Comprehensive Advanced Physical and General Chemistry Study Guide

Atomic Structure, Subatomic Particles, and Electron Configurations

• An atom is the smallest constituent unit of an element that can participate in a chemical reaction.

• Atoms consist of three primary subatomic particles: protons, neutrons, and electrons.

• Fundamental Subatomic Properties:

  • Proton ($p$): Relative mass = $1$, Relative charge = $+1$.
  • Electron ($e^-$): Relative mass = $\frac{1}{1840}$, Relative charge = $-1$.
  • Neutron ($n$): Relative mass = $1$, Relative charge = $0$.

• Atomic Relationships and Neutrality:

  • In an electrically neutral atom: $\text{Number of protons} = \text{Number of electrons}$.
  • The atomic number ($Z$) of an element is uniquely defined by its number of protons.
  • Mass number (nucleon number, $A$): $\text{Mass number} = \text{Number of protons} + \text{Number of neutrons}$.

• Behavior in an Electric Field:

  • When a collimated beam containing protons, neutrons, and electrons passes between oppositely charged plates:
  • Electrons deflect toward the positive ($+$ ) plate because they carry a negative charge. Their deflection angle is very large owing to their negligible mass.
  • Protons deflect toward the negative ($-$ ) plate because they carry a positive charge. Their deflection angle is smaller than that of electrons due to their significantly larger mass.
  • Neutrons continue straight without deflection because they carry no electrical charge ($0$).
  • Mathematical Relationship of Deflection:
  • $\text{Angle of deflection} \propto \frac{\text{charge}}{\text{mass}}$
  • $\text{Angle of deflection} = k \times \frac{\text{charge}}{\text{mass}}$
  • The constant $k$ depends strictly on the potential difference (p.d.) applied across the positive and negative plates.

• Atomic Structural Architecture:

  • Nucleus:
  • Located at the center of the atom, containing protons and neutrons.
  • Carries a net positive charge.
  • Represents the heaviest part of the atom because it houses the high-mass nucleons.
  • An element's chemical identity is fundamentally defined by its nuclear proton count.
  • Quantum Shells:
  • Extranuclear regions housing orbiting electrons.
  • A shell represents the energetic pathway or orbital route taken by an electron as it circles the nucleus.
  • Electrons are constrained in their trajectories by continuous electrostatic forces of attraction exerted by the positively charged nucleus.
  • Each shell corresponds to a principal quantum number ($n$), numbered sequentially starting from the shell nearest to the nucleus ($n = 1, 2, 3, 4, \dots$).
  • The maximum electron capacity of any principal shell is governed by the formula $2n^2$:
    • Shell $n = 1$: $2(1)^2 = 2\text{ electrons}$
    • Shell $n = 2$: $2(2)^2 = 8\text{ electrons}$
    • Shell $n = 3$: $2(3)^2 = 18\text{ electrons}$
    • Shell $n = 4$: $2(4)^2 = 32\text{ electrons}$

• Subshells and Orbitals:

  • The number of subshells in any principal shell is equal to its shell number ($n$).
  • Subshells differ in shape, spatial extension, and energy levels, and are categorized as $s, p, d,$ and $f$:
  • Shell $1$: $1$ subshell ($1s$)
  • Shell $2$: $2$ subshells ($2s, 2p$)
  • Shell $3$: $3$ subshells ($3s, 3p, 3d$)
  • Shell $4$: $4$ subshells ($4s, 4p, 4d, 4f$)
  • Orbitals:
  • An orbital is defined as the three-dimensional region of space around the nucleus where there is the highest probability of locating an electron.
  • Any single orbital can accommodate a maximum of two electrons.
  • The two electrons within an orbital must possess opposite spins (represented diagrammatically by antiparallel arrows in boxes: $\uparrow\downarrow$) to minimize electrostatic repulsion between their identical negative charges, conferring energetic stability.
  • Orbital Capacities per Subshell:
    • $s$ subshell: $1$ orbital, maximum $2\text{ electrons}$
    • $p$ subshell: $3$ degenerate orbitals ($p_x, p_y, p_z$), maximum $6\text{ electrons}$
    • $d$ subshell: $5$ orbitals, maximum $10\text{ electrons}$
    • $f$ subshell: $7$ orbitals, maximum $14\text{ electrons}$
  • Orbital Geometry:
  • $s$ Orbitals: Completely spherical; non-directional. The diameter of the sphere increases proportionally with the principal quantum number ($n$) as distance from the nucleus increases.
  • $p$ Orbitals: Dumbbell-shaped consisting of two distinct lobes separated by a nodal plane at the nucleus. The three degenerate orbitals lie mutually perpendicular along the Cartesian axes ($p_x, p_y, p_z$) and possess equal energy in an isolated atom.

• Relative Energies of Orbitals and the Aufbau Principle:

  • Electrons within shells possess both potential and kinetic energies; larger principal quantum numbers correspond to higher energetic states.
  • As the distance from the nucleus increases, the energy gap between consecutive principal shells becomes progressively smaller.
  • This energy compression causes subshell overlap beginning at shell $n = 4$, such that the $4s$ subshell possesses lower energy than the $3d$ subshell and is therefore populated first.

• Rules for Constructing Electronic Configurations:

  • Aufbau Principle: Electrons fill orbitals of the lowest available energy levels before occupying higher levels ($1s \rightarrow 2s \rightarrow 2p \rightarrow 3s \rightarrow 3p \rightarrow 4s \rightarrow 3d \rightarrow 4p \dots$).
  • Hund's Rule: When filling degenerate orbitals (such as the three $p$ orbitals), electrons occupy them singly with parallel spins before pairing occurs, minimizing inter-electronic repulsion.
  • Complete filling of a subshell must occur before proceeding to populate the next subshell.

