Definitions [3]
Atomic Spectra are the spectra of the electromagnetic radiation emitted or absorbed by an electron during transitions between different energy levels within an atom.
Define the term Electronic configuration.
Electronic configuration of an atom is defined as the distribution of its electrons in orbitals.
The arrangement of electrons in various shells, subshells, and orbitals is called electronic configuration.
Written as: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ … etc.
Formulae [5]
Radius of the n-th Bohr Orbit (General):
\[r_n=\frac{\varepsilon_0n^2h^2}{\pi mZe^2}\]
\[\mathrm{i.e.,}r_n\propto n^2\mathrm{and}r_n\propto\frac{1}{Z}\]
Radius of n-th orbit for Hydrogen-like atom:
\[r_n=0.53\left(\frac{n^2}{Z}\right)\mathrm{\r{A}}\]
\[v_n=\frac{nh}{2\pi mr_n}\]
\[r=\frac{n^2h^2}{4\pi^2mkZe^2}\]
\[v=\frac{2\pi kZe^2}{nh}\]
For hydrogen atom (Z = 1):
\[v=\frac{2\pi ke^2}{nh}=\alpha\frac{c}{n}\]
where α is the fine structure constant and \[\alpha=\frac{1}{137}.\]
Total Energy of Electron in n-th Orbit (General):
- \[E_n=\frac{-Z^2me^4}{8\varepsilon_0^2n^2h^2}\]
Total Energy (Alternate form):
- \[E_n=-\frac{2\pi^2mk^2Z^2e^4}{n^2h^2}\]
Total Energy for Hydrogen-like Atom (Simplified):
- \[E_n=-13.6\frac{Z^2}{n^2}\mathrm{eV},\quad n=1,2,3\ldots\]
Key Points
Atoms are made up of three fundamental subatomic particles — electron, proton, and neutron. Their discovery was a milestone in understanding atomic structure.
Discovery Timeline:
| Particle | Year | Scientist | Experiment |
|---|---|---|---|
| Electron | 1897 | J.J. Thomson | Cathode ray tube experiment — cathode rays are streams of tiny, negatively charged particles |
| Proton | 1911 | Ernest Rutherford | Alpha-particle scattering on gold foil — hydrogen nucleus identified and renamed proton |
| Neutron | 1932 | James Chadwick | Nuclear reaction: bombardment of beryllium with alpha-particles produced neutral, massive particles |
Properties of Subatomic Particles
| Particle | Symbol | Absolute Charge (C) | Relative Charge | Mass (kg) | Mass (u) | Approx. Mass (u) |
|---|---|---|---|---|---|---|
| Electron | e⁻ | −1.6022 × 10⁻¹⁹ | −1 | 9.10938 × 10⁻³¹ | 0.00054 | 0 |
| Proton | p+ | +1.6022 × 10⁻¹⁹ | +1 | 1.6726 × 10⁻²⁷ | 1.00727 | 1 u |
| Neutron | no | 0 | 0 | 1.67493 × 10⁻²⁷ | 1.00867 | 1 u |
Atoms are made up of three fundamental subatomic particles — electron, proton, and neutron. Their discovery was a milestone in understanding atomic structure.
Discovery Timeline:
| Particle | Year | Scientist | Experiment |
|---|---|---|---|
| Electron | 1897 | J.J. Thomson | Cathode ray tube experiment — cathode rays are streams of tiny, negatively charged particles |
| Proton | 1911 | Ernest Rutherford | Alpha-particle scattering on gold foil — hydrogen nucleus identified and renamed proton |
| Neutron | 1932 | James Chadwick | Nuclear reaction: bombardment of beryllium with alpha-particles produced neutral, massive particles |
Properties of Subatomic Particles
| Particle | Symbol | Absolute Charge (C) | Relative Charge | Mass (kg) | Mass (u) | Approx. Mass (u) |
|---|---|---|---|---|---|---|
| Electron | e⁻ | −1.6022 × 10⁻¹⁹ | −1 | 9.10938 × 10⁻³¹ | 0.00054 | 0 |
| Proton | p+ | +1.6022 × 10⁻¹⁹ | +1 | 1.6726 × 10⁻²⁷ | 1.00727 | 1 u |
| Neutron | no | 0 | 0 | 1.67493 × 10⁻²⁷ | 1.00867 | 1 u |
Atoms are made up of three fundamental subatomic particles — electron, proton, and neutron. Their discovery was a milestone in understanding atomic structure.
