What calorimetry measures, and why it is a temperature measurement
You cannot measure heat directly. What you can measure is temperature, and calorimetry is the trick that converts one into the other. Run a reaction inside an insulated vessel of known contents, watch the temperature move, and the heat released or absorbed follows from how much material warmed and by how much.
The conversion factor is the specific heat capacity, c: the energy needed to raise one gram of a substance by one degree. Water's value, 4.184 J/(g·°C), is unusually large — that is why oceans moderate climate, why water is the coolant of choice, and why aqueous calorimetry gives modest temperature changes even for vigorous reactions.
Multiply mass by specific heat by temperature change and you have the heat in joules: q = mcΔT. Divide by the moles of whatever limited the reaction and you have the molar enthalpy, the number that goes in a table and can be compared with a literature value.
The sign convention causes most of the confusion. q as computed here is the heat gained by the contents of the calorimeter. If the temperature rose, the contents gained energy, which means the reaction gave it up, so the reaction is exothermic and its ΔH is negative. The minus sign in ΔH = −q/n encodes exactly that handover.
Two instruments dominate. A coffee-cup calorimeter is an open polystyrene cup at constant atmospheric pressure, so the heat it measures is directly the enthalpy change, ΔH. A bomb calorimeter is a sealed steel vessel at constant volume, so it measures the internal energy change ΔU, and converting to ΔH requires a correction of ΔnRT for any change in the moles of gas.
Each term, and where the calorimeter constant comes in
m is the mass that changed temperature, not the mass of reactant. In a coffee-cup neutralisation where you mix 50.0 mL of acid with 50.0 mL of base, the mass that warms is the whole 100 g of combined solution. Using the mass of the acid alone halves your answer.
c is the specific heat of that mixture. For dilute aqueous solutions the universal approximation is to use water's 4.184 J/(g·°C). It is an approximation: 1 M sodium chloride solution is nearer 3.9, and concentrated solutions lower still. For work below about 1 M the error this introduces is a few percent, smaller than the thermometer error in most student experiments.
ΔT is final minus initial. Its sign carries the physics, so do not take an absolute value. And because it is a difference, degrees Celsius and kelvin are interchangeable here — a rise of 6.8 °C is a rise of 6.8 K. This is the one place in thermochemistry where you are allowed to work in Celsius.
C_cal is the heat capacity of the apparatus. The cup, the lid, the stirrer and the thermometer all warm along with the solution, and that energy is real but invisible in mcΔT. You measure C_cal in a separate calibration: add a known amount of hot water to a known amount of cold water in the same cup, compute how much heat went missing relative to a perfect calorimeter, and divide by the temperature change. Typical polystyrene-cup values are small, roughly 10–50 J/°C; a bomb calorimeter is thousands. Note that a calorimeter constant always increases the magnitude of the computed heat, whichever way the temperature moved: q_total = (mc + C_cal)ΔT, so adding C_cal scales the same ΔT by a larger factor. It never changes the sign of the result, and it always makes the reported |ΔH| larger.
n is the limiting reactant. Enthalpies are quoted per mole of a specified species, and that species must be the one that ran out. Mixing 0.050 mol of acid with 0.060 mol of base gives 0.050 mol of reaction; dividing by 0.060 understates the magnitude of ΔH by 17%.
Worked example: neutralising 50.0 mL of 1.00 M HCl with 50.0 mL of 1.00 M NaOH
You mix 50.0 mL of 1.00 M hydrochloric acid with 50.0 mL of 1.00 M sodium hydroxide in a polystyrene cup. Both solutions start at 21.0 °C. The temperature peaks at 27.8 °C.
- Find the mass that warmed. The combined volume is 100.0 mL, and treating the dilute solution as water at 1.00 g/mL gives m = 100.0 g.
- Choose the specific heat. Dilute aqueous solution, so c = 4.184 J/(g·°C).
- Compute ΔT. 27.8 − 21.0 = +6.8 °C. Positive, so the mixture warmed.
- Compute the heat. q = 100.0 × 4.184 × 6.8 = 2845.1 J, or 2.845 kJ.
- Find the limiting reactant. 0.0500 L × 1.00 M = 0.0500 mol of HCl and 0.0500 mol of NaOH. They are stoichiometrically equal, so n = 0.0500 mol of reaction.
- Convert to molar enthalpy. ΔH = −2845.1 J ÷ 0.0500 mol = −56 902 J/mol = −56.9 kJ/mol.
The accepted value for the enthalpy of neutralisation of a strong acid by a strong base is about −57.1 kJ/mol, so this experiment lands within half a percent — which is better than it deserves, because ignoring the calorimeter constant and any heat lost to the room both push the measured magnitude down. Those two errors partly offset the fact that a peak temperature is always slightly lower than the true adiabatic maximum.
Now add a calibration. Suppose the cup was measured at C_cal = 25.0 J/°C. Then q_cal = 25.0 × 6.8 = 170 J, the total becomes 3015.1 J, and ΔH = −60.3 kJ/mol. That is now further from the literature value, which tells you something useful: the uncorrected agreement was partly luck, and a serious measurement needs both the calibration and a cooling-curve extrapolation.
