Definitions [14]
The atoms of different elements which have the same mass number A, but different atomic number Z, are called isobars.
The atoms of the same element, having same atomic number Z, but different mass number A, are called isotopes.
OR
Atoms having the same atomic number (Z) but different mass numbers (A).
The tiny unit (packet or quantum) of radiant energy having energy equal to hvhv, where hh is Planck's constant and vv is the frequency of radiation, is called a Photon.
The phenomenon of emission of electrons from a metal surface, when radiation of appropriate frequency is incident on it, is called the Photoelectric Effect.
Define photoelectric effect.
The phenomenon of emission of electrons from a metal surface when radiation of appropriate frequency is incident on it is known as the photoelectric effect.
Atomic Spectra are the spectra of the electromagnetic radiation emitted or absorbed by an electron during transitions between different energy levels within an atom.
The spectrum consisting of bright lines on a dark background, emitted when an atomic gas is excited at low pressure by passing an electric current through it, is called the Emission Line Spectrum.
- The collection of different spectral lines obtained due to transition of an electron in hydrogen atom from upper energy levels to lower energy levels is called the Hydrogen Spectrum.
- The hydrogen spectrum consists of specific wavelengths of light emitted by hydrogen atoms. When transition of an electron in a hydrogen atom occurs between energy levels, it emits or absorbs photons of certain wavelengths, creating a series of lines known as the hydrogen spectrum.
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.
Define the term atomic number.
The number of protons in the nucleus is known as the atomic number of the element and is denoted by Z.
The number of protons in the nucleus of an atom, which is characteristic of a chemical element and determines its place in the periodic table. Atomic number is also equal to the number of electrons in an atom.
Define the term mass number.
The total number of neutrons and protons in the nucleus is called the mass number of the element and is denoted by A.
The atomic number of an atom is equal to the number of protons in its nucleus (which is same as the number of electrons in a neutral atom).
The mass number of an atom is equal to the total number of nucleons (i.e., the sum of the number of protons and the number of neutrons) in its nucleus.
Formulae [8]
E = hv
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):
- \[E_n=-13.6\frac{Z^2}{n^2}\mathrm{eV},\quad n=1,2,3\ldots\]
\[\Delta E=h\nu=E_i-E_f\]
\[\frac{1}{\lambda_{\mathrm{vac}}}=R_H\left[\frac{1}{n_1^2}-\frac{1}{n_2^2}\right]\]
where \[R_{H}=1.097\times10^{7}\mathrm{m}^{-1}\] (Rydberg constant)
Key Points
- The word atom comes from the Greek word atomos meaning uncuttable or indivisible.
- Generally, the size of an atom is about 10⁻¹⁰ m (this distance is the average distance between the nucleus and the outermost shell carrying electrons).
- Atomic number (Z) = Number of protons = Number of electrons (in neutral atom).
- Atomic mass = Number of neutrons + Number of protons.
- Atomic mass = Equivalent mass × Valency = 6.4 × Specific heat (cal).
- Discovered by J. J. Thomson (1897) using the cathode ray tube experiment
- Charge = –1.6 × 10⁻¹⁹ C
- Mass = 9.109 × 10⁻³¹ kg (very small)
- Cathode rays travel in straight lines
- Deflected by electric and magnetic fields (proves negative charge)
- Discovered by E. Goldstein using a discharge tube (canal rays)
- Charge = +1.6 × 10⁻¹⁹ C
- Mass = 1.673 × 10⁻²⁷ kg
- Present in the nucleus of an atom
- Determines the atomic number (Z) of an element
- Discovered by James Chadwick (1932)
- Charge = 0 (neutral)
- Mass = 1.675 × 10⁻²⁷ kg (almost equal to proton)
- Present in the nucleus along with protons
- Responsible for isotopes and atomic mass
Reaction: \[_4^9Be+_2^4He\to_6^{12}C+_0^1n\]
- 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.
Isobars are atoms of different elements that have the same mass number but different atomic numbers.
Same in isobars:
- Mass number (A)
- Number of nucleons
Different in isobars:
- Atomic number (Z)
- Number of protons, electrons, and neutrons
- Electronic configuration
- Position in periodic table
- Chemical properties
Examples: \[_{18}Ar^{40}\mathrm{and}_{19}K^{40}\]
Isotopes are atoms of the same element that have the same atomic number but different mass numbers (different number of neutrons).