• Ground-State Configurations of the First 30 Elements:

  • $_1\text{H}: 1s^1$
  • $_2\text{He}: 1s^2$
  • $_3\text{Li}: 1s^2\,2s^1$
  • $_4\text{Be}: 1s^2\,2s^2$
  • $_5\text{B}: 1s^2\,2s^2\,2p^1$
  • $_6\text{C}: 1s^2\,2s^2\,2p^2$
  • $_7\text{N}: 1s^2\,2s^2\,2p^3$
  • $_8\text{O}: 1s^2\,2s^2\,2p^4$
  • $_9\text{F}: 1s^2\,2s^2\,2p^5$
  • $_{10}\text{Ne}: 1s^2\,2s^2\,2p^6$
  • $_{11}\text{Na}: 1s^2\,2s^2\,2p^6\,3s^1$
  • $_{12}\text{Mg}: 1s^2\,2s^2\,2p^6\,3s^2$
  • $_{13}\text{Al}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^1$
  • $_{14}\text{Si}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^2$
  • $_{15}\text{P}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^3$
  • $_{16}\text{S}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^4$
  • $_{17}\text{Cl}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^5$
  • $_{18}\text{Ar}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6$
  • $_{19}\text{K}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^1$
  • $_{20}\text{Ca}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^2$
  • $_{21}\text{Sc}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^1\,4s^2$
  • $_{22}\text{Ti}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^2\,4s^2$
  • $_{23}\text{V}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^3\,4s^2$
  • $_{24}\text{Cr}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^5\,4s^1$ (Anomalous: Promotion of one $4s$ electron to $3d$ provides a half-filled $3d^5$ subshell, conferring extra stability)
  • $_{25}\text{Mn}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^5\,4s^2$
  • $_{26}\text{Fe}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^6\,4s^2$
  • $_{27}\text{Co}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^7\,4s^2$
  • $_{28}\text{Ni}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^8\,4s^2$
  • $_{29}\text{Cu}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^{10}\,4s^1$ (Anomalous: Promotion of one $4s$ electron to $3d$ produces a fully-filled $3d^{10}$ subshell, conferring extra stability)
  • $_{30}\text{Zn}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^{10}\,4s^2$

• Electronic Configuration of Ions:

  • An ion is formed when an atom loses or gains electrons to achieve an energetically stable noble gas configuration.
  • Anions: Formed when non-metal atoms gain electrons. Because non-metals possess high electronegativities and multiple valence electrons, relatively little energy is required to complete their valence shell:
  • $\text{O}: 1s^2\,2s^2\,2p^4 \rightarrow \text{O}^{2-}: 1s^2\,2s^2\,2p^6$
  • $\text{P}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^3 \rightarrow \text{P}^{3-}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6$
  • $\text{S}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^4 \rightarrow \text{S}^{2-}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6$
  • $\text{Cl}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^5 \rightarrow \text{Cl}^-: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6$
  • Cations: Formed when metal atoms lose electrons from the outermost principal quantum shell and outermost orbital:
  • $\text{Mg}: 1s^2\,2s^2\,2p^6\,3s^2 \rightarrow \text{Mg}^{2+}: 1s^2\,2s^2\,2p^6$
  • Transition Metal Ion Formation Rule: Electrons in the $4s$ subshell are lost before any electrons from the $3d$ subshell:
  • $_{21}\text{Sc}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^1\,4s^2 \rightarrow \text{Sc}^{3+}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6$
  • $_{24}\text{Cr}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^5\,4s^1 \rightarrow \text{Cr}^{3+}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^3$
  • $_{30}\text{Zn}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^{10}\,4s^2 \rightarrow \text{Zn}^{2+}: 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^{10}$

Ionisation Energy and Periodic Trends

• Definitions of Ionisation Energy (IE):

  • Ionisation Energy: The energy required to remove the most loosely bound electron (outermost electron) from a gaseous particle.
  • First Ionisation Energy ($IE_1$): The energy required to remove one mole of electrons from one mole of gaseous atoms of an element to form one mole of gaseous unipositive ions ($+1$ charge):
  • $\text{Mg}(g) \rightarrow \text{Mg}^+(g) + e^- \quad (1\text{st } IE\text{ of Mg})$
  • $\text{Ca}(g) \rightarrow \text{Ca}^+(g) + e^- \quad (1\text{st } IE\text{ of Ca})$
  • $\text{F}(g) \rightarrow \text{F}^+(g) + e^- \quad (1\text{st } IE\text{ of F})$
  • $\text{Na}(g) \rightarrow \text{Na}^+(g) + e^- \quad (1\text{st } IE\text{ of Na})$
  • Second Ionisation Energy ($IE_2$): The energy required to remove one mole of electrons from one mole of gaseous unipositive ions to form one mole of gaseous dipositive ions ($+2$ charge):
  • $\text{Mg}^+(g) \rightarrow \text{Mg}^{2+}(g) + e^- \quad (2\text{nd } IE\text{ of Mg})$
  • $\text{Zn}^+(g) \rightarrow \text{Zn}^{2+}(g) + e^- \quad (2\text{nd } IE\text{ of Zn})$
  • $\text{Ca}^+(g) \rightarrow \text{Ca}^{2+}(g) + e^- \quad (2\text{nd } IE\text{ of Ca})$
  • Higher Ionisation Equations:
  • $1\text{st } IE\text{ of Chromium: } \text{Cr}(g) \rightarrow \text{Cr}^+(g) + e^-$
  • $3\text{rd } IE\text{ of Aluminium: } \text{Al}^{2+}(g) \rightarrow \text{Al}^{3+}(g) + e^-$
  • $5\text{th } IE\text{ of Manganese: } \text{Mn}^{4+}(g) \rightarrow \text{Mn}^{5+}(g) + e^-$

• Endothermic Nature of Ionisation:

  • Removing an electron from a gaseous species is always endothermic ($IE > 0$) because electrostatic attractions between the positively charged nucleus and the negative electron must be overcome.

