Bohr’s Atomic Model

 

Bohr's model for hydrogen-like system

To address the drawbacks of the Rutherford model and combining some basic laws of classical physics with principles of quantum theory, Niels Bohr proposed his famous atomic model in 1913. This model successfully explains various properties of atoms, such as their stability, spectral characteristics, and ionization energies, etc. Because of this contribution, he was awarded Nobel Prize in 1922. It mainly deals with the hydrogen and hydrogen-like systems such as He⁺, Li²⁺, Be³⁺, etc. where only want electron is present outside the nucleus.

Basic Postulates of the Model

1. The electrons are revolving in some specified circular orbits around the nucleus where the whole positive charge is concentrated. This permitted orbits are restricted by the quantum condition that the angular momentum of the electron in the circular path about the nucleus must be an integral multiple of h/2π, where h is the plank constant, i.e. angular momentum = mur = n(h/2π), where n = 1, 2, 3, 4,..... m = mass of the electron; r = radius of the orbit; and u = tangential velocity of revolving electron.

2. When an electron is revolving in a permitted orbit, it will neither except nor radiate any energy. This situation is called stationary state. It is worth mentioning that this postulate ignores the demand of electromagnetic theory according to which for the revolving charged particle, energy will be radiated in the form of electromagnetic radiation.

3. When an electron jumps from an orbit of energy E_i to another orbit of energy E_f, a definite amount of energy in the form of radiation is emitted or absorb depending upon whether the final state is at lower or higher energy compared to the initial one respectively. The frequency (ν) of radiation is given by:

hν = E_f - E_i

4. For all other considerations, the model will obey the laws of classical physics, such as: For the revolving electron in a circular orbit in a stationary state, the electrostatic Coulombic force of attraction between the nucleus and electron is balanced by centrifugal force (mu²/r) due to angular motion of electron is obtained from Newton's law of motion.

Expression for the Radius of Bohr’s Orbit

Let an electron of charge e and mass m be revolving with the tangential velocity u in a circular orbit of r_n around the nucleus having charge Ze. For hydrogen and different hydrogen- like systems, such as He⁺, Li²⁺, Be³⁺,….. the values of Z are 2, 3, 4….respectively.

For the equilibrium condition, the electrostatic Coulombic force of attraction is balanced by the centrifugal force, as the gravitational force between the electron and the nucleus is negligible. The Coulombic force of attraction (F_n) is given by,

F_e = 1/4πε₀(Ze.e/r_n²) = 1/4πε₀(Ze²/r_n²)

Here, ε₀ is called absolute permittivity of the medium, i.e. vacuum in this case. Its value is 8.85x 10^-12 C²N^-1m^-2.

The centrifugal force (Fc) is given by,

Fc = mu²/r_n

In equilibrium condition F_e = mu²/r_n

In equilibrium condition, F_e = F_c, i.e.

1/4πε₀ × Ze²/r_n² = mu²/r_n

Or, r_n = 1/4πε₀ (Ze²/mu²)…………………………………………………………..(1)

According to the Bohr’s quantum restriction,

mur_n = n(h/2π);

u = nh/ (mr_n2π)……………………………………………………………………….(2)

So from this equation (1) and (2)

r_n = radius of n-th Bohr’s orbit = n²h²ε₀/πme²Z = (4πε₀)n²h²/4mπ²e²Z = r₁n² (SI unit)

In CGS system

Fe = Ze²/r_n²    and 1/4πε₀ = 1,

r_n = n²h²/4π²mZe², (CGS unit)………………………………………………………(3)

 

Expression for Energy of the Revolving Electron in Bohr’s Orbit

The total energy (E_n) of the electron is the sum of potential and kinetic energy where kinetic energy = ½ mu². The potential energy of the electron is given by the work done to bring the electron from infinity where potential energy is zero to r_n. Thus

Potential energy = = ⁿ∫_∞ Fedr  = ⁿ∫_∞ 1/4πε₀(Ze²/r² )dr = - Ze²/4πε₀r_n

The negative sign indicates that the work is done on the electron. Thus,

E_n = ½ mu²- Ze²/4πε₀r_n = ½ (Ze²/4πε₀r_n) - Ze²/4πε₀r_n

    = - Ze²/8πε₀r_n

Now, using the value of r_n from (1) we get

E_n = -me⁴Z²/8ε₀²n²h² = E₁/n², in SI unit

En = -2π²mZ²e⁴/n²h² = E1/n², in CGS unit

Frequency of radiation when an electron jumps from one level to another

Now, if an electron jumps from an outer n_i-th orbit to an inner n_f-th orbit, the frequency of the radiated energy is given by according to Bohr’s postulate,

hν = En_i - En_f

    = Z²e⁴m/8ε₀²h²(1/n_i² – 1/n_f²)

