Thermodynamics • Topic 2 of 3

Enthalpy Changes & Hess’s Law

The standard enthalpy of reaction $\Delta_r H^\circ$ is the enthalpy change when reactants in their standard states (pure form at 1 bar and a stated temperature, usually 298 K) convert completely to products in their standard states. It is calculated from standard enthalpies of formation using

$\Delta_r H^\circ=\sum \nu_p\,\Delta_f H^\circ(\text{products})-\sum \nu_r\,\Delta_f H^\circ(\text{reactants})$

where $\nu$ are stoichiometric coefficients. By convention the standard enthalpy of formation of an element in its most stable form (e.g. $\text{O}_2(g)$, graphite) is zero.

Several named enthalpy changes appear repeatedly. The enthalpy of formation $\Delta_f H^\circ$ is for forming one mole of a compound from its elements. The enthalpy of combustion $\Delta_c H^\circ$ is for complete combustion of one mole in oxygen (always negative). The enthalpy of neutralisation is the heat released when one mole of water forms from acid–base reaction; for strong acid + strong base it is a near-constant $-57.1$ kJ/mol because only $\text{H}^++\text{OH}^-\rightarrow\text{H}_2\text{O}$ occurs.

Other key terms: enthalpy of atomisation $\Delta_a H^\circ$ (breaking one mole of substance into gaseous atoms), enthalpy of sublimation $\Delta_{sub} H^\circ$ (solid to gas directly) and ionisation enthalpy (removing an electron from a gaseous atom). These building blocks let us assemble cycles for changes that cannot be measured directly.

The cornerstone is Hess’s law of constant heat summation: the total enthalpy change for a reaction is the same whether it occurs in one step or in several steps. This follows directly because $H$ is a state function — the change depends only on initial and final states, not the path. Practically, we add up the enthalpies of a sequence of steps (reversing a step changes the sign; multiplying a step multiplies its $\Delta H$) to obtain the enthalpy of an overall reaction.

Bond enthalpy is the energy needed to break one mole of a particular bond in the gaseous state. Because breaking bonds absorbs energy and forming bonds releases it, an estimate of reaction enthalpy is

$\Delta_r H=\sum(\text{bond enthalpies of bonds broken})-\sum(\text{bond enthalpies of bonds formed})$

The lattice enthalpy $\Delta_{lattice} H^\circ$ of an ionic solid is the enthalpy change when one mole of the solid separates into gaseous ions. It cannot be measured directly, so we obtain it from a Born–Haber cycle — a Hess’s-law cycle that links sublimation, dissociation, ionisation, electron gain and lattice formation back to the standard enthalpy of formation, allowing us to solve for the one unknown lattice enthalpy.

Hess law enthalpy cycle for carbon to carbon dioxideC(s) + O₂(g)CO₂(g)CO(g) + ½O₂ΔH₁ = −393.5 kJΔH₂ΔH₃ = −283.0ΔH₁ = ΔH₂ + ΔH₃ (Hess law)
1
Worked Example
Given $\Delta_f H^\circ$: $\text{CO}_2(g)=-393.5$, $\text{H}_2\text{O}(l)=-285.8$, $\text{CH}_4(g)=-74.8$ kJ/mol. Find $\Delta_c H^\circ$ of methane: $\text{CH}_4+2\text{O}_2\rightarrow\text{CO}_2+2\text{H}_2\text{O}$.
Solution
  1. $\Delta_r H^\circ=[\Delta_f H(\text{CO}_2)+2\Delta_f H(\text{H}_2\text{O})]-[\Delta_f H(\text{CH}_4)+2(0)]$.
  2. $=[-393.5+2(-285.8)]-[-74.8]$.
  3. $=(-965.1)-(-74.8)=-890.3$ kJ.

Answer: $\Delta_c H^\circ=-890.3\ \text{kJ/mol}$.

2
Worked Example
Using Hess’s law find $\Delta_f H^\circ$ of CO. Given: $\text{C}+\text{O}_2\rightarrow\text{CO}_2$, $\Delta H=-393.5$ kJ; $\text{CO}+\tfrac12\text{O}_2\rightarrow\text{CO}_2$, $\Delta H=-283.0$ kJ.
Solution
  1. Target: $\text{C}+\tfrac12\text{O}_2\rightarrow\text{CO}$.
  2. Add equation 1 and the reverse of equation 2: $\Delta H=(-393.5)+(+283.0)$.
  3. $\Delta_f H^\circ(\text{CO})=-110.5$ kJ.

