Heats of Combustion and Gravimetric Energy Density of Four Fuels: 21-Mat-A1 Thermodynamics, May 2017, Q5

Question 5: Heats of Combustion and Gravimetric Energy Density of Four Fuels (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Standard enthalpies of formation at 25 °C for methanol, propane, heptane, octane, and the complete-combustion products CO₂(g) and H₂O(l), all in kJ/mol (table above the questions on the source page).

Given data — formation enthalpies, 298 K
Species	ΔH°_f (kJ mol⁻¹)
CH₃OH	−201
C₃H₈	−105
C₃H₁₆	−224
C₈H₁₈	−259
CO₂	−394
H₂O	−286

Find. $\Delta H_c$ per mole for each fuel, then (converting through molar mass) which fuel releases the most heat per unit mass.

Approach. Write the balanced complete-combustion reaction for each fuel ($C_nH_m+\left(n+\tfrac{m}{4}\right)O_2\to nCO_2+\tfrac{m}{2}H_2O$), apply Hess's law with $\Delta H^\circ_f(O_2)=0$, then divide each molar heat of combustion by the fuel's molar mass to compare on a mass basis.

(a) Methanol. $CH_3OH+\tfrac32O_2\to CO_2+2H_2O$: $$\Delta H_c=\bigl[\Delta H_f(CO_2)+2\Delta H_f(H_2O)\bigr]-\Delta H_f(CH_3OH)=(-394-572)-(-201)=\boxed{-765\ \text{kJ}\,\text{mol}^{-1}}.$$
(b) Propane. $C_3H_8+5O_2\to 3CO_2+4H_2O$: $$\Delta H_c=\bigl[3(-394)+4(-286)\bigr]-(-105)=(-1182-1144)+105=\boxed{-2221\ \text{kJ}\,\text{mol}^{-1}}.$$
(c) Heptane. $C_7H_{16}+11O_2\to 7CO_2+8H_2O$: $$\Delta H_c=\bigl[7(-394)+8(-286)\bigr]-(-224)=(-2758-2288)+224=\boxed{-4822\ \text{kJ}\,\text{mol}^{-1}}.$$
(d) Octane. $C_8H_{18}+12.5O_2\to 8CO_2+9H_2O$: $$\Delta H_c=\bigl[8(-394)+9(-286)\bigr]-(-259)=(-3152-2574)+259=\boxed{-5467\ \text{kJ}\,\text{mol}^{-1}}.$$
(e) Heat per unit mass. Dividing each $|\Delta H_c|$ by the fuel's molar mass ($M_{CH_3OH}=32.04$, $M_{C_3H_8}=44.10$, $M_{C_7H_{16}}=100.20$, $M_{C_8H_{18}}=114.23$ g mol⁻¹): $$\dfrac{765}{32.04}=23.9,\quad \dfrac{2221}{44.10}=\boxed{50.4},\quad \dfrac{4822}{100.20}=48.1,\quad \dfrac{5467}{114.23}=47.9\ \ (\text{kJ}\,\text{g}^{-1}).$$ Propane delivers the most heat per unit mass, even though octane releases more heat per mole — propane's higher H:C ratio (more energy-dense C–H bonds relative to its molar mass) outweighs the larger molar heat of the bigger hydrocarbons.

Fuel	ΔH_c (kJ/mol)	M (g/mol)	Heat per mass (kJ/g)
Methanol CH₃OH	−765	32.04	23.9
Propane C₃H₈	−2221	44.10	50.4 (highest)
Heptane C₃H₁₆	−4822	100.20	48.1
Octane C₈H₁₈	−5467	114.23	47.9

Concept & Theory

Topic. Heats of combustion from formation enthalpies, and the distinction between molar and gravimetric (per-mass) energy density.

Key relations. $\Delta H_c=\sum\nu_i\Delta H_{f,i}(\text{products})-\sum\nu_i\Delta H_{f,i}(\text{reactants})$ (Hess's law); gravimetric heating value $=|\Delta H_c|/M$.

Why this works. Formation enthalpies are defined relative to the same reference elements in their standard states, so any reaction's enthalpy is exactly the difference of its species' formation enthalpies regardless of mechanism. Ranking fuels by molar heat of combustion answers "heat per mole," a different question from "heat per kilogram" — the two rankings can (and here do) disagree.

Common pitfall. Reporting octane as the "best" fuel because it has the largest molar $|\Delta H_c|$, without dividing through by molar mass to answer the "per unit weight" question actually asked.

Source. Gaskell, Introduction to the Thermodynamics of Materials, Ch. 6 (heats of reaction/combustion from tabulated formation data).

Question 5 of 7: Heats of Combustion and Gravimetric Energy Density of Four Fuels

Question 5: Heats of Combustion and Gravimetric Energy Density of Four Fuels (20 marks)