Mix 50.0 mL of 1.00 mol/L hydrochloric acid with only 40.0 mL of 1.00 mol/L sodium hydroxide. In our worked calorimeter, the reaction releases about 2.284 kJ of heat. Dividing by the 0.0400 mol that actually reacts gives −57.1 kJ per mole of neutralisation. Dividing by all 0.0500 mol of acid instead gives an incorrect −45.7 kJ/mol.
The temperature rise determines an estimated quantity of heat only after an energy balance. It cannot tell you which amount of substance belongs in the denominator. Stoichiometry supplies that answer. This guide separates reaction heat, solution heat, apparatus heat and reaction extent, then shows the different mistakes each can conceal.
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Choose a route or read all twelve chapters. Every teaching chapter remains open and the chapter index expands by theme.
Define the heat and moles: Chapters 1–2. Start this route.
Work the calculation: Chapters 3–5. Start this route.
Find the error: Chapters 6–7. Start this route.
Test assumptions: Chapters 8–10. Start this route.
Practise and continue: Chapters 11–12. Start this route.
Chapter index · Worked practice · Thermochemistry subject root
Chapter index
Define the heat and moles · Chapters 1–2
Work the calculation · Chapters 3–5
Find the error · Chapters 6–7
Test assumptions · Chapters 8–10
Practise and continue · Chapters 11–12
Heat q is an amount of energy transferred, measured in joules or kilojoules. Enthalpy change for a reaction under specified conditions is commonly reported per mole of reaction as written in the balanced equation. A temperature rise can help estimate q; it does not by itself provide the molar value.
In a simplified constant-pressure calorimeter with negligible heat exchange with the wider room, energy conservation gives qrxn + qsolution + qcal ≈ 0. For an exothermic reaction, qrxn is negative while the surroundings warm and absorb positive heat.
The model must also identify how much chemical reaction occurred. Dividing the correctly estimated q by the wrong number of moles produces a wrong molar enthalpy even if every temperature and heat-capacity figure is accurate.
Our illustrative chemical equation is HCl(aq) + NaOH(aq) → NaCl(aq) + H2O(l). Its one-to-one coefficients mean one mole of reaction consumes one mole of acid and one mole of base. For an ideal strong-acid, strong-base system the net ionic equation is H+(aq) + OH−(aq) → H2O(l).
Let ξ represent reaction extent in moles for the written equation. If 0.0400 mol of hydroxide is consumed completely, ξ = 0.0400 mol. A reactant added in excess has not all reacted, so its starting amount is not ξ.
For 2A + B → products, consuming 0.060 mol A corresponds to ξ = 0.060/2 = 0.030 mol, if the reaction behaves as written. This coefficient adjustment is vital whenever a laboratory reports enthalpy ‘per mole of reaction’ rather than ‘per mole of a particular substance’.
Volume must be expressed in litres when multiplying molarity in mol/L. Acid amount n(HCl) = 0.0500 L × 1.00 mol/L = 0.0500 mol. Base amount n(NaOH) = 0.0400 L × 1.00 mol/L = 0.0400 mol.
Since the balanced reaction consumes them one-to-one, sodium hydroxide is limiting. At complete reaction, 0.0400 mol HCl reacts, leaving 0.0100 mol of acid unconsumed. The reaction extent ξ is therefore 0.0400 mol, not the 0.0500 mol originally added and not the sum of reactant amounts.
The solution left in the vessel can still be acidic even though neutralisation has happened. That is not a paradox; it is an ordinary consequence of adding unequal chemical amounts.
Assume the acid and base begin at the same temperature and the relevant corrected rise is ΔT = +4.60 K. Approximate the mixed liquid volume as 90.0 mL, its density as 1.00 g/mL and the solution specific heat as 4.184 J/(g·K). The illustrative solution mass is then 90.0 g.
Assume the instrument’s measured heat capacity is Ccal = 120 J/K and that the calorimeter itself warms by 4.60 K. The reaction completes, no appreciable material escapes and external losses are neglected for the core arithmetic. These are teaching assumptions, not claims that an ordinary cup is perfectly insulated.
Different laboratory vessels and solution compositions have different calorimeter constants, heat capacities, losses and sensor responses. Record which inputs were measured and which were approximated.
The solution absorbs qsolution = mcΔT = (90.0 g)(4.184 J/(g·K))(4.60 K) = 1732.176 J. The calorimeter absorbs qcal = CcalΔT = (120 J/K)(4.60 K) = 552 J.
Under the stated heat balance, qrxn ≈ −(1732.176 + 552) J = −2284.176 J = −2.284176 kJ. This is the total reaction heat for the amount that reacted in the example. The negative sign indicates heat released by the reacting system.