• Successive Ionisation Energies:

  • Successive ionisation energies increase progressively ($1\text{st} < 2\text{nd} < 3\text{rd} \dots$).
  • As each electron is removed, the ionic charge becomes increasingly positive while the nuclear charge remains constant.
  • The effective attraction experienced by the remaining electrons increases, pulling them closer to the nucleus and requiring substantially more energy for subsequent removals:
  • $\text{Mg}(g) \rightarrow \text{Mg}^+(g) + e^- \quad \Delta H = +736\,kJ\,mol^{-1}$
  • $\text{Mg}^+(g) \rightarrow \text{Mg}^{2+}(g) + e^- \quad \Delta H = +1450\,kJ\,mol^{-1}$
  • $\text{Mg}^{2+}(g) \rightarrow \text{Mg}^{3+}(g) + e^- \quad \Delta H = +7740\,kJ\,mol^{-1}$
  • The sharp jump between the 2nd and 3rd ionisation energies of magnesium reflects the removal of an electron from an inner, complete principal quantum shell ($2p$), closer to the nucleus with reduced shielding.

• Factors Affecting Ionisation Energy:

  • Atomic Radius (Size of the Atom):
  • Larger atoms exhibit weaker electrostatic attraction between the nucleus and outermost valence electrons.
  • $IE \propto \frac{1}{\text{Distance between nucleus and outermost electron}}$
  • Magnitude of Nuclear Charge (Number of Protons):
  • Higher nuclear charge exerts a stronger electrostatic pull on valence electrons, demanding more energy for removal.
  • $IE \propto \text{Nuclear charge}$
  • Shielding Effect (Screening):
  • Inner shell electrons repel valence electrons, counteracting nuclear pull.
  • $IE \propto \frac{1}{\text{Shielding effect}}$

• Specific Comparative Examples:

  • Sodium ($_{11}\text{Na}$) vs. Potassium ($_{19}\text{K}$):
  • Despite potassium possessing a higher nuclear charge ($19$ protons vs. $11$ protons), its first $IE$ is lower than that of sodium.
  • Potassium's valence electron resides in the $4s$ subshell, which is further from the nucleus and shielded by more inner shells than the $3s$ electron of sodium.
  • Magnesium ($_{12}\text{Mg}$) vs. Calcium ($_{20}\text{Ca}$):
  • $Ca < Mg$. Calcium's valence electron is in the $4s$ subshell, experiencing greater distance and shielding compared to the $3s$ valence electrons in magnesium.
  • Lithium ($_3\text{Li}$) vs. Sodium ($_{11}\text{Na}$):
  • $Na < Li$. Sodium's outermost electron occupies the $3s$ subshell, situated further from the nucleus and experiencing greater inner-shell shielding than lithium's $2s$ electron.
  • Nitrogen ($_7\text{N}$) vs. Oxygen ($_8\text{O}$):
  • Both elements have valence electrons in the $2p$ subshell experiencing identical shielding, and oxygen has a higher nuclear charge ($8$ protons vs. $7$ protons).
  • However, oxygen exhibits a lower first $IE$ than nitrogen.
  • In nitrogen, the $2p$ subshell contains three singly occupied orbitals ($2p_x^1\,2p_y^1\,2p_z^1$). In oxygen, one orbital contains paired electrons ($2p_x^2\,2p_y^1\,2p_z^1$).
  • The inter-electronic repulsion between paired electrons in the same orbital elevates their energy, allowing one of the paired electrons to be removed more easily.
  • Phosphorus ($_{15}\text{P}$) vs. Sulfur ($_{16}\text{S}$):
  • Sulfur has a lower first $IE$ than phosphorus despite higher nuclear charge.
  • Inter-electronic spin-pair repulsion in the doubly occupied $3p$ orbital of sulfur lowers the energy required to remove the electron relative to the half-filled $3p^3$ subshell of phosphorus.

• Periodic Trends in First Ionisation Energy:

  • Down a Group: First $IE$ decreases.
  • Principal quantum shells increase, increasing atomic radius.
  • Inner electron shells multiply, substantially increasing shielding.
  • The combined effects of increased distance and shielding completely outweigh the increase in nuclear charge, resulting in a reduced effective nuclear charge on valence electrons.
  • Across a Period (Left to Right): First $IE$ generally increases.
  • Valence electrons enter the same principal shell, maintaining approximately constant shielding.
  • Nuclear charge increases progressively with atomic number, causing effective nuclear charge to rise and pulling valence electrons closer to the nucleus.
  • Exceptions Across a Period:
  • Group 2 to Group 13 (e.g., $\text{Mg}$ vs. $\text{Al}$): Aluminium's first $IE$ is lower than magnesium's because aluminium's outermost electron occupies a $3p$ subshell, which is higher in energy, further from the nucleus, and shielded by the inner $3s^2$ pair. Magnesium's $3s^2$ subshell is completely filled, conferring extra stability.
  • Group 15 to Group 16 (e.g., $\text{P}$ vs. $\text{S}$): Sulfur's first $IE$ drops below phosphorus's due to spin-pair repulsion in the $3p^4$ configuration.

Chemical Bonding and Molecular Structure

• A chemical bond represents the electrostatic force of attraction holding particles together in a chemical substance.

• Major classes include ionic, covalent, coordinate (dative), and metallic bonds.

