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Hess’s Law of Constant Heat Summation | Born-Haber Cycle | Lattice Energy

Hess’s Law of Constant Heat Summation

Born-Haber Cycle · Lattice Energy · Thermochemistry · State Function Principle

Hess’s Law states that if a chemical transformation can occur through several pathways, whether in a single step or in several stages, the total heat change remains constant regardless of the technique employed to effectuate the change. This is a direct consequence of the law of conservation of energy applied to chemical reactions.

\[ \Delta H_{\text{total}} = \sum \Delta H_{\text{steps}} \]

Enthalpy is a state function, meaning its change depends only on the initial and final states, not on the path taken.

General Principle: Multiple Routes, One Destination

Consider the transformation of substance A into substance B. According to Hess’s Law, the enthalpy change (ΔH) will be identical regardless of the complexity of the reaction path.

  • Route 1 (Direct): A → B (ΔH = ΔH₁)
  • Route 2 (Two-Step): A → C → B (ΔH = ΔH₂ + ΔH₃)
  • Route 3 (Multi-Step): A → D → E → B (ΔH = ΔH₄ + ΔH₅ + ΔH₆)
\[ \Delta H_{\text{direct}} = \sum \Delta H_{\text{steps}} \]

Practical Example: Formation of Carbon Dioxide (CO₂)

Carbon dioxide can be produced in one single step or through a two-stage process via carbon monoxide.

Direct Route: Carbon reacts with oxygen to form CO₂ directly. \[ \text{C(s)} + \text{O}_2(g) \rightarrow \text{CO}_2(g) \quad \Delta H = -393.5\ \text{kJ/mol} \]
Indirect Route (Two Steps):
Step i: \(\text{C(s)} + \frac{1}{2}\text{O}_2(g) \rightarrow \text{CO}(g) \quad \Delta H = -110.5\ \text{kJ}\)
Step ii: \(\text{CO}(g) + \frac{1}{2}\text{O}_2(g) \rightarrow \text{CO}_2(g) \quad \Delta H = -283.0\ \text{kJ}\)
Total: \(\text{C(s)} + \text{O}_2(g) \rightarrow \text{CO}_2(g) \quad \Delta H = -110.5 + (-283.0) = -393.5\ \text{kJ}\)

The result matches the direct route, confirming Hess’s Law.

Hess's Law pathway diagram showing direct and indirect routes
Fig. 1: Hess’s Law – enthalpy change is independent of the path taken.

Born-Haber Cycle for Ionic Compounds

The Born-Haber cycle is a thermochemical cycle used primarily to calculate the lattice energy of ionic compounds, which is otherwise difficult to measure directly. Named after Max Born and Fritz Haber, it applies Hess’s Law to the formation of an ionic solid from its constituent elements. The cycle treats the overall enthalpy of formation as a multi-step process involving several distinct energy changes, including enthalpy of atomization, ionization energy, electron affinity, and sublimation.

\[ \Delta H_f^\circ = \Delta H_{\text{at}}^\circ + \Delta H_{\text{sub}}^\circ + \text{IE} + \text{EA} + \Delta H_{\text{lattice}} \]

Where:

  • ΔHf° = standard enthalpy of formation
  • ΔHat° = atomization energy
  • ΔHsub° = sublimation energy
  • IE = ionization energy
  • EA = electron affinity
  • ΔHlattice = lattice energy (negative for formation of crystal, but often defined as positive for dissociation)

Step-by-Step Calculation for NaCl

The direct synthesis of NaCl from sodium metal and chlorine gas has an enthalpy change of -411 kJ (standard enthalpy of formation).

\[ \text{Na(s)} + \frac{1}{2}\text{Cl}_2(g) \rightarrow \text{NaCl(s)} \quad \Delta H^\circ = -411\ \text{kJ/mol} \]

The indirect pathway consists of five steps:

Step 1: Sublimation of sodium metal \[ \text{Na(s)} \rightarrow \text{Na(g)} \quad \Delta H_1 = +108\ \text{kJ/mol} \]
Step 2: Dissociation of chlorine molecules \[ \frac{1}{2}\text{Cl}_2(g) \rightarrow \text{Cl(g)} \quad \Delta H_2 = +121\ \text{kJ/mol} \] (Note: Bond dissociation enthalpy of Cl₂ is 242 kJ/mol, so for half mole it is 121 kJ)
Step 3: Ionization of sodium atoms \[ \text{Na(g)} \rightarrow \text{Na}^+(g) + e^- \quad \Delta H_3 = +495\ \text{kJ/mol} \]
Step 4: Electron affinity of chlorine \[ \text{Cl(g)} + e^- \rightarrow \text{Cl}^-(g) \quad \Delta H_4 = -348\ \text{kJ/mol} \]
Step 5: Formation of crystal lattice from gaseous ions \[ \text{Na}^+(g) + \text{Cl}^-(g) \rightarrow \text{NaCl(s)} \quad \Delta H_5 = -U \] where \(U\) is the lattice energy (positive for dissociation).

According to Hess’s Law, the sum of the enthalpy changes of the five steps must equal the direct formation enthalpy:

\[ \Delta H_1 + \Delta H_2 + \Delta H_3 + \Delta H_4 + \Delta H_5 = \Delta H_f^\circ(\text{NaCl}) \]
\[ 108 + 121 + 495 – 348 – U = -411 \]
\[ (108 + 121 + 495 – 348) – U = -411 \]
\[ 376 – U = -411 \]
\[ -U = -411 – 376 = -787 \quad \Rightarrow \quad U = +787\ \text{kJ/mol} \]

The lattice energy of NaCl is therefore 787 kJ/mol (positive indicates energy required to separate one mole of solid into gaseous ions).

Born-Haber cycle for NaCl showing energy changes
Fig. 2: Born-Haber cycle for sodium chloride. The cycle visualizes the enthalpy changes from elements to gaseous ions to solid crystal.

Summary of Key Equations

\[ \Delta H_{\text{reaction}} = \sum \Delta H_{\text{products}} – \sum \Delta H_{\text{reactants}} \quad \text{(via Hess’s Law)} \]
\[ \Delta H_f^\circ(\text{ionic solid}) = \Delta H_{\text{sub}} + \frac{1}{2}\Delta H_{\text{diss}} + \text{IE} + \text{EA} + \Delta H_{\text{lattice}} \]
Complete Lecture: Hess’s Law & Born-Haber Cycle (Urdu/Hindi)

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© 2026 — Comprehensive guide on Hess’s Law of Constant Heat Summation and the Born-Haber Cycle.

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