065 – Hess’s Law

Calorimetry is how chemistry puts a number on heat. A vessel is insulated so that whatever energy a process releases or absorbs stays in its contents, the temperature of those contents is read before and after, and the energy follows from the mass, the specific heat capacity and the temperature change. Numbers obtained this way are what a process engineer uses to size a cooling jacket, what a food laboratory reports as a calorie count, and what a lime or cement plant uses to budget the fuel needed to drive carbon dioxide out of limestone.

The measurement is worth making because enthalpy is a state function: the heat exchanged in going from one set of substances to another depends only on what those substances are, not on the route taken between them. That is Hess’s law. It means a reaction too slow, too violent or too impure to measure directly can still be obtained, by adding the enthalpies of any sequence of steps that starts and finishes in the right place. It also means a measured heat says nothing on its own — it has to be divided by the amount of substance responsible for it, so every calorimetric result begins with a mole count and a limiting reagent.

In this laboratory you will run five processes in the same calorimeter and record the temperature before and after each one. Two are physical: ethanol mixing with water, and ammonium chloride dissolving in water. They exchange heat in opposite directions. One is a control in which nothing measurable happens. The other two are chemical: calcium carbonate attacked by hydrochloric acid, and hydrochloric acid neutralised by sodium hydroxide. You will convert each temperature change into an enthalpy per mole, compare the five against one another, and then use Hess’s law to show that the smallest of the five is the small difference between two much larger numbers.

Educational Goals

Calorimetric technique

• Charge a calorimeter, close the lid, stir the contents and take an initial and a final temperature in the correct order.
• Explain what the lid and the stirrer each contribute, and why a reading taken before stirring is not the temperature of the mixture.

Measuring an amount of substance

• Deliver 100 mL of a liquid with a graduated cylinder and weigh a powder onto a tared weighing boat to the nearest 0.01 g.
• Convert a volume and a concentration, or a mass and a molar mass, into a number of moles before any energy is calculated.

Turning a temperature change into an enthalpy

• Apply q = m × c × ΔT to the contents of the calorimeter and then ΔH = −q / n to obtain a molar enthalpy with the right sign.
• Identify the limiting reagent first, and use its mole count rather than the larger one.

Telling physical heat effects from chemical ones

• Classify each of the five processes as mixing, dissolution or reaction, and say in each case which bonds or intermolecular forces are broken and which are formed.
• Use the terms exothermic and endothermic with the sign convention that goes with them, rather than as labels for hot and cold.

Applying Hess’s law

• Construct a two-step route to a reaction whose enthalpy is hard to measure, and show that the two steps add to the one-step value.
• Recognise spectator ions cancelling, and explain why every strong acid neutralised by every strong base gives the same molar enthalpy.

Reading a control experiment

• State what Experiment 3 establishes even though nothing happens in it, and why the interpretation of Experiment 4 depends on it.

Judging how far a calorimetric result can be trusted

• Explain why the specific heat capacity of the contents, not that of water, is the one that belongs in the calculation.

Protocol

Experiment 1 : Water + Alcohol

1. Measure 100 mL of distilled water using the graduated cylinder.
2. Pour the contents of the graduated cylinder into the calorimeter.
3. Immerse the tip of the digital thermometer in the liquid to take its temperature.
4. The initial temperature of the water will appear in the results table.
5. Measure 100 mL of ethanol using the graduated cylinder.
6. Pour the contents of the graduated cylinder into the calorimeter.
7. Attach the lid onto the calorimeter.
8. Press the green stirrer button on the calorimeter lid and let it stir for at least 10 seconds.
9. Insert the digital thermometer into the lid of the calorimeter.
10. The temperature of the mixture will appear in the results table.
11. Stop the agitator by pressing the red button.
12. Remove the thermometer from the calorimeter lid.
13. Remove the lid of the calorimeter and empty its contents into the recovery bin.
14. Rinse the calorimeter with distilled water and empty its contents into the recovery tank.
15. Rinse the graduated cylinder with distilled water and empty its contents into the recovery tank.

