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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.
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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
Linear momentum of an electron in Bohr orbit of H-atom (principal quantum number n) is proportional to ______.
Answer in brief.
State the postulates of Bohr’s atomic model.
Starting with 𝑟 = `(ε_0h^2n^2)/(pimZe^2),` Show that the speed of an electron in nth orbit varies inversely to principal quantum number.
State Bohr's second postulate for the atomic model. Express it in its mathematical form.
State any two limitations of Bohr’s model for the hydrogen atoms.
Calculate the longest wavelength in the Paschen series.
(Given RH =1.097 ×107 m-1)
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.
Bohr model is applied to a particle of mass 'm' and charge 'q' is moving in a plane under the influence of a transverse magnetic field 'B. The energy of the charged particle in the nth level will be (h = Planck's constant).
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?
With the increase in principal quantum number, the energy difference between the two successive energy levels ____________.
The ratio of the velocity of the electron in the first orbit to that in the second orbit is ____________.
If the ionisation potential of helium atom is 24.6 volt, the energy required to ionise it will be ____________.
When hydrogen atom is in its first excited level, its radius is how many time its ground state radius?
According to Bohr's theory, the expression for the kinetic and potential energy of an electron revolving in an orbit is given respectively by ______.
If the speed of an electron of hydrogen atom in the ground state is 2.2 x 106 m/s, then its speed in the third excited state will be ______.
The minimum energy required to excite a hydrogen atom from its ground state is ____________.
In hydrogen atom, during the transition of electron from nth outer orbit to first Bohr orbit, a photon of wavelength `lambda` is emitted. The value of 'n' is [R =Rydberg's constant] ____________.
When an electron in hydrogen atom jumps from third excited state to the ground state, the de-Broglie wavelength associated with the electron becomes ____________.
The electron of mass 'm' is rotating in first Bohr orbit of radius 'r' in hydrogen atom. The orbital acceleration of the electron in first orbit is ______.
(b =Planck's constant)
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 ______
The third line of the Balmer series, in the emission spectrum of the hydrogen atom, is due to the transition from the ______.
The speed of an electron in ground state energy level is 2.6 × 106 ms-1, then its speed in third excited state will be ______.
In Bohr’s atomic model, speed and time period of revolution of an electron in n = 3 level are respectively.
Find the momentum of the electron having de Broglie wavelength of 0.5 A.
