Advertisements
Advertisements
Question
Classically, an electron can be in any orbit around the nucleus of an atom. Then what determines the typical atomic size? Why is an atom not, say, a thousand times bigger than its typical size? The question had greatly puzzled Bohr before he arrived at his famous model of the atom that you have learnt in the text. To simulate what he might well have done before his discovery, let us play as follows with the basic constants of nature and see if we can get a quantity with the dimensions of length that is roughly equal to the known size of an atom (~ 10−10 m).
(a) Construct a quantity with the dimensions of length from the fundamental constants e, me, and c. Determine its numerical value.
(b) You will find that the length obtained in (a) is many orders of magnitude smaller than the atomic dimensions. Further, it involves c. But energies of atoms are mostly in non-relativistic domain where c is not expected to play any role. This is what may have suggested Bohr to discard c and look for ‘something else’ to get the right atomic size. Now, the Planck’s constant h had already made its appearance elsewhere. Bohr’s great insight lay in recognising that h, me, and e will yield the right atomic size. Construct a quantity with the dimension of length from h, me, and e and confirm that its numerical value has indeed the correct order of magnitude.
Advertisements
Solution
(a) Charge on an electron, e = 1.6 × 10−19 C
Mass of an electron, me = 9.1 × 10−31 kg
Speed of light, c = 3 × 108 m/s
Let us take a quantity involving the given quantities as `("e"^2/(4piin_0 "m"_"e""c"^2))`.
Where,
∈0 = Permittivity of free space
And `1/(4pi in_0)` = 9 × 109 N m2 C−2
The numerical value of the taken quantity will be:
`1/(4pi in_0) xx "e"^2/("m"_"e""c"^2)`
= `9 xx 10^9 xx (1.6 xx 10^(-19))^2/(9.1 xx 10^(-31) xx (3 xx 10^8)^2`
= 2.81 × 10−15 m
Hence, the numerical value of the taken quantity is much smaller than the typical size of an atom.
(b) Charge on an electron, e = 1.6 × 10−19 C
Mass of an electron, me = 9.1 × 10−31 kg
Planck’s constant, h = 6.63 × 10−34 Js
Let us take a quantity involving the given quantities as `(4pi in_0 ("h"/(2pi))^2)/("m"_"e" "e"^2)`.
Where,
∈0 = Permittivity of free space
And , `1/(4pi in_0)` = 9 × 109 N m2 C−2
The numerical value of the taken quantity will be:
`4pi in_0 xx ("h"/(2pi))^2/("m"_"e""e"^2)`
= `1/ (9 xx 10^9) xx ((6.63 xx 10^(-34))/(2 xx 3.14))^2/(9.1 xx 10^-31 xx (1.6 xx 10^-19)^2)`
= 0.53 × 10−10 m
Hence, the value of the quantity taken is of the order of the atomic size.
APPEARS IN
RELATED QUESTIONS
A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. What series of wavelengths will be emitted?
If Bohr’s quantisation postulate (angular momentum = nh/2π) is a basic law of nature, it should be equally valid for the case of planetary motion also. Why then do we never speak of quantisation of orbits of planets around the sun?
Find the wavelength of the electron orbiting in the first excited state in hydrogen atom.
Which wavelengths will be emitted by a sample of atomic hydrogen gas (in ground state) if electrons of energy 12.2 eV collide with the atoms of the gas?
What will be the energy corresponding to the first excited state of a hydrogen atom if the potential energy of the atom is taken to be 10 eV when the electron is widely separated from the proton? Can we still write En = E1/n2, or rn = a0 n2?
In which of the following transitions will the wavelength be minimum?
Let An be the area enclosed by the nth orbit in a hydrogen atom. The graph of ln (An/A1) against ln(n)
(a) will pass through the origin
(b) will be a straight line with slope 4
(c) will be a monotonically increasing nonlinear curve
(d) will be a circle
Find the radius and energy of a He+ ion in the states (a) n = 1, (b) n = 4 and (c) n = 10.
A hydrogen atom emits ultraviolet radiation of wavelength 102.5 nm. What are the quantum numbers of the states involved in the transition?
(a) Find the first excitation potential of He+ ion. (b) Find the ionization potential of Li++ion.
A group of hydrogen atoms are prepared in n = 4 states. List the wavelength that are emitted as the atoms make transitions and return to n = 2 states.
Whenever a photon is emitted by hydrogen in Balmer series, it is followed by another photon in Lyman series. What wavelength does this latter photon correspond to?
Show that the ratio of the magnetic dipole moment to the angular momentum (l = mvr) is a universal constant for hydrogen-like atoms and ions. Find its value.
A hydrogen atom in ground state absorbs a photon of ultraviolet radiation of wavelength 50 nm. Assuming that the entire photon energy is taken up by the electron with what kinetic energy will the electron be ejected?
Electrons are emitted from an electron gun at almost zero velocity and are accelerated by an electric field E through a distance of 1.0 m. The electrons are now scattered by an atomic hydrogen sample in ground state. What should be the minimum value of E so that red light of wavelength 656.3 nm may be emitted by the hydrogen?
The Balmer series for the H-atom can be observed ______.
- if we measure the frequencies of light emitted when an excited atom falls to the ground state.
- if we measure the frequencies of light emitted due to transitions between excited states and the first excited state.
- in any transition in a H-atom.
- as a sequence of frequencies with the higher frequencies getting closely packed.
In the Auger process an atom makes a transition to a lower state without emitting a photon. The excess energy is transferred to an outer electron which may be ejected by the atom. (This is called an Auger electron). Assuming the nucleus to be massive, calculate the kinetic energy of an n = 4 Auger electron emitted by Chromium by absorbing the energy from a n = 2 to n = 1 transition.
