A single electron orbit around a stationary nucleus of charge `+ Ze` where Z is a constant and e is the magnitude of the electronic charge. It requires`47.2 eV` to excite the electron from the second bohr orbit to the third bohr orbit. Find
(i) The value of Z
(ii) The energy requiredto nucleus the electron from the third to the fourth bohr orbit
(iii) The wavelength of the electronmagnetic radiation required to remove the electron from the first bohr orbit to inlinity
(iv) The energy potential energy potential energy and the angular momentum of the electron in the first bohr orbit
(v) The radius of the first bohr orbit (The ionization energy of hydrogen atom ` = 13.6 eV ` bohr radius `= 5.3 xx 10^(-11) matre` velocity of light `= 3 xx 10^(8) m//sec` planks 's constant ` = 6.6 xx 10^(-34)` jules - sec )

Explanation

The energy of the electron in nth orbit of hydrogen-like atom is given by
`E_(n_ = - (Z^(2) R h c)/(n^(2))` (i)
a. The energy required to excite the electron from second `(n = 2)` Bohr orbit to the third `(n = 3)` Bohr orbit is given by
`Delta E = E_(3) - E_(2) = - (Z^(2) R h c)/(3^(2)) - ((-Z^(2) R h c)/(3^(2)))`
`= Z^(2) R h c ((1)/(4) - (1)/(9)) = (5)/(36) Z^(2) R h c`
Given , ionization energy of hydrogen atom
`= R h c = 13.6 e V` and `Delta E = E_(3) - E_(2) = 47.2 e V`
Thus, we have `47.2 e V = (5)/(36) Z^(2) xx 13.6 e V`
`Z^(2) = (36)/(5) xx (47.2)/(13.6) = 25 implies Z = 5`
The energy required to excite the electron from the third to fourth orbit is given by
` E_(4) - E_(3) = - (Z^(2) R h c)/(4^(2)) - ((-Z^(2) R h c)/(3^(2)))`
`= Z^(2) R h c ((1)/(3^(2)) - (1)/(4^(2)))`
`= 5^(2) xx (13.6 e V) xx (7)/(144) = 16.63 e V`
c. The ionization energy of atom of the required to remove the electron from the first bohr orbit to `oo` is
`E_(oo) - E_(1) = - (Z^(2) R h c)/((oo)^(2)) - ((-Z^(2) R h c)/(1^(2)))`
`= Z^(2) R h c`
If `lambda` is the wavelength of corresponding electromagnetic radiation, then
`(hc)/(lambda) = Z^(2) R h c`
i.e., `(hc)/(lambda) = 5^(2) xx 13.6 e V = 340 e V`
`:. lambda = (hc)/(340 e V) = (6.63 xx 10^(-34) xx 3 xx 10^(8))/(340 xx 1.6 xx 10^(-19))`
`= 36.056 xx 10^(-10) m - 36.056 Å`
d. Kinetic energy of electron in the first Bohr's orbit .
`E K_(1) = - E_(1) = 340 e V`
Potential energy of electron in the first Bohr's orbit .
`U_(1) = 2 E_(1) = - 2 xx 340 V = - 680 e V`
Angular momentum of electron in the first Bohr's orbit .
`= n (h)/(2 pi) = 1 (h)/(2 pi) = (6.63 xx 10^(-34))/(2 pi)`
`= 1.0546 xx 10^(-34)J s`
e. Radius of the first bohr orbit for given atom
`= ((epsilon_(0) h^(2) n^(2))/(pi m Ze ^(2))) _(n = 1) = ((epsilon_(0) h^(2)// pi me^(2)))/(Z)`
`= ("Radius of first bohr orbit of hydrogen")/(Z)`
`= (5.3 xx 10^(-11))/(5) m = 1.06 xx 10^(-11) m`