Comprehensive Study Guide on Atomic Structure, Bohr's Model, and Quantum Mechanics

Fundamental Atomic Theory and Subatomic Particles

• Dalton's Atomic Theory (1808):

  • All matter consists of indivisible particles called atoms, which are the ultimate particles of matter.
  • All atoms of a given element are identical in mass and properties.
  • Chemical compounds are formed by combinations of two or more different types of atoms.

• Cathode Ray Discharge Tube Experiment:

  • Cathode rays originate at the cathode and move toward the anode.
  • Rays are invisible to the naked eye; their movement and behavior are observed using fluorescent materials.
  • Rays travel in straight lines in the absence of electric or magnetic fields.
  • Rays deflect toward positive potentials in the presence of electric or magnetic fields, confirming they consist of negatively charged particles called electrons.
  • Properties of cathode rays (electrons) are independent of the electrode material and the gas present in the tube, establishing electrons as fundamental constituents of all atoms.

• Characteristics of the Electron:

  • Discovery & Charge: Absolute charge was determined by R. A. Millikan using the Oil Drop Experiment (1906–1914) as $-1.6 \times 10^{-19}\,\text{C}$ (or $-4.8 \times 10^{-10}\,\text{esu}$). The relative charge is $-1$.
  • Charge-to-Mass Ratio ($e/m$): Measured by J. J. Thomson in 1897 using a cathode ray tube with mutually perpendicular electric and magnetic fields. Greater charge or higher electric/magnetic field voltage increases particle deflection. The value of $e/m$ is $1.75 \times 10^{11}\,\text{C\,kg}^{-1}$ (or $1.76 \times 10^8\,\text{C\,g}^{-1}$).
  • Absolute Mass: Calculated using charge and charge-to-mass ratio:     $\text{Mass} = \frac{e}{e/m} = \frac{1.602 \times 10^{-19}\,\text{C}}{1.76 \times 10^8\,\text{C\,g}^{-1}} = 9.10 \times 10^{-28}\,\text{g} = 9.10 \times 10^{-31}\,\text{kg}$
  • Relative Mass: $\frac{1}{1840}\,\text{a.m.u.}$

• Characteristics of the Proton:

  • Discovery: Discovered by E. Goldstein in a modified cathode ray tube producing positive streams of particles called canal rays. The lightest and smallest positive ion obtained from hydrogen gas is defined as a proton.
  • Absolute Mass: $1.6 \times 10^{-24}\,\text{g}$.
  • Relative Mass: $1\,\text{a.m.u.}$
  • Absolute Charge: $+1.6 \times 10^{-19}\,\text{C}$.
  • Relative Charge: $+1$.

• Characteristics of the Neutron:

  • Discovery: Discovered by James Chadwick by bombarding a thin sheet of beryllium with alpha ($\alpha$) particles, causing the emission of electrically neutral particles called neutrons.
  • Absolute Mass: Equal to the mass of a proton ($1\,\text{a.m.u.}$ relative mass).
  • Charge: Electrically neutral ($0\,\text{C}$).

• Early Atomic Models:

  • J. J. Thomson's Atomic Model: First atomic model; known variously as the apple pie model, plum pudding model, or watermelon model.
  • Rutherford's Atomic Model: Based on the alpha ray scattering experiment. Compared atomic structure to the solar system (solar system model) and determined the existence and relative scale of the nucleus.

• Atomic Classifications and Terminology:

  • Atomic Number ($Z$): Defined as the number of protons in the nucleus of an atom, which equals the number of electrons in a neutral atom.
  • Mass Number ($A$): Defined as the sum of protons ($Z$) and neutrons ($n$):     $A = Z + n$
  • Isotopes: Atoms possessing the same atomic number ($Z$) but different mass numbers ($A$). Examples include:
  • Hydrogen isotopes: Protium ($^1\text{H}$), Deuterium ($^2\text{H}$ or $\text{D}$), and Tritium ($^3\text{H}$ or $\text{T}$).
  • Carbon isotopes: $^{12}\text{C}$, $^{13}\text{C}$, and $^{14}\text{C}$.
  • Isobars: Atoms possessing the same mass number ($A$) but different atomic numbers ($Z$). Example: $^{14}\text{C}$ and $^{14}\text{N}$.
  • Isotones: Atoms containing the same number of neutrons ($n$) but different numbers of protons ($Z$). Example: $^{36}_{16}\text{S}$, $^{37}_{17}\text{Cl}$, $^{38}_{18}\text{Ar}$, $^{39}_{19}\text{K}$, and $^{40}_{20}\text{Ca}$ are isotones because each contains $20$ neutrons.

Electromagnetic Radiation and Quantum Phenomena

• Key Developments Leading to Bohr's Model:

  • Dual character of electromagnetic radiation (possessing both wave-like and particle-like properties).
  • Quantization of electronic energy levels.

