) `
`:. E_(2) = - (4R_(h))/(4) = - R_(n) (:' E_(2) = (E_(1))/(n^(2)))`
`:. E_(2) - E_(1) = 3R_(6) = 3 xx 2.18 xx 10^(-19) J`
`E_(2) - E_(1) =(hc)/(lambda)`
`:. lambda = (6.626 xx 10^(-34) xx 3 xx 10^(8))/(3 xx 2.18 xx 10^(-19))`
`= 303.89 xx 10^(-19) m`
`303.89 Å`
b. Radius `(r_(1))` of H-like atom
`= (r_(H))/(Z) = (0.529 xx 10^(-8))/(2)`
`= 2.645 xx 10^(-9) cm `