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.

(i) Atom, as a whole, is electrically neutral and therefore contains equal amount of positive and negative charges.

(ii) In Thomson’s model, an atom is a spherical cloud of positive charges with electrons embedded in it.

2. Rutherford’s atomic model:

(i) In Rutherford’s model, most of the mass of the atom and all its positive charge are concentrated in a tiny nucleus (typically one by ten thousand the size of an atom), and the electrons revolve around it.

(ii) Rutherford's model predicts that atoms are unstable because the accelerated electrons revolving around the nucleus must spiral into the nucleus. This contradicts the stability of matter.

(iii) It cannot explain the characteristic line spectra of atoms of different elements.

3. Spectral series:

The atomic hydrogen emits a line spectrum consisting of various series. The frequency of any line in a series can be expressed as
Lyman series: $f= Rc\left(\frac{1}{1^2}-\frac{1}{n^2}\right); n=\mathrm{2,3},4,\dots$
Balmer series: $f= Rc\left(\frac{1}{2^2}-\frac{1}{n^2}\right); n=\mathrm{3,4},5,\dots$
Paschen series: $f= Rc\left(\frac{1}{3^2}-\frac{1}{n^2}\right); n=\mathrm{4,5},6,\dots$
Brackett series: $f= Rc\left(\frac{1}{4^2}-\frac{1}{n^2}\right); n=\mathrm{5,6},7,\dots$
Pfund series: $f= Rc\left(\frac{1}{5^2}-\frac{1}{n^2}\right); n=\mathrm{6,7},8,\dots$

4. Bohr's atomic model:

(i) Niels Bohr proposed a model for hydrogenic (single electron) atoms. He introduced three postulates and laid the foundations of quantum mechanics.

(ii) When an electron transits from higher energy level to lower energy level a photon is emitted having energy equal to the energy difference between the initial and final states. The frequency of the emitted photon is then given by: $hv= E_i-E_f$ 

(iii) An atom absorbs radiation of the same frequency the atom emits, in which case the electron is transferred to an orbit with a higher value of n.$E_i+hv= E_f$

(iv) For the electron revolving in a stationary orbit is $L= nh/2\pi$ where n is an integer called a quantum number.

(v) In a hydrogen atom, an electron revolves in certain stable orbits (called stationary orbits) without the emission of radiant energy.

(vi) For a hydrogen atom radius of nth energy level, $r_n= \left(\frac{n^2}{m}\right){\left(\frac{h}{2\pi }\right)}^2\frac{4\pi {\epsilon }_0}{e^2}$

(vii) Energy of the electron in nth energy level $E_n= -\frac{m{e}^4}{8n^2{\epsilon }_0^2{h}^2}= -13.6\frac{1}{{\mathrm{n}}^2} \mathrm{e}\mathrm{V}$

(viii) Bohr’s model is applicable only to hydrogenic (single electron) atoms.

5. de Broglie's explanation of bohr's second postulate of quantisation:

de Broglie’s hypothesis that electrons have a wavelength $\lambda = h/mv$ gave an explanation for Bohr’s quantised orbits by bringing in the wave particle duality. The orbits correspond to circular standing waves in which the circumference of the orbit equals a whole number of wavelengths.