Discovery Timeline:
| Particle | Year | Scientist | Experiment |
|---|---|---|---|
| Electron | 1897 | J.J. Thomson | Cathode ray tube experiment — cathode rays are streams of tiny, negatively charged particles |
| Proton | 1911 | Ernest Rutherford | Alpha-particle scattering on gold foil — hydrogen nucleus identified and renamed proton |
| Neutron | 1932 | James Chadwick | Nuclear reaction: bombardment of beryllium with alpha-particles produced neutral, massive particles |
Properties of Subatomic Particles
| Particle | Symbol | Absolute Charge (C) | Relative Charge | Mass (kg) | Mass (u) | Approx. Mass (u) |
|---|---|---|---|---|---|---|
| Electron | e⁻ | −1.6022 × 10⁻¹⁹ | −1 | 9.10938 × 10⁻³¹ | 0.00054 | 0 |
| Proton | p+ | +1.6022 × 10⁻¹⁹ | +1 | 1.6726 × 10⁻²⁷ | 1.00727 | 1 u |
| Neutron | no | 0 | 0 | 1.67493 × 10⁻²⁷ | 1.00867 | 1 u |
- Proposed by J. J. Thomson in 1904 after the discovery of electrons.
- The atom is a uniform sphere of positive charge.
- Electrons are embedded within this sphere.
- The positive charge is spread evenly throughout the atom.
- Total positive charge = total negative charge, so the atom is neutral.
- The model explained the presence of electrons in atoms.
- It did not include a nucleus in the atom.
- It failed to explain Rutherford’s results from the gold foil experiment.
- Proposed by Ernest Rutherford in 1911 based on the gold foil (α-particle scattering) experiment.
- Most α-particles passed straight through, showing that the atom is mostly empty space.
- Some α-particles were deflected, indicating the presence of a positively charged centre.
- Very few α-particles were deflected at large angles or bounced back, proving a dense nucleus.
- All the positive charge and most of the mass are concentrated in a tiny nucleus (~10⁻¹⁵ m).
- Electrons revolve around the nucleus in circular orbits.
- The electrostatic force of attraction between nucleus and electrons keeps them in orbit.
- Limitation: Could not explain stability of atom and line spectra of hydrogen.
- Electromagnetic radiation possesses both particle and wave properties.
- Wave characteristics are shown when it propagates (interference, diffraction).
- Particle characteristics are shown when interacting with matter (photoelectric effect, black body radiation).
Key Wave Relations:
\[c=\lambda\times\nu\quad\mathrm{or}\quad\nu=\frac{c}{\lambda}\quad\mathrm{and}\quad\frac{1}{\lambda}=\bar{\nu}\]
- c = speed of light = 3 × 10⁸ m/s
- λ = wavelength (m)
- ν = frequency (Hz or s⁻¹)
- \[\bar{\nu}\] = wave number (cm⁻¹)
Electromagnetic Spectrum (Increasing Wavelength →)
Cosmic rays → γ-rays → X-rays → UV → Visible → IR → Microwaves → Radio waves
- The radius of Bohr's orbit is proportional to \[n^{2}\] and inversely proportional to Z.
- For hydrogen (Z = 1), the ground state (n = 1) radius is 0.53 Å, known as Bohr's radius.
- The velocity of an electron decreases as the orbital number (n) increases.
- For hydrogen, orbital speed of electron equals \[\alpha\frac{c}{n}\], where \[\alpha=\frac{1}{137}\].
- The total energy of an electron in any orbit is negative, indicating a bound state.
- For hydrogen-like atoms, the energy of an electron in the n-th orbit is \[-13.6\frac{Z^2}{n^2}\mathrm{~eV}.\] .
- Could not explain the fine structure (splitting) of the spectral lines of hydrogen.
- Failed to explain the spectra of multi-electron atoms.
- Could not explain the splitting of spectral lines in a magnetic field (Zeeman effect) and an electric field (Stark effect).
- Failed to explain the formation of molecules and chemical bonding.
- Inconsistent with Heisenberg’s Uncertainty Principle.
- Could not explain the intensity of spectral lines.
- Proposed by Louis de Broglie in 1924: just as light shows both particle and wave nature, matter in motion also exhibits wave-like behaviour.
- The wave associated with a moving particle is called the matter wave or de Broglie wave.
- de Broglie wavelength equation: \[\lambda=\frac{h}{mv}=\frac{h}{p}\]
where m = mass of particle, v = velocity, p = momentum.
- For an electron: λ = h/mv or λ = h/p
- Stated by Werner Heisenberg in 1927.
- It is impossible to determine the exact position and exact momentum (velocity) of an electron simultaneously.
- Significant only for microscopic objects; negligible for macroscopic objects.
Mathematical Expression:
or equivalently:
Schrödinger Wave Equation:
Schrödinger developed the fundamental equation of quantum mechanics which incorporates the wave-particle duality of matter:
where H = Hamiltonian operator, Ψ (psi) = wave function, E = total energy of the system.
- Wave function (ψ): The solution of this equation has no physical significance by itself.
- ψ²: Probability density — gives the probability of finding an electron at a point within the atom.
