CBSE Class 11 Chemistry Structure Of Atom Notes Set 08

These revision notes on Structure of Atom for CBSE Class 11 Chemistry cover the nature of electromagnetic radiation, Planck's quantum theory, the photoelectric effect, atomic spectra of hydrogen, Bohr's model with its formulas, de Broglie's relation, Heisenberg's uncertainty principle, the quantum mechanical model, quantum numbers, nodes, shapes and energies of orbitals and the rules for writing electronic configurations. Use them for quick revision before attempting the MCQs and online tests of this chapter.

The nature of electromagnetic radiation and the experimental results regarding atomic spectra played an important role in the development of Bohr's model of the atom.

1. Electromagnetic Wave Theory

According to Maxwell's electromagnetic wave theory, energy is emitted continuously from any source in the form of radiations travelling in wave form and associated with electric and magnetic fields oscillating perpendicular to each other and to the direction of radiation.

Characteristics of a Wave

(i) Wavelength (λ) Distance between any two consecutive crests or troughs. It is expressed in Å, nm, pm or m.
(ii) Frequency (ν) Number of waves passing through a point in one second. Units are hertz (Hz) or s⁻¹.
(iii) Velocity (c) Distance travelled by the wave in one second. Units are m s⁻¹.
\( c = \nu \times \lambda \;\Rightarrow\; \lambda = \frac{c}{\nu} \)
(iv) Wave number (\( \bar{\nu} \)) Reciprocal of wavelength.
\( \bar{\nu} = \frac{1}{\lambda} \)
Its commonly used unit is cm⁻¹ (not the SI unit).

2. Particle Nature of Electromagnetic Radiation: Planck's Quantum Theory

1. The ideal body, which emits and absorbs radiations of all frequencies, is called a black body and the radiation emitted by such a body is called black body radiation.
2. According to Planck's quantum theory, the radiant energy emitted or absorbed by an atom is in small discrete packets of energy known as quantum; in the case of light, the quantum of energy is called a photon.
\( E = h\nu \) or \( E = h\frac{c}{\lambda} \)
where h = Planck's constant = 6.63 × 10⁻³⁴ J s and E = energy of a photon or quantum.
3. If n is the number of quanta of a particular frequency and E_T is the total energy, then E_T = nhν.

3. Photoelectric Effect

1. The phenomenon of ejection of electrons from a metal surface when light of a certain frequency strikes its surface is called the photoelectric effect.
2. For each metal there is a characteristic minimum frequency, known as the threshold frequency (ν₀), below which the photoelectric effect is not observed.
3. The number of electrons ejected is proportional to the intensity or brightness of light.
4. Following the principle of conservation of energy, the kinetic energy of the ejected electron is given by the equation
\( h\nu = h\nu_0 + \frac{1}{2} m_e v^2 \)

Dual Nature of Electromagnetic Radiation

Light possesses both particle-like and wave-like properties. Whenever radiation interacts with matter, it displays particle-like properties, in contrast to the wave-like properties it shows while propagating.

4. Atomic Spectra

1. The pictorial representation of the arrangement of various types of electromagnetic radiation in increasing order of wavelength (or decreasing order of frequency) is known as a spectrum.
2. The spectrum of radiation emitted by a substance that has absorbed energy, in increasing order of wavelength or decreasing order of frequency, is called an emission spectrum.
3. An absorption spectrum is like the photographic negative of an emission spectrum.
4. The line spectra of hydrogen lie in three regions of the electromagnetic spectrum, i.e. infrared, visible and UV region.
5. The set of lines in the visible region is known as the Balmer series.
6. The general formula for the wave number of lines of the different series of the hydrogen spectrum is
\( \bar{\nu} = 109677 \left[ \frac{1}{n_1^2} - \frac{1}{n_2^2} \right] \text{ cm}^{-1} \)
where n₁ = 1, 2, … and n₂ = n₁ + 1, n₁ + 2, …
7. The value 109677 cm⁻¹ is called the Rydberg constant.
8. The first five series of lines, corresponding to n₁ = 1, 2, 3, 4, 5, are known as the Lyman, Balmer, Paschen, Brackett and Pfund series respectively.

