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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.
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 brief.
State the postulates of Bohr’s atomic model.
The radii of Bohr orbit are directly proportional to ______
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)
If aO is the Bohr radius and n is the principal quantum number then, state the relation for the radius of nth orbit of the electron in terms of Bohr radius and principal quantum number.
State any two limitations of Bohr’s model for the hydrogen atoms.
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 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 ____________.
How many moles of electrons are required for reduction of 9 moles of Cr3+ to Cr?
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 ______.
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 ______.
In hydrogen emission spectrum, for any series, the principal quantum number is n. Corresponding maximum wavelength λ is ______.
(R = Rydberg's constant)
When an electron in hydrogen atom is excited from its 3rd to 5th stationary orbit, tbe change in angular momentum of electron is (Planck's constant: h = 6.62 x 10-34 Js) ____________.
Ratio of centripetal acceleration for an electron revolving in 3rd orbit to 5th orbit of hydrogen atom is ______.
The minimum energy required to excite a hydrogen atom from its ground state is ____________.
If a charge on the body is 1 nC, then how many electrons are present on the body?
If n is principal quantum number and r is the radius of the orbit in which electron revolves around nucleus, then its kinetic energy is ____________.
An electron makes a transition from an excited state to the ground state of a hydrogen like atom. Out of the following statements which one is correct?
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 relation between magnetic moment of revolving electron 'M' and principle quantum number 'n' is ______.
The orbital frequency of an electron in the hydrogen atom ______.
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 ______.
Let Ee and Ep represent the kinetic energy of electron and photon, respectively. If the de-Broglie wavelength λp of a photon is twice the de-Broglie wavelength λe of an electron, then `E_p/E_e` is ______.
(speed of electron = `c/100`, c = velocity of light)
