A single electron orbits a stationary nucleus of charge `+ Ze`, where `Z` is a constant and `e` is the magnitude of electronic charge . It requires `47.2 eV` to excite . Find
a the value of `Z`
b the energy required to excite the electron from the third to the fourth Bohr orbit.
the wavelength of electromagnetic rediation required to remove the electron from the first Bohr orbit to infinity.
d Find the `KE,PE`, and angular momentum of electron in the first Bohr orbit.
e the redius of the first Bohr orbit
[The ionization energy of hydrogen atom `= 13.6 eV` Bohr radius `= 5.3 xx 10^(_11) m`, "velocity of light" `= 3 xx 10^(-8)jm s ^(-1)`, Planck's constant `= 6.6 xx 10^(-34)j - s]`

Explanation

The energy required to excite the electron from `n_(1)` to `n_(2)` orbit revolving aroung the nucleus with charge `+Ze` is given by `E_(n_(2)) - E_(n_(1)) = (me^(4))/(8epsi_(0)^(2))[1/n_(1)^(2) - (1)/(n_(2)^(2))]Z^(2) rArr E_(n_(2)) - E_(n_(1)) Z^(2) xx (13.5)[1/(n_(1)^(2))- (1)/(n_(2)^(2))]`
(i) Since `47.2` eV energy is required to excite the electron from `n_(1) = 2` to `n_(2) = 3` orbit
`47.2 = Z^(2) xx 13.6 [(1)/(2^(2)) - (1)/3^(2)] rArr Z^(2) = (47.2 xx 36)/(13.6 xx 5) = 24.968 ~~ 25 rARr Z = 5`
(ii)The energy required to excite the electron from `n_(1) = 3` to `n_(2) = 4` is given by
`E_(4) - E_(3) = 13.6 Z^(2) = 5[1/(3^(2)) - (1)/(4^(2))] = (25 xx 13.6 xx 7)/(144) = 16.53 eV`
(iii) The energy required to remove the electron from the first Bohr orbit to infinty `(oo)` is given by
`E_(oo) - E_(3) = 13.6 xx Z^(2)[1/(1^(2)) - (1)/(oo^(2))] = 13.6 xx 25 eV = 340 eV`
In order to calculate the wavelength of radiation we use Bhor's ferqunecy relation
`hf = (hc)/(lambda) = 13.6 xx 25 xx (1.6 xx 10^(-19))J rArr lambda = ((6.6 xx 10^(-34)) xx 10^(8) xx 3)/(13.6 xx 25 xx (1.6 xx 10^(-19))) = 36.397Å`
(iv) `K.E. 1/2 mv_(1)^(0) = 1/2 xx (Ze^(2))/(4pig epsi_(0)r_(1)) = 543.4 xx 10^(-19)J , P.E. = -2 xK.E. = - 1086.8 xx 10^(-19)J`
Angular momentum `= mv_(1)r_(1) = (h)/(2pi) = 1.05 xx 10^(-34) Js`
The radius `r_(1)` of the first bohr orbit is given by
`r_(1) = (epsi_(0)h^(2))/(pime^(2)) *(1)/(Z) = (0.53 xx 10^(-10))/(5) (':' (epsi_(0)h^(2))/(pime^(2)) = 0.53 xx 10^(-10)m) = 1.106 xx 10^(-10) m = 0.106Å`