In this chapter, we will explore Bohr’s atomic model and its postulate. We will also see the derivation of radius of Bohr’s orbit, velocity of an electron, and energy of an electron in each orbit.
Rutherford’s nuclear model of the atom simply proposed that the atom consists of a dense, positively charged nucleus at its center and the negatively charged electrons were present outside the nucleus. It didn’t say anything about how or where those electrons are arranged around the nucleus. Moreover, this model also could not explain why electrons did not fall into the nucleus due to the electrostatic force of attraction.
In order to overcome the objections of Rutherford’s atomic model and to explain the hydrogen spectrum, a Danish physicist Niels Bohr in 1913 put forwarded a new quantum mechanical model of atom to improve Rutherford’s atomic model. This model was based on the Planck’s quantum theory and the spectrum of hydrogen atom.
Niels Bohr was the first to explain quantitatively the general features of the structure of hydrogen atom and its spectrum (i.e. spectral lines). He developed an atomic model for hydrogen and hydrogen like one electron species on the basis of Planck’s concept of quantisation of energy.
Bohr’s atomic model successfully explained the discrete spectral lines of hydrogen by proposing that electrons revolve in specific quantized orbits around the nucleus and emit or absorb energy only when transitioning between these orbits.
Important Postulates of Bohr’s Atomic Model
The main postulates of Bohr’s atomic model for hydrogen atom are as follows:
(1) An atom consists of three particles: electrons, protons, and neutrons. Electrons have a negative charge, protons have a positive charge, and neutrons have no charge, i.e. they are neutral. The atom on the whole is electrically neutral due to the presence of an equal number of negative electrons and positive protons.
(2) The atom has a nucleus which contains all the protons and neutrons. The size of the nucleus is very small and is present at the center of the atom.
(3) Negatively charged electrons revolve around the nucleus in a circular path of fixed radius and energy. These paths are called orbits. These orbits are arranged concentrically around the nucleus, as shown in the below figure (a).
(4) Each orbit in Bohr’s atomic model is associated with a definite amount of energy, which remains constant as long as the electron stays in that orbit. Hence, these orbits are called energy levels or shells. These energy levels or shells are represented in two ways: either by the letters K, L, M, N, . . . .etc. or by the numbers 1, 2, 3, 4, 5, . . .etc. as shown in the above figure (a). These integers are termed quantum numbers of orbits. The energy levels or shells are counted from the center outwards.
(5) As long as the electron remains in a specific orbit, it does not emit (i.e. radiate or lose) or gain (i.e. absorb) energy. It means that the energy of an electron in that orbit is constant. Therefore, these orbits are also called stationary states or stationary energy levels. This concept of stationary energy levels explains the stability of an atom because, according to Bohr’s model of atom, an electron cannot lose energy gradually and fall into the nucleus.
(6) When an electron is present in the lowest energy level or stationary state (which is also closest to the nucleus), we say it’s in the ground state or normal state. In the ground state, the potential energy of the electron is minimum. Therefore, this state is the most stable state of the atom.
(7) Now, if the energy from some external source is supplied to the electron, it absorbs the supplied energy in quanta or photons and jumps to a higher energy level. For example, when an electron is in the ground state with n =1, absorbs one quantum or photon of energy (h𝜈; h = Planck’s constant, and 𝜈 = frequency of the emitted or absorbed radiation), it moves to the higher energy level with n = 2, as shown in the above figure (b).
The amount of absorbed energy is equal to the difference in energy between the two energy levels. Thus, when the electron jumps from energy level 1 to energy level 2, the amount of energy absorbed by the electron is given by:
ΔE = E2 – E1 = h𝜈
Where, E1 and E2 are the energies associated with the electron in energy levels 1 and 2, respectively.
The electron present in the energy levels n = 2, 3, 4, . . . . etc. is said to be in an excited state.
(8) The electron in the excited state (i.e. electron in energy level 2, as in the above case) has a tendency to come back to the ground state (i.e. energy level 1). In doing so, the excited electron emits the same amount of energy in the form of a photon of radiation. Thus, when the excited electron comes down from energy level 2 to energy level 1 (i.e. ground state), it radiates the absorbed energy. The amount of energy radiated by the excited electron is given by:
ΔE = E2 – E1 = h𝜈
Where, E1 and E2 are the energies associated with the electron in energy levels 1 and 2, respectively.