• Properties of Electromagnetic Radiation:

  • Oscillating electric and magnetic fields generated by accelerating charged particles are perpendicular to each other and perpendicular to the wave's direction of propagation.
  • Electromagnetic waves do not require a physical medium and travel through a vacuum.
  • Array of electromagnetic radiations differing in frequency or wavelength forms the electromagnetic spectrum.
  • Six Core Properties:
  • Wavelength ($\lambda$): The distance between two consecutive crests or troughs. Expressed in $\text{cm}$, $\text{Å}$ ($1\,\text{Å} = 10^{-10}\,\text{m} = 10^{-8}\,\text{cm}$), $\mu\text{m}$, or $\text{nm}$.
  • Frequency ($\nu$): The number of wave cycles passing a fixed point per second. Units are cycles per second ($\text{cps}$) or Hertz ($\text{Hz}$).
  • Velocity ($c$): The distance traversed by a wave in one second. All electromagnetic waves travel at $c = 3 \times 10^{10}\,\text{cm\,s}^{-1}$ ($3 \times 10^8\,\text{m\,s}^{-1}$) in vacuum.
  • Wave Number ($\bar{\nu}$): The reciprocal of wavelength ($\bar{\nu} = \frac{1}{\lambda}$), representing the number of wavelengths per centimeter or meter. Expressed in $\text{cm}^{-1}$ or $\text{m}^{-1}$.
  • Amplitude ($a$): The height of a crest or depth of a trough. Determines radiation intensity.
  • Time Period ($T$): Time taken for one complete wave vibration cycle:       $T = \frac{1}{\nu}$       Expressed in seconds per cycle.

• Black Body Radiation:

  • Ordinary objects absorb a portion of incident radiation while reflecting and transmitting the rest.
  • An ideal body that absorbs 100% of all incident radiant energy falling upon it is defined as a black body.

• Photoelectric Effect:

  • Discovered by H. Hertz in 1887.
  • Ejection of electrons from a metal surface when light of suitable frequency strikes it.
  • The number of ejected electrons is directly proportional to light intensity.
  • Threshold Frequency ($\nu_0$): The characteristic minimum light frequency required to eject electrons. Below this frequency, no photoelectric effect occurs regardless of light intensity.

• Light Dispersion and Spectra:

  • White light (from the sun or incandescent lamps) consists of seven component colors: Violet, Indigo, Blue, Green, Yellow, Orange, Red (VIBGYOR).
  • Dispersion: The splitting of white light into its seven constituent colors when passed through a prism.
  • Spectrum: The array of seven colors extending continuously from red to violet.
  • Atomic Spectrum: Light emitted by excited atoms, molecules, or ions returning to lower energy states.

Quantum Theory of Radiation

• Planck's Quantum Theory:
  • Presented by Max Planck in 1901 to explain black body radiation and the photoelectric effect.
  • Hot bodies emit or absorb radiant energy discontinuously in tiny packets called quanta (singular: quantum).
  • The energy of each quantum is proportional to the radiation frequency:     $E \propto \nu$$E = h\nu$
  • Relation with wavelength and light speed:     $E = \frac{hc}{\lambda}$$\lambda = \frac{hc}{E}$$\nu = \frac{c}{\lambda}$
  • Planck's Constant ($h$) Values:     $h = 6.624 \times 10^{-27}\,\text{erg}\cdot\text{sec}$$h = 6.626 \times 10^{-34}\,\text{J}\cdot\text{s}$

Bohr's Model of the Atom

• Postulates of Bohr's Atomic Model:

  • Electrons orbit the nucleus in specific circular paths called orbits, shells, or energy levels.
  • Each orbit has a fixed energy and is labeled K, L, M, N… corresponding to principal quantum numbers $n = 1, 2, 3, 4\dots$.
  • Energy increases with distance from the nucleus ($E_1 < E_2 < E_3 < E_4\dots$).
  • An electron does not radiate energy while revolving within a specific orbit. These are termed stationary orbits or stationary energy states.
  • Energy changes occur discontinuously during orbit jumps: absorbing energy when moving lower-to-higher, emitting energy when moving higher-to-lower:     $\Delta E = E_2 - E_1 = h\nu$
  • Principle of Quantization of Angular Momentum: Electrons move only in orbits where orbital angular momentum ($mvr$) is an integral multiple of $\frac{h}{2\pi}$:     $mvr = \frac{nh}{2\pi}$     Where $n = 1, 2, 3, 4\dots$ is the principal quantum number, $m$ is electron mass, $v$ is velocity, and $r$ is orbit radius.

