Bohr's atom model
Following some serious objections to Rutherford's atom model, Bohr suggested some modifications in the atom model. The modified atom model is called Bohr's atom model whose basic postulates are described below :
(i) The central part of the atom called nucleus, contains whole of positive charge and almost whole of the mass of atom. Electrons revolve round the nucleus in fixed circular orbits.
(ii) Electrons are capable of revolving only in certain fixed orbits, called stationary orbits or permitted orbits. In such orbits they do not radiate any energy.
(iii) When revolving in permitted orbit an electron possesses angular momentum 'L' ( = mvr) which is an integral multiple of h/2π
i.e., L = n . h/2π
where 'n' is an integral and 'h' is planck's constant.
(iv) Electrons are capable of changing the orbits. On absorbing energy they move to a higher orbit while emission of energy takes place when electrons move to a lower orbit. If 'f' is the frequency of radiant energy,
hf = W₂ − W₁
Here
W₁ = enerɡy of electron in lower orbit
and
W₂ = enerɡy of electron in hiɡher orbit
(v) All the laws of mechanics can be applied to electron revolving in a stable orbit while they are not applicable to an electron in transition.
Bohr's theory of atomic structure
Consider an electron of mass 'm' moving round a nucleus having a charge + Ze (Fig. 1) where 'Z' is the atomic number. If 'r' is the radius of orbit, centripetal force 'Fe' required by the electron.
 |
| Fig. 1. Electron revolving around a nucleus. |
Fe = k (mv²/r) ... (1)
The force is provided by the electrostatic attraction between the nucleus and the electron. Electrostatic force 'Fe' is given by
Fe = (1/4πɛ₀) × (q₁ × q₂/r²)
Here q₁ = Ze and q₂ = e,
Fe = (1/4πɛ₀) × (Ze × e/r²)
or Fe = (1/4πɛ₀) × (Ze²/r²) ... (2)
From equations (1) and (2), we get
mv²/r = (1/4πɛ₀) × (Ze²/r²)
∴ mv² = (1/4πɛ₀) × (Ze²/r) ... (3)
According to the basic postulate of bohr's atomic model,
L = mvr = n . h/2π ... (4)
(i) Orbital velocity of electron
Dividing equations (3) and (4), we get
mv²/mvr = (1/4πɛ₀) × [(Ze²/ r) × (2π/nh)]
∴ Vn = (1/4πɛ₀) × (2π Ze²/nh) ... (5)
'Vn' denotes the velocity of electrons in nth orbit.
Equation (5) indicates that :
(i) For a particular orbit (n = constant), orbital velocity of electron varies directly as the atomic number of the substance, i.e., Vn ∝ Z.
(ii) For a particular element (Z = constant), orbital velocity of the electron varies inversely as the order of the orbit
i.e., Vn ∝ 1/n
Let v₁ be the velocity of electron in the first orbit (n = 1)
v₁ = 1/4πɛ₀ ... (6)
Dividing equations (5) by (6), we get
Vn/v₁ = 1/n
∴ Vn = v₁/n ... (7)
(ii) Radius of orbit of electron
From equation (4)
v = nh/2πmr
Substituting for 'v' in equation (3), we get
m (n²h²/4π²m²r²) = (1/4πɛ₀) × (Ze²/r)
∴ r = 4πɛ₀ × (n²h²/4π²mZe²) ... (8)
r = (ɛ₀/π) × (n²h²/mZe²) ... (9)
Since 'n' can have only integral values, only those orbits are possible which have radii corresponding to n = 1, 2, 3, ...... Thus, the orbits are quantised.
Equation (8) indicates that :
(i) Radius of a particular orbit of electron (n = constant) varies inversely as the atomic number i.e., r ∝ 1/Z. It means heavier the element, shorter is the radius of orbit.
(ii) For a particular element (Z = constant) radii of different orbits vary directly as the square of the order of orbit i.e., r ∝ n². It means the outer radii will be more spaced apart than the inner ones.
(iii) Energy of electron. As an electron revolves around the nucleus in a stable orbit, it possesses energy which is composed of two energies.
(a) Kinetic energy. It is the energy possessed by the electron by virtue of its motion in the orbit.
