Bohr’s Original Quantum Theory of Spectra

Regarded as one of the greatest discoveries, Niels Bohr’s original quantum theory of spectra was one of the most revolutionary and successful theory of atomic dynamics.

Both Thomson and Rutherford recognized that the electrons must revolve about the nucleus in order to avoid falling into it. They along, with Bohr, realized that according to Maxwell’s theory, accelerated charges revolving with orbital frequency f should radiate light waves of frequency f. Unfortunately, pushed to its logical conclusion, this classical model leads to failure. As the electron radiates energy, its orbit radius steadily decreases and its frequency of revolution increases. This leads to an ever-increasing frequency of emitted radiation and an ultimate catastrophic collapse of the atom as the electron plunges into the nucleus.

These deductions of electrons falling into the nucleus and a continuous emission spectrum from elements were boldly circumvented by Bohr. He simply hypothesized that classical radiation theory, which had been confirmed by Hertz’s detection of radio waves using large circuits, did not hold for atomic-sized systems. Furthermore, he drew on the work of Planck and Einstein as sources of the correct theory of atomic systems. He overcame the problem of a classical electron that continually lost energy by applying Plank’s ideas of quantized energy levels to orbiting atomic electrons. Thus he suggested that electrons in atoms are generally confined to certain stable, non-radiating energy levels and orbits known as stationary states. He employed Einstein’s concept of the photon to arrive at an expression of the frequency of the light when the electron jumps from one stationary state to another. Therefore, if ΔE is the separation of two possible electronic stationary states, then ΔE = hf, where h is the Planck’s constant and f is the frequency of the emitted light regardless of the frequency of the electron’s orbital motion. In this manner, by combining certain principles of classical mechanics with new quantum principles of light emission, Bohr arrived at a theory of the atom that agreed outstandingly with experiment!

Also Read: Bohr’s Correspondence Principle

Bohr’s Model of the Hydrogen Atom

Let’s consider the diagram below representing Bohr’s model of the hydrogen atom:

The following as the basic ideas of the Bohr Theory as it applies to an atom of hydrogen:

• The electron moves in circular orbits about the proton under the influence of the Coulomb force of attraction as illustrated in the figure above.
• Only certain orbits are stable. These stable orbits are ones in which the electron does not radiate. Thus the energy is fixed or stationary in time, and ordinary classical mechanics may be used to describe the electron’s motion in these stable orbits.
• Radiation is emitted by the atom when the electron “jumps” from a more energetic initial stationary state to a less energetic lower state. This “jump” cannot be visualized or treated classically. In particular, the frequency f of the photon emitted in the jump is independent of the frequency of the electron’s orbital motion. Instead, the frequency of the light emitted is related to the change in the atom’s energy and is given by the Plank-Einstein formula:  E_i – E_f = hf,where, E_i is the energy of the initial state, E_f is the energy of the final state, and Ei > E_f.
• The size of the allowed electron orbits is determined by an additional quantum condition imposed on the electron’s orbital angular momentum. Namely, the allowed orbits are those for which the electron’s orbital angular momentum about the nucleus is an integral multiple of h = h/2 .

             m_evr = nh        n = 1, 2, 3, . . .  (Equation 1.0)

We can use the above postulates to calculate the allowed energy levels and emission wavelengths of the hydrogen atom. The electrical potential energy of the system shown in figure above is given by:

U = qV = -ke²/r, where k is the Coulomb constant.

Therefore, the total energy of the atom, which contains both kinetic and potential energy terms, is:

Applying the Newton’s second law to this system, we can notice that the Coulomb attractive force on the electron, ke²/r², must equal to the product of mass and the centripetal acceleration of the electron, that is:

From the above expression we can find the kinetic energy to be:

Substituting the value of K into Equation 1.1 gives the total energy of the atom as:

Notice that the total energy is negative, indicating a bound electron-proton system. This implies that energy in the amount of ke²/2r must be added to the atom to remove the electron to infinity and leave it motionless. An expression for r, the radius of the electron orbit, may be obtained by eliminating v between Equation 1.0 and Equation 1.3

Equation 1.5 shows that only particular orbits are allowed and that these preferred orbits follow from the non-classical step of requiring the electron’s angular momentum to be an integral multiple of h. The smallest radius occurs for n = 1, and is termed to as the Bohr radius denoted a₀. The value for the Bohr radius is:

The actuality that Bohr’s theory gave a value for a₀ in good agreement with the experimental size of hydrogen without any empirical calibration of orbit size was regarded as a remarkable achievement for this theory.

The quantization of the orbit radii immediately leads to energy quantization. This can be seen by substituting r_n = n²a₀ into Equation 1.4, giving for the allowed energy levels:

Inserting numerical values into Equation 1.7, (recall k is the Coulomb constant given by 8.988 x 10⁹ Nm²/C² and e is the magnitude of the electronic charge = 1.602 x 10^-19 C), we get:

The integers, n corresponding to the discrete or quantized, values of the atom’s energy have the unique name quantum numbers. Quantum numbers are fundamental to quantum theory and in general refer to the set of integers that label the discrete values of key atomic quantities, such as energy and angular momentum. The lowest stationary, or non-radiating, state is called the ground state, has n = 1, and has an energy E₁ = – 13.6 eV. The next state, or first excited state, has n = 2 and an energy E₂ = E₁/2² = – 3.4 eV. This is illustrated in the energy-level diagram Figure 1.1 below showing the energies of these discrete energy states and the corresponding quantum numbers.

The uppermost level, corresponding to n = ∞ or r = ∞ and E = 0, represents the state for which the electron is removed from the atom and is motionless. The minimum energy needed to ionize the atom i.e. to completely remove an electron in the ground state from the proton’s influence is called the ionization energy. As shown in Figure 1.1, the ionization energy for the hydrogen based on Bohr’s calculation is 13.6 eV. This represented another key achievement for the Bohr Theory, since the ionization energy for the hydrogen had already been measured to be precisely 13.6 eV.

We can apply Equation 1.7 with Bohr’s third assumption [refer to Bohr’s ideas as applied to hydrogen atom above] to calculate the frequency of the photon emitted when the electron jumps from an outer orbit to an inner orbit:

Since the quantity actually measured is wavelength, it is appropriate to convert frequency to wavelength using c =fλ so that we get:

The outstanding actuality is that the theoretical expression, Equation 2.0 is identical to Balmer’s empirical relation:

Provided that the combination of constants ke²/2a₀hc is equal to the experimentally established Rydberg constant, R = 1.0973732 x 10⁷ m^-1. After Bohr confirmed the agreement of these two quantities to a precision of about 1%, it was recognized as the greatest achievement of his quantum theory of hydrogen.

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