Calorimetry: measuring heats of reaction

Where does “energy 1700 kJ per 100 g” on a food label come from? Originally the food was dried, pressed into a pellet, and ignited in a steel bomb full of high-pressure oxygen while someone watched how much 2 kg of surrounding water warmed. A thermometer cannot read “heat”; it reads temperature. To turn a temperature change into heat you must know the heat capacity of whatever is being warmed — and you must block every crack through which heat can slip away. That is calorimetry.

The heat needed to warm an object by 1 K is its heat capacity $C$ (J/K). It grows with the amount of material, so we compare substances with the specific heat $c$ per gram (J g⁻¹ K⁻¹) or the molar heat capacity per mole. Water’s specific heat, 4.18 J g⁻¹ K⁻¹, is unusually large among common substances — about nine times that of iron. That is why oceans steady the climate, why radiators carry hot water rather than hot oil, and why a calorimeter is usually filled with water: a small temperature change stands for a lot of heat.

Two nested polystyrene cups, a lid, a temperature probe and a stirrer make a very good calorimeter. The reaction runs in an open cup at constant pressure, so what it measures is $q_p = \Delta H$. Conservation of heat says that everything the reaction releases is absorbed by the solution and the cup:

where $C_\text{cal}$ is the calorimeter constant — the heat capacity of cups, stirrer and probe. It must be calibrated first: pour hot water of known temperature into cold water in the cup; part of the heat the hot water gives up warms the cold water, and the “missing” part went into the cup.

The second problem is heat leakage. Even a good cup exchanges heat with the room, so the highest reading is always a little lower than it would be if the reaction were instant and leak-free. The remedy: extend the baseline before mixing and the slow straight-line drift after mixing back to the moment of mixing; the gap between the two lines at that moment is the corrected $\Delta T$.

Heats of combustion are measured in a bomb calorimeter: about 1 g of sample is pressed into a pellet, sealed in a thick steel bomb with 30 bar of oxygen, immersed in about 2 kg of water, and fired electrically. The bomb is rigid and sealed, so the volume is fixed, no expansion work is done, and what it measures is $q_V = \Delta U$. The instrument’s total heat capacity (water plus bomb, about 10 kJ/K) is calibrated by burning benzoic acid, whose certified energy of combustion is −26.434 kJ/g.

Once $\Delta_\text{c}U$ is known, the last lesson’s $\Delta H = \Delta U + \Delta n(\mathrm{g})RT$ gives $\Delta_\text{c}H$. Sugars consume as much O₂ as the CO₂ they make, $\Delta n(\mathrm{g}) = 0$, and the two are equal; naphthalene or octane lose gas on burning, so their $\Delta H$ is more negative than $\Delta U$.