Comprehensive Study Notes on Atomic Structure

Introduction to Subatomic Particles and Cathode Ray Experiments

• Subatomic particles are the building blocks of an atom. The primary particles discussed are electrons, protons, and neutrons.

• The Cathode Ray Experiment was the foundational study for discovering the electron.

• Conditions for the Cathode Ray Discharge Tube:

  • Gas is placed at extremely low pressure, approximately $10^{-4}\,mmHg$.

  • High voltage is applied, typically between 10,0\int_0^{\infty}\!\int_0^{\infty}\!\placeholder{}\,dx\,dx to $20,000\,V$.

• Logic for Low Pressure:

  • At low pressure, gas molecules are far apart, increasing the mean free path for electrons.

  • Electrons ejected from the cathode strike gas particles and ionize them, creating a bundle of electrons without excessive scattering.

  • If gas density were high, electrons would lose momentum through frequent collisions, preventing the generation of a clear beam.

• Source of Cathode Rays:

  • The primary source is the cathodic metal material itself.

  • Secondary electrons are generated via the ionization of gas atoms in the tube.

• Zinc Sulfide ($ZnS$) Coating:

  • This phosphor material is used behind the anode. When electrons strike it, it produces a characteristic green glow.

• Influence of Gas Nature:

  • Cathode rays are a bundle of electrons. Consequently, their properties remain constant regardless of the identity of the gas used in the tube ($A$, $B$, $C$, etc.).

• Properties of Cathode Rays:

  • Travel in straight lines (casting shadows of objects in their path).

  • Composed of negatively charged particles (attracted to positive plates in an electric field).

  • Possess mass and kinetic energy (can rotate a light mica paddle wheel).

  • Generate X-rays when striking heavy metals like Molybdenum ($Mo$), Copper ($Cu$), or Tungsten ($W$).

  • Ionize the gases through which they pass.

• Specific Charge of Electrons ($e/m$):

  • Calculated as $\text{Charge} / \text{Mass} = -1.6 \times 10^{-19}\,C / 9.1 \times 10^{-31}\,kg$.

  • The constant value is $-1.76 \times 10^{11}\,C\,kg^{-1}$.

  • This value is universal for all atoms, identifying the electron as a fundamental constituent of matter.

Millikan’s Oil Drop Experiment

• This experiment was designed to determine the charge of a single electron.

• Apparatus Components:

  • A chamber with two metallic plates (top positive, bottom negative).

  • An atomizer to break oil into a fine mist.

  • An ionization source (X-rays) to ionize the air and provide electrons to be captured by falling oil drops.

  • A telescope to observe the motion of the droplets.

• Procedure:

  • Fine droplets fall through a tiny hole in the positive plate.

  • Electrons from ionized gas attach to the droplets.

  • The electrical potential between the plates is adjusted until the upward electrical force ($qE$) balances the downward gravitational force ($mg$), stabilizing the droplet.

• Conclusion on Quantization:

  • Millikan found that the charge ($q$) on any droplet was always an integral multiple of a constant minimum value.

  • Minimum electronic charge observed: $-1.6 \times 10^{-19}\,C$.

  • Formula: $q = n \times e$, where $n$ is an integer ($1, 2, 3 \dots$). Individuals cannot have fractional charges like $3.5e$.

Anode Rays and Neutron Discovery

• Anode Ray Experiment (Canal Rays):

  • Conducted using a perforated cathode.

  • While cathode rays move toward the anode, positive ions (residual gas ions) move toward the cathode.

  • These rays pass through the cathode perforations and create a pink or red glow on the glass.

  • Unlike cathode rays, the nature of anode rays depends entirely on the gas inside the tube because they are ionized gas atoms.

• Discovery of the Proton:

  • When hydrogen gas is used, the resulting positive ion is the smallest possible cathion ($H^+$).

  • This particle is called the proton.

  • The $e/m$ ratio for anode rays varies with the gas and is maximum for hydrogen due to its small mass.

• Discovery of the Neutron:

  • Discovered by James Chadwick by bombarding Beryllium with alpha particles.

  • Nuclear Reaction: $^9_4Be + ^4_2He \rightarrow ^{12}_6C + ^1_0n$.

  • The neutron is a neutral particle with a mass slightly greater than that of a proton ($1.675 \times 10^{-27}\,kg$).

Properties of Charge and its Physics in Atomic Structure

• Charge is Quantized: $Q = ne$.

• Types of Charge:

  • Positive ($+1.6 \times 10^{-19}\,C$) via electron loss.