• Structural Shell Diagram of Bohr's Model:   

• Radii of Bohr Orbits ($r$):

  • For an electron of mass $m$ and charge $e$ revolving with tangential velocity $v$ around a nucleus with charge $Ze$ at radius $r$:
  • Electrostatic attraction force (Coulomb's Law):     $F_{\text{electrostatic}} = \frac{K Z e^2}{r^2}$     Where $K = \frac{1}{4\pi\varepsilon_0} = 9 \times 10^9\,\text{N}\cdot\text{m}^2\cdot\text{C}^{-2}$ in SI units ($K = 1$ in C.G.S. units).
  • Centripetal force balancing electrostatic attraction:     $F_{\text{centripetal}} = \frac{m v^2}{r}$
  • Equating forces:     $\frac{K Z e^2}{r^2} = \frac{m v^2}{r} \implies v^2 = \frac{K Z e^2}{r m} \quad \text{--- (i)}$
  • From angular momentum quantization:     $v = \frac{n h}{2\pi m r} \quad \text{--- (ii)}$
  • Substituting equation (ii) into equation (i):     $\frac{n^2 h^2}{4\pi^2 m^2 r^2} = \frac{K Z e^2}{r m}$$r = \frac{n^2 h^2}{4\pi^2 m K Z e^2} \quad \text{--- (iii)}$
  • Substituting physical constants ($h = 6.625 \times 10^{-34}\,\text{J}\cdot\text{s}$, $\pi = 3.14$, $m = 9.1 \times 10^{-31}\,\text{kg}$, $e = 1.6 \times 10^{-19}\,\text{C}$, $K = 9 \times 10^9\,\text{N}\cdot\text{m}^2\cdot\text{C}^{-2}$):     $r = \frac{n^2 \times (6.625 \times 10^{-34})^2}{4 \times (3.14)^2 \times (9.1 \times 10^{-31}) \times (1.6 \times 10^{-19})^2 \times (9 \times 10^9)}$$r = \frac{0.529 \times n^2}{Z}\,\text{Å}$
  • For Hydrogen Atom ($Z = 1$):     $r = 0.529 \times n^2\,\text{Å} = 0.529 \times n^2 \times 10^{-10}\,\text{m} = 0.529 \times n^2 \times 10^{-8}\,\text{cm}$

• Electrostatic Forces on Electron in Orbit:   

• Energy of an Electron ($E$):

  • Total Energy $E = \text{Kinetic Energy (K.E.)} + \text{Potential Energy (P.E.)}$
  • $\text{K.E.} = \frac{1}{2} m v^2 = \frac{K Z e^2}{2r}$
  • $\text{P.E.} = -\frac{K Z e^2}{r}$
  • Total Energy equation:     $E = \frac{K Z e^2}{2r} - \frac{K Z e^2}{r} = -\frac{K Z e^2}{2r}$
  • Substituting $r = \frac{n^2 h^2}{4\pi^2 m K Z e^2}$ into energy formula:     $E = -\frac{2\pi^2 Z^2 K^2 m e^4}{n^2 h^2} \quad \text{--- (iii)}$
  • For Hydrogen Atom ($Z = 1$):     $E = -\frac{2\pi^2 K^2 m e^4}{n^2 h^2}$
  • Substituting constant values:     $E = -\frac{2 \times (2.14)^2 \times (9 \times 10^9)^2 \times (9.1 \times 10^{-31}) \times (1.6 \times 10^{-19})^4}{n^2 \times (6.625 \times 10^{-34})^2}$
  • Standard Unit Values for Hydrogen Atom ($Z = 1$):     $E_n = -\frac{13.6}{n^2}\,\text{eV/atom}$$E_n = -\frac{313.6}{n^2}\,\text{kcal/mol}$$E_n = -\frac{1312}{n^2}\,\text{kJ/mol}$
  • Unit Conversion Factors:     $1\,\text{J} = 6.2419 \times 10^{18}\,\text{eV}$$1\,\text{eV} = 23.06\,\text{kcal/mol}$

• Limitations of Bohr's Atomic Model:

  • Unable to explain the spectra of multi-electron species.
  • Failed to explain fine spectral line structures observed in high-resolution spectroscopy.
  • Failed to explain atomic spectra for elements other than hydrogen.
  • Failed to explain line splitting in electric fields (Stark effect) or magnetic fields (Zeeman effect).
  • Failed to explain chemical bonding and molecule formation from constituent atoms.

Hydrogen Emission Spectrum and Spectral Series

• Hydrogen Spectrum Mechanics:

  • Passing an electric discharge through hydrogen gas ($\text{H}_2$) at low pressure emits bright light comprising series of lines across various spectral regions.
  • Hydrogen spectral emission lines diagram:     

• General Rydberg Equation:   $\bar{\nu} = \frac{1}{\lambda} = R \left[ \frac{1}{n_1^2} - \frac{1}{n_2^2} \right]\,\text{cm}^{-1}$   Where $n_1$ is lower orbit number, $n_2$ is higher orbit number ($n_2 > n_1$), and Rydberg constant $R = 109677\,\text{cm}^{-1} = 1.09677 \times 10^7\,\text{m}^{-1}$.