Experiment 2 : Water + NH₄Cl

1. Measure 100 mL of distilled water using the graduated cylinder.
2. Pour the contents of the graduated cylinder into the calorimeter.
3. Immerse the tip of the digital thermometer in the liquid to take its temperature.
4. The initial temperature of the water will appear in the results table.
5. Weigh approximately 9.2g (6 mL) of powdered ammonium chloride using the weighing boat.
6. Pour the contents of the weighing boat into the calorimeter.
7. Attach the lid onto the calorimeter.
8. Press the green agitator button on the calorimeter lid and let it stir for at least 10 seconds.
9. Insert the digital thermometer into the calorimeter lid.
10. The temperature of the mixture will appear in the results table.
11. Stop the agitator by pressing the red button.
12. Remove the thermometer from the calorimeter lid.
13. Remove the lid of the calorimeter and empty its contents into the recovery bin.
14. Rinse the calorimeter with distilled water and empty its contents into the recovery tank.
15. Rinse the graduated cylinder with distilled water and empty its contents into the recovery tank.

Experiment 3 : Water + CaCO₃

1. Measure 100 mL of distilled water using the graduated cylinder.
2. Pour the contents of the graduated cylinder into the calorimeter.
3. Immerse the tip of the digital thermometer in the liquid to take its temperature.
4. The initial temperature of the water will appear in the results table.
5. Weigh approximately 9.5g (3.5 mL) of powdered calcium carbonate using the weighing boat.
6. Pour the contents of the weighing boat into the calorimeter.
7. Attach the lid to the calorimeter.
8. Press the green agitator button on the calorimeter lid and let it stir for at least 10 seconds.
9. Insert the digital thermometer into the calorimeter lid.
10. The temperature of the mixture will appear in the results table.
11. Stop the agitator by pressing the red button.
12. Remove the thermometer from the calorimeter lid.
13. Remove the lid of the calorimeter and pour the liquid contents into the black recovery bin and transfer the solids using the spatulas.
14. Rinse the calorimeter with distilled water and empty its contents into the recovery tank.
15. Rinse the graduated cylinder with distilled water and empty its contents into the recovery tank.

Experiment 4 : HCl + CaCO₃

1. Measure 100 mL of hydrochloric acid (HCl) 2 M using the graduated cylinder.
2. Pour the contents of the graduated cylinder into the calorimeter.
3. Immerse the tip of the digital thermometer in the liquid to take its temperature.
4. The initial temperature of the hydrochloric acid will appear in the results table.
5. Weigh approximately 9.5g (3.5 mL) of powdered calcium carbonate using the weighing boat.
6. Pour the contents of the weighing boat into the calorimeter.
7. Attach the lid to the calorimeter.
8. Insert the digital thermometer into the calorimeter lid.
9. Start the stopwatch.
10. Press the green stirrer button on the calorimeter lid and let it stir.
11. Observe the reaction occurring in the temperature vs. time graph.
12. When the reaction is complete (the temperature will have reached a plateau) stop the stopwatch.
13. Stop the agitator by pressing the red button.
14. Remove the thermometer from the calorimeter lid.
15. Remove the lid of the calorimeter and empty its contents into the recovery bin.
16. Rinse the calorimeter with distilled water and empty its contents into the recovery tank.
17. Rinse the graduated cylinder with distilled water and empty its contents into the recovery tank.

Note : The reaction speed is accelerated 10x.

Experiment 5 : HCl + NaOH

1. Measure 100 mL of 1M NaOH using the graduated cylinder.
2. Pour the contents of the graduated cylinder into the calorimeter.
3. Immerse the tip of the digital thermometer into the liquid to take its temperature.
4. The initial temperature of the water will appear in the results table.
5. Measure 100 mL of 1M HCl using the graduated cylinder.
6. Pour the contents of the graduated cylinder into the calorimeter.
7. Attach the lid to the calorimeter.
8. Press the green stirrer button on the calorimeter lid and let it stir for at least 10 seconds.
9. Insert the digital thermometer into the calorimeter lid.
10. The temperature of the mixture will appear in the results table.
11. Stop the agitator by pressing the red button.
12. Remove the thermometer from the calorimeter lid.
13. Remove the lid of the calorimeter and empty its contents into the recovery bin.
14. Rinse the calorimeter with distilled water and empty its contents into the recovery tank.
15. Rinse the graduated cylinder with distilled water and empty its contents into the recovery tank.