5. Bohr's Model

The main postulates of this model are as follows:
1. The energy of an electron is quantised. The orbits or stationary states of the electron are numbered n = 1, 2, 3, … or K, L, M, … These integral numbers are known as the principal quantum number.
2. The energy of an electron in an orbit is constant, i.e. it does not change with time.
3. The difference in energy ΔE is given by
\( \Delta E = h\nu \;\Rightarrow\; \nu = \frac{\Delta E}{h} = \frac{E_2 - E_1}{h} \)
where E₁ and E₂ are the energies of the lower and higher allowed energy states respectively.
4. The angular momentum (mvr) of an electron in a given stationary state is given as
\( mvr = \frac{nh}{2\pi} \) (n = 1, 2, 3, …)

Significance of Bohr's Model

This model is useful to explain the following facts:
(i) Calculation of Bohr radius According to Bohr's model, the radius of the n^th orbit is given by
\( r_n = \frac{52.9\, n^2}{Z} \text{ pm} = \frac{0.0529\, n^2}{Z} \text{ nm} \)
where Z is the nuclear charge.
(ii) Calculation of energy of the electron The energy of the electron in the n^th orbit is
\( E_n = -R_H \left( \frac{1}{n^2} \right) \) (n = 1, 2, 3, …)
where R_H = Rydberg constant = \( \frac{2\pi^2 m e^4}{h^2} \) = 2.18 × 10⁻¹⁸ J (for n = 1).
For the n^th orbit of a hydrogen-like species of atomic number Z,
\( E_n = -\frac{2.18 \times 10^{-18}}{n^2} Z^2 \text{ J per atom} = -\frac{13.6}{n^2} Z^2 \text{ eV per atom} \approx -\frac{1312}{n^2} Z^2 \text{ kJ mol}^{-1} \)
(iii) Calculation of frequency and wave number for a transition The energy gap between the two orbits is given by
\( \Delta E = E_f - E_i \)
\( \bar{\nu} = \frac{\nu}{c} = R_H \left( \frac{1}{n_i^2} - \frac{1}{n_f^2} \right) = 109677 \left( \frac{1}{n_i^2} - \frac{1}{n_f^2} \right) \text{ cm}^{-1} \)
where n_i and n_f stand for the initial and final orbits.
For an absorption spectrum, n_f > n_i. For an emission spectrum, n_i > n_f.

Limitations of Bohr's Model

It fails to account for the finer details of the hydrogen atom spectrum observed by using spectroscopic techniques. It could not explain the ability of atoms to form molecules by chemical bonds.

6. Towards Quantum Mechanical Model of an Atom

1. In view of the shortcomings of Bohr's model, two important developments, i.e. the dual behaviour of matter and Heisenberg's uncertainty principle, came into existence.
2. de Broglie explained the dual nature of the electron (i.e. wave nature as well as particle nature). His hypothesis is given by
\( \lambda = \frac{h}{mv} = \frac{h}{p} \)
where λ = wavelength, v = velocity of the particle, m = mass of the particle and p = momentum.
3. Heisenberg's uncertainty principle This principle is a result of the dual nature of matter and radiation. According to it, it is impossible to determine at any given instant both the momentum and the position of subatomic particles, like the electron, simultaneously.
\( \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \) or \( \Delta x \cdot \Delta v_x \geq \frac{h}{4\pi m} \)
4. Heisenberg's uncertainty principle rules out the existence of definite paths or trajectories of electrons and other similar particles.
5. The effect of Heisenberg's uncertainty principle is significant only for the motion of microscopic objects and is negligible for macroscopic objects. This is because for milligram-sized or heavier objects, the associated uncertainties are of hardly any real consequence.
6. In the case of a microscopic object like an electron, Δv·Δx obtained is much larger and such uncertainties are of real consequence.
7. The precise statement of the position and momentum of electrons has to be replaced by statements of the probability that the electron has at a given position and momentum.

7. Quantum Mechanical Model of Atom

1. The branch of science that takes into account this dual behaviour of matter (i.e. wave nature as well as particle nature) is called quantum mechanics.
2. Schrödinger derived an equation for an electron which describes the wave motion of an electron along any three axes. For a system (such as an atom or molecule) whose energy does not change with time, the Schrödinger equation is written as \( \hat{H}\psi = E\psi \), where \( \hat{H} \) is a mathematical operator called the Hamiltonian.
3. The important features of the quantum mechanical model of the atom are:
(i) The energy of electrons in atoms is quantised.
(ii) Both the exact position and exact velocity of an electron in an atom cannot be determined simultaneously.
(iii) An atomic orbital is the wave function ψ for an electron in an atom.
(iv) The probability of finding an electron at a point within an atom is proportional to the square of the orbital wave function, |ψ|², at that point, which is known as the probability density and is always positive.
From the value of |ψ|² at different points within an atom, it is possible to predict the region around the nucleus where the electron will most probably be found.

