STRUCTURE OF ATOM in One Shot: All Concepts - JEE Chemistry

Structure of Atom

Discovery and Properties of Cathode and Anode Rays

Cathode rays are streams of negatively charged particles (electrons) emitted from the cathode in a discharge tube at low pressure and high voltage. Anode rays (canal rays) are streams of positive ions moving towards the cathode. The $e/m$ ratio for cathode rays is always fixed, as they are always electrons, while for anode rays it varies depending on the gas used.

• Cathode rays:
  • Originate from cathode material and ionized gas molecules.
  • Move in straight lines (cast shadows).
  • Deflected towards positive terminal in electric/magnetic…

Structure of Atom

Discovery and Properties of Cathode and Anode Rays

Cathode rays are streams of negatively charged particles (electrons) emitted from the cathode in a discharge tube at low pressure and high voltage. Anode rays (canal rays) are streams of positive ions moving towards the cathode. The $e/m$ ratio for cathode rays is always fixed, as they are always electrons, while for anode rays it varies depending on the gas used.

• Cathode rays:
  • Originate from cathode material and ionized gas molecules.
  • Move in straight lines (cast shadows).
  • Deflected towards positive terminal in electric/magnetic fields (negatively charged).
  • Can rotate a paddle wheel (have mass).
  • Generate X-rays when striking heavy metals.
  • $e/m$ ratio is constant for all gases.
• Anode rays:
  • Composed of positive ions (e.g., $A^+$, $B^+$).
  • $e/m$ ratio depends on the ion's mass and charge.
  • Cause glow behind the cathode.

Properties of Electric Charge

Charge is quantized, comes in two types, and follows specific laws. These properties are essential for understanding atomic structure and related calculations.

• Quantization: $q = n \cdot e$, where $n$ is an integer.
• Types: Positive ($+1.6 \times 10^{-19}$ C) and negative ($-1.6 \times 10^{-19}$ C).
• Coulomb's Law: Force between two charges at distance $r$ is $F = k \frac{q_1 q_2}{r^2}$, $k = 9 \times 10^9$ N·m²/C².
• Potential Energy: $U = k \frac{q_1 q_2}{r}$; positive for repulsion, negative for attraction.
• Kinetic Energy on Acceleration: If a charge $q$ is accelerated through potential $V$, $KE = qV$; velocity $v = \sqrt{2qV/m}$.

Atomic Models: Thomson, Rutherford, and Bohr

Early atomic models attempted to explain atomic structure and stability. Each model introduced key concepts and had specific limitations.

• Thomson Model (Plum Pudding): Atom is a sphere of positive charge with embedded electrons.
• Rutherford Model:
  • Atom is mostly empty space.
  • Positive charge and mass concentrated in a small nucleus ($\sim 10^{-15}$ m).
  • Electrons revolve around nucleus; centrifugal force balances electrostatic attraction.
  • Could not explain atomic stability or line spectra.
• Bohr Model:
  • Electrons revolve in specific stationary orbits without emitting energy.
  • Angular momentum quantized: $mvr = n\frac{h}{2\pi}$.
  • Energy absorbed/emitted only during transitions between orbits: $\Delta E = E_{n_2} - E_{n_1}$.

Bohr Model: Key Formulas and Results

Bohr's model provides explicit formulas for radius, velocity, and energy of electrons in hydrogen-like atoms. These are valid only for single-electron systems.

• Radius: $r_n = 0.529 \frac{n^2}{Z}$ Å
• Velocity: $v_n = 2.18 \times 10^6 \frac{Z}{n}$ m/s
• Total Energy: $E_n = -13.6 \frac{Z^2}{n^2}$ eV
• Kinetic Energy: $|E_n|$
• Potential Energy: $2E_n$
• Time Period: $T \propto \frac{n}{Z^2}$
• Frequency: $f \propto \frac{Z^2}{n}$
• Acceleration: $a \propto \frac{Z^4}{n^4}$
• Applicability: Only for single-electron systems (H, He$^+$, Li$^{2+}$, etc.)

Electromagnetic Radiation and Dual Nature

Electromagnetic radiation exhibits both wave and particle nature. Key properties and formulas are essential for calculations involving atomic transitions.

• Wave Properties:
  • Wavelength ($\lambda$): Distance between two consecutive crests/troughs.
  • Frequency ($\nu$): Number of waves passing a point per second.
  • Speed: $c = \lambda \nu$, $c = 3 \times 10^8$ m/s.
  • Wave number: $\bar{\nu} = 1/\lambda$
• Electromagnetic Spectrum Order:
  • Wavelength: Radio > Microwave > Infrared > Visible > Ultraviolet > X-ray > Gamma
  • In visible: $\lambda_{red} > \lambda_{violet}$; $\nu_{violet} > \nu_{red}$
• Particle Nature (Planck's Quantum Theory):
  • Energy of photon: $E = h\nu = \frac{hc}{\lambda}$
  • Planck's constant: $h = 6.626 \times 10^{-34}$ J·s
  • Number of photons: $N = \frac{\text{Total Energy}}{\text{Energy per photon}}$
• Photoelectric Effect:
  • Electrons ejected only if incident photon energy $>$ work function ($\phi$).
  • $KE_{max} = h\nu - \phi$
  • Number of electrons ejected $\propto$ intensity; $KE_{max} \propto$ frequency.