 ν = Z²e⁴m/8ε₀²h³(1/n_i² – 1/n_f²)

Hence, if it is expressed in terms of w_n (=1/λ), it is called wave number and its given by,

w_n = 1/λ = ѵ/c = Z²e⁴m/8ε₀²ch³(1/n_i² – 1/n_f²) = RZ²(1/n_i² – 1/n_f²)

Where c = velocity of light, R = Rydberg constant = e⁴m/8ε₀²ch³

It leads to En = -RchZ²/n²

In CGS system,

w_n = 2π²mZ²e⁴/ch³(1/n_i² – 1/n_f²)

So, Rydberg constant, R = 2π²mZ²e⁴/ch³

Numerical Values of Radius and Energy in Bohr’s Atom

In hydrogen-like systems, the expression of radius (r_n) for the n-th orbit is given by,

r_n = n²h²ε₀/πme²Z (SI unit)

Using the values of,

ε₀ = 8.85 × 10^-12 C²N^-1m^-2, h = 6.625 × 10^-34 Js, m = 9.11 × 10^-31 kg, e = 1.6 × 10^-19 C

Using these values in the above expression we get

 r_n = (n²/Z) × 0.53 ×10^-10 m = (n²/Z) × 53 pm

In CGS system,

r_n = n²h²/4π²mZe²

   = (n²/Z) × 0.53 ×10^-8 cm

So for hydrogen-like systems (e.g. He⁺, Li²⁺, Be³⁺, Ne⁹⁺, etc.) is given by r₁ which can be expressed in terms of r_1(H) as follows:

r(H) = 53 pm

r(He⁺) = 53/2 pm = 26.5 pm (Z = 2 for He⁺)

r(Li²⁺) = 53/3 pm = 17.6 pm (Z = 3 for Li²⁺)

r(Ne⁹⁺) = 53/10 pm = 5.3 pm ( Z = 10 for Ne⁹⁺)

Energy expression of an electron in hydrogen like system in the nth orbit is given by

E_n = -me⁴Z²/8ε₀²n²h² = -RchZ²/n² (in SI unit)

E_n = -21.8 × 10^-19Z²/n² J/atom

    = -21.8×10^-19× 6.023×10²³ Z²/n²

= -1313.3Z²/n² KJ/mole

For hydrogen atom, in the ground state n = 1 and Z = 1. Thus, E₁ = -1313.3 KJ/mole

Similarly in CGS system, En = -21.8×10^-12 (Z²/n²) erg/atom

En = -13.6(Z²/n²) eV/atom, (1 erg = 6.25 × 10¹¹ eV)

So for hydrogen, E₁ = -13.6 eV/ atom

Bohr’s Theory and the Spectral Lines in Hydrogen-like System

Qualitative orbital diagram for hydrogen and hydrogen like system and origin of different spectral lines

Origin of the emission spectral lines at least in the case of hydrogen (H) and hydrogen like systems (i.e., He⁺, Li²⁺, Be³⁺….etc.) can be quantitatively explained in the light of Bohr’s theory. If the electron falls to the first orbit (n = 1) from any higher orbit, the Lyman series is generated. According to the Bohr’s theory, the energy difference between the orbits in which the electron transition occurs is fixed and it leads to a certain frequency. For a particular system, the frequency of the spectral line depends on the values of n_f and n_i. Similarly, for Balmer, Paschen, Brackett and Pfund series the values of n_f are 2, 3, 4 and 5 respectively.

The mathematical formulation for the frequency (ν) and wave number (w_n) for different spectral lines in different series are obtained from Bohr’s theory as follows,

ν = Z²e⁴m/8ε₀²h³|(1/n_i² – 1/n_f²)| = 3.3 ×10¹⁵×Z²|(1/n_i² – 1/n_f²)| cycles/sec

w_n = RZ²|(1/n_i² – 1/n_f²)| = 109.7Z²|(1/n_i² – 1/n_f²)| m^-1

For Lyman series, n_f = 1, and ni = 2 (α-line), 3 (β-line), 4 (γ-line), in the series]

Similarly for

Balmer series, n_f = 2, and n_i = 3, 4, 5, 6….