If 'v' is the velocity of electron,
K.E. = 1/2 (mv²)
Substituting for 'mv²' from equation (3), we get
K.E. = 1/2 (1/4πɛ₀) × (Ze²/r)
(b) Potential energy. It is the energy possessed by the electron by virtue of its position near the nucleus. Potential energy of two charges q₁ and q₂ separated a distance 'r' apart is
P.E. = (1/4πɛ₀) × (q₁q₂/r)
Here q₁ = Ze and q₂ = -e
∴ P.E. = (1/4πɛ₀) × (Ze × (-e)/r)
or P.E. = - (1/4πɛ₀) × (Ze²/r)
(c) Total energy. Total energy 'W'of an electron revolving round the nucleus is
W = K.E. + P.E.
= ½ (1/4πɛ₀) × (Ze²/r) - (1/4πɛ₀) × (Ze²/r)
= (1/4πɛ₀) × (Ze²/r) × (½ - 1)
= - ½ × (1/4πɛ₀) × (Ze²/r)
Substituting for 'r' from equation (8), we get
W = - ½ (1/4πɛ₀) × [Ze²/(ɛ₀/n) (n²h²/mZe²)]
W = - (1/8ɛ₀²) × (Z²me⁴/n²h²) ... (10)
Energy of an electron revolving in an orbit is negative. It means that electron is bound to the nucleus.
Equation (10) indicates that :
(i) energy of an electron revolving in a particular orbit (n = constant) varies directly as the square of the atomic number of the atom. Because of the negative sign an electron in nth orbit of a heavier element is less energetic than that of lighter element in nth orbit.
(ii) Energy of an electron of a particular element (Z = constant) varies inversely as the square of the order of the orbit. Again, because of negative sign the electrons in he outer orbits of an element are more energetic than those in inner orbits.
(iii) Frequency, wavelength and wave number of radiation. When an electron jumps from a higher orbit to a lower one, the difference of the energy is emitted in the form of a radiation. If the electron jumps from an orbit (n = n₂) of energy 'W₂' to one (n = n₁) of energy 'W₁', according to the basic postulate of Bohr's theory,
hf = W₂ - W₁
W₁ = - (1/4πɛ₀)² × (2π²Z²me⁴/n₁²h²)
and W₂ = - (1/4πɛ₀)² × (2π²Z²me⁴/n₂²h²)
∴ hf = [- (1/4πɛ₀)² × (2π²Z²me⁴/n₂²h²)] - [- (1/4πɛ₀)² × (2π²Z²me⁴/n₁²h²)]
∴ f = (1/4πɛ₀)² × (2π²Z²me⁴/h²) × [1/n₁² - 1/n₂²]
or f = (1/4πɛ₀)² × (2π²Z²me⁴/h³) × [1/n₁² - 1/n₂²] ... (11)
If 'λ' is the wavelength of radiation,
f = c/λ
where 'c' is the velocity of radiation (light).
∴ c/λ = (1/4πɛ₀)² × (2π²Z²me⁴/h³) × [1/n₁² - 1/n₂²]
∴ 1/λ = (1/4πɛ₀)² × (2π²Z²me⁴/ch³) × [1/n₁² - 1/n₂²] ... (12)
Since 1/λ = ̄f (wave number of radiation)
∴ ̄f = (1/4πɛ₀)² × (2π²Z²me⁴/ch³) × [1/n₁² - 1/n₂²] ... (13)
or ̄f = RZ² (1/n₁² - 1/n₂²)
where R = (1/4πɛ₀)² × (2π²me⁴)/ch³ is called Rydberg's constant.
∴ R = (1/4πɛ₀)² × (2π²me⁴/ch³)
or R = (1/8ɛ₀²) × (me⁴/ch³)
Value of R is 10973731 m-1 or 1.0973731 ×107 m-1
In S.I. (k = 1/4πɛ₀)
̄f = (1/8ε₀²) × (Z²me⁴/ch³) (1/n₁² - 1/n₂²) ... (14)
Equation (14) indicates that :
(i) Wave number and hence the wavelength of emitted radiation depends upon the order of two orbits between which the transition takes place.
(ii) For a particular transition (n₁ and n₂ constant) the radiation emitted by a heavier element possesses greater wave number and hence the smaller wavelength.
Bohr's theory of hydrogen atom
Hydrogen atom has atomic number 'Z' as one. It contains one electron revolving round the nucleus. Putting Z = 1, we get quantities connected with hydrogen atom.
