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
What is the maximum number of emission lines when the excited electron of an H atom in n = 6 drops to the ground state?
How many electrons in an atom may have the following quantum numbers?
n = 4, `m_s = -1/2`
In accordance with the Bohr’s model, find the quantum number that characterises the earth’s revolution around the sun in an orbit of radius 1.5 × 1011 m with orbital speed 3 × 104 m/s. (Mass of earth = 6.0 × 1024 kg)
Using Bohr's postulates, derive the expression for the orbital period of the electron moving in the nth orbit of hydrogen atom ?
Using Bohr’s postulates, derive the expression for the frequency of radiation emitted when electron in hydrogen atom undergoes transition from higher energy state (quantum number ni) to the lower state, (nf).
When electron in hydrogen atom jumps from energy state ni = 4 to nf = 3, 2, 1, identify the spectral series to which the emission lines belong.
The electron in hydrogen atom is initially in the third excited state. What is the maximum number of spectral lines which can be emitted when it finally moves to the ground state?
The numerical value of ionization energy in eV equals the ionization potential in volts. Does the equality hold if these quantities are measured in some other units?
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.
According to Maxwell's theory of electrodynamics, an electron going in a circle should emit radiation of frequency equal to its frequency of revolution. What should be the wavelength of the radiation emitted by a hydrogen atom in ground state if this rule is followed?
Radiation coming from transition n = 2 to n = 1 of hydrogen atoms falls on helium ions in n = 1 and n = 2 states. What are the possible transitions of helium ions as they absorbs energy from the radiation?
A parallel beam of light of wavelength 100 nm passes through a sample of atomic hydrogen gas in ground state. (a) Assume that when a photon supplies some of its energy to a hydrogen atom, the rest of the energy appears as another photon. Neglecting the light emitted by the excited hydrogen atoms in the direction of the incident beam, what wavelengths may be observed in the transmitted beam? (b) A radiation detector is placed near the gas to detect radiation coming perpendicular to the incident beam. Find the wavelengths of radiation that may be detected by the detector.
The earth revolves round the sun due to gravitational attraction. Suppose that the sun and the earth are point particles with their existing masses and that Bohr's quantization rule for angular momentum is valid in the case of gravitation. (a) Calculate the minimum radius the earth can have for its orbit. (b) What is the value of the principal quantum number n for the present radius? Mass of the earth = 6.0 × 10−24 kg. Mass of the sun = 2.0 × 1030 kg, earth-sun distance = 1.5 × 1011 m.
State any two Bohr’s postulates and write the energy value of the ground state of the hydrogen atom.
Draw energy level diagram for a hydrogen atom, showing the first four energy levels corresponding to n=1, 2, 3 and 4. Show transitions responsible for:
(i) Absorption spectrum of Lyman series.
(ii) The emission spectrum of the Balmer series.
Answer the following question.
Calculate the orbital period of the electron in the first excited state of the hydrogen atom.
When the electron orbiting in hydrogen atom in its ground state moves to the third excited state, show how the de Broglie wavelength associated with it would be affected.
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)
The binding energy of a H-atom, considering an electron moving around a fixed nuclei (proton), is B = `- (Me^4)/(8n^2ε_0^2h^2)`. (m = electron mass). If one decides to work in a frame of reference where the electron is at rest, the proton would be moving around it. By similar arguments, the binding energy would be
B = `- (Me^4)/(8n^2ε_0^2h^2)` (M = proton mass)
This last expression is not correct because ______.
The Bohr model for the spectra of a H-atom ______.
- will not be applicable to hydrogen in the molecular from.
- will not be applicable as it is for a He-atom.
- is valid only at room temperature.
- predicts continuous as well as discrete spectral lines.
The de Broglie wavelength of an electron in the first Bohr’s orbit of hydrogen atom is equal to ______.