Paschen series, n_f = 3 and n_i = 4, 5, 6, 7…

Brackett series n_f = 4 and n_i = 5, 6, 7….

Pfund series n_f = 5 and n_i = 6, 7, 8….

Here it is noteworthy that the more corrected values will be obtained if the correction for the finite mass of the nucleus is taken into consideration. However, a fairly good conformity between the calculated and experimental values of ν and w_n gives a sound basis of Bohr’s model. For a particular spectral line in hydrogen- like systems (e.g. He⁺, Be²⁺, Li³⁺, etc.) we get:

w_Be²⁺/w_H = λ_H/λ_Be²⁺ =  (R_Be²⁺/R_H)×16, (Z =1 for H, Z = 4 for Be²⁺)

               =16 (taking R_Be²⁺ = R_H  i.e. the nuclei are supposed to remain at rest)

Similarly, λ_H/λ_Li³⁺ = 9 (Z = 3 for Li³⁺)

 

Ionization and Resonance Potential for Hydrogen and Hydrogen-Like System in the Light of Bohr’s Theory

Resonance and excitation potential is the energy required to excite the electron in an isolated atom or ion (hydrogen-like system, in the present consideration) in gaseous condition from its ground state (n =1) to its next higher orbit (n = 2). Correctly, it is called 1^st resonance or excitation potential. Similarly, the transitions, n = 1 to n = 3 and n = 1 to n = 4 give the measures of 2^nd and 3^rd resonance potentials. In the case of ionization potential, it involves the excitation from n = 1 to n = ∞. Here, it is worth mentioning that the parameters are of energy expression, but they contain the term potential. The term potential is induced mistakenly, because to measure the excitation energy, it is very often imparted by applying a potential gradient. From Bohr’s theory, the following expressions (without considering the nuclear motion) come,

1^st Resonance potential (RP) = E_n2 –E_n1 = Z²e⁴m/8ε₀²h²|(1/n_i² – 1/n_f²)|

                                                               = Z²e⁴m/8ε₀²h²|(1/1² – 1/2²)|

                                                               = 16.35×10^-19 joules/atom or 10.02 eV/ atom

 

Ionization Potential (IP) = E_∞ - E₁

                                       = Z²e⁴m/8ε₀²h²|(1/1² – 1/∞²)|

                                       = 21.8 × 10^-19 Z² joules/atom

It leads to IP = 13.6 eV/ atom = 1313Z² KJ/mole,

For the hydrogen atom (Z = 1), the experimental values are in good conformity with the predicted values from Bohr’s theory.

 

Bohr’s Theory of Correspondence principle

In the microscopic world dealing with the fundamental particles, the principles of quantum physics are not very often governed by the classical physics, but in the macroscopic world which is in our sense, both of them lead to the identical conclusion. This is verified in many cases, such as: in the theory of relativity, theory of radiation and wave property of matter, etc. The same thing can also be proved in Bohr’s theory.

Bohr’s postulates appear to ignore the demand of classical physics for the revolving electron, but in 1923 Bohr established that for larger values of quantum number n, both the quantum theory and classical theory lead to the same result. This is known as Bohr’s correspondence principle.

According to the classical electromagnetic theory, the revolving electron in a circular orbit radiates energy whose frequency is given by the frequency (f) of its revolution and its harmonics (i.e. integral multiple of the frequency of revolution).

Thus, f = u/(2πr); (where u gives the linear velocity of the revolving electron)

In Bohr’s theory,

mu²/r = Ze²/4πε₀r² or u = e(Z/4πε₀rm)^1/2

Hence, f = u/2πr = e/2πr(Z/4πε₀rm)^1/2 = e/2π(Z/4πε₀r³m)^1/2

              = e/2π(Z/4πε₀m)^1/2(πme²Z/n²h²ε₀)^3/2, [ putting r = n²h²ε₀/πme²Z]

              = 2RcZ²/n³, R = e⁴m/8ε₀²ch³

Now let us consider

ν = Z²e⁴m/8ε₀²h³(1/n_i² – 1/n_f²) = RcZ²(1/n_i² – 1/n_f²)

Now, let us consider the case for n_i = n and n_f = n-p, where p = 1, 2, 3……,

Hence, ν = Z²e⁴m/8ε₀²h³[(1/(n-p)² – 1/n²)] = Z²e⁴m/8ε₀²h³[(2pn-p²/n²(n-p)²)]

When n>>>>> p, 2pm-p² = 2pm; and (n-p)² = n²

ν = Z²e⁴m/8ε₀²h³(2p/n³) = 2RcZ²p/n³

When p =1, f and ν becomes identical and the frequency of radiation is the frequency of revolution. For p = 2, 3, 4 …. Harmonics of the fundamental frequency are obtained. Thus, for the very high values of n, of the order of 10⁴, both the results are identical, but for the smaller values of n, there appears a serious discrepancy.