- The region where the probability of finding an electron is maximum = atomic orbital.
Quantum Numbers:
Four quantum numbers together describe the complete "address" of every electron in an atom:
| Quantum Number | Symbol | What it describes | Allowed Values |
|---|---|---|---|
| Principal | n | Shell size & energy level | 1, 2, 3, 4, … (positive integers) |
| Azimuthal (Subsidiary) | l | 3D shape of orbital; angular momentum | 0 to (n−1) |
| Magnetic Orbital | mₗ | Spatial orientation of orbital | −l to +l (including 0) |
| Spin | mₛ | Direction of electron spin | +½ (clockwise) or −½ (anticlockwise) |
| Orbitals | Shape of Orbitals | Design of Orbitals | Angular Nodes | Radial Nodes |
|---|---|---|---|---|
| s | Spherical | s | 0 | n − 1 |
| p | Dumbbell | \[P_{x^{\prime}}P_{y^{\prime}}P_{z}\] | 1 | n − 2 |
| d | Double dumbbell | \[\mathrm{d_{xy^{\prime}}d_{yz^{\prime}}d_{zx^{\prime}}d_{x^{2}-y^{2}},d_{z^{2}}}\] | 2 | n − 3 |
| f | Complex | \[\mathrm{f_{xyz},f_{x(y^2-z^2)^{\prime}}f_{y(z^2-x^2)^{\prime}}f_{z(x^2-y^2)^{\prime}}f_{x^3},f_{y^3},f_{z^3}}\] | 3 | n − 4 |
Aufbau Principle:
Electrons fill orbitals in order of increasing energy. The energy order follows the (n + l) rule:
-
n + l rule: Lower value of (n + l) → lower energy. If (n + l) is the same for two orbitals, the one with lower n has lower energy.
-
Filling order: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s …
Hund's Rule of Maximum Multiplicity:
Electrons never pair up in orbitals of the same subshell until each orbital in that subshell has at least one electron (is singly occupied).
Pauli's Exclusion Principle:
No two electrons in an atom can have the same set of all four quantum numbers. As a result, each orbital can hold a maximum of 2 electrons with opposite spins.
Special Stability: Cr and Cu
- Chromium (Cr, Z=24): Expected 1s²2s²2p⁶3s²3p⁶3d⁴4s² → Actual: 1s²2s²2p⁶3s²3p⁶3d⁵4s¹ (half-filled 3d is extra stable)
- Copper (Cu, Z=29): Expected 1s²2s²2p⁶3s²3p⁶3d⁹4s² → Actual: 1s²2s²2p⁶3s²3p⁶3d¹⁰4s¹ (fully-filled 3d is extra stable)
Half-filled and fully-filled sets of degenerate orbitals have extra stability.
Nodes in Orbitals:
- Radial nodes = n − l − 1
- Angular nodes = l
- Total nodes = n − 1
- Number of nodal planes = l
Completely filled (d¹⁰, f¹⁴) and half-filled (d⁵, f⁷) subshells are extra stable.
Electrons may rearrange to achieve this stability, leading to exceptions to Aufbau principle
e.g.,
- Cr: 3d⁵ 4s¹
- Cu: 3d¹⁰ 4s¹
Reasons for Extra Stability:
1. Symmetry: Half-filled and fully filled orbitals have symmetrical distribution → leads to greater stability
2. Exchange Energy: More parallel spins → more exchange interactions → higher exchange energy → more stability
Concepts [25]
- Subatomic Particles
- Subatomic Particles
- Charge to Mass Ratio of Electron
- Charge on the Electron
- Subatomic Particles
- Atomic Models
- J. J. Thomson’s Atomic Model
- Advantage and Limitations of Thomson’s Atomic Model
- Lord Rutherford’s Atomic model
- Limitations of Rutherford’s Atomic Model
- Wave Nature of Electromagnetic Radiation
- Evidence for the Quantized Electronic Energy Levels - Atomic Spectra
- Bohr’s Model for Hydrogen Atom
- Limitations of Bohr's Model
- Towards Quantum Mechanical Model of the Atom
- Dual Behaviour of Matter: De Broglie's relationship
- Heisenberg’s Uncertainty Principle
- Quantum Mechanical Model of Atom
- Quantum Mechanical Model of the Atom - Orbitals and Quantum Numbers
- Quantum Mechanical Model of the Atom - Concept of Shells and Subshells
- Quantum Mechanical Model of the Atom - Shapes of Atomic Orbitals
- Quantum Mechanical Model of the Atom - Energies of Orbitals
- Quantum Mechanical Model of the Atom - Filling of Orbitals in Atom
- Quantum Mechanical Model of the Atom - Electronic Configuration of Atoms
- Quantum Mechanical Model of the Atom - Stability of Completely Filled and Half Filled Subshells