• Summary Table of Spectral Series:

  • Lyman Series:
  • Lower Orbit ($n_1$): $1$
  • Higher Orbits ($n_2$): $2, 3, 4, 5, 6, 7\dots$
  • Spectral Region: Ultraviolet
  • Wave Number Equation: $\bar{\nu} = R \left[ \frac{1}{1^2} - \frac{1}{n_2^2} \right]$
  • Balmer Series:
  • Lower Orbit ($n_1$): $2$
  • Higher Orbits ($n_2$): $3, 4, 5, 6, 7\dots$
  • Spectral Region: Visible (only series visible to naked human eye)
  • Wave Number Equation: $\bar{\nu} = R \left[ \frac{1}{2^2} - \frac{1}{n_2^2} \right]$
  • Paschen Series:
  • Lower Orbit ($n_1$): $3$
  • Higher Orbits ($n_2$): $4, 5, 6, 7\dots$
  • Spectral Region: Near Infrared
  • Wave Number Equation: $\bar{\nu} = R \left[ \frac{1}{3^2} - \frac{1}{n_2^2} \right]$
  • Brackett Series:
  • Lower Orbit ($n_1$): $4$
  • Higher Orbits ($n_2$): $5, 6, 7\dots$
  • Spectral Region: Middle Infrared
  • Wave Number Equation: $\bar{\nu} = R \left[ \frac{1}{4^2} - \frac{1}{n_2^2} \right]$
  • Pfund Series:
  • Lower Orbit ($n_1$): $5$
  • Higher Orbits ($n_2$): $6, 7\dots$
  • Spectral Region: Far Infrared
  • Wave Number Equation: $\bar{\nu} = R \left[ \frac{1}{5^2} - \frac{1}{n_2^2} \right]$

Worked Solved Problems

• Problem 1: Higher Orbit Calculation in Lyman Series

  • Statement: The wavelength of a spectral line emitted by hydrogen atom in the Lyman Series is $\frac{16}{15R}\,\text{cm}$. Determine the value of $n_2$ (where $R = 109677\,\text{cm}^{-1}$).
  • Solution:
  • Equation for Lyman series ($n_1 = 1$):       $\bar{\nu} = \frac{1}{\lambda} = R \left[ \frac{1}{1^2} - \frac{1}{n_2^2} \right]$
  • Given wavelength $\lambda = \frac{16}{15R}\,\text{cm} \implies \frac{1}{\lambda} = \frac{15R}{16}$.
  • Equating values:       $\frac{15R}{16} = R \left[ \frac{1}{1^2} - \frac{1}{n_2^2} \right]$$\frac{15}{16} = 1 - \frac{1}{n_2^2}$$\frac{1}{n_2^2} = 1 - \frac{15}{16} = \frac{1}{16} \implies n_2^2 = 16 \implies n_2 = 4$

• Problem 2: Orbit Radius Calculation for $\text{Li}^{2+}$ Ion

  • Statement: Calculate the radius of Bohr's 3rd orbit in $\text{Li}^{2+}$ ion.
  • Solution:
  • Bohr orbit radius equation:       $r = \frac{n^2 h^2}{4\pi^2 m K Z e^2}$
  • Substituting $n = 3$ and parameters for lithium ($Z = 4$ in given formulation):       $r = 1.07\,\text{Å}$

• Problem 3: Wavelength Determination from Photon Energy

  • Statement: The energy of a photon is $3 \times 10^{-12}\,\text{ergs}$. Calculate its wavelength in nanometers ($\text{nm}$).
  • Solution:
  • Energy wavelength relationship:       $E = \frac{hc}{\lambda} \implies \lambda = \frac{hc}{E}$
  • Substituting constants $h = 6.62 \times 10^{-27}\,\text{erg}\cdot\text{sec}$, $c = 3 \times 10^{10}\,\text{cm/s}$, $E = 3 \times 10^{-12}\,\text{erg}$:       $\lambda = \frac{6.62 \times 10^{-27}\,\text{erg}\cdot\text{sec} \times 3 \times 10^{10}\,\text{cm/s}}{3 \times 10^{-12}\,\text{erg}}$$\lambda = 6.62 \times 10^{-5}\,\text{cm}$
  • Converting to nanometers ($1\,\text{cm} = 10^7\,\text{nm}$):       $\lambda = 6.62 \times 10^{-5} \times 10^7\,\text{nm} = 662\,\text{nm}$