  • Negative ($-1.6 \times 10^{-19}\,C$) via electron gain.

• Coulombic Force ($F$):

  • $F = \frac{k q_1 q_2}{r^2}$.

  • $k = 9 \times 10^9\,N\,m^2\,C^{-2}$.

• Potential Energy ($PE$):

  • $PE = \frac{k q_1 q_2}{r}$.

  • Negative for attractive forces (e.g., electron-nucleus).

  • Positive for repulsive forces (e.g., electron-electron).

• Velocity of an Accelerated Charge:

  • If a particle with charge $q$ is accelerated from rest through a potential difference of $V$ volts, its potential energy ($qV$) converts to kinetic energy ($\frac{1}{2} m v^2$).

  • $v = \text{Velocity} = \text{SQRT}(\frac{2 q V}{m})$.

  • To yield results in $m/s$, use SI units: charge in Coulombs, potential in Volts, mass in kg.

• Stopping Potential:

  • The voltage required to stop a moving charged particle by applying reverse polarity.

  • Calculated using the same equation: $v = \text{SQRT}(\frac{2 q V_{stop}}{m})$.

Closest Distance of Approach (CDA)

• The scenario involves an alpha particle ($He^{2+}$) fired with velocity $v_0$ toward a stationary nucleus of atomic number $Z$.

• Principle: Total Energy Conservation ($TE_A = TE_B$).

  • Initial point ($A$) at infinity: $PE = 0$, $KE = \frac{1}{2} m_{\alpha} v_0^2$.

  • Final point ($B$) (point of momentary rest): $KE = 0$, $PE = \frac{k (2e)(Ze)}{R_0}$.

• Formula for CDA ($R_0$) for Alpha Particles:

  • $R_0 = \frac{4 k Z e^2}{m_{\alpha} v_0^2}$.

• General Formula for any particle of charge $q$:

  • $R_0 = \frac{2 k q Z e}{m v_0^2}$.

• Proportionality:

  • $R_0 \text{ is directly proportional to } Z$.

  • $R_0 \text{ is inversely proportional to } v_0^2$.

• Relation between Proton and Alpha CDA:

  • Mass of Alpha ($m_{\alpha}$) is approximately $4 \times$ mass of proton ($m_p$).

  • Charge of Alpha ($q_{\alpha}$) is $2e$, Charge of Proton ($q_p$) is $e$.

  • For the same velocity $v_0$, the ratio of CDA ($R_{\alpha} / R_p$) is roughly $1/2$.

Early Atomic Models and Rutherford’s Theory

• Thomson’s Model (Plum Pudding/Watermelon):

  • Atom is a sphere of positive charge with electrons embedded like seeds.

  • Proven incorrect by alpha particle scattering experiments.

• Rutherford’s Scattering Experiment:

  • Alpha particles bombarded a thin gold foil.

  • Observations:

    • Most particles pass straight through (atom is mostly empty space).

    • Few deviate at small angles (positive charge is concentrated).

    • One in millions returns $180^{\circ}$ (nucleus is extremely small and dense).

• Rutherford’s Conclusions:

  • The entire mass and positive charge reside in the center, called the nucleus.

  • Atomic Radius: approximately $10^{-8}\,cm$ ($10^{-10}\,m$).

  • Nuclear Radius: approximately $10^{-13}\,cm$ ($10^{-15}\,m$ or $1\,Fermi$).

  • Difference in size is roughly a factor of $10^5$.

• Calculation of Nuclear Radius ($R$):

  • $R = R_0 \times A^{1/3}$.

  • $R_0$ is a constant ($1.1 \times 10^{-13}\,cm$ to $1.44 \times 10^{-13}\,cm$).

  • $A$ is the mass number.

  • Example: For $A = 125$, and $R_0 = 1.3\,Fermi$, $R = 1.3 \times (125)^{1/3} = 6.5\,Fermi$.

Electromagnetic Radiations (EMR)

• EMR consists of oscillating electric and magnetic field vectors that are perpendicular to each other and to the direction of propagation.

• Characteristics of Waves:

  • Wavelength ($\lambda$): Distance between two consecutive crests or troughs.

  • Frequency ($\nu$): Number of waves passing through a point in one second. Units: Cycles per second ($cps$), $Hertz (Hz)$, or $s^{-1}$.

  • Velocity ($c$): All EMR travels at simple speed of light $c = 3 \times 10^8\,m/s$.

  • Relationship: $c = \nu \times \lambda$.

