UNIT 2.3 ATOMIC STRUCTURE - BOHR MODEL

LEC: 7       Evidence for the Quantized Electronic Energy Levels: Atomic spectra

The restriction of any property to discrete values is called quantization. The speed of light depends upon the nature of the medium through which it passes. As a result, the beam of light is deviated or refracted from its original path as it passes from one medium to another.

It is observed that when a ray of white light is passed through a prism, the wave with shorter wavelength (the violet light) bends more than the one with a longer wavelength (The light of red). Since ordinary white light consists of waves with all the wavelengths in the visible range, a ray of white light is spread out into a series of coloured bands called spectrum. Such a spectrum is called continuous spectrum because violet merges into blue, blue into green and so on.

When electromagnetic radiation interacts with matter, atoms and molecules may absorb energy and reach to a higher energy state. With higher energy, these are in an unstable state. For returning to their normal (more stable, lower energy states) energy state, the atoms and molecules emit radiations in various regions of the electromagnetic spectrum.

Emission and Absorption Spectra

The spectrum of the visible light, as discussed above, was continuous as all wavelengths (red to violet) of the visible light are represented in the spectra. The emission spectra of atoms in the gas phase, on the other hand, do not show a continuous spread of wavelength from red to violet, rather they emit light only at specific wavelengths with dark spaces between them. Such spectra are called line spectra or atomic spectra because the emitted radiation is identified by the appearance of bright lines in the spectra

Atoms, molecules or ions that have absorbed radiation are said to be “excited”. To produce an emission spectrum, energy is supplied to a sample by heating it or irradiating it and the wavelength

(or frequency) of the radiation emitted, as the sample gives up the absorbed energy, is recorded.

An absorption spectrum is like the photographic negative of an emission spectrum. The spectrum of radiation emitted by a substance that has absorbed energy is called an emission spectrum.

A continuum of radiation is passed through a sample which absorbs radiation of certain wavelengths. The missing wavelength which corresponds to the radiation absorbed by the matter, leave dark spaces in the bright continuous spectrum this is called absorption spectum. The study of emission or absorption spectra is referred to as spectroscopy.

Line emission spectra are of great interest in the study of electronic structure. Each element has a unique line emission spectrum. The characteristic lines in atomic spectra can be used in chemical analysis to identify unknown atoms in the same way as finger prints are used to identify people.

Of all the elements, hydrogen atom has the simplest line spectrum. Line spectrum becomes more and more complex for heavier atom. There are however certain features which are common to all line spectra, i.e there is regularity in the line spectrum of each element.

BOHR’S MODEL FOR HYDROGEN ATOM

Neils Bohr (1913) was the first to explain quantitatively the general features of hydrogen atom structure and its spectrum. Though the theory is not the modern quantum mechanics, it can still be used to rationalize many points in the atomic structure and spectra. Bohr’s model for hydrogen atom is based on the following postulates:

i) The electron in the hydrogen atom can move around the nucleus in a circular path of fixed radius and energy. These paths are called orbits, stationary states or allowed energy states. These orbits are arranged concentrically around the nucleus.

ii) The energy of an electron in the orbit does not change with time. However, the electron will move from a lower stationary state to a higher stationary state when required amount of energy is absorbed by the electron or energy is emitted when electron moves from higher stationary state to lower stationary state The energy change does not take place in a continuous manner.

iii) The frequency of radiation absorbed or emitted when transition occurs between two stationary states that differ in energy by ΔE is given by

Where E₁ and E₂ are the energies of the lower and higher allowed energy states respectively. This expression is commonly known as Bohr’s frequency rule.

iv) An electron can move only in those orbits for which its angular momentum is integral multiple of  h/2π  that is why only certain fixed orbits are allowed.) The angular momentum of an electron in a given stationary state can be expressed as in equation 

The details regarding the derivation of energies of the stationary states used by Bohr, are quite complicated and will be discussed in higher classes. However, according to Bohr’s theory for hydrogen atom:

a) The stationary states for electron are numbered n = 1,2,3………. These integral numbers are known as Principal quantum numbers.

b) The radii of the stationary states are expressed as r_n = 52.9 x n² pm
Thus the radius of the first stationary state, called the Bohr radius, is 52.9 pm. Normally the electron in the hydrogen atom is found in this orbit (that is n=1). As n increases the value of r will increase. In other words the electron will be present away from the nucleus.

c) The most important property associated with the electron, is the energy of its stationary state. It is given by the expression

where R_H is called Rydberg constant and its value is 2.18×10^–18 J. The energy of the lowest state, also called as the ground state, is E₁ = -2.18×10^-18 J. and for n = 2, will be, E₂ = -2.18×10^-18 ( 1/2²) = -0.545×10^-18 J.

