Unit 2: Structure of Atom Study Guide for Structure of the Atom

Introduction to Atomic Structure

• Historical Perspective: The concept of the atom was proposed by early Indian and Greek philosophers around $400\,B.C.$ They believed that matter, if subdivided continually, would yield a final particle that is "uncuttable" or "non-divisible." This gave rise to the term "atom," derived from the Greek word "a-tomio."

• Dalton’s Atomic Theory (1808): Proposed by John Dalton, a British schoolteacher, this theory provided a scientific basis for the atom. It considered the atom the ultimate, indivisible particle of matter. Dalton's theory successfully explained the law of conservation of mass, the law of constant composition, and the law of multiple proportions.

• Failure of Dalton’s Theory: It could not explain why certain substances, like glass or ebonite, become electrically charged when rubbed with silk or fur. Experiments in the late $19^{th}$ and early $20^{th}$ centuries eventually established that atoms are composed of sub-atomic particles: electrons, protons, and neutrons.

Discovery of Sub-atomic Particles

• Fundamental Rule of Charges:

  • Like charges repel each other.

  • Unlike charges attract each other.

Discovery of the Electron

• Electrolysis (1830): Michael Faraday showed that passing electricity through an electrolyte solution results in chemical reactions and the deposition of matter at electrodes, suggesting the particulate nature of electricity.

• Cathode Ray Discharge Tube Experiments (Mid-1850s): Faraday and others studied electrical discharge in partially evacuated tubes equipped with metal electrodes (cathode and anode). Highlights include:

  • Conditions: Discharge is observed only at very low pressures and very high voltages.

  • Observations: Current flows from the negative electrode (cathode) to the positive electrode (anode) as a stream of particles.

  • Detection: A hole in the anode and a phosphorescent coating (zinc sulphide, $ZnS$) behind it allowed detection; rays striking the coating created a bright spot.

• Results of Cathode Ray Experiments:

  • Rays start at the cathode and move to the anode.

  • They are invisible but detectable via fluorescent/phosphorescent materials (e.g., television picture tubes).

  • In the absence of fields, they travel in straight lines.

  • In electrical or magnetic fields, they behave like negatively charged particles.

  • Their characteristics are independent of the electrode material and the gas used, proving they are basic constituents of all atoms.

Charge to Mass Ratio of Electron ($e/m_e$)

• J.J. Thomson (1897): Measured the ratio by applying perpendicular electrical and magnetic fields to electrons in a cathode ray tube.

• Factors Affecting Deflection:

  1. Magnitude of charge: Higher negative charge results in greater interaction and deflection.

  2. Mass: Lighter particles deflect more.

  3. Field Strength: Deflection increases with higher voltage or magnetic field strength.

• Calculated Value:   $e/m_e = 1.758820 \times 10^{11}\,C\,kg^{-1}$

  • Where $m_e$ is the mass of the electron in $kg$ and $e$ is the magnitude of the charge in coulombs ($C$).

Charge on the Electron

• Millikan’s Oil Drop Experiment (1906-1914): R.A. Millikan determined the charge of an electron by observing the motion of oil droplets in an electrical condenser.

• Key findings:

  • Air was ionized by X-rays, and droplets acquired charge via collision with ions.

  • Charge ($q$) on any droplet is always an integral multiple of the elementary charge ($e$): $q = n e$ (where $n = 1, 2, 3...$).

  • Charge value: $-1.602176 \times 10^{-19}\,C$.

• Mass of the Electron: Calculated using the $e/m_e$ ratio and the charge:   $m_e = 9.1094 \times 10^{-31}\,kg$

Discovery of Protons and Neutrons

• Canal Rays (Protons): Modified cathode ray tubes revealed positively charged particles moving in the opposite direction.

  • Characteristics depend on the gas used: they are gaseous ions.

  • Mass is dependent on the gas type.

  • Smallest and lightest positive ion (from hydrogen) is called the proton, characterized in $1919$.

