Lattice Energy and Energetics Study Notes

Core Definitions and Learning Outcomes

• Standard Enthalpy Change of Atomisation ($\Delta_{at}H$): The enthalpy change when one mole of gaseous atoms is formed from an element in its standard state, measured at standard conditions ($298\,K$ and $100\,kPa$).

• Electron Affinity (EA): The energy change when each atom in one mole of atoms in the gaseous state gains an electron to form a $-1$ ion.

• Lattice Energy ($\Delta_{LE}H$): The energy change when one mole of an ionic compound is formed from its ions in the gaseous state (measured at a stated temperature, usually $298\,K$, and pressure, usually $100\,kPa$).

• Enthalpy Change of Solution ($\Delta_{sol}H$): The enthalpy change when one mole of an ionic solid dissolves in water to form an infinitely dilute solution.

• Enthalpy Change of Hydration ($\Delta_{hyd}H$): The enthalpy change when one mole of an ion in its gaseous state is completely hydrated by water.

Lattice Energy Fundamentals

• In covalent compounds, bond enthalpies measure the bond strength. In ionic compounds, the equivalent measurement is lattice energy.

• Example Equations:

  • $Na^+(g) + Cl^-(g) \rightarrow NaCl(s)$

  • $\Delta_{LE}H = -780\,kJ\,mol^{-1}$

  • $Mg^{2+}(g) + 2Cl^-(g) \rightarrow MgCl_2(s)$

  • $\Delta_{LE}H = -2526\,kJ\,mol^{-1}$

• Lattice Enthalpy vs. Lattice Energy: At A Level, these terms are treated as synonymous.

Factors Affecting Lattice Energy Magnitude

Lattice energies are determined by four primary factors:

1. Magnitude of Charges on the Ions: Higher charges result in stronger electrostatic attractions.

2. Sum of the Ionic Radii: Smaller ions can get closer together, increasing the force of attraction.

3. Type of Lattice Structure.

4. Extent of Covalent Interactions: The presence of covalency increases the magnitude (makes it more negative).

Case Study: Magnesium Chloride vs. Sodium Chloride The lattice energy of $MgCl_2$ is significantly higher than $NaCl$ because:

• Magnesium ($Mg^{2+}$) has twice the charge of sodium ($Na^+$).

• There are twice as many chloride ions, leading to more cation-to-anion interactions.

• The ionic radius of $Mg^{2+}$ is smaller than $Na^+$, resulting in a shorter distance between ion centers.

Data Table: Lattice Energies of Various Compounds

Atomisation and Electron Affinity Details

Standard Enthalpy Change of Atomisation Examples:

• $C(s) \rightarrow C(g)$ ($\Delta_{at}H = +717\,kJ\,mol^{-1}$)

• $Na(s) \rightarrow Na(g)$ ($\Delta_{at}H = +107\,kJ\,mol^{-1}$)

• $\frac{1}{2}H_2(g) \rightarrow H(g)$ ($\Delta_{at}H = +218\,kJ\,mol^{-1}$)

Electron Affinity Trends:

• 1st Electron Affinity: Usually negative (exothermic). Exceptions include noble gases where the new electron must enter a new quantum shell, experiencing repulsion from existing valence electrons.

  • $Cl(g) + e^- \rightarrow Cl^-(g)$ ($1^{st} EA = -349\,kJ\,mol^{-1}$)

  • $O(g) + e^- \rightarrow O^-(g)$ ($1^{st} EA = -325\,kJ\,mol^{-1}$)

• 2nd Electron Affinity: Always positive (endothermic). Energy is required to overcome the electrostatic repulsion between the incoming electron and the already negative ion.

Born-Haber Cycles

Born-Haber cycles apply Hess's Law to calculate lattice energy, which cannot be measured directly. They relate the enthalpy of formation ($\Delta_f H$) to the enthalpy of atomisation ($\Delta_{at} H$), ionisation energies (IE), electron affinities (EA), and lattice energy ($\Delta_{LE} H$).

Example: Calculating Lattice Energy for Sodium Chloride ($NaCl$) Data:

• $\Delta_f H[NaCl(s)] = -411\,kJ\,mol^{-1}$

• $\Delta_{at} H[Na(s)] = +107\,kJ\,mol^{-1}$

• $1^{st} IE[Na(g)] = +496\,kJ\,mol^{-1}$

• $\Delta_{at} H[Cl_2(g)] = +122\,kJ\,mol^{-1}$

• $EA[Cl(g)] = -349\,kJ\,mol^{-1}$

Applying Hess's Law: $+107 + 496 + 122 + (-349) + \Delta_{LE}H = -411$ $\Delta_{LE}H = -411 - 107 - 496 - 122 + 349 = -787\,kJ\,mol^{-1}$

Experimental vs. Theoretical Lattice Energy

• Experimental Lattice Energy: Calculated indirectly via Born-Haber cycles based on experimental data.

