Atomic Structure : Bohr model of hydrogen atom, limitations of Bohr’s model

Bohr’s atomic model.

(1)     This model was based on the quantum theory of radiation and the classical law of physics. It gave new idea of atomic structure in order to explain the stability of the atom and emission of sharp spectral lines.

(2)     Postulates of this theory are :

(i)     The atom has a central massive core nucleus where all the protons and neutrons are present. The size of the nucleus is very small.

(ii)    The electron in an atom revolve around the nucleus in certain discrete orbits. Such orbits are known as stable orbits or non – radiating or stationary orbits.

(iii)   The force of attraction between the nucleus and the electron is equal to centrifugal force of the moving electron.

         Force of attraction towards nucleus = centrifugal force

(iv)   An electron can move only in those permissive orbits in which the angular momentum (mvr) of the electron is an integral multiple of h/2π Thus, \[\text{mvr}=n\frac{\text{h}}{\text{2}\pi }

Where, m = mass of the electron, r = radius of the electronic orbit, v = velocity of the electron in its orbit.

(v)    The angular momentum can be  \[\frac{h}{2\pi },\,\,\,\frac{2h}{2\pi },\,\,\,\frac{3h}{2\pi },......\,\,\frac{nh}{2\pi }. This principal is known as quantization of angular momentum. In the above equation ‘n’ is any integer which has been called as principal quantum number. It can have the values = 1, 2, 3, ——- (from the nucleus). Various energy levels are designed as K (= 1), L(= 2), M(= 3) ——- etc. Since the electron present in these orbits is associated with some energy, these orbits are called energy levels.

(vi)   The emission or absorption of radiation by the atom takes place when an electron jumps from one stationary orbit to another.

(vii) The radiation is emitted or absorbed as a single quantum (photon) whose energy hv is equal to the difference in energy ΔE of the electron in the two orbits involved. Thus, hv = ΔE

Where ‘h’ =Planck’s constant, v = frequency of the radiant energy. Hence the spectrum of the atom will have certain fixed frequency.

(viii) The lowest energy state (= 1) is called the ground state. When an electron absorbs energy, it gets excited and jumps to an outer orbit. It has to fall back to a lower orbit with the release of energy.

 

(3)     Advantages of Bohr’s theory

(i)     Bohr’s theory satisfactorily explains the spectra of species having one electron, viz. hydrogen atom, He+, Li2+ etc.

(ii)    Calculation of radius of Bohr’s orbit : According to Bohr, radius of orbit in which electron moves is

\[r=\left[ \frac{{{h}^{2}}}{4{{\pi }^{2}}m{{e}^{2}}k} \right].\frac{{{n}^{2}}}{Z}

where, n = Orbit number, m =Mass number 9.1 × 10−31/kg = Charge on the electron 1.6 × 10−19   Z = Atomic number of element, k = Coulombic constant 9 × 109 Mn2c−2

After putting the values of mekh, we get.

\[{{r}_{n}}=\frac{{{n}^{2}}}{Z}\times 0.529{\AA}\ or\ {{r}_{n}}=\frac{{{n}^{2}}}{Z}\times 0.529nm

(a)    For a particular system [e.g., H, He+ or Li+2]

r ∞ n2 [Z = constant]

Thus we have   \[\frac{{{r}_{1}}}{{{r}_{2}}}=\frac{n_{1}^{2}}{n_{2}^{2}}   i.e., r1 : r2 : r3…….. : 1 : 4 : 9…….r1 < r2 < r3

(b)    For particular orbit of different species

\[r\propto \frac{1}{Z} [Z =constant] Considering A and B species, we have  \[\frac{{{r}_{A}}}{{{r}_{B}}}=\frac{{{Z}_{B}}}{{{Z}_{A}}}

Thus, radius of the first orbit H, He+, Li+2 and Be+3 follows the order: H > He+ > Li+2 > Be+2

(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}}

For H atom, \[{{V}_{n}}=\frac{2.188\times {{10}^{8}}}{n}cm.\,\,{{\sec }^{-1}}

(a)    For a particular system [H, He+ or Li+2]

\[\text{V}\propto \frac{\text{1}}{\text{n}} [Z = constant] Thus, we have, \[\frac{{{V}_{1}}}{{{V}_{2}}}=\frac{{{n}_{2}}}{{{n}_{1}}}

The order of velocity is

\[{{V}_{1}}>{{V}_{2}}>{{V}_{3}}.........\,\,\,or\,\,\,\,{{V}_{1}}:{{V}_{2}}:{{V}_{3}}..........:1:\frac{1}{2}:\frac{1}{3}........