• Neutrons: Discovered by James Chadwick in $1932$ by bombarding a thin sheet of beryllium with $\alpha$-particles. He observed neutral particles with a mass slightly greater than that of a proton.

Properties of Fundamental Particles

• Electron ($e$):

  • Absolute Charge: $-1.602176 \times 10^{-19}\,C$

  • Mass: $9.109382 \times 10^{-31}\,kg$ ($0.00054\,u$)

• Proton ($p$):

  • Absolute Charge: $+1.602176 \times 10^{-19}\,C$

  • Mass: $1.6726216 \times 10^{-27}\,kg$ ($1.00727\,u$)

• Neutron ($n$):

  • Absolute Charge: $0$

  • Mass: $1.674927 \times 10^{-27}\,kg$ ($1.00867\,u$)

Early Atomic Models

Thomson Model of Atom (1898)

• Proposed that the atom is a uniform sphere of positive charge with electrons embedded in it.

• Known as the "plum pudding," "raisin pudding," or "watermelon" model.

• Crucial aspect: Mass of the atom is assumed to be uniformly distributed.

• Result: Explained overall neutrality but failed to align with future experimental results.

Radioactivity and Different Radiations

• X-rays: Discovered by Wilhelm R”entgen ($1895$). High penetrating power; produced when electrons strike dense metal targets; not deflected by fields.

• Radioactivity: Discovered by Henri Becquerel ($1896$). Emission of radiation by elements.

• Types of rays (Rutherford):

  • $\alpha$-rays: High energy particles with $+2$ charge and $4$ units of atomic mass (helium nuclei).

  • $\beta$-rays: Negatively charged particles similar to electrons.

  • $\gamma$-rays: Neutral high-energy radiation (short wavelength).

• Penetrating power: \alpha < \beta < \gamma (ratio $1:100:1000$).

Rutherford’s Nuclear Model of Atom

• Alpha Scattering Experiment: Hans Geiger and Ernest Marsden bombarded a thin gold foil ($\sim 100\,nm$) with $\alpha$-particles.

• Observations:

  • Most particles passed undeflected.

  • A small fraction was deflected by small angles.

  • Very few ($\sim 1$ in $20,000$) bounced back ($180^{\circ}$ deflection).

• Conclusions:

  • Most of the atom is empty space.

  • Positive charge and mass are concentrated in a very small volume (nucleus).

  • Atomic radius is $\sim 10^{-10}\,m$, while nuclear radius is $\sim 10^{-15}\,m$. (Analogy: If a cricket ball is the nucleus, the atomic radius is $5\,km$).

• Structure proposed:

  • Electrons move around the nucleus in circular paths called orbits (solar system analogy).

  • Electrons and nucleus are held by electrostatic forces.

Atomic and Mass Numbers

• Atomic Number ($Z$): Number of protons in the nucleus. In a neutral atom, this also equals the number of electrons.

• Mass Number ($A$): Total number of nucleons (protons + neutrons).   $A = Z + n$

• Isotopes: Atoms with the same $Z$ but different $A$ (different number of neutrons). Chemical properties remain the same.

  • Examples: Hydrogen isotopes are Protium ($^{1}_{1}H$), Deuterium ($^{2}_{1}D$), and Tritium ($^{3}_{1}T$).

• Isobars: Atoms with the same $A$ but different $Z$.

  • Examples: $^{14}_{6}C$ and $^{14}_{7}N$.

Drawbacks of the Rutherford Model

• Instability: According to Maxwell’s electromagnetic theory, an accelerated charged particle (like an electron in orbit) must emit radiation. This loss of energy would cause the orbit to shrink, and the electron would spiral into the nucleus in $10^{-8}\,s$.

• Electronic Structure: The model does not describe how electrons are distributed or what their energies are.

Developments Leading to the Bohr Model

Wave Nature of Electromagnetic Radiation

• James Maxwell (1870): Proposed that accelerating charged particles produce oscillating electric and magnetic fields called electromagnetic radiation.