What does the negative electronic energy (En) for hydrogen atom mean?

The energy of the electron in a hydrogen atom has a negative sign for all possible orbits, This negative sign means that the energy of the electron in the atom is lower than the energy of a free electron at rest. A free electron at rest is an electron that is infinitely far away from the nucleus and is assigned the energy value of zero. Mathematically, E_∞=0. As the electron gets closer to the nucleus (as n decreases), E_n becomes larger in absolute value and more and more negative. The most negative energy value is given by n=1 which corresponds to the most stable orbit. We call this the ground state.
When the electron is free from the influence of nucleus, the energy is taken as zero. the electron in this situation is associated with the stationary state of Principal Quantum number n = ∞ and is called as ionized hydrogen atom. When the electron is attracted by the nucleus and is present in orbit n, the energy is emitted and its energy is lowered. That is the reason for the presence of negative sign in energy equation and depicts its stability relative to the reference state of zero energy and n = ∞.

d) Bohr’s theory can also be applied to the ions containing only one electron, similar to that present in hydrogen atom. For example, He⁺ Li²⁺, Be³⁺ and so on. The energies and radii of the stationary states associated with these kinds of ions (also known as hydrogen like species) are given by the expression.

where Z is the atomic number and has values 2, 3 for the helium and lithium atoms respectively. From the above equations, it is evident that the value of energy becomes more negative and that of radius becomes smaller with increase of Z . This means that electron will be tightly bound to the nucleus.

e) It is also possible to calculate the velocities of electrons moving in these orbits.

qualitatively the magnitude of velocity of electron increases with increase of positive charge on the nucleus and decreases with increase of principal quantum number.

Explanation of Line Spectrum of Hydrogen

Line spectrum observed in case of hydrogen atom, can be explained quantitatively using Bohr’s model. According to assumption 2, radiation (energy) is absorbed if the electron moves from the orbit of smaller Principal quantum number to the orbit of higher Principal quantum number, whereas the radiation (energy) is emitted if the electron moves from higher orbit to lower orbit. 

The energy gap between the two orbits is given by equation

The frequency (ν ) associated with the absorption and emission of the photon can be evaluated by 

In case of absorption spectrum, n_f > n_i and the term in the parenthesis is positive and energy is absorbed. On the other hand in case of emission spectrum n_i > n_f , Δ E is negative and energy is released.

Limitations of Bohr’s Model

Bohr’s model of the hydrogen atom was no doubt an improvement over Rutherford’s nuclear model, as it could account for the stability and line spectra of hydrogen atom and hydrogen like ions (for example, He⁺, Li²⁺ , Be³⁺, and so on). However, Bohr’s model was too simple to account for the following points.

i) It fails to account for the finer details (doublet, that is two closely spaced lines) of the hydrogen atom spectrum observed by using sophisticated spectroscopic techniques. This model is also unable to explain the spectrum of atoms other than hydrogen, for example, helium atom which possesses only two electrons.

 ii) Further, Bohr’s theory was also unable to explain the splitting of spectral lines in the presence of magnetic field (Zeeman effect) or an electric field (Stark effect).

iii) It could not explain the ability of atoms to form molecules by chemical bonds. In other words, taking into account the points mentioned above, one needs a better theory which can explain the salient features of the structure of complex atoms.

iv) It violates the Heisenberg Uncertainty Principle because it considers electrons to have both a known radius and orbit.

v) The Bohr Model provides an incorrect value for the ground state orbital angular momentum.

vi) The wave character of the electron is not considered in Bohr model.