Advertisements
Advertisements
प्रश्न
State the postulates of Bohr’s atomic model. Hence show the energy of electrons varies inversely to the square of the principal quantum number.
Advertisements
उत्तर
Bohr’s three postulates are:
- In a hydrogen atom, the electron revolves around the nucleus in a fixed circular orbit with constant speed.
- The radius of the orbit of an electron can only take certain fixed values such that the angular momentum of the electron in these orbits is an integral multiple of `"h"/(2π)`, h being the Planck’s constant.
- An electron can make a transition from one of its orbits to another orbit having lower energy. In doing so, it emits a photon of energy equal to the difference in its energies in the two orbits.
Expression for the energy of an electron in the nth orbit of Bohr’s hydrogen atom:
- Kinetic energy:
Let, me = mass of the electron
rn = radius of nth orbit of Bohr’s hydrogen atom
vn = velocity of electron
−e = charge of the electron
+e = charge on the nucleus
Z = a number of electrons in an atom.
According to Bohr’s first postulate,
`("m"_"e""V"_"n"^2)/"r"_"n" = 1/(4piepsilon_0) xx ("Ze"^2)/("r"_"n"^2)`
where, `epsilon_0` is permittivity of free space.
∴ `"m"_"e""v"_"n"^2 = "Z"/(4piepsilon_0) xx "e"^2/"r"_"n"` ….(1)
The revolving electron in the circular orbit has linear speed and hence it possesses kinetic energy.
It is given by, K.E = `1/2 "m"_"e""v"_"n"^2`
∴ K.E = `1/2 xx ("Z"/(4piepsilon_0) xx "e"^2/"r"_"n")` ….[From equation (1)]
∴ K.E = `"Ze"^2/(8piepsilon_0"r"_"n")` .…(2) - Potential energy:
The potential energy of the electron is given by, P.E = V(−e)
where,
V = electric potential at any point due to charge on the nucleus
− e = charge on the electron.
In this case,
∴ P.E = `1/(4piepsilon_0) xx "e"/"r"_"n" xx (-"Ze")`
∴ P.E = `1/(4piepsilon_0) xx (-"Ze"^2)/"r"_"n"`
∴ P.E = −`("Ze"^2)/(4piepsilon_0"r"_"n")` ….(3) - Total energy:
The total energy of the electron in any orbit is its sum of P.E and K.E.
∴ T.E = K.E + P.E
= `("Ze"^2/(8piepsilon_0"r"_"n")) + (-"Ze"^2/(4piepsilon_0"r"_"n"))` ….[From equations (2) and (3)]
∴ T.E = `-"Ze"^2/(8piepsilon_0"r"_"n")` ….(4) - But, rn = `((epsilon_0"h"^2)/(pi"m"_"e""Ze"^2)) xx "n"^2`
Substituting for rn in equation (4),
∴ T.E = `−1/(8piepsilon_0) xx "Ze"^2/(((epsilon_0"h"^2)/(pi"m"_"e""Ze"^2))"n"^2)`
= `-1/(8piepsilon_0) xx ("Z"^2"e"^2pi"m"_"e""e"^2)/(epsilon_0"h"^2"n"^2)`
∴ T.E = −`("m"_"e""Z"^2"e"^4)/(8epsilon_0^2"h"^2) xx 1/"n"^2` ….(5)
⇒ T.E. ∝ `1/"n"^2`
संबंधित प्रश्न
Answer in one sentence:
Name the element that shows the simplest emission spectrum.
According to Bohr's second postulate, the angular momentum of the electron is the integral multiple of `h/(2pi)`. The S.I unit of Plank constant h is the same as ______
The speed of electron having de Broglie wavelength of 10 -10 m is ______
(me = 9.1 × 10-31 kg, h = 6.63 × 10-34 J-s)
The linear momentum of the particle is 6.63 kg m/s. Calculate the de Broglie wavelength.
State Bohr's second postulate for the atomic model. Express it in its mathematical form.
The angular momentum of an electron in the 3rd Bohr orbit of a Hydrogen atom is 3.165 × 10-34 kg m2/s. Calculate Plank’s constant h.
Obtain an expression for wavenumber, when an electron jumps from a higher energy orbit to a lower energy orbit. Hence show that the shortest wavelength for the Balmar series is 4/RH.
The radius of electron's second stationary orbit in Bohr's atom is R. The radius of the third orbit will be ______
The wavelength of the first line in Balmer series in the hydrogen spectrum is 'λ'. What is the wavelength of the second line in the same series?
Which of the following statements about the Bohr model of the hydrogen atom is FALSE?
When the electron in hydrogen atom jumps from fourth Bohr orbit to second Bohr orbit, one gets the ______.
For a certain atom when the system moves from 2E level to E, a photon of wavelength `lambda` is emitted. The wavelength of photon produced during its transition from `(4"E")/3` level to E is ____________.
Taking the Bohr radius as a0= 53 pm, the radius of Li++ ion in its ground state, on the basis of Bohr's model, will be about ______.
The total energy of an electron in an atom in an orbit is -3.4 eV. Its kinetic and potential energies are, respectively ______.
In the nth orbit, the energy of an electron `"E"_"n"= -13.6/"n"^2"eV"` for hydrogen atom. The energy required to take the electron from first orbit to second orbit will be ____________.
For an electron, discrete energy levels are characterised by ____________.
In hydrogen spectnun, the wavelengths of light emited in a series of spectral lines is given by the equation `1/lambda = "R"(1/3^2 - 1/"n"^2)`, where n = 4, 5, 6 .... And 'R' is Rydberg's constant.
Identify the series and wavelenth region.
The minimum energy required to excite a hydrogen atom from its ground state is ____________.
In Bohr's model of hydrogen atom, which of the following pairs of quantities are quantized?
In any Bohr orbit of hydrogen atom, the ratio of K.E to P.E of revolving electron at a distance 'r' from the nucleus is ______.
The radius of orbit of an electron in hydrogen atom in its ground state is 5.3 x 10-11 m After collision with an electron, it is found to have a radius of 13.25 x 10-10 m. The principal quantum number n of the final state of the atom is ______.
The ground state energy of the hydrogen atom is -13.6 eV. The kinetic and potential energy of the electron in the second excited state is respectively ______
An electron of mass m and charge e initially at rest gets accelerated by a constant electric field E. The rate of change of de-Broglie wavelength of this electron at time t ignoring relativistic effects is ______.
Ultraviolet light of wavelength 300 nm and intensity 1.0 Wm−2 falls on the surface of a photosensitive material. If one percent of the incident photons produce photoelectrons, then the number of photoelectrons emitted from an area of 1.0 cm2 of the surface is nearly ______.
Calculate the radius of the first Bohr orbit in the hydrogen atom.
Find the momentum of the electron having de Broglie wavelength of 0.5 A.