• Ionic Bonding:

  • The electrostatic force of attraction operating between oppositely charged ions.
  • Formed by the complete transfer of electrons from a metal atom to a non-metal atom.
  • Dot-and-Cross Formulations:
  • $NaCl$: Metal $\text{Na}\,(2.8.1) \rightarrow [\text{Na}]^+$; Non-metal $\text{Cl}\,(2.8.7) \rightarrow [\text{Cl}]^-$.
  • $MgO$: Metal $\text{Mg}\,(2.8.2) \rightarrow [\text{Mg}]^{2+}$; Non-metal $\text{O}\,(2.6) \rightarrow [\text{O}]^{2-}$.
  • $Na_2O$: Two $[\text{Na}]^+$ cations donate to one $[\text{O}]^{2-}$ anion.
  • $CaS$: Metal $\text{Ca}\,(2.8.8.2) \rightarrow [\text{Ca}]^{2+}$; Non-metal $\text{S}\,(2.8.6) \rightarrow [\text{S}]^{2-}$.
  • $KF$: Metal $\text{K}\,(2.8.8.1) \rightarrow [\text{K}]^+$; Non-metal $\text{F}\,(2.7) \rightarrow [\text{F}]^-$.
  • Ionic vs. Atomic Radii:
  • Cation Radius: A cation is smaller than its parent neutral atom ($\text{Na}^+ < \text{Na}$). Loss of valence electrons removes the outermost shell, reducing electron-electron repulsion and shielding; the unchanged nuclear charge pulls the remaining electron cloud closer to the nucleus.
  • Anion Radius: An anion is larger than its parent neutral atom ($\text{O}^{2-} > \text{O}$). Gaining electrons increases inter-electronic repulsion without increasing nuclear charge, causing expansion of the electron cloud.
  • Factors Determining Ionic Bond Strength:
  • Ionic Radius: Smaller ions pack more closely together in the crystal lattice, leading to stronger electrostatic forces of attraction.
  • Ionic Charge Magnitude: Higher charges on the interacting ions generate exponentially stronger electrostatic attractions.

• Covalent Bonding:

  • Formed by the sharing of electron pairs between atoms, typically between non-metals.
  • Can form between a metal and a non-metal if the difference in their electronegativities is very small.
  • Defined as the electrostatic attraction between the positively charged nuclei of the bonded atoms and the shared pair of electrons.
  • Dot-and-Cross Examples:
  • $H_2$: $1\text{ bond pair, } 0\text{ lone pairs}$
  • $HCl$: $1\text{ bond pair, } 3\text{ lone pairs on Cl}$
  • $H_2O$: $2\text{ bond pairs, } 2\text{ lone pairs on O}$
  • $NH_3$: $3\text{ bond pairs, } 1\text{ lone pair on N}$
  • $CO_2$: Two double bonds ($4\text{ shared pairs total}$, $2\text{ lone pairs on each O}$ )
  • $CCl_4$: $4\text{ bond pairs, } 3\text{ lone pairs on each Cl}$
  • Octet Deficiencies and Expansions:
  • Incomplete Octets: Compounds like $AlCl_3$ (central aluminium surrounded by $6$ valence electrons) and $BeCl_2$ (central beryllium surrounded by $4$ valence electrons) have incomplete octets; they are electron-deficient and reactive.
  • Expanded Octets: Phosphorus in $PCl_5$ possesses $10$ valence electrons; Sulfur in $SF_4$ has $10$ electrons, and in $SF_6$ has $12$ electrons. Elements in Period 3 and beyond can expand their octets because they possess accessible, empty $3d$ orbitals that can participate in bonding.
  • Molecular Existence Anomalies:
    • $SF_6\text{ vs. } OF_6$: $SF_6$ exists because sulfur has empty $3d$ orbitals capable of accommodating $12$ bonding electrons. Oxygen ($n=2$) lacks $d$ orbitals and cannot expand its octet beyond $8$ electrons; hence $OF_6$ cannot exist.
    • $PCl_5\text{ vs. } PI_5$: Phosphorus has a small atomic radius and can accommodate five chlorine atoms. Iodine atoms have significantly larger atomic radii, making it impossible to fit five bulky iodine atoms around the small central phosphorus atom due to severe steric hindrance.

• Coordinate (Dative) Covalent Bonding:

  • A covalent bond in which both shared electrons are provided by only one of the participating atoms.
  • Formation Conditions:
  • The donor particle must possess at least one lone pair of electrons.
  • The acceptor particle must possess at least one vacant orbital in its valence shell.
  • Examples:
  • Ammonium Ion ($NH_4^+$): Formed when ammonia reacts with an acid ($H^+$):       $NH_3 + H^+ \rightarrow NH_4^+$       The nitrogen atom in ammonia donates its lone pair into the empty $1s$ orbital of the hydrogen ion.
  • Oxonium (Hydronium) Ion ($H_3O^+$): Formed when water coordinates with an acid ($H^+$):       $H_2O + H^+ \rightarrow H_3O^+$
  • Aluminium Chloride - Ammonia Adduct ($AlCl_3 \cdot NH_3$): The lone pair on ammonia's nitrogen coordinates into the empty orbital of the electron-deficient aluminium atom in $AlCl_3$.
  • Aluminium Chloride Dimer ($Al_2Cl_6$): In solid aluminium chloride, bonding is intermediate. At $178\,^\circ\text{C}$, it sublimes to form vapor consisting of $Al_2Cl_6$ dimers, where two bridging chlorine atoms each donate a lone pair to adjacent electron-deficient aluminium atoms. At high temperatures ($800\,^\circ\text{C}$), the dimer dissociates completely into trigonal planar $AlCl_3(g)$.