  • Wave Number ($\bar{\nu}$): reciprocal of wavelength, $1/\lambda$. Also $\bar{\nu} = \nu / c$.

• Electromagnetic Spectrum Sequence (Increasing Wavelength):

  • Gamma rays < X-rays < UV < Visible < Infrared < Microwaves < Radio waves.

• Visible Region:

  • Violet ($400\,nm$) to Red ($750\,nm$).

  • Red has the longest wavelength and lowest frequency.

  • Violet has the shortest wavelength and highest frequency.

• Conversion Units:

  • $1\,nm = 10^{-9}\,m$.

  • $1\,\text{Å} = 10^{-10}\,m$.

  • $1\,pm = 10^{-12}\,m$.

Dual Nature of Light and Planck’s Quantum Theory

• Diffraction and interference support the wave nature of light.

• Black Body Radiation and the Photoelectric Effect support the particle nature of light.

• Planck’s Quantum Theory:

  • Energy is emitted or absorbed in discrete packets called "quanta" or "photons."

  • Energy of 1 photon: $E = h\nu = \frac{hc}{\lambda}$.

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

• Energy of 1 Mole of Photons:

  • $E_{total} = N_A \times \frac{hc}{\lambda}$.

• Applications in Chemistry:

  • Photochemical reactions (1 bond requires 1 photon to dissociate).

  • Total energy absorbed = (Number of photons) $\times$ (Energy of one photon).

Photoelectric Effect

• Ejection of electrons (photoelectrons) from a metal surface when light of suitable frequency strikes it.

• Threshold Frequency ($\nu_0$):

  • The minimum frequency required for electron ejection. Each metal has a unique $\nu_0$.

• Work Function ($W_0$ or $\Phi$):

  • Minimum energy required to eject an electron. $W_0 = h\nu_0$.

• Einstein’s Photoelectric Equation:

  • $h\nu = W_0 + KE_{max}$.

  • $KE_{max} = h\nu - h\nu_0 = \frac{hc}{\lambda} - \frac{hc}{\lambda_0}$.

• Key Observations:

  • If \nu < \nu_0, no ejection occurs regardless of intensity.

  • Kinetic Energy depends on the frequency of incident light, not intensity.

  • Photoelectric Current (number of photoelectrons) depends on the intensity (brightness) of light.

  • Ejection is instantaneous; no time lag.

• Potential and Momentum Relations:

  • $KE = \frac{p^2}{2m}$, where $p$ is momentum.

  • $1\,eV = 1.6 \times 10^{-19}\,J$.

  • To find wavelength in Å from energy in eV: $\lambda (\text{Å}) = \frac{12400}{E (eV)}$.

Bohr Model of Hydrogen Atom

• Postulates:

  • Electrons revolve in circular stationary orbits without emitting energy.

  • Angular momentum is quantized: $mvr = \frac{nh}{2\beta}$.

  • Energy is absorbed to move to a higher level; energy is emitted to fall to a lower level ($\Delta E = h\nu$).

• Mathematical Relations for Orbit $n$:

  • Radius ($r_n$): $r_n = 0.529 \times \frac{n^2}{Z}\,\text{Å}$.

  • Velocity ($v_n$): $v_n = 2.18 \times 10^6 \times \frac{Z}{n}\,m/s$.

  • Total Energy ($E_{total}$): $E_n = -13.6 \times \frac{Z^2}{n^2}\,eV/atom$.

• Energy Relations:

  • $KE = -TE = - \frac{1}{2} PE$.

  • Total energy is negative, representing an attractive, stable state.

• Dynamic Parameters:

  • Time Period ($T$): $T \text{ is proportional to } \frac{n^3}{Z^2}$.

  • Frequency of Revolution ($f$): $f \text{ is proportional to } \frac{Z^2}{n^3}$.

  • Centripetal Acceleration ($a$): $a \text{ is proportional to } \frac{Z^3}{n^4}$.

• Single Electron Systems:

  • Bohr's model applies only to systems with one electron ($H$, $He^+$, $Li^{2+}$, $Be^{3+}$, etc.). It fails for multi-electron systems due to inter-electron repulsions.

Hydrogen Spectrum and Transitions

• Rydberg Equation:

  • $\bar{\nu} = \frac{1}{\lambda} = R Z^2 \times (\frac{1}{n_1^2} - \frac{1}{n_2^2})$.

  • Rydberg constant ($R$): $109,677\,cm^{-1}$.