The above two points (7) and (8) can be summarized as below:
There is no change in the energy of the electrons with time as long as they revolve in the same energy level. Hence, an atom is stable. The change in the energy of an electron occurs when it moves from a lower energy level to a higher energy level or comes down from a higher energy level to a lower energy level.
When an electron absorbs (or gains) the required amount of energy, it moves from a lower energy level or stationary state and when an electron comes down from a higher energy level to a lower energy level, it loses energy. However, the energy change does not take place in a continuous manner, but it takes place in definite packets (quanta or photons) of energy as proposed by Max Planck.
The quantum or photon of energy absorbed or emitted by an electron is the difference in the energy between the lower and higher energy levels of the atom.
ΔE = Ehigh – Elow = hν, where h is Planck’s constant and ν the frequency of a photon emitted or absorbed energy.
(9) According to Bohr’s atomic model, an electron cannot move in all orbits. It can move only in that orbit in which the angular momentum (mvr) of an electron orbiting around the nucleus is an integral multiple of Planck’s constant divided by 2π, such as 2h/2π, 3h/2π, . . . . . , nh/2π. Thus, the angular momentum of a moving electron is given by
mvr = nh/2π
Here,
- m = mass of the electron
- v = velocity of the electron
- h = Planck’s constant = 6.626 * 10-34 Js
- r = radius of the orbit in which the electron is moving.
- n = integral number, 1, 2, 3, . . . . etc. which has been called principal quantum number by Bohr. The n denotes the number of orbit in which the electron is moving. For example, the value of n = 1, 2, 3, 4, . . . denotes to the K, L, M, and N orbits, respectively starting from the nucleus.
This equation mvr = nh/2π means the angular momentum of the orbiting electron is quantised. That indicates that the magnitude of angular momentum is always a whole number, and cannot be in the fraction. This postulate introduced the concept of the quantisation of angular momentum, which is a fundamental principle in quantum mechanics.
Meaning of Quantisation
According to Bohr’s atomic model, the energy of an electron in an atom is quantised. Quantisation means that a physical quantity, such as energy, cannot vary continuously to have any arbitrary values, but can change only discontinuously to have some specific discrete values.
For example, when a person moves down a staircase, his energy change discontinuously. He can have only certain definite values of energy corresponding to the energies of various steps. In this case, the energy of the person is quantised.
On the other hand, when a person moves down a ramp, his energy change continuously. He can have any value of energy corresponding to any point on the ramp. In this case, energy is not quantised.
10) In an atom, the centrifugal force acting on the moving electron balances the attractive force between the electron and nucleus.
Calculation of Radius of Bohr’s Orbit
With the help of the above postulates and some classical laws of Physics, Niel Bohr was able to find out the radius of each orbit of the hydrogen atom, energy associated with each orbit and wavelength of the radiation emitted or absorbed during the transitions between orbits. So, let’s calculate the radius of orbit of Bohr’s atomic model of atom.
Let us consider an electron of charge e revolving around a nucleus of charge Ze, where Z is the atomic number and e the charge on a proton. Let m be the mass of the electron, r the radius of the orbit in which electron is revolving and ν the tangential velocity of the revolving electron, as shown in the below figure.
According to Coulomb’s law, the electrostatic force of attraction (F) between the nucleus and the moving electron = KZe * e /r2 where, K = constant = 1/4ℼε0 = 9 * 109 Nm2/C2
And centrifugal force acting on the electron = mv2/r
Bohr assumed that both opposing forces must be balancing each other exactly to keep the electron in the orbit. Thus,
mv2/r = KZe * e /r2 (i.e. at equilibrium)
Then, v2 = KZe2/mr —-(1)
According to one of the postulates of Bohr’s theory, angular momentum of the revolving electron is given by the expression:
mvr = nh/2ℼ
v = nh/2ℼ*mr
Squaring both sides, we get
v2 = n2h2/4ℼ2m2r2 – – – – – – (2)
From equation (1) and (2);
KZe2/mr = n2h2/4ℼ2m2r2
On solving, we get:
Radius of nth orbit r = n2h2/4ℼ2mKZe2
This is the equation of the radius of the nth orbit in a hydrogen like atom.
In C.G.S unit, the value of K = 1. Therefore, r = n2h2/4ℼ2mZe2 – – – – (3)
Since the value of h, m and e had been determined experimentally. Therefore, substitute the value of h = 6.62 * 10-27 erg sec, m = 9.1 * 10-28 g, and e = 1.6 * 10-19 C in (4), we have
On putting the value of e, h, and m, we get:
r = 0.529 * n2 / Z in Å unit.