• Valence Bond Theory and Orbital Overlap:

  • Covalent bonds form when outer-shell atomic orbitals containing single unpaired electrons overlap, pairing the electrons.
  • Sigma ($\sigma$) Bonds:
  • Formed by direct head-on (axial) overlap of atomic orbitals ($s-s, s-p,\text{ or } p-p$).
  • All single covalent bonds are sigma bonds.
  • Sigma bonds are stronger than pi bonds because head-on overlap allows greater spatial overlap of atomic orbitals along the internuclear axis.
  • Pi ($\pi$) Bonds:
  • Formed by the sideways (lateral) overlap of parallel $p$ orbitals above and below the internuclear plane.
  • A pi bond forms only when multiple bonds exist between two atoms: a double bond contains $1\sigma + 1\pi$, and a triple bond contains $1\sigma + 2\pi$.
  • Deducing Sigma and Pi Bond Counts:
  • $H-C \equiv N$: $2\sigma, 2\pi$
  • Ethane ($C_2H_6$): $7\sigma, 0\pi$
  • Propanoic acid ($CH_3CH_2COOH$): $10\sigma, 1\pi$
  • Sulfuric acid ($H_2SO_4$ with two $S=O$ and two $S-OH$): $6\sigma, 2\pi$
  • But-2-enal derivative ($CH_3-CH=CH-CHO$): $10\sigma, 2\pi$

• Valence Shell Electron Pair Repulsion (VSEPR) Theory:

  • Electron pairs surrounding a central atom repel each other electrostatically and adopt a geometry that maximizes spatial separation, minimizing repulsion and giving the lowest energy.
  • Hierarchy of Repulsion:$\text{lone pair - lone pair} > \text{lone pair - bond pair} > \text{bond pair - bond pair}$
  • Lone pairs are held closer to the central nucleus than bonding pairs, producing a more concentrated, localized electron cloud that exerts greater repulsive force, reducing adjacent bond angles.
  • Rules for Predicting Molecular Geometries:
  • Total electron pairs: $\frac{\text{Valence } e^- \text{ of central atom} + \text{electrons needed by peripheral atoms}}{2}$
  • Double bonds are treated as single bonding domains for geometry prediction.
  • For oxoanions/oxo-compounds (where terminal oxygen is double-bonded): $\text{Electron pairs} = \frac{\text{Valence } e^- + \text{needed } e^-}{2} - (\text{Number of terminal oxygen atoms})$

• VSEPR Geometries and Bond Angles:

  • $2\text{ Bond Pairs}, 0\text{ Lone Pairs}$: Linear, bond angle = $180^\circ$. Examples: $BeCl_2, CO_2, NO_2^+$.
  • $3\text{ Bond Pairs}, 0\text{ Lone Pairs}$: Trigonal planar, bond angle = $120^\circ$. Examples: $AlCl_3, BH_3, BCl_3, CO_3^{2-}, NO_3^-$.
  • $4\text{ Bond Pairs}, 0\text{ Lone Pairs}$: Tetrahedral, bond angle = $109.5^\circ$. Examples: $CH_4, CCl_4, SiCl_4, SO_4^{2-}, NH_4^+$.
  • $3\text{ Bond Pairs}, 1\text{ Lone Pair}$: Trigonal pyramidal, bond angle = $107^\circ$. The lone pair pushes the three bonding pairs closer together. Examples: $NH_3, PCl_3, H_3O^+$.
  • $2\text{ Bond Pairs}, 2\text{ Lone Pairs}$: Bent / Non-linear / V-shaped, bond angle = $104.5^\circ$. Two lone pairs exert strong mutual repulsion, reducing the angle from tetrahedral. Examples: $H_2O, H_2S, H_2Se, F_2O$.
  • $5\text{ Bond Pairs}, 0\text{ Lone Pairs}$: Trigonal bipyramidal, bond angles = $90^\circ, 120^\circ, 180^\circ$. Example: $PCl_5$.
  • $6\text{ Bond Pairs}, 0\text{ Lone Pairs}$: Octahedral, bond angle = $90^\circ$. Example: $SF_6$.
  • $4\text{ Bond Pairs}, 2\text{ Lone Pairs}$: Square planar, bond angle = $90^\circ$. The two lone pairs occupy opposite axial positions ($180^\circ$ apart) to minimize repulsion. Examples: $XeF_4, IF_4^-$.

• Worked Examples of Shape Determinations:

  • $H_2O$: $\frac{2 + 6}{2} = 4\text{ pairs}$ ($2\text{ bp}, 2\text{ lp}$) $\rightarrow$ Bent ($104.5^\circ$).
  • $NH_3$: $\frac{5 + 3}{2} = 4\text{ pairs}$ ($3\text{ bp}, 1\text{ lp}$) $\rightarrow$ Trigonal pyramidal ($107^\circ$).
  • $NH_4^+$: $\frac{5 + 4 - 1}{2} = 4\text{ pairs}$ ($4\text{ bp}$) $\rightarrow$ Tetrahedral ($109.5^\circ$).
  • $SF_6$: $\frac{6 + 6(1)}{2} = 6\text{ pairs}$ ($6\text{ bp}$) $\rightarrow$ Octahedral ($90^\circ$).
  • $CH_3^+$: $\frac{4 + 3 - 1}{2} = 3\text{ pairs}$ ($3\text{ bp}$) $\rightarrow$ Trigonal planar ($120^\circ$).
  • $F_2O$: $\frac{2 + 6}{2} = 4\text{ pairs}$ ($2\text{ bp}, 2\text{ lp}$) $\rightarrow$ Bent ($104.5^\circ$).
  • $CO_2$: $\frac{4 + 2(2)}{2} - 2 = 2\text{ pairs}$ ($2\text{ bp}$) $\rightarrow$ Linear ($180^\circ$).
  • $CO_3^{2-}$: $\frac{4 + 3(2) + 2}{2} - 3 = 3\text{ pairs}$ ($3\text{ bp}$) $\rightarrow$ Trigonal planar ($120^\circ$).
  • $SO_2$: $\frac{6 + 2(2)}{2} - 2 = 3\text{ pairs}$ ($2\text{ bp}, 1\text{ lp}$) $\rightarrow$ Bent.
  • $SO_3^{2-}$: $\frac{6 + 3(2) + 2}{2} - 3 = 4\text{ pairs}$ ($3\text{ bp}, 1\text{ lp}$) $\rightarrow$ Trigonal pyramidal.

