Bohr's atomic model: Evidence in the favour of Bohr's theory, limitation of Bohr's atomic model.

Evidences in favour of Bohr's theory:

1. Estimation of ratio of mass of electron to mass of system.

The value of Rydberg's constant for H-atom and He atome are given by, $$R_H= \frac{R_\infty}{1+\frac{m}{M}}$$ Where, m= mass of electron and M= mass of proton $$R_{He}= \frac{R_\infty}{1+\frac{m}{M_{He}}}$$ now, $$\frac{R_{He}}{R_H}= \frac{1+\frac{m}{M}}{1+\frac{m}{M_{He}}}$$ $$or, R_{He}+R_{He}\frac{m}{M_{He}}= R_H+ R_H\frac{m}{M}$$ since \(M_{He}\approx 4M\) $$or, R_{He}-R_H= R_H\frac m M - R_{He}\frac{m}{4m}$$ $$or, R_{He}-R_H=\frac m M \biggl(R_H-\frac{RH_e}{4}\biggr)$$ $$or, \frac{m}{M}= \frac{R_{He}-R_H}{R_H-\frac{R_{He}}{4}}\dotsm(1)$$

By substituting the value of Rydberge constant \(R_{He}=10972240m^{-1}\). We can calculate , \(\frac mM\)

$$\therefore\frac mM = \frac{10972240-10967770}{10972240-\frac{10967770}{4}}$$

$$\frac mM\approx\frac{1}{1836}$$

This ratio is almost equall to the ratio calculated by other methods, it support the validity of Bohr's theory.

2. Wave number corresponding to single ionized Hellium atom.

Singley ionized Hellium atom is similar to that of hydrogen atom except Z= 2 for \(He^+\)

The wave number corresponding to singly ionized Hellium atom is given by,

$$\nu=\frac{1}{\lambda}= R_z.Z^2\biggl(\frac{1}{n_1^2}-\frac{1}{n_2^2}\biggr)\dotsm(1)$$

$$= R_z\bigg(\frac{1}{n_1^2}-\frac{1}{n_2^2}\biggr)\cdot 4$$

So the spectrom corresponding to singlely ionized atom is similar that of hydrogen atom except the wave number of corresponding lines in \(He^+\) is almost 4 times that of H- atom. Which is experimentally valid.

3. Discovery of isotope hydrogen ( H- atom) i.e [ Deuterium and tritium]

According to Bohr's theory for the first line of Balmar series \(n_1=2\) and \(n_2=3\)

For H-atom ( ordinary hydrogen )

$$\frac{1}{\lambda_H}= \frac{R_\infty}{\biggl(1+\frac{m}{M_H}\biggr)}\biggl(\frac{1}{2^2}-\frac{1}{3^2}\bigg)$$

$$\frac{1}{\lambda_H}=\frac{R_\infty}{\biggl(1+\frac{m}{M_H}\biggr)}\frac{5}{36}$$

$$or, \frac{1}{\lambda_H}=\frac{5}{36}\frac{R_\infty}{\biggl(1+\frac{m}{M_H}\biggr)}\dotsm(1)$$

For Deuterium,

$$\frac{1}{\lambda_D}=\frac{R_\infty}{\biggl(1+\frac{m}{M_D}\biggr)}\cdot\biggl(\frac{1}{2^2}-\frac{1}{3^2}\bigg)$$

$$\frac{1}{\lambda_D}=\frac{5R_\infty}{36}\cdot\frac{1}{1+\frac{m}{2m}}$$

$$\lambda_D<\lambda_H$$

Hence all the wavelength corresponding to deuterium is shifted toward shorter wavelength side thean the wavelength in ordinary H- atom. Experimentally it is found that H-\(\alpha\) line of ordinary H- atom has wavelength,

$$\lambda H_\alpha= 6563A^\circ$$

and of deuterium $$\lambda_{D_\alpha}= 6561A^\circ$$

This supports the existance of Bohr's theory.

*Separation of the first line of the Balmar sereis inspectrum of ordinary hydrogen and tritium.Given, \(R_\infty\)= 10 and \(\frac mM= \frac{1}{1836}\).

Solution:

For first line of Balmar series \(n_1\)=2 and \(n_2\)=3

$$\frac{1}{\lambda_H}=\frac{R_\infty}{\bigg(1+\frac mM\biggr)}\biggl(\frac{1}{2^2}-\frac{1}{3^2}\biggr)$$

$$=\frac{5}{36}\frac{R_\infty}{\biggl(1+\frac mM\biggr)}$$

$$or, \lambda_H=\frac{36}{5}\frac{\biggl(1+\frac mM\biggr)}{R_\infty}\dotsm(1)$$

where, m= mass of electron

M= mass of proton

For tritium

$$\frac{1}{\lambda_T}=\frac{R_\infty}{\bigg(1+\frac m{M_T}\biggr)}\biggl(\frac{1}{2^2}-\frac{1}{3^2}\biggr)$$

$$or, \lambda_T= \frac{36}{5}R_\infty\biggl(1+\frac{m}{3M}\biggr)\;\;\;\; [ M_T\approx2M ]\dotsm(2)$$