• Spectral Series:

  • Lyman: $n_1 = 1$, $n_2 = 2, 3 \dots \infty$ (UV region).

  • Balmer: $n_1 = 2$, $n_2 = 3, 4 \dots \infty$ (Visible region).

  • Paschen: $n_1 = 3$, $n_2 = 4, 5 \dots \infty$ (Infrared region).

  • Brackett: $n_1 = 4$, $n_2 = 5, 6 \dots \infty$ (Infrared region).

  • Pfund: $n_1 = 5$, $n_2 = 6, 7 \dots \infty$ (Infrared region).

  • Humphrey: $n_1 = 6$, $n_2 = 7, 8 \dots \infty$ (Far Infrared region).

• Number of Spectral Lines in a Sample:

  • For transition from level $n$ to ground state: $\frac{n(n-1)}{2}$.

  • For transition between levels $n_2$ and $n_1$: $\frac{\Delta n(\Delta n + 1)}{2}$, where $\Delta n = n_2 - n_1$.

• Shortest and Longest Lines:

  • Shortest Line = Highest Energy = Transition from $\infty$ to $n_1$.

  • Longest Line = Lowest Energy = Transition from $n_{1+1}$ to $n_1$ (First line of the series).

Advanced Atomic Energy Definitions

• Ground State ($n=1$).

• Ionization Energy ($IE$):

  • Energy to move electron from ground state to $\infty$.

  • Value: $+13.6 Z^2\,eV$.

• Ionization Potential ($IP$):

  • Potential to accelerate electron for ionization. $IP = 13.6 Z^2\,Volts$.

• Excitation Energy:

  • Energy required to raise electron from Ground state to an Excited state (n > 1).

  • First Excitation: $2 \text{ to } 1$; Second Excitation: $3 \text{ to } 1$.

• Binding Energy or Separation Energy:

  • Energy to remove an electron from its current orbit to infinity.

  • $BE = -E_{total} \text{ of that orbit}$.

De Broglie Wavelength and Heisenberg Uncertainty

• De Broglie Hypothesis:

  • Matter, like light, has a dual nature. $\lambda = \frac{h}{p} = \frac{h}{mv}$.

  • Kinetic Energy connection: $\lambda = \frac{h}{\text{SQRT}(2mKE)}$.

  • For an electron accelerated by $V$ volts: $\lambda = \text{SQRT}(\frac{150}{V})\, \text{Å} \approx \frac{12.27}{\text{SQRT}(V)} \,\text{Å}$.

  • Quantized orbits: $n\lambda = 2\beta r$.

• Heisenberg Uncertainty Principle:

  • It is impossible to determine position and momentum of a subatomic particle simultaneously with absolute precision.

  • $\Delta x \times \Delta p \beta \frac{h}{4\beta}$.

  • Alternate form: $\Delta E \times \Delta t \beta \frac{h}{4\beta}$.

  • Effect of high energy photons: Watching an electron requires short wavelength light (high energy), which changes the electron's momentum.

Quantum Mechanical Model of the Atom

• Orbitals:

  • Three-dimensional regions where the probability of finding an electron is maximum (>90\%).

  • Wave function (\u03C8): Amplitude of the electron wave. No direct physical significance.

  • Probability Density (\u03C8\u00B2): Relative probability of finding an electron at a point.

• Nodes:

  • Points/surfaces where probability density is zero.

  • Radial Nodes: $n - l - 1$.

  • Angular Nodes: $l$.

  • Total Nodes: $n - 1$.

• Quantum Numbers:

  • Principal ($n$): Determines shell, size, energy.

  • Azimuthal ($l$): Determines subshell, shape ($s=0$, $p=1$, $d=2$, $f=3$). Range: $0 \text{ to } n-1$.

  • Magnetic ($m_l$): Determines orientation. Range: $-l \text{ to } +l$.

  • Spin ($s$): Internal spin of electron ($+1/2$ or $-1/2$).

• Orbital Angular Momentum: $\text{SQRT}(l(l+1)) \times \frac{h}{2\beta}$.

• Fillings Rules:

  • Pauli Exclusion: No two electrons can have identical sets of four quantum numbers.

  • Aufbau Principle: Electrons fill lower energy orbitals first based on $(n+l)$ value.

  • Hund’s Rule: Degenerate orbitals fill singly with parallel spin before pairing begins.

• Exchange Energy:

  • Energy released when electrons with the same spin exchange positions within a subshell.

  • Half-filled and full-filled configurations are extra stable due to higher exchange energy and symmetry.