Electronegativity, Bond Polarity, and Intermolecular Forces

• Electronegativity:

  • The power of an atom to attract the shared pair of electrons in a covalent bond toward itself.
  • Governing Factors:
  • Atomic Radius: Smaller atoms pull shared electrons closer to their nucleus, giving higher electronegativity.
  • Shielding: Fewer shielding shells permit a stronger effective nuclear pull on shared electrons.
  • Nuclear Charge: A higher positive charge in the nucleus increases the attraction for shared electron pairs.
  • Periodic Trends:
  • Down a group: Electronegativity decreases because increases in atomic radius and shielding outweigh the added nuclear charge.
  • Across a period: Electronegativity increases because atomic radius decreases, shielding remains constant, and nuclear charge increases.
  • Most electronegative elements in decreasing order: $\text{F} > \text{O} > \text{N} \approx \text{Cl}$.

• Bond Polarity and Molecular Dipoles:

  • Differences in electronegativity generate bond dipoles: the more electronegative atom acquires a partial negative charge ($\delta-$) and the less electronegative acquires a partial positive charge ($\delta+$).
  • A large electronegativity difference leads to ionic character; a small or zero difference results in covalent character.
  • Pure non-polar bonds occur between identical elements or elements with nearly identical electronegativities ($H-H, O=O, F-F, Cl-Cl, C-H$).
  • Molecular Polarity:
  • A molecule is polar only if it possesses polar bonds arranged asymmetrically such that individual bond dipoles do not cancel.
  • Highly symmetrical molecules with polar bonds are non-polar overall because their bond dipoles cancel out:
    • Linear $Y-X-Y$ ($CO_2, Cl-Be-Cl$) $\rightarrow$ Non-polar.
    • Trigonal planar $AY_3$ ($BF_3$) $\rightarrow$ Non-polar; but $AB_2Y$ is polar.
    • Tetrahedral $AY_4$ ($CCl_4$) $\rightarrow$ Non-polar; but $AY_3X$ ($CHCl_3$) is polar.
    • Trans-1,2-dichloroethene $\rightarrow$ Non-polar; Cis-1,2-dichloroethene $\rightarrow$ Polar.
    • Trans-diamminedichloroplatinum(II) $\rightarrow$ Non-polar; Cisplatin $\rightarrow$ Polar.
    • 1,4-dichlorobenzene $\rightarrow$ Non-polar; 1,2-dichlorobenzene $\rightarrow$ Polar.
    • 1,3,5-trichlorobenzene $\rightarrow$ Non-polar; 1,2,3-trichlorobenzene $\rightarrow$ Polar.

• Impact of Polarity on Bond Strength and Reactivity:

  • Bond Strength: Polar bonds are generally stronger than equivalent non-polar bonds because electrostatic attractions between dipoles reinforce the covalent sharing ($C \equiv O > N \equiv N$).
  • Halogen Halide Trend: Along the series $H-F, H-Cl, H-Br, H-I$, the electronegativity of the halogen decreases, bonds become less polarized, and bond strength progressively decreases.
  • Reactivity: Polar molecules are generally more chemically reactive than non-polar molecules because their dipoles attract incoming charged species and electrophiles/nucleophiles. For example, carbon monoxide ($C \equiv O$) is more reactive than dinitrogen ($N \equiv N$) because of its dipole moment.
  • Factors Influencing Covalent Bond Strength:
  • Bond polarity (greater polarization increases electrostatic attraction, strengthening the bond).
  • Number of shared electron pairs (triple bonds are stronger than double bonds, which are stronger than single bonds).
  • Atomic radii of bonded atoms (smaller atoms allow closer orbital overlap, yielding stronger, shorter bonds).

• Intermolecular Forces (Van der Waals Forces):

  • Forces of attraction operating between separate molecules (distinct from intramolecular covalent bonds within a molecule).
  • Permanent Dipole-Dipole (p.d.d.) Forces: Electrostatic attractions between molecules containing permanent dipoles ($HCl, CHCl_3$).
  • Induced Dipole-Dipole (i.d.d.) Forces (London Dispersion Forces):
  • The primary intermolecular attraction in non-polar molecules.
  • Formed by temporary imbalances in electron distribution, generating instantaneous dipoles that induce complementary dipoles in neighboring molecules.
  • Factors Affecting i.d.d. Strength:
    • Electron Density / Number of Electrons: Molecules with larger electron clouds polarize more easily, producing stronger, more extensive i.d.d. forces.
    • Surface Area of Contact: Molecules with larger surface areas exhibit stronger i.d.d. forces. Unbranched alkanes pack more efficiently and have greater surface contact than branched isomers, giving them higher boiling points (e.g., butane boils higher than methylpropane).
  • Relative Strength of Forces:$\text{Hydrogen bonding} > \text{Permanent dipole-dipole} > \text{Induced dipole-dipole}$

• Hydrogen Bonding:

