12.1 Choose the correct alternative from the clues given at the end of
the each statement:
(a) The size of the atom in Thomson’s model is ………. the atomic
size in Rutherford’s model. (much greater than/no different
from/much less than.)
(b) In the ground state of ………. electrons are in stable equilibrium,
while in ………. electrons always experience a net force.
(Thomson’s model/ Rutherford’s model.)
(c) A classical atom based on ………. is doomed to collapse.
(Thomson’s model/ Rutherford’s model.)
(d) An atom has a nearly continuous mass distribution in a ……….
but has a highly non-uniform mass distribution in ……….
(Thomson’s model/ Rutherford’s model.)
(e) The positively charged part of the atom possesses most of the
mass in ………. (Rutherford’s model/both the models.)


12.2 Suppose you are given a chance to repeat the alpha-particle
scattering experiment using a thin sheet of solid hydrogen in place
of the gold foil. (Hydrogen is a solid at temperatures below 14 K.)
What results do you expect?

12.3 What is the shortest wavelength present in the Paschen series of
spectral lines?


12.4 A difference of 2.3 eV separates two energy levels in an atom. What
is the frequency of radiation emitted when the atom make a
transition from the upper level to the lower level?


12.5 The ground state energy of hydrogen atom is –13.6 eV. What are the
kinetic and potential energies of the electron in this state?


12.6 A hydrogen atom initially in the ground level absorbs a photon,
which excites it to the n = 4 level. Determine the wavelength and
frequency of photon.


12.7 (a) Using the Bohr’s model calculate the speed of the electron in a
hydrogen atom in the n = 1, 2, and 3 levels. (b) Calculate the orbital
period in each of these levels.


12.8 The radius of the innermost electron orbit of a hydrogen atom is
5.3×10–11 m. What are the radii of the n = 2 and n =3 orbits?


12.9 A 12.5 eV electron beam is used to bombard gaseous hydrogen at
room temperature. What series of wavelengths will be emitted?


12.10 In accordance with the Bohr’s model, find the quantum number
that characterises the earth’s revolution around the sun in an orbit
of radius 1.5 × 1011 m with orbital speed 3 × 104m/s. (Mass of earth = 6.0 × 1024 kg.)


ADDITIONAL EXERCISES

12.11 Answer the following questions, which help you understand the
difference between Thomson’s model and Rutherford’s model better.
(a) Is the average angle of deflection of α-particles by a thin gold foil
predicted by Thomson’s model much less, about the same, or
much greater than that predicted by Rutherford’s model?
(b) Is the probability of backward scattering (i.e., scattering of
α-particles at angles greater than 90°) predicted by Thomson’s
model much less, about the same, or much greater than that
predicted by Rutherford’s model?
(c) Keeping other factors fixed, it is found experimentally that for
small thickness t, the number of α-particles scattered at
moderate angles is proportional to t. What clue does this linear
dependence on t provide?
(d) In which model is it completely wrong to ignore multiple
scattering for the calculation of average angle of scattering of
α-particles by a thin foil?

12.12 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.


12.13 Obtain an expression for the frequency of radiation emitted when a
hydrogen atom de-excites from level n to level (n–1). For large n,
show that this frequency equals the classical frequency of revolution
of the electron in the orbit.


12.14 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, 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 (~ 10–10m).
(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 recognizing 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.

12.15 The total energy of an electron in the first excited state of the
hydrogen atom is about –3.4 eV.
(a) What is the kinetic energy of the electron in this state?
(b) What is the potential energy of the electron in this state?
(c) Which of the answers above would change if the choice of the
zero of potential energy is changed?

12.16 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?


12.17 Obtain the first Bohr’s radius and the ground state energy of a
muonic hydrogen atom [i.e., an atom in which a negatively charged
muon (μ– ) of mass about 207me orbits around a proton].