(i) Radius of orbit. From equation (9), we get
r = (ɛ₀/π) × (n²h²/me²)
(ii) Energy of electron. From equation (10), we get
W = - (1/8ɛ₀²) × (me⁴/n²h²)
(iii) Frequency, wavelength and wave number of radiation.
Putting Z = 1 in equation (11), (12) and (13),
f = (1/8ɛ₀²) × (me⁴/h³) × (1/n₁² - 1/n₂²)
1/λ = (1/8ɛ₀²) × (me⁴/ch³) × (1/n₁² - 1/n₂²)
̄f = (1/8ɛ₀²) × (me⁴/ch³) × (1/n₁² - 1/n₂²)
Hydrogen Spectrum
As transition of electron takes place from a higher orbit to a lower orbit, difference of energy is radiated in the form of radiation. The wavelength of the radiation depends upon the initial and final orbit within which the transition takes place. Accordingly a number of series are emitted. Each series is composed of a number of lines (Fig. 2).
_copy_7932x9600.jpg) |
| Fig. 2. Production of hydrogen spectrum. |
(i) Lymen series. This is a series in which all the lines correspond to transition of electrons from a higher excited state to orbit having n = 1, i.e., n₁ = 1 and n₂ = 2, 3, 4 ... wave number of lines constituting 'Lymen series' are given by
̄f = R [1/(1²) - 1/n²]
where n = 2, 3, 4, .....
and 'R' is the Rydberg's constant.
This series lies /observed in ultraviolet region
(ii) Balmer series. This is a series in which all the lines correspond to transition of electrons from higher excited state to the orbit having n = 2
i.e., n₁ = 2, n₂ = 3, 4, 5, ....
Therefore, wave numbers of lines constituting 'Balmer series' are given by
̄f = R (1/2² - 1/n²) where n = 3, 4, 5, ...
First member of this series corresponds to the transition of electron from 3rd to 2nd orbit.
∴ n₁ = 2, n₂ = 3
∴ 1/λ = R (1/2² - 1/3²) = R (1/4 - 1/9)
or 1/λ = 5R/36 ∴ λ = 36/5R
Substituting R = 1.09737 × 10
λ = 6563 ̊A
The limiting case of this series is given by n₂ = ∞
∴ 1/λ = R (1/2²) = R/4
∴ λ = 4/R = 3646 ̊A
The value of wavelength indicates that the series lies in the visible region.
(iii) Paschen series. This is a series in which all the lines correspond to transition of electrons from a higher excited state to the orbit having n = 3,
i.e., n₁ = 3, and n₂ = 4, 5, 6, 7, .....
∴ Wave-numbers of lines constituting 'Paschen series' are given by,
̄f = R (1/3² - 1/n²) where n = 4, 5, 6, 7, ....
This series lies/observed in infrared region
(iv) Brackett series. This is a series in which all the lines correspond to transition of electrons from a higher excited state to the orbit having n = 4,
i.e., n₁ = 4, and n₂ = 5, 6, 7, ......
Therefore, wave number of lines constituting 'Brackett series' are given by,
̄f = R (1/4² - 1/n²) where n = 5, 6, 7, ....
This series lies/observed in infrared region.
(v) P-fund series. This is a series in which all the lines correspond to the transition of electrons from a higher excited state to the orbit having n = 5,
i.e., n₁ = 5 and n₂ =6, 7, 8, .....
̄f = R (1/5² - 1/n²) where n = 6, 7, 8, ....
This series lies/observed in infrared region.
Energy Level of Hydrogen Atom
The energy 'W' of an electron revolving round the nucleus is
W = - [(1/8ɛ₀²) × (me⁴/n²h²),
m = 9.1 × 10-31 kg
e = 1.59 × 10-19 C,
h = 6.67 × 10-34 joule sec.
For the innermost orbit n = 1. Energy 'W₁' of electron in the innermost orbit is given by
W₁ = - (9 × 109)²
2 × (3.142)² × 9.1 × 10-31 × (1.59 × 10-19)⁴/(6.67 × 10-34)²
= - 21.78 × 10-19 J.
Since 1eV = 1.6 × 10-19 J.