Same in isotopes:
- Atomic number (Z)
- Number of protons and electrons
- Electronic configuration
- Position in periodic table
- Chemical properties (nearly identical)
Different in isotopes:
- Mass number (A)
- Number of neutrons
- Physical properties
Examples: \[_1H^1and_1H^2\]
Two key developments provided the foundation for Bohr's model:
(i) Wave-Particle Duality of Electromagnetic Radiation
Electromagnetic radiation has a dual nature — it behaves both as a wave and as a stream of particles called photons. Each photon carries energy:
where h = Planck's constant = 6.626 × 10⁻³⁴ J·s and ν = frequency.
Key wave properties:
-
Wavelength (λ): Distance between two consecutive crests or troughs
-
Frequency (ν): Number of waves passing a given point per second (unit: Hz or s⁻¹)
-
Wave number \[(\bar{\nu})\]: Number of wavelengths per unit length = 1/λ (unit: cm⁻¹ or m⁻¹)
Relation between speed, frequency, and wavelength:
\[c=\nu\lambda\quad\Rightarrow\quad\nu=\frac{c}{\lambda}\]
Longer wavelength → smaller frequency → lower energy of radiation.
(ii) Quantisation of Energy
Results of atomic spectra showed that atoms absorb or emit energy only in discrete amounts. This gave evidence that energy is quantised — it comes in fixed packets (quanta).
Evolution of Quantum Theory (Timeline):
Classical Theory (matter = particles, radiation = waves)
↓
Einstein & Planck Energy is Quantised
↓
Bohr Line Spectra (Bohr's H atom model)
↓
de Broglie Matter has Wave Nature
↓
Heisenberg Uncertainty Principle
↓
Schrödinger Quantum Theory (matter & radiation both have wave-particle duality)
- 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
- Proposition:
Energy is emitted in packets (quanta).
At higher frequencies, the energy of a packet is large. - Planck assumed that atoms behave like tiny oscillators that emit electromagnetic radiation only in discrete packets of energy E = hv, where v is the frequency of the oscillator.
- The emissions occur only when the oscillator makes a jump from one quantized level of energy to another of lower energy.
- This model of Planck formed the basis for explaining the observations of the photoelectric effect.
- When light of frequency ≥ threshold frequency (ν₀) falls on a metal surface, electrons are emitted.
- If ν < ν₀ → no emission, regardless of intensity.
- The threshold frequency (ν₀) varies with metal.
- Energy equation: \[h\nu=h\nu_0+\frac{1}{2}mv^2\]
- Maximum kinetic energy depends only on frequency, not on intensity.
- Number of emitted electrons depends on intensity (for ν ≥ ν₀).
- Emission of electrons is instantaneous (no time lag).
- Increasing frequency increases the kinetic energy of electrons.
- Increasing intensity increases the number of electrons, not their energy.
- The photoelectric effect proves the particle (quantum) nature of light.
- 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 \[-13.6\frac{Z^2}{n^2}\mathrm{~eV}.\] .
- Lyman series — transitions to n = 1; region: ultraviolet
- Balmer series — transitions to n = 2; region: visible
- Paschen series — transitions to n = 3; region: infrared
- Brackett series — transitions to n = 4; region: infrared
- Pfund series — transitions to n = 5; region: infrared
- The spectrum of hydrogen is important as most of the universe is made of hydrogen.
- Balmer series involves transitions starting/ending with the first excited state (n = 2) of hydrogen.
- 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
- The structure of an atom and its nucleus was developed from the discovery of electrons by J.J. Thomson and alpha particle scattering experiments by Rutherford.
- An atom consists of electrons, protons, and neutrons, with protons and neutrons in the nucleus and electrons revolving in stationary orbits.
- The maximum number of electrons in a shell is given by 2n², and the shells are named K, L, M, N, O, P, and Q.
Concepts [31]
- History of Atom
- Electrons
- Protons
- Neutrons
- Atomic Models
- J. J. Thomson’s Atomic Model
- Lord Rutherford’s Atomic model
- Drawbacks of Rutherford Atomic Model
- Atomic Number (Z) and Mass Number (A)
- Isobars
- Isotopes
- Developments Leading to the Bohr’s Atomic Model
- Wave Nature of Electromagnetic Radiation
- Particle Nature of Electromagnetic Radiation: Planck's Quantum Theory of Radiation
- The Photoelectric Effect
- Evidence for the Quantized Electronic Energy Levels - Atomic Spectra
- Bohr’s Model for Hydrogen Atom
- Hydrogen Spectrum
- 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 - 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
- Quantum Mechanical Model of the Atom - Concept of Shells and Subshells
- Structure of the Atom and Nucleus