For the hydrogen atom (Z= 1), the radius of the first orbit (n = 1) is approximately 0.529 Å, known as the Bohr radius. For higher orbits such as n = 2, 3,, 4 . . . . , the radius increases proportionally to n2.
Calculation of Velocity of an Electron in Bohr’s Orbit
Velocity of the revolving electron in nth orbit is given by
mvr = nh/2ℼ
v = nh/2ℼmr —– (1)
Putting the value of r in equation (1)
Then, Velocity v = nh * 4ℼ2mZe2 / 2ℼmn2h2
Hence, velocity of the revolving electron (v) = 2ℼZe2/nh
On putting the values of ℼ, e, and h, we will get:
v = 2.188 * 108 * Z/n in Cmsec-1
In this way, velocity of the revolving electron in an orbit is directly proportional to atomic number (Z) and inversely proportional to quantum number (n) of the energy level.
Note:
(1) For a fixed value of n, v ∝ Z, r ∝ 1/Z
(2) For fixed value of Z, v ∝ 1/n, r ∝ n2
Calculation of Energy of an Electron in Each Orbit
The total energy of a revolving electron in a particular orbit is
T. E. = Kinetic Energy (K.E.) + Potential Energy (P.E.)
Here, K.E. of an electron = mv2/2
And the P.E. of an electron = -KZe2/r
Hence, T.E. = mv2/2 + -KZe2/r ——– (1)
We know that mv2/r = KZe2/r2
or, mv2 = KZe2/r
Substituting the value of mv2 in the equation (1), we get
T.E. = KZe2/2r – KZe2/r = -KZe2/2r
So, T.E. = -KZe2/2r
In C.G.S unit, K = 1
∴ T.E. = -Ze2/2r – – – – – – (2)
Substituting the value of r in the equation (2), we get
T.E. = -Ze2 * 4ℼ2Ze2m/n2h2 = -2ℼ2Z2e4m/n2h2
Thus, the total energy of an electron in nth orbit is given by
En = -2ℼ2Z2e4m/n2h2 – – – – – – (3)
For hydrogen atom, Z = 1.
So, En = -2ℼ2e4m/n2h2
Putting the value of ℼ, m, e, and h,
E = -2 * (3.14)2 * (9.1 * 10-31) * (1.6 * 10-19)4 / n2 * (6.625 * 10-34)2
or E = -2.179 * 10-18 / n2 J per atom
or E = -13.6 / n2 in eV per atom (1J = 6.2419 * 1018 eV)
or E = -313.6 / n2 kcal/mol (1eV = 23.06 kcal/mol)
or E = -1312 / n2 KJmol-1
In general, En = -13.6 * Z2/ n2 in eV
Significance of Negative Value of Energy
It is assumed that the energy of an electron at infinity is arbitrarily to be zero. This state is called zero-energy state. When an electron moves and comes under the influence of nucleus from infinity, it does some work and spends its energy in this process.
As a result, the energy of the electron decreases and it becomes less than zero. This means that the energy of an electron acquires a negative value, indicating that it is now bound to the nucleus. The more negative the energy, the stronger the electron is bound to the nucleus.
The total energy of an electron in Bohr’s atomic model is inversely proportional to the square of the principal quantum number “n” with a negative sign. The electron has minimum energy in the first orbit and its energy increases when the value of n increases, i.e. it becomes less negative.
When the value of n is zero, the electron can have a maximum energy value of zero. Here, the zero energy means the electron is no longer bound to the nucleus, i.e. it is not under attraction towards nucleus.
Calculation of Orbital Frequency for an Electron
Orbital frequency is the number of revolutions per second by an electron in a shell or orbit. We can calculate as follows:
Number of revolutions per sec by an electron in a shell = Velocity/circumference = v/2ℼr
ν = (2ℼZe2/nh) / 2ℼ (n2h2/4ℼ2mKZe2)
or, ν = (2ℼZe2 * 4ℼ2mKZe2 ) / 2ℼn3h3
or, ν = 4ℼ2 KZ2e3/2ℼn3h3
or, ν = 2ℼKZ2e3 / n3h3
In C.G.S. unit, K = 1.
or, ν = 2ℼZ2e3 / n3h3
On putting the value of e and h, we get
ν = 6.65 * 1015 * Z2 / n3
Time period of revolution of an electron in nth orbit Tn = 2ℼr/vn
Tn = 1.5 * 10-16 *n3 / Z2 second
