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.
What is the energy of an electron in a hydrogen atom for n = ∞?
Using de Broglie’s hypothesis, obtain the mathematical form of Bohr’s second postulate.
Calculate the longest wavelength in the Paschen series.
(Given RH =1.097 ×107 m-1)
Calculate the wavelength for the first three lines in the Paschen series.
(Given RH =1.097 ×107 m-1)
Calculate the shortest wavelength in the Paschen series if the longest wavelength in the Balmar series is 6563 Ao.
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.
How the linear velocity 'v' of an electron in the Bohr orbit is related to its quantum number 'n'?
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 ______.
In Bohr's model of hydrogen atom, the period of revolution of the electron in any orbit is proportional to ______.
The binding energy of an electron in nth orbit of the hydrogen atom is given by `"E"_"n" = 13.6/"n"^2 "eV."` The energy required to knock an electron from the second orbit in eV will be ____________.
For an electron, discrete energy levels are characterised by ____________.
The acceleration of electron in the first orbit of hydrogen atom is ______.
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 ______.
In Bohr model, speed of electron in nth orbit of hydrogen atom is ______. (b = Planck's constant, n = principal quantum number, ∈0 is the permittivity of free space, e = electronic charge)
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] ____________.
Using Bohr's quantization condition, what is the rotational energy in the second orbit for a diatomic molecule. (I = moment of inertia of diatomic molecule, h = Planck's constant)
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 ______.
When an electron in hydrogen atom revolves in stationary orbit, it ______.
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)
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.
Compute the shortest and the longest wavelength in the Lyman series of hydrogen atom.