8. Atomic Orbitals and Quantum Numbers

The set of four numbers which gives complete information about the energy, shape, orientation and spin of an electron in an atom is called the set of quantum numbers. There are four types of quantum numbers:

(i) Principal Quantum Number (n)

It determines the size and, to a large extent, the energy of the orbital. It is always a positive integer, n = 1, 2, 3, …
1. The principal quantum number also identifies the shell.
2. All the orbitals of a given value of n constitute a single shell of the atom and are represented by the following letters:
n = 1, 2, 3, 4, …
Shell = K, L, M, N, …
3. The sub-shells are the orientations and shapes of the orbitals, labelled in order as s, p, d, f.
4. Number of sub-shells in a shell = shell number (n).
5. The first sub-shell has 1 orbital. Each successive sub-shell adds 2 more orbitals (1, 3, 5, 7, etc.).
6. Each orbital can hold only 2 electrons of opposite spin.

(ii) Azimuthal Quantum Number (l)

It determines the angular momentum of the electron and is also known as the orbital angular momentum or subsidiary quantum number. For a given value of n, the possible values of l are l = 0, 1, 2, …, (n − 1). For example, if n = 1, the only value of l is 0.

Angular momentum of the electron in an orbital \( = \sqrt{l(l+1)}\,\frac{h}{2\pi} = \sqrt{l(l+1)}\,\hbar \)
It also tells about the shape of the orbitals of a subshell.
Energy of subshells (same shell): s < p < d < f.

(iii) Magnetic Quantum Number (m_l)

It describes the behaviour of the electron in a magnetic field (orientation of the orbital). For a given value of l (subshell), 2l + 1 values of m_l are possible:
m_l = −l, −(l − 1), −(l − 2), …, 0, …, (l − 2), (l − 1), l

(iv) Spin Quantum Number (s)

It tells about the spinning motion of the electron, i.e. clockwise or anticlockwise. For a given value of m_l,
s = +\( \frac{1}{2} \) and −\( \frac{1}{2} \).
It helps to explain the magnetic properties of substances.
Spin angular momentum \( = \sqrt{s(s+1)}\,\frac{h}{2\pi} \)

Useful Relations

1. Number of subshells in the n^th shell = n
2. Number of orbitals in a subshell = 2l + 1
3. Maximum number of electrons in a subshell = 2(2l + 1)
4. Number of orbitals in the n^th shell = n²
5. Maximum number of electrons in the n^th shell = 2n²

9. Node and Nodal Plane

A node is a region of space where the probability of finding an electron is zero.
(i) (n − l − 1) = radial nodes
(ii) l = angular or planar nodes
(iii) (n − 1) = total nodes

10. Shape of Atomic Orbitals

1. s-orbitals Spherical in shape.
2. p-orbitals Dumb-bell in shape.
3. d-orbitals Four d-orbitals have a cloverleaf shape with four lobes. Three orbitals have lobes between the axes and are called d_xy, d_yz and d_xz. The fourth has lobes along the axes and is called d_x²−y². The fifth has a dumb-bell shape with a doughnut-shaped electron cloud (ring) at the centre and is called d_z².

11. Energies of Orbitals

The lower the value of (n + l) for an orbital, the lower is its energy. If two orbitals have the same value of (n + l), the orbital with the lower value of n will have the lower energy.

12. Rules for Filling of Electrons in Orbitals

Filling of electrons into the orbitals of different atoms takes place according to the following rules:
(i) Aufbau principle In the ground state of atoms, the orbitals are filled in order of their increasing energies:
1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, …
(ii) Pauli's exclusion principle No two electrons in an atom can have the same set of four quantum numbers. It can also be stated as "only two electrons may exist in the same orbital and these electrons must have opposite spin". The maximum number of electrons in the shell with principal quantum number n is equal to 2n².
(iii) Hund's rule of maximum multiplicity It states that "pairing of electrons in the orbitals belonging to the same subshell (p, d or f) does not take place until each orbital belonging to that subshell has got one electron each, i.e. it is singly occupied."

13. Stability of Half-filled and Completely Filled Orbitals

1. The ground state electronic configuration of an atom corresponds to the lowest energy state and gives higher stability.
2. The electronic configuration of most atoms follows the basic rules. However, certain elements such as Cr and Cu do not follow the rules.
3. The extra stability of half-filled and fully filled electronic configurations can be explained in terms of symmetry and exchange energy, with the help of Hund's rule.

14. Electronic Configuration of Elements

1. The arrangement of electrons in the various shells, sub-shells and orbitals of an atom is termed its electronic configuration.
2. It is written in terms of nl^x, where n indicates the shell, l indicates the subshell and x indicates the number of electrons present in the subshell.