Anticipated Outcomes

Five processes, one calorimeter, and a temperature change to be converted into an enthalpy in each case. The table gathers what each experiment should give. The heat capacity column is the capacity of the calorimeter contents, which is not the same from one experiment to the next, and it is the reason the five temperature changes are not in proportion to the five molar enthalpies.

From a temperature change to an enthalpy

Every one of the five results is obtained the same way, in two steps. First the heat that entered or left the contents of the calorimeter, q = m × c × ΔT, where m is the mass of everything in the vessel and c its specific heat capacity. Then the molar enthalpy, ΔH = −q / n, where n is the number of moles of the substance the enthalpy is being quoted per, and the minus sign converts “the contents got warmer” into “the process released energy”. Experiment 5 worked through: 200 mL of mixed solution has a mass of about 203 g and a specific heat capacity of about 4.16 J/(g·K), so the contents hold 203 × 4.16 = 845 J/K; a rise of 6.4 °C therefore corresponds to q = 845 × 6.4 = 5.41 kJ; and since 100 mL of 1 mol/L acid contains 0.100 mol, ΔH = −5.41 / 0.100 = −54 kJ per mole of HCl.

The step students most often get wrong is the heat capacity. It is tempting to use 4.18 J/(g·K) for everything, because most of what is in the vessel is water. In Experiment 1 that gives the wrong answer by nearly a quarter: the contents are 100 g of water at 4.18 plus 78.9 g of ethanol at 2.44 J/(g·K), so the capacity is 100 × 4.18 + 78.9 × 2.44 = 418 + 192 = 610 J/K, whereas 178.9 g of water would be 748 J/K. Using the water figure with the observed 8.0 °C would return 6.0 kJ instead of 4.88 kJ and a molar enthalpy of −3.5 rather than −2.85 kJ/mol. The same caution applies, less severely, to the concentrated salt solutions of Experiments 2, 4 and 5, whose specific heat capacities run about 4 to 7 per cent below that of pure water.

Experiment 1 — ethanol and water: a physical process, and it warms

Nothing reacts here. No bond inside a water molecule or an ethanol molecule is broken, and no new substance is made; what changes is which molecules are hydrogen-bonded to which. Breaking the hydrogen-bond network of pure water and the weaker network of pure ethanol costs energy, and forming water–ethanol hydrogen bonds returns it. The ethanol hydroxyl group is a good donor and acceptor, and the small ethanol molecule fits into water’s network without disrupting much of it, so slightly more energy comes back than went in and the mixture warms. 100 mL of ethanol is 78.9 g, or 1.71 mol at M = 46.07 g/mol; with the contents’ capacity of 610 J/K, a rise of 8.0 °C is 4.88 kJ, which is 2.85 kJ per mole of ethanol. Expressed per mole of the whole mixture — 5.55 mol of water plus 1.71 mol of ethanol, so 7.26 mol — that is 0.67 kJ/mol, which sits squarely inside the published excess enthalpy of mixing for this system, whose minimum is about 0.8 kJ/mol near an ethanol mole fraction of 0.2. The figure on this page is therefore a realistic one, not a convenient one.

The same intermolecular fit shows up in the volume, and it is worth measuring. The excess volume of mixing for water and ethanol at this composition is about −0.9 cm³ per mole of mixture, so 7.26 mol contract by roughly 7 mL: pouring 100 mL into 100 mL gives about 193 mL, not 200. A student who reads the level in the calorimeter and finds it short has not spilled anything. The contraction and the warming are two faces of one fact, that the mixed liquid is more tightly held together than either pure liquid was.

