Exhaustive Notes on Periodic Properties: Atomic Radius, Ionization Energy, Electron Affinity, Electronegativity, and Thermodynamic Properties

Size of Atom and Atomic Radius

• Definition of Atomic Radius: In simple terms, it is considered the maximum distance of the outermost spherical electron cloud from the center of an atom's nucleus. If an atom is assumed to be a sphere, then its atomic radius is the radius of that sphere.
• Theoretical Limitation (Heisenberg's Uncertainty Principle): In reality, determining the exact shape or radius of a free atom is impossible because the location of electrons is uncertain. Therefore, atomic radius is estimated based on the distance between the nuclei of two atoms in a chemical bond.
• Atomic Size/Radius Definition: The distance between the nucleus of an atom and the electron cloud present in the outermost shell is called the atomic radius.
• Periodic Trend in Atomic Size:
  • Across a Period (Left to Right): The atomic size decreases with the increase in atomic number. Although an additional proton is added to the nucleus and an additional electron is added to the same energy level, the extra electron does not provide enough shielding. This results in a stronger effective nuclear charge ($Z_{eff}$), pulling the electrons closer and decreasing the size.
  • Example (2nd Period): From Lithium ($Li$) to Fluorine ($F$), the number of protons increases. Fluorine ($F$) has the highest number of protons in its period, resulting in the strongest nuclear attraction and the smallest atomic size.
• Group Trend in Atomic Size:
  • Down a Group (Top to Bottom): The atomic size increases. As we move down, a new energy level/shell is added. Although the nuclear charge increases, the addition of new shells increases the distance between the nucleus and the outermost electrons, leading to a net decrease in attraction and an increase in radius.
  • Example (Group 1): From $Li$ to $Cs$, the size increases as follows: $Li$ ($0.152 \text{ nm}$), $Na$ ($0.186 \text{ nm}$), $K$ ($0.227 \text{ nm}$), $Rb$ ($0.248 \text{ nm}$), $Cs$ ($0.265 \text{ nm}$).

Types of Atomic Radii

Depending on the bonding environment, radii are categorized into four types:

• Covalent Radius:
  • Definition: Half of the distance between the nuclei of two atoms bonded by a single covalent bond in a molecule.
  • Calculation: For a $C-C$ bond in diamond, the bond distance is measured at $0.154 \text{ nm}$ ($154 \text{ pm}$). The covalent radius of Carbon is $\frac{154}{2} = 77 \text{ pm}$ or $0.077 \text{ nm}$. For a $Cl-Cl$ bond, the distance is $198 \text{ pm}$, so the radius of Chlorine is $99 \text{ pm}$.
• Metallic Radius:
  • Definition: Half of the inter-nuclear distance between two adjacent metal atoms in a metallic lattice.
  • Comparison: Metallic radius is generally $10-20\%$ larger than the covalent radius. In covalent bonds, electron sharing pulls atoms closer. In metal lattices, electrons revolve collectively around many atoms, resulting in a slightly weaker bond and larger inter-atomic distance.
  • Example: In the Sodium ($Na$) lattice, the distance between adjacent atoms is $372 \text{ pm}$, making the metallic radius $\frac{372}{2} = 186 \text{ pm}$.
• Ionic Radius:
  • Definition: The distance from the center of the nucleus until which its attraction extends to the outermost electron level of another ion.
  • Cation Size: Formed when an atom loses electrons. Cations are always smaller than their parent atoms because the loss of an entire energy level often occurs and the effective nuclear charge increases on the remaining electrons ($Na > Na^+$).
  • Anion Size: Formed when an atom gains electrons. Anions are larger than their parent atoms. Although no new orbit is added, increased electron-electron repulsion causes the electron cloud to expand ($Cl < Cl^-$). The anion size can sometimes be nearly double the atomic size.
• Van der Waals Radius:
  • Definition: Half of the distance between the nuclei of two atoms in adjacent molecules in a solid, which are attracted by weak Van der Waals forces. It also applies to noble (inert) gases in the solid state.
  • Rank of Radii Size: Van der Waals Radius > Metallic Radius > Covalent Radius.

Ionization Energy (IE)