Then ΔHrxn ≈ qrxn/ξ = −2.284176 kJ / 0.0400 mol ≈ −57.1 kJ/mol. The result describes the equation and the approximated conditions, not a newly measured reference enthalpy.
| Calculation | Value |
|---|---|
| Starting HCl | 0.0500 mol |
| Limiting NaOH / reaction extent | 0.0400 mol |
| Solution heat | +1732.176 J |
| Calorimeter heat | +552 J |
| Reaction heat | −2284.176 J |
| Molar enthalpy | −57.1 kJ per mole of reaction |
If someone divides the same −2.284176 kJ by all 0.0500 mol acid charged, the result is −45.7 kJ/mol. That calculation incorrectly assumes every mole of acid reacted even though base ran out first. Relative to −57.1 kJ/mol, it understates the magnitude by 20%.
An equally tempting but chemically meaningless denominator is 0.0900 mol, the sum of starting acid and base. The quoted heat is not energy per mole of all dissolved reactants; the numerator belongs to 0.0400 mol of the balanced reaction.
This is a denominator error, separate from heat loss or thermometer precision. More decimal places in the measured temperature cannot repair it.
If the analyst neglects a genuinely non-negligible calorimeter heat capacity and counts only the solution heat, qrxn would be estimated as −1.732176 kJ. Dividing by the correct 0.0400 mol gives only about −43.3 kJ/mol for this constructed experiment.
That is a numerator error: a heat-absorbing part of the surroundings was omitted. Using excess acid moles is a denominator error. It is possible for two mistakes to counteract one another partly and yield a deceptively plausible number. A rigorous worked solution keeps the two audits separate.
The companion calorimeter heat-capacity article explains in more detail why water alone may not capture the full energy transfer.
A peak reading is not necessarily the corrected reaction temperature rise. During mixing and after the reaction, the vessel can exchange heat with its surroundings; sensors have response times; stirring and concentration affect the observed trace. The experimental protocol should state how ΔT was inferred.
Heat-capacity approximations for pure water may be imperfect for solutions, particularly concentrated mixtures. The calorimeter constant can change when parts are replaced. Reaction completeness, competing processes and mixing heats may also matter.
Temperature is an observation, not proof that a limiting reagent reacted fully or that an idealised net ionic process was isolated perfectly. When precision matters, support stoichiometric extent with suitable chemistry and controls.
Now consider a different fictional equation: 2A + B → products. Starting amounts are 0.060 mol A and 0.050 mol B. Compare potential extents: 0.060/2 = 0.030 mol from A and 0.050/1 = 0.050 mol from B. A limits the reaction at ξ = 0.030 mol.
If the measured total reaction heat is −3.00 kJ, the enthalpy per mole of reaction as written is −3.00/0.030 = −100 kJ/mol reaction. Dividing by 0.060 mol of A consumed gives −50 kJ per mole of A consumed. Both ratios can have meaning if correctly named; they are not interchangeable labels for the same molar reaction enthalpy.
Writing the reporting basis beside each energy unit prevents the equation coefficients from disappearing during calculation.
Does a bigger temperature rise always mean a larger molar enthalpy? No. It also depends on mass, heat capacities, reaction extent and losses. Is the limiting reagent always the smallest starting mole quantity? No. Compare each reagent amount after dividing by its stoichiometric coefficient.
If the solution warms, is qrxn positive? Under the standard sign convention for a heat-releasing reaction, qrxn is negative and the surroundings have positive heat change. Can a perfect calculation prove a laboratory result? No; it proves only the arithmetic implied by the stated measurements and assumptions.
Can qp be treated as ΔH? For the defined system at constant pressure with the usual appropriate conditions, heat at constant pressure is associated with enthalpy change; experimental calibration and additional forms of work must still be considered.
Practice A: Mix 60.0 mL of 0.800 mol/L HCl with 30.0 mL of 1.00 mol/L NaOH. Starting amounts are 0.0480 and 0.0300 mol. The base limits, ξ = 0.0300 mol. If qrxn = −1.71 kJ, then the model gives ΔH ≈ −57.0 kJ/mol.
Practice B: return to the original heat −2.284176 kJ. Divide by 0.0500 mol excess acid: −45.7 kJ/mol. Explain, in a full sentence, why that denominator represents charged rather than reacted moles.
Practice C: starting from only qsolution, list the evidence you need before adding qcal and claiming a molar enthalpy: instrument heat capacity, relevant ΔT, insulating and mixing assumptions and the reaction extent.
Ledger one is chemical amount: balance the equation, convert volume and molarity to moles, identify the limiting reagent and calculate ξ. Ledger two is heat flow: define the system, estimate all heat-absorbing surroundings, apply the correct signs and units, and note losses or approximations.
Only after both ledgers close do we divide qrxn by ξ and write kJ/mol of the specified reaction. This method transfers from neutralisation to dissolution, precipitation and other solution calorimetry problems, although each experiment has its own chemistry and heat terms.
Continue at The Gold Standard of Thermochemistry, then compare the Secondary 4 calorimetry heat-loss explanation to see how the experimental limitations appear at an earlier learning level.
Sources and connected reading
The worked solution is an original educational model. Numbers are deliberately constructed; results are not empirical measurements or claimed reference enthalpies.