• Theoretical Lattice Energy: Calculated using electrostatic theory equations (e.g., Born-Landé, Born-Mayer, or Kapustinskii).

Theoretical Assumptions:

1. Ions are in direct contact.

2. Ions are perfectly spherical.

3. Charge is evenly distributed (ions act as point charges).

Interpreting Differences:

• If values match (e.g., Sodium halides), the compound is close to "100% ionic."

• Large differences (e.g., Silver halides) indicate the ionic model is insufficient because the bonding has significant covalent character.

• Covalency makes the experimental lattice energy more negative (stronger bonding) than the theoretical value.

Polarisation and Fajan's Rules

Covalent character in ionic substances arises from the polarisation of the anion by the cation. The cation distorts the electron cloud of the anion, causing it to move closer or overlap.

Fajan’s Rules for Increased Polarisation:

1. Cation: High charge and small size (high charge density or "polarising power").

2. Anion: High charge and large size (more easily "polarisable").

Examples:

• $Mg^{2+}-Cl^-$ has more covalent character than $Na^+-Cl^-$ because $Mg^{2+}$ has a higher charge density.

• $AgI$ has more covalent character than $AgF$ because the large iodide ion ($I^-$) is more easily polarised than the small fluoride ion ($F^-$).

Enthalpy of Solution and Hydration

Enthalpy Change of Solution ($\Delta_{sol}H$):

• Involves breaking the lattice (endothermic) and hydrating gaseous ions (exothermic).

• $\Delta_{sol}H = -\Delta_{LE}H + \sum \Delta_{hyd}H$

• Measured at infinite dilution (where further dilution releases no more heat). This is found via extrapolation because it cannot be measured directly.

Interactions in Hydration:

• Cations: Ion-dipole interactions with the $\delta-$ Oxygen of water. Transition metals may form dative covalent bonds.

• Anions: Ion-dipole interactions with $\delta+$ Hydrogen and hydrogen bonding using anion lone pairs.

Trends in $\Delta_{hyd}H$:

• Increases (becomes more negative) with increasing ionic charge.

• Increases (becomes more negative) with decreasing ionic radius (higher charge density correlates to more negative $\Delta_{hyd}H$).

Entropy and Solubility

Substances are more likely to be soluble if $\Delta_{sol}H$ is exothermic. However, many compounds with endothermic $\Delta_{sol}H$ still dissolve due to entropy.

The Entropy Equation: $\Delta S_{total} = -\frac{\Delta H}{T} + \Delta S_{system}$

• Transition to Solution: Disordered solution state and an increase in the number of particles result in a positive $\Delta S_{system}$.

• For endothermic salts, solubility depends on whether $T \times \Delta S_{system}$ overcomes the endothermic $\Delta H$. Increasing temperature makes $\Delta S_{total}$ more positive, which makes the reaction feasible (\Delta G < 0).

Questions & Discussion

Checkpoint 12B.1

• Q: Why is $\Delta_{LE}H$ for $NaF$ larger (more negative) than $KCl$?

• A: $Na^+$ is smaller than $K^+$ and $F^-$ is smaller than $Cl^-$. The smaller inter-ionic distance in $NaF$ leads to stronger electrostatic attractions.

• Q: Why is $\Delta_{LE}H$ for $CaO$ four times larger than $KF$?

• A: Inter-ionic distances are similar ($0.240\,nm$ vs $0.271\,nm$), but the product of charges for $Ca^{2+}O^{2-}$ is $2 \times 2 = 4$, compared to $1 \times 1 = 1$ for $KF$.

Checkpoint 12B.2

• Q: Why do $CaF_2$ and $AgF$ differ in agreement between theoretical and experimental values?

• A: $CaF_2$ is close to 100% ionic. $AgF$ has significant covalent bonding, which electrostatic theory does not account for.

• Q: Calculate ordering of polarising power for $Mg^{2+}$, $Al^{3+}$, $Li^{2+}$, $Na^+$, $Ca^{2+}$, $K^+$.

• A: Order (based on charge/$radius^2$): K^+ < Na^+ < Ca^{2+} < Li^+ < Mg^{2+} < Al^{3+}.

Checkpoint 12B.3

• Q: Explain what happens to the thermometer when $1\,g$ of $NaF$ dissolves in $250\,cm^3$ of water ($\Delta_{sol}H = +0.3\,kJ\,mol^{-1}$).

• A: No measurable change. The temperature decrease is too small for a standard thermometer (graduated in $1\,^{\circ}C$) to register.