Reading the sign and judging the magnitude
Read the sign first. A negative ΔH means the process gave out heat: combustion, neutralisation, most precipitations, the dissolution of calcium chloride or concentrated sulfuric acid. A positive ΔH means it took heat in: the dissolution of ammonium nitrate, most evaporation and melting, the endothermic barium hydroxide-ammonium thiocyanate demonstration that freezes a block of wood to a beaker.
Then judge the magnitude against the class of process. Neutralisation of a strong acid by a strong base is about −57 kJ/mol regardless of which acid and base you pick, because the reaction is always H⁺ + OH⁻ → H₂O. Weak acids give a smaller magnitude because some energy is consumed dissociating them. Heats of solution are usually within ±50 kJ/mol. Combustion enthalpies are hundreds to thousands of kilojoules per mole — methane is −890 kJ/mol — and if a coffee-cup result comes out in that range, check the moles figure.
Precision in this experiment is dominated by the thermometer. A ±0.1 °C reading uncertainty on a 6.8 °C change is 1.5%; on a 0.5 °C change it is 20%. The sensitivity table above shows exactly what your own reading error costs. The practical remedy is to make ΔT larger by using more concentrated solutions, not by using more solution — doubling the concentration doubles both the heat and the mass-independent temperature rise, while doubling the volume raises both q and m and leaves ΔT unchanged.
Systematic errors nearly all push in the same direction. Heat leaks to the room during the run, the vessel absorbs heat you did not account for, and you read the peak after some cooling has already occurred. All three make the measured magnitude too small. The standard corrections are to plot temperature against time before and after mixing and extrapolate both straight lines back to the mixing instant, and to calibrate the vessel.
Specific heat capacities at or near 25 °C
| Material | c, J/(g·°C) | Comment |
|---|---|---|
| Water (liquid) | 4.184 | The reference value; use for dilute aqueous solutions |
| Ice (0 °C) | 2.09 | Roughly half the liquid value |
| Water vapour | ~2.0 | At constant pressure, near 100 °C |
| Ethanol | 2.44 | |
| Methanol | 2.53 | |
| Glycerol | 2.43 | |
| Air | 1.005 | At constant pressure |
| Aluminium | 0.897 | Highest of the metals in this table |
| Sodium chloride (solid) | 0.864 | |
| Glass (borosilicate) | ~0.75 | Varies with composition |
| Iron | 0.449 | |
| Copper | 0.385 | |
| Silver | 0.235 | |
| Gold | 0.129 | |
| Lead | 0.128 | Lowest of the metals in this table |
Metal values follow the Dulong-Petit pattern: the molar heat capacity of a solid element is close to 25 J/(mol·K), so the specific heat falls as the atomic mass rises.
Errors and assumptions this calculation hides
- Using the reactant mass instead of the solution mass. The whole mixture warms. In a 50 mL + 50 mL neutralisation, m is 100 g.
- Dividing by the wrong moles. ΔH is quoted per mole of limiting reactant. Identify which reagent runs out before you divide.
- Assuming the calorimeter absorbs nothing. The cup, lid, stirrer and thermometer all take heat. Calibrate C_cal and enter it, or accept that your magnitude is biased low.
- Reading the peak temperature straight off the thermometer. Heat leaks out during mixing, so the observed peak is below the adiabatic value. Extrapolate the cooling line back to the mixing time.
- Treating a concentrated solution as water. The 4.184 approximation is good below about 1 M; a 5 M sodium chloride solution is nearer 3.3 J/(g·°C), some 20% lower, and its density is no longer 1.00 g/mL either.
- Confusing constant-pressure and constant-volume measurements. A coffee cup gives ΔH directly; a bomb gives ΔU, and converting needs ΔH = ΔU + ΔnRT for the change in moles of gas.
- Ignoring a phase change inside the temperature range. If anything melts, boils or dissolves during the run, the latent heat is absorbed with no temperature change, and q = mcΔT misses it completely.
How this connects to the rest of thermochemistry
Calorimetry is the experimental root of the thermochemical tables. Every standard enthalpy of formation in a data book traces back, directly or through Hess's law, to a measured temperature change in a calibrated vessel. Once those values are tabulated, you no longer need the experiment: the enthalpy of reaction calculator combines formation enthalpies to predict ΔH for a reaction nobody has ever run in a calorimeter.
Enthalpy alone does not tell you whether a reaction will happen. That requires entropy as well, through ΔG = ΔH − TΔS; the Gibbs free energy calculator combines your measured ΔH with an entropy change to give the spontaneity and the equilibrium constant. Plenty of endothermic processes run spontaneously — ammonium nitrate dissolving in water is the classic example — because the entropy gain outweighs the enthalpy cost.
Two adjacent measurements use the same instrument and the same q = mcΔT arithmetic. A heat of solution measures the energy change on dissolving a salt, which combines lattice and hydration terms. A heat of fusion or vaporisation measures a phase change and appears as heat absorbed with no temperature change at all, which is why those runs need a mass-based rather than a temperature-based analysis. The colligative consequences of dissolving that solute — the freezing point depression and boiling point elevation — are separate effects that depend on particle count rather than on energy.
If your reaction involves gases and you need to convert between the volume you measured and the moles you divide by, use the ideal gas law calculator first, then bring the mole count back here.