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.
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.
APPEARS IN
RELATED QUESTIONS
Find the frequency of revolution of an electron in Bohr’s 2nd orbit; if the radius and speed of electron in that orbit is 2.14 × 10-10 m and 1.09 × 106 m/s respectively. [π= 3.142]
Using Bohr's postulates, derive the expression for the total energy of the electron in the stationary states of the hydrogen atom ?
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.
A parallel beam of light of wavelength 100 nm passes through a sample of atomic hydrogen gas in ground state. (a) Assume that when a photon supplies some of its energy to a hydrogen atom, the rest of the energy appears as another photon. Neglecting the light emitted by the excited hydrogen atoms in the direction of the incident beam, what wavelengths may be observed in the transmitted beam? (b) A radiation detector is placed near the gas to detect radiation coming perpendicular to the incident beam. Find the wavelengths of radiation that may be detected by the detector.
The light emitted in the transition n = 3 to n = 2 in hydrogen is called Hα light. Find the maximum work function a metal can have so that Hα light can emit photoelectrons from it.
Light from Balmer series of hydrogen is able to eject photoelectrons from a metal. What can be the maximum work function of the metal?
Radiation from hydrogen discharge tube falls on a cesium plate. Find the maximum possible kinetic energy of the photoelectrons. Work function of cesium is 1.9 eV.
When the electron orbiting in hydrogen atom in its ground state moves to the third excited state, show how the de Broglie wavelength associated with it would be affected.
What is the energy in joules released when an electron moves from n = 2 to n = 1 level in a hydrogen atom?
If the radius of first electron orbit in hydrogen atom be r then the radius of the fourth orbit ill be ______.
The wavelength of the first time line of Ballmer series is 6563 A°. The Rydberg constant for hydrogen is about:-
The simple Bohr model cannot be directly applied to calculate the energy levels of an atom with many electrons. This is because ______.
The number of times larger the spacing between the energy levels with n = 3 and n = 8 spacing between the energy level with n = 8 and n = 9 for the hydrogen atom 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:
The first ionization energy of H is 21.79 × 10-19 J. The second ionization energy of He atom is ______ × 10-19J.
What is the energy of an electron in stationary state corresponding to n = 2?
Oxygen is 16 times heavier than hydrogen. Equal volumes of hydrogen and oxygen are mixed. The ratio of speed of sound in the mixture to that in hydrogen is ______.
Write the ionisation energy value for the hydrogen atom.
The wavelength of the second line of the Balmer series in the hydrogen spectrum is 4861 Å. Calculate the wavelength of the first line of the same series.
The de Broglie wavelength of an electron in the first Bohr’s orbit of hydrogen atom is equal to ______.