Seperation between two wavelength

$$\therefore\lambda_H-\lambda_T=\frac{36}{5}R_\infty\biggl(1+\frac mM-1-\frac{m}{2M}\biggr)$$

$$=\frac{36}{5}R_\infty\cdot\frac 23\cdot\frac mM$$

$$=\frac{36}{5\cdot10973740}\cdot\frac 23 \cdot\frac{1}{1836}$$

$$=2.38A^\circ\;\;\;\;\; \lambda_T<\lambda_D<\lambda_H$$

Bohr's corresponding principle

Statement:

" For very large value of quantum number (n) the result of quantum mechanics is in agreement with the reslut of classical mechanics".

According to classical mechanics " The frequency of radiation emmited by an atom is equall to it's frequenc of revolution".

The time period of electrone revolving in \(n^{th}\) orbits is given by,

Time period= \(\frac{ Circumference\; of \;orbit}{Speed \;of electron}\)

$$T= \frac{2{\pi}r_n}{V_n}\dotsm(1)$$

Where, \(r_n\)= radius of \(n^th\) orbit= \(\frac{\epsilon_\circ n^2h^2}{\pi mZe^2}\)

\(V_n\)= speed of electron= \(\frac{Ze^2}{2\epsilon_\circ nh}\dotsm(2)\)

$$or, T=\frac{2\pi\cdot\frac{\epsilon_\circ n^2h^2}{\pi mZe^2}}{\frac{Ze^2}{2\epsilon_\circ nh}}$$

$$or, T= \frac{4\epsilon_{\circ}^2h^3n^3}{mz^2e^4}$$

$$\therefore\;\;\; Frequency=\frac 1T= \frac{mz^2e^4}{4\epsilon_{\circ}^2h^3n^3}\dotsm(3)$$

Equation (3) gives the frequency of radiation emmited or the frequency of revolution of a electron.

According to quantum mechanics ( Bohr's theory ) the energy of radiation emmited during transition of electrone from \(n^th\) level to \((n-1)^{th}\) level is given as,

$$h\nu=E_n-E_{n-1}$$

$$=\frac{-me^4z^2}{8\epsilon_{\circ}^2n^2h^2}-\biggl(-\frac{me^4z^2}{8\epsilon_{\circ}^2(n-1)^2h^2}\biggr)$$

$$=\frac{-me^4z^2}{8\epsilon_{\circ}^2n^2h^2}\biggl[\frac{1}{(n-1)^2}-\frac{1}{n^2}\biggr]$$

$$=\frac{-me^4z^2}{8\epsilon_{\circ}^2n^2h^2}\biggl[\frac{n^2-(n-1)^2}{n^2(n-1)^2}\biggr]$$

$$=\frac{me^4z^2}{8\epsilon_{\circ}^2n^2h^2}\biggl[\frac{n^2-n^2+2n+1}{n^2(n-1)^2}\biggr]$$

$$\nu=\frac{me^4z^2}{8\epsilon_{\circ}^2n^2h^3}\biggl[\frac{2n-1}{n^2(n-1)^2}\biggr]\dotsm(4)$$

For large value of n,

$$2n-1\approx2n$$ $$n-1\approx n$$ $$\nu=\frac{me^4z^2}{8\epsilon_{\circ}^2n^2h^3}\biggl[\frac{2n}{n^2-n^2}\biggr]$$

$$\nu=\frac{me^4z^2}{4\epsilon_{\circ}^2n^3h^3}\dotsm(5)$$

From equation (3)

$$Frequency=\frac 1T=\frac{me^4z^2}{4\epsilon_{\circ}^2n^3h^3}$$

From (3) and (5) we can say that frequency of radiation emmited is equll to the frequency of revolution. This proves Bohr's corresponding principle.

Failure of Bohr's atomic model:

1. Bohr's theory couldn't explain the splitting of \(H_\alpha, H_\beta, H_\gamma\)- line etc. When observed high resolving power microscope \(H_alpha\) line consist of more closely spaced line with seperation ( 0.13\(A^\circ)\).

2. The electron revolves in eleptical orbit around the nucleus. But according to Bohr's theory, orbit are always circular.

3. Bohr's theory couldn't explain splitting of spectral line in magnetic and electric field.

4. Bohr's theory couldn't explain the distribution of large number of electron in an atoms.

Reference:

Reviews of Modern Physics. Lancaster, P.A.: Published for the American Physical Society by the American Institute of Physics, 1952. Print.

Wehr, M. Russell, and James A. Richards. Physics of the Atom. Reading, MA: Addison-Wesley Pub., 1984. Print.

Young, Hugh D., and Roger A. Freedman. University Physics. Boston, MA: Pearson Custom, 2008. Print.

Adhikari, P.B. A Textbook of Physics. 2070 ed. Vol. II. Kathmandu: Sukunda Publication, 2070. Print.