∴ W1 = - (21.78× 10-19/1.6 ×10-19)
= - 13.6 eV
For the 1st excited state,
n = 2
W₂ = W₁/4 = - (13.6/4)
= - 3.4 eV
For 2nd excited state,
n = 3
W₃ = W₁/9 = − 13.6/9 eV
= − 1.51 eV.
Similarly, for other excited states
W₄ = − 0.85 eV
and W₅ = − 0.54 eV
_copy_8709x9600.jpg) |
| Fig. 3. Energy level diagram for hydrogen atoms. |
Various energy levels are shown in Fig. 3. The set of spectral lines is also shown in figure.
Excitation and Ionisation Potentials
(i) Excitation potential. Under normal conditions, the orbital electrons keep on revolving around the nucleus in fixed orbits. When exposed to incident energy from outside, the electrons absorb energy and have a tendency to shift to higher orbits. In this state the atom is said to be existing in excited state.
Excitation potential of a particular state is the minimum accelerating potential to which if the electron is subjected, it acquires just the required amount of energy to reach a desired higher orbit.
Illustration. Let us suppose we want to take the electron of hydrogen atom from first to second orbit.
Energy of electron in first orbit = - 13.6 eV
Energy of electron in second orbit = - 3.4 eV
Required amount of energy = (-3.4 eV) - (-13.6 eV) = 10.2 eV
This energy can be acquired by the electron if it is accelerated between two electrodes having a potential difference of 10.2 eV.
Therefore, the excitation potential for hydrogen atom from first to second orbit is 10.2 V.
Similarly, it can be calculated that the excitation potentials of hydrogen atom from first to third, fourth and fifth orbits are respectively 8.69 V, 9.66 V.
(ii) Ionisation potential. An electron is said to be just free if its energy has zero value. This will happen for n = ∞ only. The atom, in this stage is said to be ionised.
Ionisation potential of particular orbital electron is the minimum accelerating potential to which if the electron is subjected, it acquires just the required amount of energy to be displaced to the outermost orbit (n = ∞).
Since hydrogen possesses only one electron and it requires 13.6 eV of energy to be displayed to the outermost orbit, therefore, ionisation potential of hydrogen is 13.6 V. Atoms which possess large number of electrons possess a variety of ionisation potentials varying with the variation in the order of orbit initially occupied by the electron.
Limitations to Bohr's Theory
A close examination of hydrogen spectrum reveals that a line, in any of the series consists of a number of lines packed very close to each other. These lines can only be seen with the help of high resolving power microscope.
Presence of these lines could not be explained on the basis of Bohr's theory which assumes that orbits of electrons around the nucleus are circular. Some field modified the theory by assuming the orbits to be elliptical and was able to explain the fine structure of atom. The theory was further modified by taking into account the spin of electron and relativistic variation of mass.
Key Formulae
1. Bohr's theory of atomic structure
(i) Orbital velocity of electron
vn = (1/4πɛ₀) × (2πZe²/nh)
(ii) Radius of orbit of electron
r = (ɛ₀/π) × (n²h²/mZe²)
(iii) Energy of electron
W = - [(1/8ɛ₀²) × (Z²me⁴/n²h²)]
(iv) Frequency, wavelength and wave number of radiation
f = (1/4πɛ₀)² × (2π²Z²me⁴/h³) × [1/n₁² - 1/n₂²]
̄f = 1/λ = (1/4πɛ₀)² × (2π²Z²me⁴/ch³) × [1/n₁² - 1/n₂²]
2. Hydrogen Spectrum
(i) Lymen series :
̄f = R [1/(1)² - 1/n²]
where n = 2, 3, 4, .....
(ii) Balmer series :
̄f = R [1/(2)² - 1/n²]
where n = 3, 4, 5, .....
(iii) Paschen series :
̄f = R [1/(3)² - 1/n²]
where n = 4, 5, 6, .....
(iv) Brackett series:
̄f = R [1/(4)² - 1/n²]
where n = 5, 6, 7, .....
(v) P-fund series:
̄f = R [1/(5)² - 1/n²]
where n = 6, 7, 8, .....
3. Energy level of hydrogen atom
W₁ = - 13.6 eV, W₂ = -3.4 eV,
W₃ = -1.51 eV, W₄ = -0.85 eV,
W₅ = -0.54 eV
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