Answer: $\Delta_f H^\circ(\text{CO})=-110.5\ \text{kJ/mol}$.

3
Worked Example
Estimate $\Delta_r H$ for $\text{H}_2(g)+\text{Cl}_2(g)\rightarrow 2\text{HCl}(g)$ from bond enthalpies: $\text{H}-\text{H}=435$, $\text{Cl}-\text{Cl}=242$, $\text{H}-\text{Cl}=431$ kJ/mol.
Solution
  1. Bonds broken: 1 H–H + 1 Cl–Cl $=435+242=677$ kJ.
  2. Bonds formed: 2 H–Cl $=2(431)=862$ kJ.
  3. $\Delta_r H=677-862=-185$ kJ.

Answer: $\Delta_r H=-185\ \text{kJ}$.

4
Worked Example
For the Born–Haber cycle of NaCl: $\Delta_f H=-411$, $\Delta_{sub} H(\text{Na})=+108$, $\tfrac12 D(\text{Cl}_2)=+121$, $IE(\text{Na})=+496$, $\Delta_{eg} H(\text{Cl})=-349$ kJ/mol. Find the lattice enthalpy.
Solution
  1. $\Delta_f H=\Delta_{sub}H+\tfrac12 D+IE+\Delta_{eg}H+\Delta_{lattice}H$.
  2. $-411=108+121+496-349+\Delta_{lattice}H$.
  3. $-411=376+\Delta_{lattice}H$, so $\Delta_{lattice}H=-787$ kJ.

Answer: $\Delta_{lattice}H=-787\ \text{kJ/mol}$ (lattice formation).

5
Worked Example
The enthalpy of combustion of carbon is $-393.5$ kJ/mol and of CO is $-283.0$ kJ/mol. How much heat is released when 12 g of carbon burns completely to $\text{CO}_2$?
Solution
  1. 12 g of carbon $=12/12=1$ mol.
  2. Complete combustion to $\text{CO}_2$ releases $\Delta_c H=-393.5$ kJ/mol.
  3. Heat for 1 mol $=393.5$ kJ released.

Answer: 393.5 kJ of heat is released.

6
Worked Example
Calculate $\Delta_r H^\circ$ for $\text{N}_2(g)+3\text{H}_2(g)\rightarrow 2\text{NH}_3(g)$ given $\Delta_f H^\circ(\text{NH}_3)=-46.1$ kJ/mol.
Solution
  1. Elements N₂ and H₂ have $\Delta_f H^\circ=0$.
  2. $\Delta_r H^\circ=2\Delta_f H(\text{NH}_3)-0=2(-46.1)$.
  3. $=-92.2$ kJ.

Answer: $\Delta_r H^\circ=-92.2\ \text{kJ}$.

Key Points

  • Standard enthalpy of reaction: $\Delta_r H^\circ=\sum\nu_p\Delta_f H^\circ(\text{products})-\sum\nu_r\Delta_f H^\circ(\text{reactants})$; elements have $\Delta_f H^\circ=0$.
  • Named enthalpies: formation, combustion (always $-$), neutralisation ($-57.1$ kJ/mol for strong acid+base), atomisation, sublimation, ionisation.
  • Hess’s law: total $\Delta H$ is path-independent, so reaction enthalpies can be added/reversed/scaled like algebraic equations.
  • Bond-enthalpy estimate: $\Delta_r H=\sum(\text{bonds broken})-\sum(\text{bonds formed})$.
  • Lattice enthalpy of ionic solids is found indirectly using a Born–Haber cycle (a Hess’s-law cycle).
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Q1.The standard enthalpy of formation of an element in its most stable form is:
Explanation: By convention $\Delta_f H^\circ$ of an element in its reference state is defined as zero.
Q2.Hess’s law is a direct consequence of the fact that:
Explanation: Because $H$ is a state function, $\Delta H$ depends only on initial and final states.
Q3.For a reaction, $\Delta_r H$ estimated from bond enthalpies is given by:
Explanation: Breaking bonds absorbs energy and forming bonds releases it: $\Delta_r H=\sum(\text{broken})-\sum(\text{formed})$.
Q4.Lattice enthalpy of an ionic solid is determined experimentally using:
Explanation: Lattice enthalpy cannot be measured directly; it is obtained from the Born–Haber cycle.