UNN CHM 101 Note on Bohr’s Model of Atom (Hydrogen)

In 1913, Bohr put forward a theory based on the quantization of energy to improve upon Rutherford’s model of the structure of the atom. This theory also satisfactorily explained the line spectrum of the hydrogen atom. He postulated that;

The electrons move around the nucleus in one of the several fixed circular orbits called energy levels. These energy levels are arranged concentrically around the nucleus and are characterized by an integer n, the lowest level being given the number 1. The energy level corresponding to n=1, 2, 3, 4 … is also known as K, L, M, N… shells.

Electrons can move about only in certain orbit which has specific energies. Their movement is possible in only those orbits for which its angular momentum is an integral multiple of h/2π or mvr = nh/2π ·        where ‘n’ is an integer 1, 2, 3, 4 …n.

The energy level nearer to the nucleus has the lowest energy, whereas that farther form has maximum energy. An electron is said to be the in-ground state when it moves in the lowest energy level and is the most stable state of the atom. As long as the electrons remain in an orbit, it does not lose energy. These orbits are hence called “stationary orbits.”

Energy is emitted or absorbed when an electron moves from one level to another. Thus by absorbing one particular quantum of energy, the electron will jump from an energy level 1 to 2 or 2 to 3. It is then said to be EXCITED. The quantum of energy absorbed in each case is equal to the difference in energies of the two levels. An electron cannot have an energy that would place it in between the two permissible orbits.

When an electron moves from a higher energy E2 orbit to a lower energy (E1) orbit, the energy (ΔE = E2 – E1) is emitted in the form of a photon of frequency V such that ΔE = E2 – E1 = hv.

Limitations of Bohr’s Model

It could not explain the spectra of atoms containing more than one electron. It could not be applied to even a simple helium atom that has two electrons.

It failed to account for the splitting of lines into finer lines as observed using spectroscopes of high resolving power.

It failed to account for the splitting of spectral lines in presence of a magnetic field (Zeeman’s effect) or an electric field (stark effect).

Heisenberg’s Uncertainty Principle

Werner Heisenberg (1927) a German physicist, stated an uncertainty principle which is the consequence of a dual behaviour of matter and radiation. It states that “it is impossible to determine simultaneously the exact position and exact velocity of an electron. This principle is an important feature of wave mechanics and discusses the relationship between a pair of conjugate properties i.e. those properties that are interdependent. For example, the position and momentum of a moving article are interdependent and these conjugate properties. If the measurement of the position of the particle correspondingly becomes less precise vice versa.

The principle applies to all bodies but becomes significant when applied to bodies with small mass and is negligible in the case of large objects.

The momentum ΔP should be along the direction x for the principle to hold well. If the product of uncertainty in position ∆x and momentum along the Y direction is considered, it will be zero. i.e. Δx . ΔPy = 0.

The effect of the Heisenberg uncertainty principle is significant only for the motion of microscopic objects and is negligible for that of macroscopic objects.

Important Features of the Quantum Mechanical Model of Atom

The picture of the structure of the atom which emerges from the application of Schrodinger’s equation is given by the Quantum Mechanical Model of the atom. Its main features are;

The electrons in atoms can have only certain specific values, hence it is said that the energy of electrons on atoms is quantized (i.e. can only have certain specific values).

The existence of a quantized electronic energy level is a direct result of the wave-like properties of electrons and are allowed solutions of the Schrodinger wave equation.

From Heisenberg’s uncertainty principle, it is established that both the exact position and exact velocity of an electron in an atom cannot be determined simultaneously hence the concept of probability of finding the electron at different points in an atom arises.

An atomic orbital is the wave function Ψ for an electron in an atom. An atom is said to occupy an orbital, whenever it is described by a wave function. As many such wave functions are possible for an electron, there are many atomic orbitals; an orbital can contain a maximum of two electrons. In a multi-electron, the electrons are filled in various orbitals in the order of increasing energy. For every electron, there are orbital wave function characteristics of the orbital it occupies. All the information about the electron in an atom is stored in its orbital wave function Ψ and quantum mechanics makes it possible to extract this information from Ψ.

The probability of finding an electron at a point within an atom is proportional to the square of the orbital wave function i.e. Ψ2 at that point. This Ψ2 is the probability density and is always positive.

Orbitals and Quantum Numbers

A large number of orbitals are possible in an atom. Qualitatively these orbitals can be distinguished by their size, shape and orientation. An orbital of smaller size means the nucleus. Similar shape and orientation mean that there is more probability of finding the electron along with certain directions along with others. Atomic orbitals are precisely distinguished by what is known as quantum numbers. Each orbital is designated by three quantum numbers labelled as n, l, and ml.

The Principal Quantum Number ‘n’ is a positive integer with a value of n = 1, 2, 3,…

It determines the size and to a large extent the energy of the orbital. For hydrogen atom and hydrogen-like species (He+, Li2+ etc) energy and size of the orbitals depends only on ‘n’.

The principal quantum number also identifies the shell. With the increase in the value of ‘n’, the number of allowed orbits increases and are given by n2. All the orbitals of a given value of ‘n’ constitute a single shell of an atom and are represented by the following letters.

 Mr Adewole O.A

 Department of Pure and Industrial Chemistry

 University of Nigeria, Nsukka

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