Advertisements
Advertisements
Question
The gravitational attraction between electron and proton in a hydrogen atom is weaker than the Coulomb attraction by a factor of about 10−40. An alternative way of looking at this fact is to estimate the radius of the first Bohr orbit of a hydrogen atom if the electron and proton were bound by gravitational attraction. You will find the answer interesting.
Advertisements
Solution
Radius of the first Bohr orbit is given by the relation,
`"r"_1 = (4pi in_0 ("h"/(2pi))^2)/("m"_"e" "e"^2)` .................(1)
Where,
∈0 = Permittivity of free space
h = Planck’s constant = 6.63 × 10−34 Js
me = Mass of an electron = 9.1 × 10−31 kg
e = Charge of an electron = 1.9 × 10−19 C
mp = Mass of a proton = 1.67 × 10−27 kg
r = Distance between the electron and the proton
Coulomb attraction between an electron and a proton is given as:
`"F"_"C" = "e"^2/(4piin_0 "r"^2)` .............(2)
Gravitational force of attraction between an electron and a proton is given as:
`"F"_"G" = ("Gm"_"p""m"_"e")/"r"^2` .........(3)
Where,
G = Gravitational constant = 6.67 × 10−11 N m2/kg2
If the electrostatic (Coulomb) force and the gravitational force between an electron and a proton are equal, then we can write:
∴ FG = FC
`("Gm"_"p""m"_"e")/"r"^2 = "e"^2/(4piin_0 "r"^2)`
∴ `"e"^2/(4piin_0) = "Gm"_"p""m"_"e"` ........(4)
Putting the value of equation (4) in equation (1), we get:
`"r"_1 = ("h"/(2pi))^2/("Gm"_"p""m"_"e"^2)`
= `(((6.63 xx 10^(-34))/(2xx3.14))^2)/(6.67 xx 10^(-11) xx 1.67 xx 10^(-27) xx (9.1 xx 10^(-31))^2) ~~ 1.21 xx 10^(29) "m"`
It is known that the universe is 156 billion light years wide or 1.5 × 1027 m wide. Hence, we can conclude that the radius of the first Bohr orbit is much greater than the estimated size of the whole universe.
RELATED QUESTIONS
What is the maximum number of emission lines when the excited electron of an H atom in n = 6 drops to the ground state?
Explain, giving reasons, which of the following sets of quantum numbers are not possible.
- n = 0, l = 0, ml = 0, ms = + ½
- n = 1, l = 0, ml = 0, ms = – ½
- n = 1, l = 1, ml = 0, ms = + ½
- n = 2, l = 1, ml = 0, ms = – ½
- n = 3, l = 3, ml = –3, ms = + ½
- n = 3, l = 1, ml = 0, ms = + ½
If the photon of the wavelength 150 pm strikes an atom and one of its inner bound electrons is ejected out with a velocity of 1.5 × 107 ms–1, calculate the energy with which it is bound to the nucleus.
State Bohr's postulate to define stable orbits in the hydrogen atom. How does de Broglie's hypothesis explain the stability of these orbits?
Using Bohr’s postulates for hydrogen atom, show that the total energy (E) of the electron in the stationary states tan be expressed as the sum of kinetic energy (K) and potential energy (U), where K = −2U. Hence deduce the expression for the total energy in the nth energy level of hydrogen atom.
When a photon stimulates the emission of another photon, the two photons have
(a) same energy
(b) same direction
(c) same phase
(d) same wavelength
A neutron having kinetic energy 12.5 eV collides with a hydrogen atom at rest. Nelgect the difference in mass between the neutron and the hydrogen atom and assume that the neutron does not leave its line of motion. Find the possible kinetic energies of the neutron after the event.
Consider a neutron and an electron bound to each other due to gravitational force. Assuming Bohr's quantization rule for angular momentum to be valid in this case, derive an expression for the energy of the neutron-electron system.
What is the energy in joules released when an electron moves from n = 2 to n = 1 level in a hydrogen atom?
Consider two different hydrogen atoms. The electron in each atom is in an excited state. Is it possible for the electrons to have different energies but same orbital angular momentum according to the Bohr model? Justify your answer.
According to Bhor' s theory the moment of momentum of an electron revolving in second orbit of hydrogen atom will be.
The ratio of the ionization energy of H and Be+3 is ______.
An ionised H-molecule consists of an electron and two protons. The protons are separated by a small distance of the order of angstrom. In the ground state ______.
- the electron would not move in circular orbits.
- the energy would be (2)4 times that of a H-atom.
- the electrons, orbit would go around the protons.
- the molecule will soon decay in a proton and a H-atom.
Find the ratio of energies of photons produced due to transition of an election of hydrogen atom from its (i) second permitted energy level to the first level. and (ii) the highest permitted energy level to the first permitted level.
A hydrogen atom in its first excited state absorbs a photon of energy x × 10-2 eV and exited to a higher energy state where the potential energy of electron is -1.08 eV. The value of x is ______.
Orbits of a particle moving in a circle are such that the perimeter of the orbit equals an integer number of de-Broglie wavelengths of the particle. For a charged particle moving in a plane perpendicular to a magnetic field, the radius of the nth orbital will therefore be proportional to:
In Bohr's theory of hydrogen atom, the electron jumps from higher orbit n to lower orbit p. The wavelength will be minimum for the transition ______.
The total energy of an electron in the nth orbit of the hydrogen atom is proportional to ______.
Using Bohr’s Theory of hydrogen atom, obtain an expression for the velocity of an electron in the nth orbit of an atom.
Calculate the radius of the second orbit of He+.
