Bohr's Atomic Model

**Category : **JEE Main & Advanced

Bohr retained the essential features of the Rutherford model of the atom. However, in order to account for the stability of the atom he introduced the concept of the stationary orbits. The Bohr postulates are,

(1) An atom consists of positively charged nucleus responsible for almost the entire mass of the atom (This assumption is retention of Rutherford model).

(2) The electrons revolve around the nucleus in certain permitted circular orbits of definite radii.

(3) The permitted orbits are those for which the angular momentum of an electron is an intergral multiple of \[h/2\pi \] where \[h\] is the Planck?s constant. If \[m\] is the mass and \[v\] is the velocity of the electron in a permitted orbit of radius \[r,\] then

\[L=mvr=\frac{nh}{2\Pi }\]; \[n=1\], 2, 3, ??\[\infty \]

Where \[L\] is the orbital angular momentum and \[n\] is the number of orbit. The integer \[n\] is called the principal quantum number. This equation is known as the Bohr quantization postulate.

(4) When electrons move in permitted discrete orbits they do not radiate or lose energy. Such orbits are called stationary or non-radiating orbits. In this manner, Bohr overcame Rutherford?s difficulty to account for the stability of the atom. Greater the distance of energy level from the nucleus, the more is the energy associated with it. The different energy levels were numbered as 1,2,3,4 .. and called as \[K,\,L,\,M,\,N,\]?. etc.

(5) Ordinarily an electron continues to move in a particular stationary state or orbit. Such a state of atom is called ground state. When energy is given to the electron it jumps to any higher energy level and is said to be in the excited state. When the electron jumps from higher to lower energy state, the energy is radiated. Advantages of Bohr?s theory (i) Bohr?s theory satisfactorily explains the spectra of species having one electron, viz. hydrogen atom, \[H{{e}^{+}},L{{i}^{2+}}\]etc. (ii) Calculation of radius of Bohr?s orbit : According to Bohr, radius of nth orbit in which electron moves is \[{{r}_{n}}=\left[ \frac{{{h}^{2}}}{4{{\pi }^{2}}m{{e}^{2}}k} \right].\frac{{{n}^{2}}}{Z}\]

Where, \[n=\]Orbit number, \[m=\]Mass number \[\left[ 9.1\times {{10}^{-31}}kg \right]\,,\]\[e=\]Charge on the electron \[\left[ 1.6\times {{10}^{-19}} \right]\] \[Z=\]Atomic number of element, k = Coulombic constant \[\left[ 9\times {{10}^{9}}N{{m}^{2}}{{c}^{-2}} \right]\]

After putting the values of m,e,k,h, we get.

\[{{r}_{n}}=\frac{{{n}^{2}}}{Z}\times 0.529\overset{{}^\circ }{\mathop{\text{A}}}\,\]

(iii) **Calculation of velocity of electron**

\[{{V}_{n}}=\frac{2\pi {{e}^{2}}ZK}{nh},\,{{V}_{n}}={{\left[ \frac{Z{{e}^{2}}}{mr} \right]}^{1/2}}\];\[{{V}_{n}}=\frac{2.188\times {{10}^{8}}Z}{n}cm.{{\sec }^{-1}}\]

(iv) **Calculation of energy of electron in Bohr's** orbit Total energy of electron = K.E. + P.E. of electron \[=\frac{kZ{{e}^{2}}}{2r}-\frac{kZ{{e}^{2}}}{r}=-\frac{kZ{{e}^{2}}}{2r}\]

Substituting of r, gives us \[E=\frac{-2{{\pi }^{2}}\,m{{Z}^{2}}{{e}^{4}}{{k}^{2}}}{{{n}^{2}}{{h}^{2}}}\] Where, n=1, 2, 3???.\[\infty \]

Putting the value of m, e, k, h,\[\pi \]we get

\[E=21.8\times {{10}^{-12}}\times \frac{{{Z}^{2}}}{{{n}^{2}}}erg\,per\,atom\]

\[=-21.8\times {{10}^{-19}}\times \frac{{{Z}^{2}}}{{{n}^{2}}}J\,per\,atom\,(1J=\text{1}{{\text{0}}^{\text{7}}}erg)\]

\[E=-13.6\times \frac{{{Z}^{2}}}{{{n}^{2}}}eV\,per\,atom\text{(1eV}=\text{1}\text{.6}\times \text{1}{{\text{0}}^{-19}}J)\]

\[=-13.6\times \frac{{{Z}^{2}}}{{{n}^{2}}}k.cal/mole\] (1 cal = 4.18J) or \[\frac{-1312}{{{n}^{2}}}{{Z}^{2}}kJmo{{l}^{-1}}\]

