JEE Chemistry Structure Of Atom Notes Set 02

Fundamental Particles and Atomic Models

Cathode rays, produced when a discharge tube at low pressure is run at very high voltage, are streams of electrons. Their charge to mass ratio, \( e/m = 1.76 \times 10^{11}\ \text{C kg}^{-1} \), is the same whichever gas fills the tube. Anode (canal) rays are positive particles; their e/m depends on the gas and is greatest for hydrogen, whose positive particle is the proton. The neutron was found by Chadwick in 1932 by bombarding beryllium with alpha particles.
Masses: electron \( 9.109\times10^{-31} \) kg, proton \( 1.673\times10^{-27} \) kg (1.00727 u), neutron \( 1.675\times10^{-27} \) kg (1.00867 u); 1 u = \( 1.66\times10^{-27} \) kg. Charge is quantised, \( Q = ne \) with \( e = 1.602\times10^{-19} \) C.
Thomson's plum-pudding model fails because Rutherford's alpha scattering showed most of the atom is empty with a tiny, dense, positive nucleus. The nucleus has diameter about \( 10^{-13} \) cm against about \( 10^{-8} \) cm for the atom, and its radius is \( R = R_0 A^{1/3} \) with \( R_0 \approx 1.1\text{ to }1.44\times10^{-13} \) cm.
Distance of closest approach of an alpha particle (energy \( \tfrac12 mv^2 \)) to a nucleus of atomic number Z: all kinetic energy becomes electrostatic potential energy, so \( r_{min} = \dfrac{2Ze^2}{4\pi\varepsilon_0\cdot\frac12 mv^2} = \dfrac{4KZe^2}{mv^2} \).

Atomic Number, Mass Number and Related Terms

Atomic number Z is the number of protons (equal to electrons in a neutral atom). Mass number A = protons + neutrons. Isotopes share Z, isobars share A, isotones share the number of neutrons, isoelectronic species share the number of electrons. Isodiaphers have the same difference n - p; isosters have the same number of atoms and electrons (N2 and CO).
Forming an ion changes only the extranuclear electrons: Z, A and the neutron count stay the same.

Wave and Particle Nature of Radiation

For any wave, \( c = \nu\lambda \) and wave number \( \bar\nu = 1/\lambda \). Electromagnetic waves all travel at \( 3\times10^{8} \) m/s in vacuum. Order of increasing wavelength: gamma rays, X-rays, ultraviolet, visible, infrared, microwaves, radio waves.
Black-body radiation and the photoelectric effect could not be explained by waves. Planck proposed that energy is exchanged only in packets (quanta): \( E = h\nu = hc/\lambda \), \( h = 6.626\times10^{-34} \) J s. Useful: \( E(\text{eV}) \approx 12400/\lambda(\text{\AA}) \) and 1 eV = \( 1.602\times10^{-19} \) J.
Photoelectric effect: no electrons are ejected below the threshold frequency \( \nu_0 \), whatever the intensity. Above it, emission is instantaneous, the number of electrons rises with intensity, and the kinetic energy rises with frequency: \( h\nu = h\nu_0 + \tfrac12 mv^2 \). The stopping potential satisfies \( eV_0 = h\nu - W_0 \).

Bohr Model of the Hydrogen-like Atom

Postulates: electrons move in stationary circular orbits without radiating; the angular momentum is quantised, \( mvr = nh/2\pi \); a photon of energy \( \Delta E = h\nu \) is absorbed or emitted in a jump between orbits.
For nuclear charge Z: \( r_n = 0.529\,\dfrac{n^2}{Z}\ \text{\AA} \), \( v_n = 2.18\times10^{6}\,\dfrac{Z}{n}\ \text{m/s} \), \( E_n = -13.6\,\dfrac{Z^2}{n^2}\ \text{eV} \) (= \( -2.18\times10^{-18}\,Z^2/n^2 \) J). Time period \( T\propto n^3/Z^2 \) and frequency of revolution \( \propto Z^2/n^3 \).
Energy relations: \( KE = -E,\ PE = 2E,\ E = -KE = PE/2 \). Ionisation energy from the ground state is \( 13.6Z^2 \) eV; the binding energy of the nth level is \( 13.6Z^2/n^2 \) eV; the first excitation energy of H is 10.2 eV.
The model works only for one-electron species (H, He+, Li2+, Be3+) and cannot explain fine structure, the Zeeman and Stark effects, or multi-electron atoms.

Hydrogen Spectrum

Rydberg's equation: \( \bar\nu = R_H Z^2\left(\dfrac{1}{n_1^2}-\dfrac{1}{n_2^2}\right) \), \( R_H = 1.097\times10^{7}\ \text{m}^{-1} \). Lines of one series share \( n_1 \): Lyman 1 (UV), Balmer 2 (visible), Paschen 3, Brackett 4, Pfund 5 (all infrared).
If electrons start from level \( n_2 \) and fall to \( n_1 \), the series has \( n_2-n_1 \) lines; when electrons can drop between any levels from level n, the total number of lines is \( n(n-1)/2 \). The series limit is the line with \( n_2=\infty \); the first line of a series has the longest wavelength. Emission spectra give bright lines; absorption spectra show dark lines at the same positions.

de Broglie Wavelength and the Uncertainty Principle

Matter has wave character: \( \lambda = h/mv = h/p \). For a particle of charge q accelerated through potential V, \( \lambda = h/\sqrt{2mqV} \); for an electron \( \lambda \approx \sqrt{150/V}\ \text{\AA} \). In a stable Bohr orbit, \( 2\pi r = n\lambda \), so n waves fit round the orbit.
Heisenberg: \( \Delta x\cdot\Delta p \ge h/4\pi \) (or \( \Delta x\cdot m\Delta v \ge h/4\pi \)), and equivalently \( \Delta E\cdot\Delta t \ge h/4\pi \). It is significant only for microscopic particles.

