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 electron from second Bohr orbit to third Bhor orbit . Find
a the value of `Z`
b the energy required to excite the electron from the third to the fourth Bohr orbit.
c. 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]`

Answer

(i) `5` (ii) `16.5 eV` (iii) `36.4 Å` (iv) `340 eV, -680 eV, (h)/(2pi)` (v) `1.06 xx 10^(-11) m`

Explanation

(i) `AE = 47.2 = 13.6Z^(2) ((1)/(4) - (1)/(9)) rArr Z = 5`
(ii) `Delta E = 13.6 xx 25 (1/9 - 1/16)`
`= (13.6 xx 25 xx 7)/(9 xx 16) eV= 16.5 eV`
(iii) `DeltaE_(l-rn) = 13.6 Z^(2)eV = (hc)/(lambda)`
`rArr 13.6 xx 25 eV rArr 13.6 xx 25 eV = (12400eV - A)/(lambda) rArr = 36.4 Å`
(iv) `(KE)_(1st "orbit") = (1)/2 mv^(2) = 1/2 mv^(2)`
`= (9.1 xx 10^(-31) xx (2.2 xx 10^(6) xx 5)^(2))/(2 xx 1.6 xx 10^(-19)) = 344 eV`
`(PE)_("max") = - 2KE = - 988 eV`
`L = nh/(2pi) (n = 1)`
(v) Radius `= (R_(0)) (n^(2))/(z) = (0.53 xx 1^(2))/(5) = 1.06 xx 10^(-11) m`