Experiment 2 — ammonium chloride: a physical process, and it cools

Dissolving an ionic solid is the sum of two large opposing terms. The lattice has to be pulled apart, which costs the lattice enthalpy, +705 kJ/mol for NH₄Cl; the separated ions are then hydrated, which returns about −690 kJ/mol. The measured enthalpy of solution is the small residue, +14.8 kJ/mol, about 2 per cent of either term — which is exactly why enthalpies of solution are hard to predict and easy to measure, and why they have to be measured. Here the residue is positive, so the process draws heat out of the water and the temperature falls. With 9.2 g, or 0.172 mol at M = 53.49 g/mol, the solution gives up 0.172 × 14.8 = 2.55 kJ; the contents are 109.2 g at about 3.90 J/(g·K), so 426 J/K, and the fall is 2550 / 426 = 6.0 °C.

Because nothing here is limiting, the size of the drop is set only by how much salt was weighed: about 0.65 °C per gram, so a gram short of the target costs a full degree of signal. Weigh as close to 9.2 g as the balance allows. The salt does dissolve completely — 9.2 g in 100 mL is 1.72 mol/L against a solubility near 37 g per 100 mL at room temperature, so the solution is at roughly a quarter of saturation and no residue should be left in the vessel. This is the principle behind an instant cold pack, which is a bag of water and a sachet of an ammonium salt.

Experiment 3 — calcium carbonate in water: the control, and why it is needed

The thermometer should not move. Calcium carbonate is not insoluble in the absolute sense — nothing is — but it is close enough that the distinction cannot be seen in a calorimeter. Its solubility product is K_sp = 3.3 × 10⁻⁹, so a saturated solution holds √K_sp = 5.7 × 10⁻⁵ mol/L of calcium ions. In 100 mL that is 5.7 × 10⁻⁶ mol, or 0.57 mg out of the 9.5 g weighed: six thousandths of one per cent. At an enthalpy of solution of about −12.3 kJ/mol the heat released is 0.07 J, and 0.07 J spread over 420 J/K of contents is a temperature rise of 0.0002 °C. A thermometer would have to be about thirty-five thousand times more sensitive than one reading whole degrees to detect it. The stirring changes nothing, because the limit is thermodynamic and not kinetic: the water is saturated within seconds and stays that way.

It would be easy to treat this as a wasted experiment, and it is the one most likely to be skipped. It should not be, because it is the reference leg of the argument that Experiment 4 rests on. Experiment 4 puts the same 9.5 g of the same powder into 100 mL of liquid and records a rise of several degrees. Without Experiment 3 a student cannot say whether that heat came from the carbonate meeting the acid or simply from the carbonate meeting a liquid. Experiment 3 removes the second possibility, and in doing so licenses the whole of the enthalpy in Experiment 4 to be assigned to the reaction. A control that returns nothing has still returned a result.

Experiment 4 — carbonate and acid, and Hess’s law at work

The reaction is CaCO₃(s) + 2 HCl(aq) → CaCl₂(aq) + H₂O(l) + CO₂(g), so two moles of acid are consumed for every mole of carbonate, and the gas leaves the vessel. Take stock before calculating anything: 9.5 g of CaCO₃ at M = 100.09 g/mol is 0.095 mol, which needs 0.190 mol of HCl, and 100 mL of 2 mol/L acid supplies 0.200 mol. The carbonate is limiting and the acid is in only 5 per cent excess, which is the tightest margin anywhere in this laboratory — weigh a gram over and the acid runs out before the powder does, and the heat measured will be short. At −14.6 kJ per mole of carbonate the reaction releases 0.095 × 14.6 = 1.39 kJ, and against contents of about 375 J/K that is a rise of 3.7 °C. That is what the contents alone would show; the thermometer reads a little less. A run recorded in the results journal starts at 21.3 °C and levels off at 24.3 °C — a rise of 3.0 °C, with the plateau reached in just under a minute on the stopwatch. The difference is the calorimeter itself taking its share of the 1.39 kJ: dividing 1.39 kJ by the observed 3.0 °C gives an effective 460 J/K, and the ≈85 J/K above the contents’ 375 J/K is the calorimeter constant made visible. Comparing the two rises is the quickest estimate of that constant this laboratory offers. The protocol notes that the simulation runs the reaction far faster than reality, so the elapsed time on screen corresponds to several minutes at the bench. Now the point of the laboratory. 14.6 kJ/mol is a small number, and the interesting thing about it is that it is small — a quarter of the neutralisation in Experiment 5, from a reaction that visibly fizzes and dissolves a solid. Hess’s law explains it, because the overall change can be reached by a route through two steps whose enthalpies are individually enormous.

