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Additional Exercises
Question 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? Answer (a) about the same The average angle of deflection of α-particles by a thin gold foil predicted by Thomson’s model is about the same size as predicted by Rutherford’s model. This is because the average angle was taken in both models. (b) much less The probability of scattering of α-particles at angles greater than 90° predicted by Thomson’s model is much less than that predicted by Rutherford’s model. (c) Scattering is mainly due to single collisions. The chances of a single collision increases linearly with the number of target atoms. Since the number of target atoms increase with an increase in thickness, the collision probability depends linearly on the thickness of the target. (d) Thomson’s model It is wrong to ignore multiple scattering in Thomson’s model for the calculation of average angle of scattering of α−particles by a thin foil. This is because a single collision causes
very little deflection in this model. Hence, the observed average scattering angle can be explained only by considering multiple scattering.
Question 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. Answer Radius of the first Bohr orbit is given by the relation,
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:
Gravitational force of attraction between an electron and a proton is given as:
Where,
Now, the relation for energy (E 2 ) of radiation at level (n − 1) is given as:
Energy (E) released as a result of de-excitation: E = E 2 −E 1 hν = E 2 − E 1 … (iii) Where, ν = Frequency of radiation emitted Putting values from equations (i) and (ii) in equation (iii), we get:
For large n, we can write
Classical relation of frequency of revolution of an electron is given as:
Where, Velocity of the electron in the nth^ orbit is given as:
v = And, radius of the nth^ orbit is given as:
r = Putting the values of equations (vi) and (vii) in equation (v), we get:
Hence, the frequency of radiation emitted by the hydrogen atom is equal to its classical orbital frequency.
Question 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−^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
0 = Permittivity of free space
And, The numerical value of the taken quantity will be:
Hence, the value of the quantity taken is of the order of the atomic size.
Question 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? Answer (a) Total energy of the electron, E = −3.4 eV Kinetic energy of the electron is equal to the negative of the total energy. K = −E = − (− 3.4) = +3.4 eV Hence, the kinetic energy of the electron in the given state is +3.4 eV. (b) Potential energy (U) of the electron is equal to the negative of twice of its kinetic energy. U = −2 K = − 2 × 3.4 = − 6.8 eV
Hence, the potential energy of the electron in the given state is − 6.8 eV. (c) The potential energy of a system depends on the reference point taken. Here, the potential energy of the reference point is taken as zero. If the reference point is changed, then the value of the potential energy of the system also changes. Since total energy is the sum of kinetic and potential energies, total energy of the system will also change.
Question 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? Answer We never speak of quantization of orbits of planets around the Sun because the angular momentum associated with planetary motion is largely relative to the value of Planck’s constant (h). The angular momentum of the Earth in its orbit is of the order of 10^70 h. This leads to a very high value of quantum levels n of the order of 10^70. For large values of n, successive energies and angular momenta are relatively very small. Hence, the quantum levels for planetary motion are considered continuous.
Question 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]. Answer
Mass of a negatively charged muon, According to Bohr’s model,
Bohr radius,
And, energy of a ground state electronic hydrogen atom,