  • A strong electrostatic attraction between a hydrogen atom covalently bound to a highly electronegative element ($F, O, N$) and the lone pair of electrons on a neighboring electronegative atom ($F, O, N$).
  • Criteria for Formation:
  • A hydrogen atom directly bonded to $F, O,$ or $N$.
  • An adjacent $F, O,$ or $N$ atom carrying at least one available lone pair.
  • Comparative Physical Properties:
  • Noble Gases (Group 18): Boiling points increase from $He$ to $Xe$ due to increasing electron counts, yielding stronger and more extensive i.d.d. forces.
  • $HCl$ vs. $Ar$: $HCl$ has a higher boiling point than argon because polar $HCl$ molecules are held by stronger permanent dipole-dipole forces, whereas non-polar argon atoms rely on weaker i.d.d. forces.
  • Hexane vs. Propane: Hexane has a higher boiling point due to its larger electron cloud and larger surface area of contact.
  • $H_2O$ vs. $HF$ vs. $NH_3$:
    • Water has the highest boiling point ($100\,^\circ\text{C}$), followed by $HF$, with $NH_3$ lowest.
    • Although fluorine is more electronegative than oxygen, each water molecule forms an average of two hydrogen bonds (possessing two hydrogen atoms and two lone pairs). Hydrogen fluoride can form only one hydrogen bond per molecule on average due to having only one hydrogen atom.
    • Ammonia has the lowest boiling point because nitrogen is the least electronegative of the three, forming weaker hydrogen bonds.
  • Ethanol vs. Ethylamine: Ethanol has a higher boiling point because oxygen is more electronegative than nitrogen, forming stronger hydrogen bonds.
  • 1,4-Benzenediol vs. 1,2-Benzenediol: 1,4-Benzenediol forms extensive intermolecular hydrogen bonds, leading to a higher melting/boiling point. In 1,2-benzenediol, adjacent hydroxyl groups form intramolecular hydrogen bonds, reducing intermolecular interactions.

Gas Laws, Ideal vs. Real Gases, and Physical States

• Kinetic Theory of Gases:

  • Describes gas particles as being in continuous, rapid, random motion, colliding with each other and with the container walls.

• Postulates of Ideal Gases:

  • Intermolecular forces of attraction between gas molecules are negligible (zero).
  • The actual volume of the gas molecules is negligible compared to the total volume occupied by the gas ($\sum V_{\text{molecules}} \ll V_{\text{container}}$).
  • Collisions between gas particles and with the container walls are perfectly elastic (no loss of kinetic energy).
  • The absolute temperature of a gas is directly proportional to the average kinetic energy of its molecules.

• The Gas Laws:

  • Boyle's Law: At constant temperature, the volume of a fixed mass of gas is inversely proportional to its pressure:     $V \propto \frac{1}{p} \implies p_1 V_1 = p_2 V_2$
  • Charles's Law: At constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute temperature ($T$ in Kelvin):     $V \propto T \implies \frac{V_1}{T_1} = \frac{V_2}{T_2}$     Absolute zero corresponds to $-273\,^\circ\text{C}$ ($0\,K$).
  • Avogadro's Law: At constant temperature and pressure, the volume occupied by a gas is directly proportional to the number of moles ($n$) of gas:     $V \propto n$

• The Ideal Gas Equation:

  • Combining the empirical gas laws yields:     $pV = nRT$
  • SI Units:
  • $p$ = Pressure in Pascals ($Pa$), where $1\,kPa = 10^3\,Pa$
  • $V$ = Volume in cubic meters ($m^3$), where $1\,dm^3 = 10^{-3}\,m^3$ and $1\,cm^3 = 10^{-6}\,m^3$
  • $n$ = Moles of gas ($mol$)
  • $T$ = Temperature in Kelvin ($K$), where $K = ^\circ\text{C} + 273$
  • $R$ = Molar gas constant = $8.31\,J\,K^{-1}\,mol^{-1}$
  • Alternative Algebraic Forms:
  • Using molar mass ($M_r$): $pV = \frac{m}{M_r}RT$
  • Gas density ($\rho = \frac{m}{V}$): $p M_r = \rho RT$
  • Molar concentration ($c = \frac{n}{V}$): $p = cRT$

• Worked Ideal Gas Calculations:

  • Volume Calculation: Volume occupied by $0.500\,mol$ of $CO_2$ at $150\,kPa$ and $19\,^\circ\text{C}$:     $p = 150 \times 10^3\,Pa, \quad T = 19 + 273 = 292\,K$$V = \frac{nRT}{p} = \frac{0.500 \times 8.31 \times 292}{150 \times 10^3} = 8.09 \times 10^{-3}\,m^3$
  • Pressure Calculation: Pressure of $4\,g$ of oxygen ($O_2$, $M_r = 32$) in a $5\,dm^3$ flask at $127\,^\circ\text{C}$:     $V = 5 \times 10^{-3}\,m^3, \quad T = 127 + 273 = 400\,K$$p = \frac{m R T}{M_r V} = \frac{4 \times 8.31 \times 400}{32 \times 5 \times 10^{-3}} = 83100\,Pa = 83.1\,kPa$
  • Volume of Methane: Volume of $272\,g$ of $CH_4$ ($M_r = 16$) at $250\,kPa$ and $54\,^\circ\text{C}$:     $p = 250 \times 10^3\,Pa, \quad T = 54 + 273 = 327\,K$$250 \times 10^3 \times V = \frac{272}{16} \times 8.31 \times 327 \implies V = 0.185\,m^3$
  • Molar Mass Determination: A $2\,dm^3$ flask holds $5.28\,g$ of gas at $200\,kPa$ and $20\,^\circ\text{C}$:     $V = 2 \times 10^{-3}\,m^3, \quad p = 200 \times 10^3\,Pa, \quad T = 20 + 273 = 293\,K$$M_r = \frac{m R T}{p V} = \frac{5.28 \times 8.31 \times 293}{(200 \times 10^3) \times (2 \times 10^{-3})} = 32.1\,g\,mol^{-1}$
  • Expansion and Compression Calculations ($p_1 V_1 = p_2 V_2$):
  • $20\,cm^3$ of $H_2$ at $200\,kPa$ expands to $90\,cm^3$ at constant temperature:       $(200 \times 10^3) \times (2 \times 10^{-5}) = p_2 \times (9 \times 10^{-5}) \implies p_2 = 44.4\,kPa$
  • A gas in a $2\,dm^3$ flask at $2\,atm$ is transferred to a $1\,dm^3$ flask:       $2 \times 2 = p_2 \times 1 \implies p_2 = 4\,atm$
  • Mixture of Gases ($p_1 V_1 + p_2 V_2 = p_3 V_3$):
  • A $2\,dm^3$ flask of $CO_2$ at $10\,kPa$ is connected to a $3\,dm^3$ flask at $8\,kPa$ ($V_3 = 5\,dm^3$):       $(10 \times 10^3)(2) + (8 \times 10^3)(3) = p_3 \times 5$$20000 + 24000 = 5 p_3 \implies p_3 = 8800\,Pa = 8.8\,kPa$