• Characteristics:

  • Fields are perpendicular to each other and to the direction of propagation.

  • Does not require a medium (travels in vacuum).

  • Speed ($c$): $3.0 \times 10^{8}\,m/s$.

• Quantities:

  • Frequency ($\nu$): Number of waves passing a point in one second ($Hz$ or $s^{-1}$).

  • Wavelength ($\lambda$): Distance between peaks ($m$, $nm$, or $\text{…}$).

  • Relationship: $c = \nu \lambda$

  • Wavenumber ($\bar{\nu}$): Number of wavelengths per unit length ($\bar{\nu} = \frac{1}{\lambda}$, unit: $m^{-1}$ or $cm^{-1}$).

Particle Nature of Radiation: Planck’s Quantum Theory

• Black-body Radiation: An ideal body that absorbs and emits all frequencies. Plotting intensity vs. wavelength shows a maximum that shifts with temperature. Classical physics failed to explain this.

• Max Planck (1900): Suggested energy is emitted or absorbed in discrete "chunks" called a quantum.

• Equation: $E = h\nu$

  • $h = 6.626 \times 10^{-34}\,J\,s$ (Planck’s constant).

Photoelectric Effect

• H. Hertz (1887): Ejection of electrons when light strikes a metal surface (e.g., $K, Rb, Cs$).

• Observations:

  • No time lag between impact and ejection.

  • Number of electrons $\propto$ light intensity.

  • Threshold Frequency ($\nu_0$): Minimum frequency required for ejection. Kinetic energy of ejected electrons $\propto$ frequency of light.

• Einstein’s Explanation (1905): Photons transfer energy to electrons.   $h\nu = h\nu_0 + \frac{1}{2}m_e v^{2}$

  • $h\nu_0 = W_0$ (Work function).

Dual Nature of Radiation

• Light behaves as a wave (interference, diffraction) and as a particle (photoelectric effect, black-body radiation).

Atomic Spectra

• Emission Spectrum: Produced by detecting radiation emitted by excited atoms/molecules as they return to lower energy states.

• Absorption Spectrum: Created when a continuum of light passes through a sample; specific wavelengths are absorbed, leaving dark spaces.

• Line Spectra: Atomic emission spectra consist of specific sharp lines rather than a continuous spread. Every element has a unique line spectrum (fingerprint).

Hydrogen Line Spectrum

• Balmer Series (1885): Formula for visible lines:   $\bar{\nu} = 109,677 \left[\frac{1}{2^{2}} - \frac{1}{n^{2}}\right]\,cm^{-1}$ ($n = 3, 4, 5...$)

• Rydberg Formula: Generalization for all series:   $\bar{\nu} = 109,677 \left[\frac{1}{n_1^{2}} - \frac{1}{n_2^{2}}\right]\,cm^{-1}$ ($109,677 = R_H$—Rydberg constant).

• Series table:

  • Lyman ($n_1=1$): Ultraviolet

  • Balmer ($n_1=2$): Visible

  • Paschen ($n_1=3$): Infrared

  • Brackett ($n_1=4$): Infrared

  • Pfund ($n_1=5$): Infrared

Bohr’s Model for Hydrogen Atom

• Postulates:

  1. Electrons move in circular orbits at fixed radii and energies (stationary states).

  2. Energy change only occurs when an electron jumps between states.

  3. Frequency Rule: $\nu = \frac{\Delta E}{h} = \frac{E_2 - E_1}{h}$.

  4. Angular Momentum Quantization: $m_e vr = \frac{nh}{2\pi}$ ($n = 1, 2, 3...$).

• Calculations for Hydrogen:

  • Radii: $r_n = n^{2} a_0$; where $a_0 = 52.9\,pm$ (Bohr radius).

  • Energy: $E_n = -R_H \left(\frac{1}{n^{2}}\right)$; where $R_H = 2.18 \times 10^{-18}\,J$.

  • Significance of Negative Energy: Lower than the energy of a free electron at rest ($E_{\infty} = 0$), denoting stability.