• Definition: The amount of energy required to remove one mole of electrons from one mole of gaseous atoms in their ground state to form one mole of unipositive gaseous ions.
• Thermochemical Nature: It is an endothermic process; the enthalpy change ($\Delta H$) is positive.
• State Requirement: It is determined in the gaseous state because, in solid/liquid states (lattice or metallic sea), atoms are interconnected by bonds. In the gaseous state, inter-molecular attraction is minimal, allowing isolated study of an atom.
• Successive Ionization Energy:
  • $1^{st} IE < 2^{nd} IE < 3^{rd} IE$.
  • Example for Magnesium: $Mg_{(g)} \rightarrow Mg^+_{(g)} + e^-$ ($\Delta H_1 = +738 \text{ kJ mol}^{-1}$); $Mg^+_{(g)} \rightarrow Mg^{2+}_{(g)} + e^-$ ($\Delta H_2 = +1450 \text{ kJ mol}^{-1}$).
  • Removing an electron from a positive ion is harder because the electron is more strongly attracted to the nucleus than in a neutral atom.
• Factors Governing Ionization Energy:
  1. Size of Atom: IE is inversely proportional to atomic size ($IE \propto \frac{1}{\text{atomic size}}$). Smaller atoms hold electrons tighter, requiring more energy.
  2. Nuclear Charge: IE is directly proportional to nuclear charge ($IE \propto \text{Nuclear charge}$). Increased protons increase attraction for outer electrons.
  3. Shielding Effect (Screening): Inner electrons act as a "curtain," shielding outer electrons from the nucleus. Order of shielding strength: $s > p > d > f$. Higher shielding leads to lower IE.
  4. Effective Nuclear Charge ($Z_{eff}$): $Z_{eff} = Z_{\text{actual}} - \text{electron shielding}$.
  5. Electronic Configuration: Elements with half-filled or fully-filled subshells are exceptionally stable and exhibit high IE. Configuration stability order: Fully-filled > Half-filled > Partially-filled.
• Anomalies in IE Trends:
  • Beryllium vs. Boron: $Be$ ($1s^2 2s^2$) has a stable fully-filled $s$-orbital. Boron ($1s^2 2s^2 2p^1$) has an unpaired p-electron which is farther from the nucleus and easier to remove. Thus, $IE_1(Be) > IE_1(B)$ ($899$ vs $800 \text{ kJ mol}^{-1}$).
  • Nitrogen vs. Oxygen: Nitrogen ($1s^2 2s^2 2p^3$) has a stable half-filled $p$-orbital. Oxygen ($1s^2 2s^2 2p^4$) has one paired electron in the p-orbital that experiences repulsion, making its removal easier. Thus, $IE_1(N) > IE_1(O)$ ($1402$ vs $1314 \text{ kJ mol}^{-1}$).

Electron Affinity (EA)

• Definition: The amount of energy released when one mole of electrons is added to one mole of gaseous atoms to form one mole of uninegative gaseous ions.
• Thermochemical Nature: It is typically an exothermic process ($\Delta H$ is negative).
• Reason for AH being Negative: Attraction by the nucleus for the incoming electron is usually stronger than the repulsion from existing electrons, releasing energy.
• Successive Electron Affinity:
  • $1^{st} EA$ is negative, but $2^{nd} EA$ and subsequent affinities are always positive.
  • Adding an electron to a negative ion requires work to overcome electrostatic repulsion ($X^- + e^- \rightarrow X^{2-}$).
  • Oxygen example: $O_{(g)} + e^- \rightarrow O^-_{(g)}$ ($\Delta H_1 = -142 \text{ kJ mol}^{-1}$); $O^-_{(g)} + e^- \rightarrow O^{2-}_{(g)}$ ($\Delta H_2 = +844 \text{ kJ mol}^{-1}$). The net process for forming $O^{2-}$ is $+702 \text{ kJ mol}^{-1}$.
• Periodic Trends in EA:
  • Period: Increases from left to right as size decreases and nuclear charge increases.
  • Group: Decreases from top to bottom as size increases.
• Factors Governing EA:
  1. Atomic Size: $EA \propto \frac{1}{\text{Size}}$.
  2. Nuclear Charge: $EA \propto \text{Nuclear charge}$.
  3. Electronic Configuration: Stability (half/fully filled) leads to lower EA. Inert gases have zero or positive EA because their shells are already full.
• Key Exceptions:
  • Chlorine vs. Fluorine: Chlorine ($Cl$) has a higher EA than Fluorine ($F$). Fluorine is very small with high electron density in its 2nd shell, leading to intense repulsion against incoming electrons. Chlorine's 3rd shell is larger, accommodating the new electron more easily.
  • Sulfur vs. Oxygen: Similarly, $EA(S) > EA(O)$.

Electronegativity

• Definition: The relative ability of an atom in a covalent bond to attract the shared pair of electrons toward itself.
• Visualizing Electronegativity: In Compound $HF$, Fluorine ($F$) is more electronegative than Hydrogen ($H$). It pulls the shared pair closer, creating a partial negative charge ($\delta^-$) on $F$ and a partial positive charge ($\delta^+$) on $H$.
• Periodic Trends:
  • Increases from left to right across a period (highest at halogens, $0$ for noble gases).
  • Decreases from top to bottom in a group.
• Scales of Measurement:
  • Pauling Scale: Most popular. Relative to H ($2.1$). Fluorine ($4.0$) is the most electronegative.
  • Millikan Method: Electronegativity = $\frac{\text{Ionization Potential (IP) + Electron Affinity (EA)}}{2}$.
• Factors Affecting Electronegativity:
  1. Size of Atom: Smaller size = higher electronegativity.
  2. Nuclear Charge: Higher nuclear charge = higher electronegativity.
  3. Oxidation State: Increasing positive oxidation state decreases the distance of the bond pair from the nucleus, increasing electronegativity. For example: $Fe$ ($1.80$) < $Fe^{2+}$ ($1.83$) < $Fe^{3+}$ ($1.96$).
  4. Hybridization: Electronegativity increases with s-character ($sp > sp^2 > sp^3$). Carbon in $sp$ ($50\% s$) is more electronegative ($3.24$) than in $sp^3$ ($25\% s, 2.48$).

