Contents
Physics Topics can also be used to explain the behavior of complex systems, such as the stock market or the dynamics of traffic flow.
What is the Bohr Model? How is the De Broglie Equation Related to Bohr’s Model?
Rutherford’s atomic model was modified by Niels Bohr. The postulates of Rutherford model which have no technical difficulties remain intact in Bohr model of atom. They are-
i) Positively charged nucleus occupies a negligible space at the centre of the atom.
ii) Negatively charged electrons revolve round the nucleus in circular orbits. If the mass, speed and radius of the orbit of an electron are m, v and r, respectively, the centripetal force necessary to revolve electrons in circular orbit,
F1 = \(\frac{m v^2}{r}\) …. (1)
iii) Electrostatic force of attraction acts between the nucleus and electrons. Now, if the atomic number of an atom be Z and charge of an electron be e, total charge of the nucleus is Ze. So, the electrostatic force of attraction between the nucleus and an electron,
F2 = \(\frac{1}{4 \pi \epsilon_0} \cdot \frac{Z e \cdot e}{r^2}\) = \(\frac{1}{4 \pi \epsilon_0} \cdot \frac{Z e^2}{r^2}\) ….. (2)
On the other hand, resolving the drawbacks of Rutherford model related to
1. stability of atom and
2. atomic line spectrum, Bohr introduced some revolutionary ideas which are not consistent with classical physics. From this, Bohr’s theory of he atom was established and the consequent structure of atom is called Bohr’s atomIc model.
Postulates of Bohr’s Theory
Bohr model is based on three postulates. These are also known as Bohr’s quantum postulates.
i) Electrons inside an atom can only revolve in some allowed orbits. When an electron revolves in an allowed orbit, It does not radiate energy. [Note that, according to classical electromagnetic theory, a rotating [i.e., accelerated] charge radiates energy]
According to this postulate, energy of an electron in any allowed orbit remains constant; hence these orbits are called stationary orbits or stable orbits.
ii) Transition of electron from one stable orbit to another is possible. During transition, an emission or an absorption of radiation occurs. Its frequency f is determined from the relation, hf = E1 ~ E2 [where, h is Planck’s constant and (E1 ~ E2) is the energy difference of the electron in the two stable orbits].
When the energy (E1) of an electron in its initial orbit is more than its energy (E2) in its final orbit, i.e., when E1 > E2, the difference in energy is converted into the energy of an emitted photon, As a result, the atom radiates energy. In this case,
E1 – E2 = hf ….. (1)
Again, if E1 < E2, an external photon should supply the difference in energy. So, in this case, the atom absorbs energy. Hence,
E2 – E1 = hf …… (2)
Equations (1) and (2) are called Bohr’s frequency conditions.
iii) The orbit, where the angular momentum of the electron is an integral multiple of \(\frac{h}{2 \pi}\), is known as a stationary or stable orbit.
Now, if the radius of any stable orbit be rn, the speed of electron in that orbit be vn, angular momentum,
Ln = momentum × radius of circular path = mvnrn
So, according to this postulate,
Ln = mvnrn = \(\frac{n h}{2 \pi}\) …. (3)
where, n = 1, 2, 3, …… This equation is called Bohr’s quantum condition, n is called the principal quantum number of electron or its orbit. These stable orbits are called Bohr orbits.
It is clear that, none of the Bohr’s postulates is consistent with the classical physics. Yet, the success of Bohr’s theory is borne out by the analysis of atomic spectrum of hydrogen and some other elements. Moreover, a clear qualitative picture related to the atomic structure of any element can be obtained from Bohr’s theory.
Hence, in spite of some inconsistency relating to some classical experiments, Bohr model is considered the basis of the atomic structure of all elements.
Bohr’s Quantum Condition from de Brog-lie’s Hypothesis
According to de Broglie’s hypothesis, a stream of any particles can be considered to be matter waves.
- If the stream of particles advances freely, the corresponding matter wave behaves as a progressive wave.
- On the other hand, if the stream of particles is confined within a definite region, then naturally the corresponding matter wave behaves as a stationary wave.
From these considerations de Broglie assumed that,
i) Matter wave of electrons confined within an atom is a stationary wave.
ii) The electronic orbits in an atom should be such that, an integral number of electron waves is present in a complete orbit [Fig.]. Otherwise, after a complete revolution, the wave will reach a different point and hence a stationary wave will not be formed.
If the radius of the circular path be r, circumference = 2πr. So, if the wavelength of electron wave be λ, then
2πr = nλ [n = 1, 2, 3, …]
Here, in Fig., four complete waves are shown in a complete orbit, i.e., n = 4.
Again, according to de Broglie’s hypothesis, If the mass of electron = m and its speed = v, then
λ = \(\frac{h}{m v}\) [h = Plank’s constant]
So, 2πr = n\(\frac{h}{m v}\) [where n = 1, 2, 3, ……..]
or, mvr = n\(\frac{h}{2 \pi}\)
For different values of n, the value of r and v will be different; if these values are taken as rn and vn for definite value of n, then
mvnrn = n\(\frac{h}{2 \pi}\) [where n = 1, 2, 3, …….]
This is Bohr’s quantum condition. So the quantum condition is consistent with the idea of de Broglie’s matter wave. But it should be kept in mind that, as an explanation of stable orbits of an atom, the above mentioned discussion is a kind of oversimplification. A true explanation can only be made with quantum mechanics.