When an electron jumps from an outer orbit (higher energy) \[{{n}_{2}}\]to an inner orbit (lower energy)\[{{n}_{1}},\]then the energy emitted in form of radiation is given by

\[\Delta E={{E}_{{{n}_{2}}}}-{{E}_{{{n}_{1}}}}=\frac{2{{\pi }^{2}}{{k}^{2}}m{{e}^{4}}{{Z}^{2}}}{{{h}^{2}}}\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)\,\] \[\Rightarrow \ \Delta E=13.6{{Z}^{2}}\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)eV/atom\] As we know that \[E=h\bar{\nu },\]\[c=\nu \lambda \]and \[\bar{\nu }=\frac{1}{\lambda }\] \[=\frac{\Delta E}{hc},\] \[=\frac{2{{\pi }^{2}}{{k}^{2}}m{{e}^{4}}{{Z}^{2}}}{c{{h}^{3}}}\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)\]

This can be represented as \[\frac{1}{\lambda }=\bar{\nu }=R{{Z}^{2}}\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)\]

Where, \[R=\frac{2{{\pi }^{2}}{{k}^{2}}m{{e}^{4}}}{c{{h}^{3}}}\]; R is known as Rydberg constant. Its value to be used is \[109678c{{m}^{-1}}.\]

The negative sign in the above equations shows that the electron and nucleus form a bound system, i.e., the electron is attracted towards the nucleus. Thus, if electron is to be taken away from the nucleus, energy has to be supplied. The energy of the electron in \[n=1\] orbit is called the ground state energy; that in the \[n=2\] orbit is called the first excited state energy, etc. When \[n=\infty \] then \[E=0\] which corresponds to ionized atom i.e., the electron and nucleus are infinitely separated \[H\to {{H}^{+}}+{{e}^{-}}\] (ionization).

(6) **Spectral evidence for quantisation** (Explanation for hydrogen spectrum on the basisof bohr atomic model)

(i) The light absorbed or emitted as a result of an electron changing orbits produces characteristic absorption or emission spectra which can be recorded on the photographic plates as a series of lines, the optical spectrum of hydrogen consists of several series of lines called Lyman, Balmar, Paschen, Brackett, Pfund and Humphrey. These spectral series were named by the name of scientist who discovered them.

(ii) To evaluate wavelength of various H-lines Ritz introduced the following expression,

\[\bar{\nu }=\frac{1}{\lambda }=\frac{\nu }{c}=R\left[ \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right]\]

Where, R is =\[\frac{2{{\pi }^{2}}m{{e}^{4}}}{c{{h}^{3}}}=\] Rydberg's constant

It's theoritical value = 109,737 cm?1 and It's experimental value = \[109,677.581c{{m}^{-1}}\] This remarkable agreement between the theoretical and experimental value was great achievment of the Bohr model.

(iii) Although H-atom consists of only one electron yet it's spectra consist of many spectral lines.

(iv) Comparative study of important spectral series of Hydrogen is shown in following table.

(v) If an electron from nth excited state comes to various energy states, the maximum spectral lines obtained will be

= \[\frac{n(n-1)}{2}.\] n= principal quantum number. As n=6 than total number of spectral lines

= \[\frac{6(6-1)}{2}=\frac{30}{2}=15.\]

(vi) Thus, at least for the hydrogen atom, the Bohr theory accurately describes the origin of atomic spectral lines.

(7) **Failure of Bohr model **

(i) Bohr theory was very successful in predicting and accounting the energies of line spectra of hydrogen i.e. one electron system. It could not explain the line spectra of atoms containing more than one electron.

(ii) This theory could not explain the presence of multiple spectral lines.

(iii) This theory could not explain the splitting of spectral lines in magnetic field (Zeeman effect) and in electric field (Stark effect). The intensity of these spectral lines was also not explained by the Bohr atomic model.

(iv) This theory was unable to explain of dual nature of matter as explained on the basis of De broglies concept.

(v) This theory could not explain uncertainty principle.