Quantum Mechanical Model and Orbitals

The Schrodinger equation gives wave functions \( \psi \); an orbital is a region of high probability of finding the electron, with \( |\psi|^2 \) the probability density. Radial nodes = \( n-l-1 \), angular (nodal) planes = \( l \), total nodes = \( n-1 \).
Shapes: s is spherical, p is dumb-bell with one nodal plane, d is mostly double dumb-bell with two nodal planes (\( d_{z^2} \) has a doughnut ring and two nodal cones). The 1s orbital has no node; 2s has one radial node; the number of maxima in the radial probability plot of ns is n.

Quantum Numbers and Electronic Configuration

n (principal) gives size and energy, with 1, 2, 3, ...; the shell has n subshells, \( n^2 \) orbitals and holds \( 2n^2 \) electrons. l (azimuthal) runs from 0 to n-1 and fixes the shape; orbital angular momentum is \( \sqrt{l(l+1)}\,\dfrac{h}{2\pi} \). m runs from -l to +l, giving \( 2l+1 \) orbitals. s is +1/2 or -1/2.
Spin only magnetic moment \( \mu = \sqrt{n(n+2)} \) B.M. with n unpaired electrons; total spin is \( n/2 \) in magnitude.
Aufbau: fill orbitals in increasing (n + l); equal (n + l) goes to lower n. Order: 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d. Pauli: no two electrons share all four quantum numbers, so an orbital holds at most two (opposite spin). Hund: degenerate orbitals fill singly with parallel spins before pairing.
Extra stability of half-filled and fully filled subshells (due to symmetry and exchange energy) gives the exceptions Cr \( [Ar]3d^5 4s^1 \), Cu \( [Ar]3d^{10}4s^1 \), Mo, Ag, Au and others. While forming cations, electrons leave the outermost (highest n) shell first, so Fe3+ is \( [Ar]3d^5 \).

Radioactivity and Nuclear Reactions

Radioactivity is a nuclear property, independent of temperature, pressure or the chemical state of the atom. Alpha rays are He nuclei, beta rays are fast electrons from the nucleus (\( n \to p + e^- \)) and gamma rays are high-energy photons. Penetration: gamma > beta > alpha; ionising power is in the reverse order.
Alpha emission (usually Z > 83) lowers A by 4 and Z by 2. Beta emission (n/p too high) raises Z by 1 at the same A. Positron emission or electron capture (n/p too low) lowers Z by 1 at the same A. Gamma emission lowers only the nuclear energy.
Group displacement law: after alpha emission the element moves two groups left; after beta emission one group right. Stability: n/p is about 1 up to Z = 20 and increases to about 1.5 for Bi-209; even-even nuclei are the most stable and odd-odd the least; magic numbers 2, 8, 20, 28, 50, 82, 126 give extra stability.
Artificial transmutation is written as target(projectile, ejected)product, for example \( \ce{^{14}_{7}N + ^{4}_{2}He -> ^{17}_{8}O + ^{1}_{1}H} \). Fission splits a heavy nucleus (as in \( \ce{^{235}_{92}U} \) with a neutron); fusion joins light nuclei and needs very high temperature. Both release energy through a loss of mass, \( E = \Delta m\,c^2 \).

Binding Energy and Radioactive Series

Mass defect is the difference between the sum of the masses of the separate nucleons and the mass of the nucleus; the binding energy is \( \Delta m\,c^2 \), with 1 u equal to 931.5 MeV. Binding energy per nucleon is highest (about 8.8 MeV) for \( \ce{^{56}_{26}Fe} \), so that nucleus is the most stable.
Natural decay chains end in stable lead or bismuth: the 4n series (thorium-232) ends at Pb-208 after 6 alpha and 4 beta emissions; the 4n+2 series (uranium-238) ends at Pb-206 after 8 alpha and 6 beta; the 4n+3 series (uranium-235) ends at Pb-207 after 7 alpha and 4 beta; the artificial 4n+1 series (neptunium) ends at Bi-209 after 8 alpha and 5 beta.

Worked Example: Photoelectric Effect

Threshold frequency is \( 6\times10^{14}\ \text{s}^{-1} \) and the light has frequency \( 1.1\times10^{15}\ \text{s}^{-1} \). Kinetic energy = \( h(\nu-\nu_0) = 6.626\times10^{-34}\times5\times10^{14} \) = \( 3.31\times10^{-19} \) J.

Worked Example: Spectral Line of He+