(b)    For a particular orbit of different species

V ∝ Z [n =constant] Thus, we have H < He+ < Li+2

(c)     For H or He+ or Li+2, we have

V1 : V2 = 2 : 1 ; V1 : V3 = 3 : 1 ; V4 = 4 : 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……….∞

Putting the value of m, e, k, h,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)=-313.6\times \frac{{{Z}^{2}}}{{{n}^{2}}}kcal./mole

(1 cal = 4.18J) or \[\frac{-1312}{{{n}^{2}}}{{Z}^{2}}kJmo{{l}^{-1}}

(a)    For a particular system[H, He+ or Li+2]

\[E\propto -\frac{1}{{{n}^{2}}} [Z =constant] Thus, we have \[\frac{{{E}_{1}}}{{{E}_{2}}}=\frac{n_{2}^{2}}{n_{1}^{2}}

The energy increase as the value of n increases

(b)    For a particular orbit of different species

E ∝ Z2 [n =constant] Thus, we have \[\frac{{{E}_{A}}}{{{E}_{B}}}=\frac{Z_{A}^{2}}{Z_{B}^{2}}

For the system H, He+, Li+2, Be+3 (n-same) the energy order is H > He+ > Li+2 > Be+3

The energy decreases as the value of atomic number Z increases.

When an electron jumps from an outer orbit (higher energy)n2 to an inner orbit (lower energy)n1 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 and  \[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 109678 cm−1

 

(4)     Quantisation of energy of electron

(i)     In ground state : No energy emission. In ground state energy of atom is minimum and for 1st orbit of H-atom, n=1.

E1 = − 13.6eV

(ii)    In excited state : Energy levels greater than n1 are excited state. i.e., for H-atom n2, n3, n4 are excited state.  For H– atom first excitation state is = n2

(iii)   Excitation potential : Energy required to excite electron from ground state to any excited state.

Ground state → Excited state

Ist excitation potential = E2 – E1 = − 3.4 + 13.6 = 10.2 eV.

IInd excitation potential =E3 – E1 = − 1.5 + 13.6 = 12.1 eV.

(iv)   Ionisation energy: The minimum energy required to relieve the electron from the binding of nucleus.

\[{{E}_{\text{ionisation}}}={{E}_{\infty }}-{{E}_{n}}=+13.6\frac{Z_{\text{eff}\text{.}}^{\text{2}}}{{{n}^{2}}}eV.

(v)    Ionisation potential\[{{V}_{\text{ionisation}}}=\frac{{{E}_{ionisation}}}{e}

(vi)   Separation energy : Energy required to excite an electron from excited state to infinity.

S.E. = E – Eexcited

(vii) Binding energy : Energy released in bringing the electron from infinite to any orbit is called its binding energy (B.E.).

Note :   Principal Quantum Number ‘n‘ = \[\sqrt{\frac{13.6}{(B.E.)}}

(5)     Spectral evidence for quantisation (Explanation for hydrogen spectrum on the basis of 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, Paschal, Bracket, Pound 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, 677.581 cm−1

This remarkable agreement between the theoretical and experimental value was great achievment of the Bohr model.

(iii)   Although H- atom consists only one electron yet it’s spectra consist of many spectral lines as shown in fig.