Read that table as a cancellation. Step 1 is calcination, the industrial decomposition of limestone to quicklime, and it is strongly endothermic: 179 kJ has to be supplied for every mole, which is why a lime kiln burns fuel. Step 2 is quicklime attacked by acid, which is strongly exothermic at −193.4 kJ/mol. Add them and CaO cancels from both sides, leaving the reaction actually carried out in the calorimeter with an enthalpy of −14.4 kJ/mol — a difference of 7 per cent between two numbers of order 190. The heat a student measures in Experiment 4 is small not because little is happening but because a great deal is happening in two directions at once. That is the single most useful thing Hess’s law does: it lets a quantity be obtained as a difference of large, separately measurable quantities when the quantity itself is awkward to isolate. Lab 060 builds the same kind of cycle for magnesium and magnesium oxide.

Both step enthalpies, and the −14.4 kJ/mol they sum to, follow from standard enthalpies of formation, so a teacher can rebuild every figure on this page from a data table rather than taking it on trust. For step 1, ΔH = [−635.1 + (−393.5)] − [−1207.6] = +179.0 kJ/mol. For step 2, ΔH = [−877.1 + (−285.8)] − [−635.1 + 2(−167.2)] = −1162.9 − (−969.5) = −193.4 kJ/mol. And directly, in one step, ΔH = [−877.1 − 285.8 − 393.5] − [−1207.6 + 2(−167.2)] = −1556.4 + 1542.0 = −14.4 kJ/mol. The three calculations use the same table and agree, which is what it means for enthalpy to be a state function.

Experiment 5 — neutralisation, and Hess’s law a second time

Written out in full the reaction is HCl(aq) + NaOH(aq) → NaCl(aq) + H₂O(l), but that equation contains two ions that do nothing. Sodium is a hydrated cation before the mixing and a hydrated cation after it; chloride likewise. Strike them out and what remains is H⁺(aq) + OH⁻(aq) → H₂O(l) — the formation of a covalent O–H bond out of two separated ions, and the only chemical event in the vessel. Both routes must give the same enthalpy, and they do. From the molecular equation, ΔH = [−407.3 + (−285.8)] − [−167.2 + (−470.1)] = −693.1 + 637.3 = −55.8 kJ/mol. From the net ionic equation, ΔH = [−285.8] − [0 + (−230.0)] = −55.8 kJ/mol. Identical, because the spectator terms that were dropped cancelled exactly — which is Hess’s law applied not to a sequence of steps but to two different descriptions of the same step.

That has a consequence worth drawing out, because it is the reason this measurement appears in every introductory course. Since the spectators cancel, the enthalpy of neutralisation does not depend on which strong acid or which strong base was used. Nitric acid and potassium hydroxide would give the same figure; so would sulfuric acid, per mole of hydrogen ion. Any pair that is fully dissociated is doing the identical reaction, and the constancy of the measured value across such pairs is the experimental evidence that strong acids really are fully dissociated. A weak acid gives a smaller figure, because part of the released energy is spent dissociating the acid first, and that difference is how an enthalpy of dissociation is obtained.

Quantitatively: 100 mL of 1 mol/L NaOH contains 0.100 mol and 100 mL of 1 mol/L HCl contains 0.100 mol, so the two are exactly equivalent, neither is in excess, and the product is 200 mL of 0.5 mol/L sodium chloride at pH 7. The value of record for this laboratory is 54 kJ released per mole, giving 5.40 kJ and a rise of 6.4 °C. For comparison, the formation-enthalpy table above returns −55.8 kJ/mol and the value usually tabulated for dilute strong acid with strong base is −57.1 to −57.3 kJ/mol; the laboratory’s 54 kJ/mol is 3 to 6 per cent below those, which is the direction and roughly the size of the shortfall an uncalibrated calorimeter produces, since heat absorbed by the vessel itself is never counted.

