Advertisements
Advertisements
प्रश्न
If a proton had a radius R and the charge was uniformly distributed, calculate using Bohr theory, the ground state energy of a H-atom when (i) R = 0.1 Å, and (ii) R = 10 Å.
Advertisements
उत्तर
In an H-atom in the ground state, the electron revolves around the point-size proton in a circular orbit of radius rB (Bohr’s radius).
As `mvr_B = h` and `(mv^2)/r_B = (-1 xx e xx e)/(4πε_0r_B^2)`
`m/r_B ((-h^2)/(m^2r_B^2)) = e^2/(4πε_0r_B^2)`
`r_B = (4πε_0 h^2)/e^2 h^2/m` = 0.53 Å
K.E. = `1/2 mv^2 = (m/2)(h/(mr_B))^2`
= `h^2/(2mr_B^2)` = 13.6 eV
PE of the electron and proton,
U = `1/(4πε_0) (e(- e))/r_B = - e^2/(4πε_0r_B)` = – 27.2 eV
The total energy of the electron, i.e.,
E = K + U = +13.6 eV – 27.2 eV = – 13.6 eV
(i) When R = 0.1 Å: R < rB (as rB = 0.51 Å) and the ground state energy is the same as obtained earlier for point-size proto 13.6 eV
(ii) When R = 10 A: R >> rB, the electron moves inside the proton (assumed to be a sphere of radius R) with new Bohr’s radius r'B
Clearly, `r_B^' = (4πε_0 h^2)/(m(e)(e^'))` .....[Replacing e2 by (e) (e’) where e’ is the charge on the sphere of radius r'B]
Since `e^' = [e/(((4pi)/3)R^3)] [((4pi)/3) r_B^('3)] = r_B^('3)/R^3`
`r_B^' = (4πε_0 h^2)/(m_e (er_B^('3) R^3)) = ((4πε_0 h^2)/(me^2))(R^3/r_B^('3))`
or `r_B^('4) = ((4πε_0 h^2)/(me^2)) R^3`
= (0.51 Å) (10 Å)3 = (510 Å4)
∴ `r_B^'` = 4.8 Å, which is less than R(= 10 Å)
KE of the electron,
`K^' = 1/2 mv^('2) = (m/2) (h^2/(m^2r_B^('2))) = h/(2mr_B^('2))`
= `(h^2/(2mr_B^2)) (r_B/r_B^')` = 13.6 eV `((0.51 Å)/(4.8 Å))^2` = 0.16 eV
Potential at a point inside the charged proton i.e.,
`V = (k_e e)/R (3 - (r_B^('2))/R^2) = k_e e ((3R^2 - r_B^('2))/R^3)`
Potential energy of electron and proton, μ = – eV
= `- e(k_ee) [(3R^2 - r_B^('2))/R^3]`
= `-(e^2/(4πε_0 r^B)) [(r_B(3R^2 - r_B^('2)))/R^3]`
= `- (27.2 eV) [((0.51 Å)(300 Å - 23.03 Å))/((1000 Å))]`
= – 3.83 eV
Total energy of the electron, E = K + U = 0.16 eV – 3.83 eV = – 3.67 eV
APPEARS IN
संबंधित प्रश्न
An electron is orbiting in 5th Bohr orbit. Calculate ionisation energy for this atom, if the ground state energy is -13.6 eV.
(i) State Bohr's quantization condition for defining stationary orbits. How does the de Broglie hypothesis explain the stationary orbits?
(ii) Find the relation between three wavelengths λ1, λ2 and λ3 from the energy-level diagram shown below.
How many electrons in an atom may have the following quantum numbers?
n = 3, l = 0
Suppose, the electron in a hydrogen atom makes transition from n = 3 to n = 2 in 10−8 s. The order of the torque acting on the electron in this period, using the relation between torque and angular momentum as discussed in the chapter on rotational mechanics is
When a photon stimulates the emission of another photon, the two photons have
(a) same energy
(b) same direction
(c) same phase
(d) same wavelength
Find the wavelength of the radiation emitted by hydrogen in the transitions (a) n = 3 to n= 2, (b) n = 5 to n = 4 and (c) n = 10 to n = 9.
A positive ion having just one electron ejects it if a photon of wavelength 228 Å or less is absorbed by it. Identify the ion.
State any two Bohr’s postulates and write the energy value of the ground state of the hydrogen atom.
A particle has a mass of 0.002 kg and uncertainty in its velocity is 9.2 × 10−6 m/s, then uncertainty in position is ≥ ____________.
(h = 6.6 × 10−34 J s)
Why was a change in the Bohr Model of atom required? Due to which important development (s), concept of movement of an electron in an orbit was replaced by, the concept of probability of finding electron in an orbital? What is the name given to the changed model of atom?
The energy of an electron in hth orbit of hydrogen atom is –13.6/n2ev energy required to excite the electron from the first orbit to the third orbit is
The ground state energy of hydrogen atoms is -13.6 eV. The photon emitted during the transition of electron from n = 3 to n = 1 unknown work function. The photoelectrons are emitted from the material with a maximum kinetic energy of 9 eV. Calculate the threshold wavelength of the material used.
Given below are two statements:
Statements I: According to Bohr's model of an atom, qualitatively the magnitude of velocity of electron increases with decrease in positive charges on the nucleus as there is no strong hold on the electron by the nucleus.
Statement II: According to Bohr's model of an atom, qualitatively the magnitude of velocity of electron increase with a decrease in principal quantum number.
In light of the above statements, choose the most appropriate answer from the options given below:
The value of angular momentum for He+ ion in the first Bohr orbit is ______.
The wavelength in Å of the photon that is emitted when an electron in Bohr orbit with n = 2 returns to orbit with n = 1 in H atom is ______ Å. The ionisation potential of the ground state of the H-atom is 2.17 × 10−11 erg.
The energy required to remove the electron from a singly ionized Helium atom is 2.2 times the energy required to remove an electron from Helium atom. The total energy required to ionize the Helium atom completely is ______.
A 100 eV electron collides with a stationary helium ion (He+) in its ground state and exits to a higher level. After the collision, He+ ions emit two photons in succession with wavelengths 1085 Å and 304 Å. The energy of the electron after the collision will be ______ eV.
Given h = 6.63 × 10-34 Js.
Write the ionisation energy value for the hydrogen atom.
The radius of the nth orbit in the Bohr model of hydrogen is proportional to ______.
How much is the angular momentum of an electron when it is orbiting in the second Bohr orbit of hydrogen atom?
