Revision: Std. XI >> Structure of Atom MAH-MHT CET (PCM/PCB) Structure of Atom

The atomic number of an element is equal to the number of protons in the nucleus.

 Atomic number (Z) = Number of protons.
                                 = Number of electrons. 

The mass number of an element is the sum of the number of protons and neutrons in the nucleus of the atom of that element.

Mass number (A) = No. of protons (p)
                             + No. of neutrons (n)

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 atoms of different elements which have the same mass number A, but different atomic number Z, are called isobars.

The atoms having different number of protons but same number of neutrons i.e., different Z and A, but same (A-Z), are called isotones. They have different number of electrons.

Electronic configuration of an atom is defined as the distribution of its electrons in orbitals.

When an electron jumps from level n_2​ to level n₁​:

\[\text{Number of emission lines}=\frac{(n_2-n_1)(n_2-n_1+1)}{2}\]

Atoms are made up of three fundamental subatomic particles — electron, proton, and neutron. Their discovery was a milestone in understanding atomic structure.

Discovery Timeline:

Properties of Subatomic Particles

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: \[_1H^1and_1H^2\]

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: \[_{18}Ar^{40}\mathrm{and}_{19}K^{40}\]

Isotones are atoms of different elements that have the same number of neutrons but different atomic and mass numbers.

Same in isotones:

• Number of neutrons

Different in isotones:

• Atomic number and mass number
• Number of protons and electrons
• Electronic configuration
• Position in periodic table

Examples: \[_1\mathrm{H}^3\mathrm{~and~}_2\mathrm{H}\mathrm{e}^4\]

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)

Emission vs. Absorption Spectrum

• Emission spectrum: Formed when atoms emit radiation upon absorbing energy. Electrons jump to higher levels, then fall back, emitting photons. Arranged in increasing order of wavelength (decreasing frequency). \[\Delta E=E_{n_2}-E_{n_1},\] where n₂ > n_1​.

• Absorption spectrum: Formed when atoms absorb specific wavelengths of radiation. Arranged in increasing order of wavelength. \[\Delta E=E_{n_2}-E_{n_1},\] where n₂ < n₁​.

Spectral Series of Hydrogen

Each series corresponds to electron transitions ending at a particular energy level:

Bohr's model applies to one-electron species such as H, He⁺, Li²⁺, etc.

Postulates of Bohr's Model:

1. Electrons revolve in fixed circular paths called stationary states, orbits, shells, or energy levels. Each has a definite, fixed energy. Energy of the electron increases as it moves away from the nucleus.

2. Angular momentum of an electron is always an integral multiple of \[\frac{h}{2\pi}:\]
   \[mvr=n\cdot\frac{h}{2\pi}\]
   where m = mass, v = velocity, r = orbital radius, n = principal quantum number.

Energy is emitted or absorbed only when an electron jumps between energy levels — not while it is in a stationary orbit. When falling from higher (E₂) to lower (E₁) energy level:

\[\Delta E=E_2-E_1=h\nu=\frac{hc}{\lambda}\]

Moving particles (like electrons) behave both as particles and as waves:

\[\lambda=\frac{h}{mv}=\frac{h}{p}\]

where p = mv = momentum of the particle.

• 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.

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.

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