• Limitations of Ideal Gas Laws and Real Gas Deviations:

  • Real gases deviate from ideality because intermolecular forces exist between molecules, and gas molecules occupy a finite volume.
  • Conditions Causing Deviation:
  • High Pressure: Molecules are compressed close together. Their physical volume becomes a significant fraction of the container volume, and intermolecular attractions increase.
  • Low Temperature: Lower kinetic energy allows attractive forces to pull molecules together, reducing collisions with container walls.
  • Conditions Approaching Ideal Behavior: Low pressure and high temperature minimize intermolecular interactions and make molecular volume negligible.
  • Pressure Difference: Real gases exert a lower pressure than predicted by the ideal gas equation because intermolecular attractions pull molecules toward one another and away from container walls.
  • Most Ideal Gas: Helium deviates least because it is a single small atom with only two electrons, giving minimal van der Waals forces.
  • Relative Deviations:
  • $CO_2$ deviates more than $H_2$ due to its larger electron cloud and stronger van der Waals forces.
  • $NH_3$ deviates more than $N_2$ due to strong hydrogen bonding.
  • Argon deviates more than neon or helium due to its larger atomic size and stronger van der Waals forces.

Solid State Structures and Carbon Allotropes

• Physical properties of substances depend on their internal structure and the nature of the chemical bonds present.

• Four structural classes exist: Giant ionic, Giant molecular, Simple molecular, and Giant metallic.

• Giant Ionic Structures:

  • Composed of a three-dimensional lattice of alternating positive and negative ions held by ionic bonds.
  • Each cation is surrounded by a set number of anions, and vice versa.
  • Coordination Number: The number of oppositely charged ions immediately surrounding a given ion.
  • Depends on the relative sizes and charges of the ions.
  • In $NaCl$, each $\text{Na}^+$ ion is coordinated by $6\text{ Cl}^-$ ions, and each $\text{Cl}^-$ is surrounded by $6\text{ Na}^+$ ions ($6:6$ coordination).
  • Physical Properties:
  • High melting and boiling points due to strong electrostatic attractions throughout the lattice.
  • Soluble in polar solvents (e.g., water), which hydrate and separate ions, but insoluble in non-polar organic solvents.
  • Electrical non-conductors in the solid state because ions are fixed in the lattice; good electrical conductors when molten or dissolved in aqueous solution because ions become mobile.
  • Hard but brittle: mechanical stress can cause layers of ions to shift, placing like charges adjacent to one another; the resulting electrostatic repulsion cleaves the crystal.

• Simple Molecular Structures:

  • Small covalent molecules held internally by strong covalent bonds, but mutually attracted by weak intermolecular forces.
  • Low melting and boiling points; electrical insulators in all states; generally insoluble in water but soluble in non-polar organic solvents.

• Giant Molecular (Macromolecular) Structures:

  • High melting and boiling points; insoluble in all solvents; typically very hard and non-conductive (except graphite).

• Allotropes of Carbon:

  • Allotropes are different structural forms of the same element existing in the same physical state.
  • Diamond:
  • Each carbon atom is covalently bonded to four other carbon atoms in an interlocking tetrahedral framework.
  • Hybridization: $sp^3$ with bond angles of $109.5^\circ$.
  • One of the hardest known substances; high melting and boiling points; electrical non-conductor; insoluble in all solvents.
  • Uses: Jewelry, cutting tools, drill bits, and industrial wire-drawing dies.
  • Graphite:
  • Carbon atoms form flat, parallel layers of interconnected hexagonal rings.
  • Hybridization: $sp^2$ with bond angles of $120^\circ$.
  • Each carbon bonds to three others; the fourth valence electron occupies an unhybridized $2p$ orbital.
  • Overlapping $2p$ orbitals form an extended delocalized $\pi$ electron cloud across each layer.
  • The carbon-carbon bonds within a layer are shorter and stronger than those in diamond due to partial double bond character.
  • Weak van der Waals forces hold adjacent layers together, allowing them to slide easily.
  • Good electrical conductor along its layers; soft and slippery; high melting and boiling points; chemically inert.
  • Uses: Dry lubricant, inert electrodes, and pencil leads.
  • Fullerenes ($C_{60}$ Buckminsterfullerene):
  • Simple molecular nanoparticle ($10^{-9}\,m$) shaped like a hollow sphere.
  • Consists of $60$ carbon atoms arranged in $20$ hexagons and $12$ pentagons.
  • Each carbon bonds to three others; hybridization is intermediate between $sp^2$ and $sp^3$, with bond angles of approximately $108^\circ$.
  • Possesses partially delocalized electrons, but delocalization is less extensive than in graphite; non-polar.
  • Low sublimation point and relatively soft due to weak van der Waals forces between molecules; poor electrical conductor; insoluble in water but slightly soluble in organic solvents.
  • More chemically reactive than diamond or graphite due to localized regions of higher electron density; reacts with $H_2, F_2, Br_2,$ and $O_2$.
  • Uses: Targeted drug delivery cages, catalysts, and specialized lubricants.