#### Exercise

**Question 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 inRutherford’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.)**

**Sol.** **(a)** No different from**(b)** Thomson’s model; Rutherford’s model**(c)** Rutherford’s model**(d)** Thomson’s model; Rutherford’s model**(e)** Both the models.

**Note:** In the Rutherford model of atom. Atom is electrically neutral sphere consisting of a very small, massive and positively charged nucleous at the centre, surrounded by revolving electrons.

**Question 2. Suppose you are given a chance to repeat the alpha-particle scattering experiment using a thin sheet solid hydrogen in place of the gold foil. (Hydrogen is a solid at temperatures below 14 K.) What results do you expect?**

**Sol.** The alpha particle scattering for large impact parameters would be the same. For smaller impact parameters the scattering would reduce considerably as hydrogen being a small atom would not be able to exert much force on the bombarding a-particle. Even for head-on collision there would be hardly any scattering for the same season.

**Question 3. What is the shortest wavelength present in the Paschen series of spectral lines?**

**Question 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?**

**Question 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?**

**Question 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.**

**Question 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.**

**Question 8. The radius of the innermost electron orbit of a hydrogen atom is 5.3 x 10 ^{-11}m.What are the radii of then= 2 and n = 3 orbits?**

**Question 9. A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. What series of wavelengths will be emitted?**

**Question 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 x 1011 m with orbital speed 3 x 104 mis. (Mass of earth= 6.0 x 1024 kg.)**

##### ADDITIONAL NCERT EXERCISES

**Question 11. Answer the following questions, which you understand the difference between Thomson’s model and Rutherford’s model better.**

**(a) Is the average angle of deflection of a-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 a -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 a-particles scattered at moderate angles is proportional tot. 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 a -particles by a thin foil?**

**Sol.**

- About the same as we are talking about average angle of deflection.
- Much less as in Thomson’s model there is no such thing as a massive central core called nucleus.
- This implies that scattering of a-particles is due to single collision only. If thickness increases then chances of single collision increases because the number of target atoms would increase. Thus, moderately scattered a-particles would increase in number.
- In Thomson’s model, positive charge is uniformly spreadout, hence there would be hardly any noticeable deflection due to single collision. Hence, multiple scattering has to be considered for average scattering angle. In Rutherford’s model most of the scattering is due to single collision hence multiple scattering can be ignored.

**Note:** In Rutherford model, most of an atom is empty space, so most alpha particles go through it undeviated or with a small deviation.

**Question 12. The gravitational attraction between electron and proton ina hydrogen atom is weaker than the coulomb attraction by a factor of about 10 ^{-4}°. 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.**

**Question 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.**

**Note:** The period of revolution of electron in Bohr sorbit is proportional to the cube of quantum number.

**Question 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 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.**

**Question 15. The total energy of an electron in the first excited state of the hydrogen 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?**

**Question 16. If Bohr’s quantisation postulate (angular momentum= nh/2 n:) 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.**

**Sol. **In Bohr’s quantisation angular momentum is proportional to h (Planck’s constant). But angular momenta associated with planetary motion are of the order of 1O70 h. This would mean n = 1 O70. For such large values of n, the differences in successive energies and angular momenta are so small that they are continuous and not discrete.

**Question 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 207 me orbits around a proton].**

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