(iv)    Comparative study of important spectral series of Hydrogen

 

S.No. Spectral series Lies in the region Transition

n2 > n1

\[{{\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 n1  = 1

n2 = 2,3,4…∞

n1 = 1 and n2 = 2

\[{{\lambda }_{\text{max}}}=\frac{4}{3R}

n1 = 1 and n2 = ∞ \[\frac{4}{3} 
(2) Balmer series Visible region n1 = 2

n2 = 3,4,5…∞

 n1 = 2 and n2  = 3

\[{{\lambda }_{\text{max}}}=\frac{36}{5R}

n1 = 2 and n2 = ∞

\[{{\lambda }_{\text{min}}}=\frac{4}{R}

\[\frac{9}{5}
(3) Paschen series Infra red region n1 = 3

n2 = 4,5,6…∞

n1 = 3 and n2 = 4

\[{{\lambda }_{\text{max}}}=\frac{144}{7R}

n1 = 3 and n2 = ∞

\[{{\lambda }_{\text{min}}}=\frac{9}{R}

\[\frac{16}{7}
(4) Brackett series Infra red region n1 = 4

 

n2 = 5,6,7…∞

n1 = 4 and n2 = 5

\[{{\lambda }_{\text{max}}}=\frac{16\times 25}{9R}

n1 = 4 and n2 = ∞

\[{{\lambda }_{\text{min}}}=\frac{16}{R}

 

 \[\frac{25}{9}
(5) Pfund series Infra red region n1 = 5

n2 = 6,7,8…∞

n1 = 5 and n2 = 6

\[{{\lambda }_{\text{max}}}=\frac{25\times 36}{11R}

n1 = 5 and n2 = ∞

\[{{\lambda }_{\text{min}}}=\frac{25}{R}

 \[\frac{36}{11}
(6) Humphrey series Far infrared region n1 = 6

n2 = 7,8…∞

n1 = 6 and n2 = 7

\[{{\lambda }_{\text{max}}}=\frac{36\times 49}{13R}

n1 = 6 and n2 = ∞

\[{{\lambda }_{\text{min}}}=\frac{36}{R}

 \[\frac{49}{13}

 

(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.

 

(6)     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.

 

Bohr – Sommerfeld’s model.

(1)     In 1915, Sommerfield introduced a new atomic model to explain the fine spectrum of hydrogen atom.

(2)     He gave concept that electron revolve round the nucleus in elliptical orbit. Circular orbits are formed in special conditions only when major axis and minor axis of orbit are equal.

(3)     For circular orbit, the angular momentum = \[\frac{nh}{2\pi } where n = principal quantum number only one component i.e. only angle changes.

(4)     For elliptical orbit, angular momentum = vector sum of 2 components. In elliptical orbit two components are,

(i)     Radial component (along the radius) = \[{{n}_{r}}\,\frac{h}{2\pi }

Where, nr = radial quantum number   

(ii)    Azimuthal component = nφ \[\frac{h}{2\pi }

Where, nφ = azimuthal quantum number

So angular momentum of elliptical orbit = \[n\frac{h}{_{r}2\pi }\,+\,{{n}_{\varphi }}\frac{h}{2\pi }

Angular momentum = \[(n{}_{r}+\,\,n{}_{\varphi })\,\frac{h}{2\pi }

(5)     Shape of elliptical orbit depends on,   

\[\frac{\text{Length}\,\,\text{of}\,\,\text{major}\,\text{axis}}{\text{Length}\,\,\text{of}\,\,\text{minor}\,\text{axis}}=\,\frac{n}{{{n}_{\varphi }}}=\frac{{{n}_{r}}+{{n}_{\varphi }}}{{{n}_{\varphi }}}     

(6)     nφ can take  all  integral  values  from  l to ‘n’ values of n depend   on the  value  of nφ. For n = 3, nφ can have values 1, 2, 3 and nr can have (n –1) to zero i.e. 2, 1 and zero respectively. 

Thus for n = 3, we have 3 paths

n                 nφ               nr                Nature of path

3                 1                 3                 elliptical

                    2                 1                 elliptical

                    3                 0                 circular

 

The possible orbits for n = 3 are shown in figure.

Thus Sommerfield showed that Bohr’s each major level was composed of several sub-levels. Therefore it provides the basis for existance of subshells in Bohr’s shells (orbits).

 

(7)     Limitation of Bohr sommerfield model :  

(i)     This model could not account for, why electrons does not absorb or emit energy when they are moving in stationary orbits.

(ii)    When electron jumps from inner orbit to outer orbit or vice –versa, then electron run entire distance but absorption or emission of energy is discontinuous.

(iii)   It could not explain the attainment of expression of \[\frac{nh}{2\pi } for angular momentum. This model could not explain Zeeman effect and Stark effect.