 

Merits of the Bohr's Atomic Model

Stability of the atoms: It explains the stability of the atoms while it is lacking in the Rutherford's model. The instability in the Rutherford's model is overcome by invoking the concept of stationery state.

Prediction of frequency of the spectral lines: The model not only explains the origin of the spectral lines in different series but also gives the mathematics formulations which nicely feet with the experimental observations at least made on the instruments of low resolving power.

Rydberg constant(R): Calculated value of R (considering the nuclear motion) nicely agrees with the experimental one.

Critical potentials- resonance and ionization potentials (Frank Hertz experiment): Bohr's formulation explains the critical potentials quantitatively for hydrogen like systems and also for other systems under some approximations. For the poly-electronics system, it requires the use of effective nuclear charge Z*e instead of the actual nuclear charge Ze. This way, the measured ionization potentials nicely fit with the predicted ones.

Explanation of Mosley's Law: It offers a theoretical basis of Mosley's Law.

Isotopic shifts: By considering the nuclear motion, the spectral shifts for the isotopes of lighter elements such as hydrogen for which mass masses of the isotopes differ significantly. These shifts are experimentally verified by this model.

Mass ratio of proton and electron (M_h/m): Corrected equation for the Rydberg constant (R_corr) by considering the nuclear motion leads to the evaluation of the ratio M_h/m from the spectroscopic studies of He⁺ and H. The ratio determined from other methods is in good agreement with this spectroscopic one.

Bohr's corresponding principal: It indicates that for very higher values of n, the atom behaves classically which is known as Bohr's corresponding principal.

 

Drawbacks of the Bohr's model:

Involvement of both classical and quantum mechanics: Here, the principles of both quantum and classical mechanics have been taken into consideration as and when required. Hence, theoretically the model cannot be supported.

Fine spectra of the spectral lines: The model can explain the origin of the spectral lines along with their frequencies when they are observed with the instruments of low resolving power. But actually each single line is consisting of a number of fine lines which are closely packed. This has been observed by using the instruments of very high resolving power. These fine spectra cannot be explained by the Bohr's theory. For example, the H_a line in Balmar series arises due to transition from n=3 to n=2, and it will lead to a change of a fixed amount of energy, ΔE = E3 - E2 and as a result, only one line of a fixed frequency will be obtained. But the spectral analysis indicates that for the above transition, different amounts of energy change (ΔE1, ΔE2, ΔE3....) occur leading to different fine lines. But these are not expected from Bohr's theory. Here it is worth mentioning that the fine lines are not due to the presence of isotopes.

Intensity of the spectral lines: Though the frequencies of the spectra lines are predicted from this theory, it cannot credit anything regarding they are relative intensities.

Poly-electronic atoms: While this theory is more or less good for the hydrogen and hydrogen like systems, such as He⁺, Li²⁺, Be³⁺, etc. Which contain only one electron, for the system of multi-electrons, this theory is not at all promising.

Quantization of angular momentum: The quantum restriction, mur = n(h/2pi) was taken arbitrarily without giving any sound explanation.

Heisenberg's uncertainty principle: Consideration of a moving electron with a definite velocity in a circular orbit of a definite radius cannot be supported by Heisenberg's uncertainty principle. According to this principal, it is impossible to determine both the velocity and the position of an electron simultaneously with a certainty.

Zeeman Effect and Stark Effect: The single line in the spectrum is found to split into a number of closely spaced lines in the presence of an external magnetic field (Zeeman effect) and electrical field (Stark effect). Such splitting cannot find any support from this theory.

Reference

1) Concise inorganic chemistry by J. D. Lee.

2) Inorganic Chemistry by James E. Huheey, Ellen A Keither, Richard L. Keither, Okhil K. Medhi.

3) Shriver and Atkins Inorganic Chemistry.