(vi) No conclusion was given for the concept of quantisation of energy

S.No. |
Spectral series |
Lies in the region |
Transition \[{{n}_{2}}>{{n}_{1}}\] |
\[{{\lambda }_{\text{max}}}=\frac{n_{1}^{2}n_{2}^{2}}{(n_{2}^{2}-n_{1}^{2})R}\] |
\[{{\lambda }_{\text{min}}}=\frac{n_{1}^{2}}{R}\] |
\[\frac{{{\lambda }_{\text{max}}}}{{{\lambda }_{\text{min}}}}=\frac{n_{2}^{2}}{n_{2}^{2}-n_{1}^{2}}\] |

(1) | Lymen series | Ultraviolet region | \[{{n}_{1}}=1\] \[{{n}_{2}}=2,3,4....\infty \] | \[{{n}_{1}}=1\text{ }\,\text{and }\,{{n}_{2}}=2\] \[{{\lambda }_{\text{max}}}=\frac{4}{3R}\] | \[{{n}_{1}}=1\,\text{ and}\,{{n}_{2}}=\infty \] \[{{\lambda }_{\text{min}}}=\frac{1}{R}\] | \[\frac{4}{3}\] |

(2) | Balmer series | Visible region | \[{{n}_{1}}=2\] \[{{n}_{2}}=3,4,5....\infty \] | \[{{n}_{1}}=2\,\text{ and }\,{{n}_{2}}=3\] \[{{\lambda }_{\text{max}}}=\frac{36}{5R}\] | \[{{n}_{1}}=2\,\text{ and }\,{{n}_{2}}=\infty \] \[{{\lambda }_{\text{min}}}=\frac{4}{R}\] | \[\frac{9}{5}\] |

(3) | Paschen series | Infra red region | n1 = 3 \[{{n}_{2}}=4,5,6....\infty \] | \[{{n}_{1}}=3\,\text{ and }\,{{n}_{2}}=4\] \[{{\lambda }_{\text{max}}}=\frac{144}{7R}\] | \[{{n}_{1}}=3\,\text{ and}\,{{n}_{2}}=\infty \] \[{{\lambda }_{\text{min}}}=\frac{9}{R}\] | \[\frac{16}{7}\] |

(4) | Brackett series | Infra red region | \[{{n}_{1}}=4\] \[{{n}_{2}}=5,6,7....\infty \] | \[{{n}_{1}}=4\,\text{ and }\,{{n}_{2}}=5\] \[{{\lambda }_{\text{max}}}=\frac{16\times 25}{9R}\] | \[{{n}_{1}}=4\,\text{ and}\,{{n}_{2}}=\infty \] \[{{\lambda }_{\text{min}}}=\frac{16}{R}\] | \[\frac{25}{9}\] |

(5) | Pfund series | Infra red region | \[{{n}_{1}}=5\] \[{{n}_{2}}=6,7,8....\infty \] | \[{{n}_{1}}=5\,\text{ and }\,{{n}_{2}}=6\] \[{{\lambda }_{\text{max}}}=\frac{25\times 36}{11R}\] | \[{{n}_{1}}=5\,\text{ and}\,{{n}_{2}}=\infty \] \[{{\lambda }_{\text{min}}}=\frac{25}{R}\] | \[\frac{36}{11}\] |

(6) | Humphrey series | Far infrared region | \[{{n}_{1}}=6\] \[{{n}_{2}}=7,8....\infty \] | \[{{n}_{1}}=6\,\text{ and }\,{{n}_{2}}=7\] \[{{\lambda }_{\text{max}}}=\frac{36\times 49}{13R}\] | \[{{n}_{1}}=6\,\text{ and}\,{{n}_{2}}=\infty \] \[{{\lambda }_{\text{min}}}=\frac{36}{R}\] | \[\frac{49}{13}\] |

*play_arrow*Composition of atom*play_arrow*Atomic number, Mass number and Atomic species*play_arrow*Atomic number, Mass number and Atomic species*play_arrow*Electromagnetic Radiations*play_arrow*Atomic spectrum - Hydrogen spectrum*play_arrow*Plum Pudding Model Of Thomson*play_arrow*Rutherford's Nuclear Model*play_arrow*Planck's Quantum Theory*play_arrow*Photoelectric Effect*play_arrow*Bohr's Atomic Model*play_arrow*Dual Nature Of Electron*play_arrow*Heisenberg's Uncertainty Principle*play_arrow*Schrodinger Wave Equation*play_arrow*Quantum Numbers*play_arrow*Rules For Filling Of Electrons In Various Orbitals*play_arrow*Rules For Filling Of Electrons In Various Orbitals*play_arrow*Electronic Configurations Of Elements

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