Advertisements
Advertisements
Question
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
Solution
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`
RELATED QUESTIONS
Answer in one sentence:
Name the element that shows the simplest emission spectrum.
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.
Derive an expression for the radius of the nth Bohr orbit for the hydrogen atom.
Calculate the wavelength for the first three lines in the Paschen series.
(Given RH =1.097 ×107 m-1)
Which of the following statements about the Bohr model of the hydrogen atom is FALSE?
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 ____________.
How many moles of electrons are required for reduction of 9 moles of Cr3+ to Cr?
In Bohr's model of hydrogen atom, the period of revolution of the electron in any orbit is proportional to ______.
In hydrogen atom, the de Broglie wavelength of an electron in the first Bohr's orbit is ____________.
[Given that Bohr radius, a0 = 52.9 pm]
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 ______.
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) ____________.
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.
Ratio of centripetal acceleration for an electron revolving in 3rd orbit to 5th orbit of hydrogen atom is ______.
If a charge on the body is 1 nC, then how many electrons are present on the body?
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?
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] ____________.
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 ______.
Electron in Hydrogen atom first jumps from third excited state to second excited state and then from second excited state to first excited state. The ratio of the wavelengths λ1 : λ2 emitted in the two cases respectively is ______.
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)
If Vn and Vp are orbital velocities in nth and pth orbit respectively, then the ratio Vp: Vn is ______.
The third line of the Balmer series, in the emission spectrum of the hydrogen atom, is due to the transition from the ______.
The value of Rydberg constant in joule is ______.
The triply ionised beryllium (Be+++) has the same electron orbital radius as that of the ground state of the hydrogen atom. The energy state (n) of triply ionised beryllium is ______.
(Z for beryllium = 4)
Calculate the energy of the electron in the ground state of the hydrogen atom. Express it in joule and in eV.
Calculate the radius of the first Bohr orbit in the hydrogen atom.
Find the ratio of radius of 1st Bohr orbit to that of 4th Bohr orbit.
The radius of the first Bohr orbit in the hydrogen atom is 0.5315 Å. The radius of the second Bohr orbit in the hydrogen atom is ______.