Set the five results side by side and the ordering is the lesson. Rearranging hydrogen bonds between two liquids that already had them costs or returns a few kilojoules per mole (Experiment 1, 2.85). Taking an ionic lattice apart and hydrating the pieces is the near-cancellation of two terms around 700 kJ/mol, leaving a residue of the same small order (Experiment 2, 14.8). Dissolving a solid whose lattice the water cannot break returns nothing at all (Experiment 3). A reaction whose two half-routes nearly cancel returns a residue of similar size (Experiment 4, 14.6). But making a covalent bond where there was none returns nineteen times what Experiment 1 does (Experiment 5, 54). Chemical change and physical change are not different in kind here — both are bond energies being traded — but they differ by an order of magnitude in how much energy is on the table.

Summary of Assignment by Grade Range

Grade 9–10

Focus: exothermic and endothermic as observable categories, and careful measurement. Students should leave able to say which of the five processes released heat, which absorbed it, and which did neither, and to use the two words correctly.

Activities: run all five experiments and record the initial and final temperature of each in a table of their own making, with the sign of every change. Sort the five into warmed, cooled and unchanged. Predict, before running Experiment 3, what a substance that does not dissolve should do to the temperature, and then say whether the prediction held. Write the word equation for Experiments 4 and 5.

Grade 11

Focus: converting a temperature change into a molar enthalpy. The step from a reading on a display to a number that can be compared with a data book is the whole of this band.

Activities: for each experiment, calculate the number of moles involved, identify the limiting reagent where there is one, apply q = m × c × ΔT and then ΔH = −q / n, and compare the result with the value quoted here. Balance the equation for Experiment 4 and show that the acid is in 5 per cent excess. Explain why the temperature drop in Experiment 2 is proportional to the mass weighed while the rise in Experiment 4 is not proportional to anything the student controls. Account for the ordering of the five molar enthalpies.

Grade 12 / College Level

Focus: Hess’s law as a working tool, and an honest error analysis. Students should be able to construct a route, defend the arithmetic from a table of formation enthalpies, and state how much of the discrepancy with published values their apparatus can account for.

Activities: reproduce the two-step calcination route to Experiment 4 and show that it sums to the one-step value; derive all three enthalpies independently from the table of standard formation enthalpies. Demonstrate that the molecular and net ionic equations of Experiment 5 give the same result, and argue from that to the claim that the enthalpy of neutralisation of any strong acid by any strong base is the same. Estimate the heat carried off by the carbon dioxide in Experiment 4. Design, without running it, a calibration step that would fix the calorimeter’s heat capacity, and say what the five results would become if it turned out to be 50 J/K. Compare this cycle with the magnesium and magnesium oxide cycle of lab 060.

Laboratory essentials

Instruments

• Calorimeter with insulated lid, motorised stirrer and green/red control buttons
• Digital thermometer
• Electronic balance
• Weighing boat
• Graduated cylinder, 250 mL (every experiment measures 100 mL; the 70 mL cylinder also supplied will not hold it)
• Spatula (Experiment 3, to transfer the undissolved carbonate)
• Stopwatch or timer (Experiment 4 only)
• Wash bottle of distilled water
• Recovery bin and recovery tank

Products

• Distilled water (100 mL in Experiments 1, 2 and 3, plus rinsing between every experiment)
• Ethanol, liquid (100 mL, Experiment 1)
• NH₄Cl, powder (9.2 g, Experiment 2)
• CaCO₃, powder (9.5 g in each of Experiments 3 and 4)
• HCl 2 mol/L, solution (100 mL, Experiment 4)
• HCl 1 mol/L, solution (100 mL, Experiment 5)
• NaOH 1 mol/L, solution (100 mL, Experiment 5)