Metallic vs. Non-Metallic Properties

• Metallic Property: The tendency of an atom to lose electrons and form cations. Increases down a group and decreases across a period.
  • Most reactive stable metal: Cesium ($Cs$). Most reactive unstable: Francium ($Fr$).
• Non-Metallic Property: The tendency to gain electrons and form anions. Decreases down a group and increases across a period.
• Metalloids (Semimetals): Elements showing properties of both metals and non-metals. Listed as: Boron ($B$), Silicon ($Si$), Germanium ($Ge$), Arsenic ($As$), Antimony ($Sb$), and Tellurium ($Te$).

Melting and Boiling Points (Thermodynamic Periodic Property)

• General Definitions:
  • Melting Point (MP): The temperature at which a solid converts to liquid (at $1 \text{ atm}$).
  • Boiling Point (BP): The temperature at which a liquid starts to boil (at $1 \text{ atm}$).
• Group 1 Metals (Alkali Metals):
  • MP and BP decrease down the group ($Li > Na > K > Rb > Cs$). As atomic size increases, electron density in the energy levels decreases, weakening the metallic bond.
  • $Li$: MP $180.5 ^\circ C$; $Cs$: MP $28.4 ^\circ C$.
• Group 17 (Halogens):
  • MP and BP increase down the group. Halogens are diatomic ($X_2$) non-polar molecules held by Van der Waals forces. As molecular mass and size increase (from $F_2$ to $I_2$), the strength of Van der Waals forces increases.
  • $F_2$ (Gaseous): MP $-233 ^\circ C$, BP $-188 ^\circ C$.
  • $I_2$ (Solid): MP $113.5 ^\circ C$, BP $184.35 ^\circ C$.
• Period 3 Elements:
  • Metals ($Na, Mg, Al$): MP and BP increase from $Na$ to $Al$ because the number of free mobile electrons increases ($Na^+$ vs $Mg^{2+}$ vs $Al^{3+}$), strengthening the metallic bond.
  • Silicon ($Si$): Highest MP and BP ($1410 ^\circ C$ / $2355 ^\circ C$) due to its three-dimensional network covalent framework structure.
  • Non-metals ($P_4, S_8, Cl_2, Ar$): MP/BP depend on Van der Waals forces. Order: $S_8 > P_4 > Cl_2 > Ar$. $S_8$ is larger and heavier than $P_4$, so it has stronger forces.
• 3d Transition Metals (d-block):
  • MP/BP depend on unpaired electrons in $3d$ and $4s$ orbitals which participate in metallic bonding.
  • IE/BP increase from $Sc$ to $V$.
  • Anomalies ($Cr$ and $Mn$): Despite having many unpaired electrons, $Cr$ and $Mn$ have lower MP/BP than expected. Their electronic configurations ($3d^5 4s^1$ and $3d^5 4s^2$) are highly stable/symmetrical, making the electrons less likely to participate in metallic bonding.
  • Late Transition Metals ($Cu, Zn$): Significant decrease in MP/BP. $Zn$ has the lowest ($MP = 420 ^\circ C$) in the series because the filled $3d^{10}$ and $4s^2$ shells are stable, leading to very weak metallic bonds.

Mathematical Examples and Problems

• Example: Ionization Energy of Hydrogen
  • Using $E = hc \cdot (\frac{1}{n_1^2} - \frac{1}{n_2^2})$ where $n_1 = 1$ and $n_2 = \infty$.
  • $E = 2.18017 \times 10^{-18} ext{ J}$. For$1 ext{ mole}$, energy needed is$1313.12 ext{ kJ mol}^{-1}.\n* **Example: Energy for Cl Anion Conversion**\n * Given affinity of Cl$=$-3.615 ext{ eV}. \n * 1 ext{ eV} = 1.602 \times 10^{-19} ext{ J}. \n * Energy for 1 ext{ mole}$of$Cl$($35.5 ext{ g}$) =$348.8 ext{ kJ}. \n * To convert 2.2187 ext{ g}$of$Cl$:$\frac{348.8 \times 2.2187}{35.5} = 21.80 ext{ kJ}.\n* **Example: Electron Affinity and Mass of Fluorine**\n * Affinity of F$=$-3.4 ext{ eV}$. Energy available =$32.806 ext{ kJ}.\n * By calculating number of atoms and moles (n = 0.10 ext{ mole}$), mass of$F$=$0.1 \times 19 = 1.9 ext{ g}$$.