Hydrogen Spectrum and Spectral Series

Hydrogen's emission spectrum consists of discrete lines grouped into series, each corresponding to electron transitions to a specific lower energy level.

• Rydberg Formula: $\frac{1}{\lambda} = RZ^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$, $R = 109677$ cm$^{-1}$
• Series:
  • Lyman: $n_1=1$, $n_2=2,3,...$ (UV region)
  • Balmer: $n_1=2$, $n_2=3,4,...$ (Visible region)
  • Paschen: $n_1=3$, $n_2=4,5,...$ (IR region)
  • Brackett: $n_1=4$, $n_2=5,6,...$ (IR)
  • Pfund: $n_1=5$, $n_2=6,7,...$ (IR)
• Energy and Wavelength:
  • Maximum energy photon: transition from $n_2 = \infty$ to $n_1$
  • Minimum energy photon: transition from $n_2 = n_1 + 1$ to $n_1$
  • $\lambda_{max} = \frac{1240}{E_{min}}$ nm, $\lambda_{min} = \frac{1240}{E_{max}}$ nm
• Number of spectral lines: For transition from $n$ to $1$, number of lines $= \frac{n(n-1)}{2}$

de Broglie Wavelength and Matter Waves

Moving microscopic particles exhibit wave-like properties, characterized by the de Broglie wavelength. Several formulas are used depending on the given data.

• General formula: $\lambda = \frac{h}{mv}$
• For electron in $n$th Bohr orbit: $\lambda = \frac{2\pi r}{n}$
• If kinetic energy $K$ is given: $\lambda = \frac{h}{\sqrt{2mK}}$
• If accelerated through $V$ volts:
  • Electron: $\lambda = \frac{12.3}{\sqrt{V}}$ Å
  • Proton: $\lambda = \frac{0.286}{\sqrt{V}}$ Å
  • Alpha particle: $\lambda = \frac{0.101}{\sqrt{V}}$ Å

Heisenberg Uncertainty Principle

The uncertainty principle states that it is impossible to simultaneously determine the exact position and momentum of a microscopic particle. This principle limits the precision of measurements at atomic scales.

• Mathematical statement: $\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$
• Alternate form: $\Delta x \cdot m\Delta v \geq \frac{h}{4\pi}$
• Implication: The more precisely one property is known, the less precisely the other can be known.

Quantum Numbers and Atomic Orbitals

Quantum numbers provide the address of an electron in an atom, specifying its energy, shape, orientation, and spin. Each quantum number has a specific role.

• Principal quantum number ($n$): Determines shell, size, and energy; number of orbitals $= n^2$; max electrons $= 2n^2$.
• Azimuthal quantum number ($l$): Determines subshell (shape); $l = 0$ (s), $1$ (p), $2$ (d), $3$ (f); number of orbitals in subshell $= 2l+1$.
• Magnetic quantum number ($m$): Orientation; values from $-l$ to $+l$.
• Spin quantum number ($s$): $+\frac{1}{2}$ or $-\frac{1}{2}$.
• Orbital angular momentum: $h/2\pi \sqrt{l(l+1)}$
• Spin angular momentum: $h/2\pi \sqrt{s(s+1)}$
• Magnetic moment: $\mu = \sqrt{n(n+2)}$ Bohr magnetons, $n$ = number of unpaired electrons.

Electronic Configuration Rules

The arrangement of electrons in atomic orbitals follows specific principles to ensure stability and uniqueness.

• Pauli Exclusion Principle: No two electrons in an atom can have all four quantum numbers the same; max two electrons per orbital with opposite spins.
• Aufbau Principle: Electrons fill orbitals in order of increasing energy (lowest $n+l$ first; if $n+l$ equal, lower $n$ first).
• Hund's Rule: Pairing occurs only after all degenerate orbitals are singly occupied; half-filled and fully filled subshells are especially stable.
• Order of filling: $1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s$

Quantum Mechanical Model and Probability Distributions

The quantum mechanical model describes electrons as wave functions, with probability distributions for their location. The Schrödinger equation governs these wave functions.

• Wave function ($\psi$): Describes amplitude; no direct physical meaning.
• Probability density ($\psi^2$): Probability of finding electron at a point.
• Radial probability distribution: $4\pi r^2 \psi^2$ gives probability in a spherical shell at distance $r$.
• Nodes: Points where probability density is zero.
  • Radial nodes: $n - l - 1$
  • Angular nodes: $l$
• s orbitals: Spherical, non-directional; $1s$ has no nodes, $2s$ has one radial node, $3s$ has two.
• p orbitals: Dumbbell-shaped, directional; $2p$ has one angular node.
• d orbitals: Double-dumbbell or cloverleaf shapes; specific orientation and node structure.

Spectral Line Calculations and Special Cases

Calculations involving spectral lines, transitions, and nodes are frequently tested. Recognizing the correct formula and applying it to the given quantum numbers is essential.

• Number of unique spectral lines for transition from $n$ to $1$: $\frac{n(n-1)}{2}$

• For transitions between $n_2$ and $n_1$: $\Delta n = n_2 - n_1$, number of lines $= \frac{\Delta n (\Delta n + 1)}{2}$

• Longest wavelength in a series: Corresponds to smallest energy transition (e.g., $n_2 = n_1 + 1$)

• Shortest wavelength: Largest energy transition ($n_2 = \infty$ to $n_1$)

• Radial and angular nodes: Use $n$ and $l$ values to determine node count for any orbital.

• Table: Quantum Numbers and Allowed Values