• Hydrogen-like Species ($He^{+}, Li^{2+}, Be^{3+}$):

  • $E_n = -2.18 \times 10^{-18} \left(\frac{Z^{2}}{n^{2}}\right)\,J$

  • $r_n = \frac{52.9 n^{2}}{Z}\,pm$

• Limitations: Fails for multi-electron atoms; cannot explain doublet splitting (Stark/Zeeman effects) or chemical bonding.

Quantum Mechanical Model of the Atom

Dual Behaviour of Matter (de Broglie)

• Matter has both particle and wave properties.

• De Broglie Relation: $\lambda = \frac{h}{mv} = \frac{h}{p}$.

• Confirmed via electron diffraction; led to the development of the electron microscope.

Heisenberg Uncertainty Principle

• It is impossible to determine simultaneously the exact position and exact momentum of a sub-atomic particle.

• Equation: $\Delta x \Delta p \geq \frac{h}{4\pi}$.

• Significance: Rules out definite paths (trajectories). Applicable only to microscopic objects.

Schrödinger Wave Equation

• Developed by Erwin Schr”dinger ($1926$), it incorporates wave-particle duality and uncertainty.

• Equation: $\hat{H}\psi = E\psi$

• Wave Function ($\psi$): Contains information about an electron in an orbital.

• $|\psi|^{2}$: Probability density of finding an electron at a point.

Quantum Numbers and Orbitals

• Principal Quantum Number ($n$): Shell (K, L, M…). Size and energy.

• Azimuthal Quantum Number ($l$): Subshell/Shape ($s, p, d, f$). Ranges from $0$ to $n-1$.

  • $l=0$ (s), $l=1$ (p), $l=2$ (d), $l=3$ (f).

• Magnetic Quantum Number ($m_l$): Orientation. Ranges from $-l$ to $+l$. Total orientations = $2l+1$.

• Spin Quantum Number ($m_s$): Intrinsic spin ($+1/2$ or $-1/2$).

Shapes of Orbitals

• s-orbitals: Spherical. Size increases with $n$. Number of radial nodes = $n-1$.

• p-orbitals: Dumb-bell shaped with two lobes. Three orientations: $p_x, p_y, p_z$.

• d-orbitals: Five orientations ($d_{xy}, d_{yz}, d_{xz}, d_{x^{2}-y^{2}}, d_{z^{2}}$). Most are double-dumb-bell.

• Nodes: Regions of zero probability.

  • Angular nodes = $l$; Radial nodes = $n - l - 1$.

  • Total nodes = $n - 1$.

Rules for Filling Orbitals

1. Aufbau Principle: Orbitals are filled in order of increasing energy based on the $(n+l)$ rule. If $(n+l)$ is equal, the orbital with lower $n$ is filled first.

2. Pauli Exclusion Principle: No two electrons in an atom can have the same four quantum numbers. Each orbital holds a maximum of two electrons with opposite spins.

3. Hund’s Rule of Maximum Multiplicity: Pairing in degenerate orbitals ($p, d, f$) starts only after each orbital is singly occupied with parallel spins.

• Stability of Half-filled/Filled Subshells: Extra stability in configurations like $d^{5}$ or $d^{10}$ (e.g., Chromium: $[Ar] 3d^{5} 4s^{1}$, Copper: $[Ar] 3d^{10} 4s^{1}$) is due to symmetry and higher exchange energy.

Questions & Discussion

• Problem 2.1: Calculate particles in $^{80}_{35}Br$. Result: $35$ protons, $45$ neutrons, $35$ electrons.

• Problem 2.5: Calculate wavenumber ($\bar{\nu}$) for $5800\,\text{…}$. Result: $\bar{\nu} = \frac{1}{5.8 \times 10^{-7}\,m} = 1.724 \times 10^{6}\,m^{-1}$.

• Problem 2.17: Orbitals for $n=3$. Result: $n^{2} = 9$ orbitals ($1$ s, $3$ p, $5$ d).