Gujarat Board Solutions for Chapter: ATOMS, Exercise 2: ADDITIONAL EXERCISES

Answer the following question, which helps you understand the difference between Thomson’s model and Rutherford’s model better.

Keeping other factors fixed, it is found experimentally that for small thickness $t$, the number of  $\alpha$-particles scattered at moderate angles is proportional to $t$. What clue does this linear dependence on $t$ provide?

Answer the following question, which helps you understand the difference between Thomson’s model and Rutherford’s model better.

In which model is it completely wrong to ignore multiple scattering for the calculation of average angle of scattering of $\alpha -$ particles by a thin foil?

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.

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.

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 $( ~{10}^{-10}\mathrm{m})$.

(a) Construct a quantity with the dimensions of length from the fundamental constants $e,m_e$ 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 is not expected to play any role. This is what may have suggested Bohr to discard 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 $h$,$m_e$ and $e$ will yield the right atomic size. Construct a quantity with the dimension of length from $h$, $m_e$and $e$ and confirm that its numerical value has indeed the correct order of magnitude.

The total energy of an electron in the first excited state of the hydrogen atom is about $-3.4 \mathrm{eV}$.

(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?

If Bohr’s quantization postulate (angular momentum $=\frac{nh}{2\pi }$) 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 quantization of orbits of planets around the sun?

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 $\left({\mu }^{-}\right)$ of mass about $207 {\mathrm{m}}_{\mathrm{e}}$, orbits around a proton]