Adding up the heat

The heat released by a reaction does not depend on the route taken, which lets you measure changes you cannot perform. Hess found this in 1840, a decade before anyone could say why it should be true.

1783 – 185015 min

By the end you should be able to

State Hess’s law, and say what makes enthalpy route-independent when heat itself is not
Work out the enthalpy of a reaction nobody can perform, from two that anyone can
Explain why the law was found a decade before the principle that explains it
Read an enthalpy level diagram, and check a cycle by taking both routes

Chemistry had learnt to weigh things, and weighing had just settled its longest argument. But every reaction plainly produced something else as well, and nobody could put it on a balance. Heat. A flame gives it out, quicklime slaked with water gives it out, and mixing acid into water gives out enough to crack the vessel if you are careless. It was obviously a quantity — twice the fuel gave twice the effect — and there was no instrument for it. Lavoisier and Laplace built one, in the winter of 1782–83. Two nested containers packed with crushed ice, with the reaction in the innermost chamber. Heat coming out of the reaction melted ice; the meltwater ran out of a tap at the bottom; you weighed the water. Every gram of it stood for a fixed quantity of heat, because melting ice takes the same amount every time. They burnt charcoal in it. They put a guinea pig in it and measured the heat of breathing, and found it agreed roughly with burning charcoal that produced the same carbon dioxide — which is a startling result to get in 1783, and Lavoisier drew the right conclusion from it. They also stated something that turns out to matter here: the heat needed to decompose a compound is equal to the heat given out when it forms. Run the change backwards and the books close. That is the first hint that these numbers are properties of the substances rather than of the operations.

Now the difficulty this lesson exists to get round. To measure the heat of a reaction, you must actually run that reaction: on its own, to completion, with nothing else happening in the vessel and no product left over that you did not intend. A great many of the reactions a chemist cares about will not oblige. Some stop halfway, leaving both sides present and a heat that belongs to a mixture. Some give more than one product. Burn carbon in a limited supply of air and you get carbon monoxide and carbon dioxide together, in a ratio that depends on the temperature, the draught and the shape of the fire. Whatever heat you measure belongs to that particular mixture on that particular day. Some need conditions no calorimeter survives — thousands of atmospheres, or hours at white heat. So the heat of those reactions is not a small quantity, or an uncertain one. It is a quantity you cannot go and get. And it is a quantity chemistry needs, because you cannot say which of two routes to a product costs less without it.

The experiment

The heat of diluting sulphuric acid, in one step and in several

Germain Henri Hess · 1840, published in Poggendorff’s Annalen der Physik und Chemie · St Petersburg

The question: If you carry out the same overall change by two different sequences of operations, do you get the same total quantity of heat?
The apparatus: Concentrated sulphuric acid, water, and a calorimeter. In one run, the acid is diluted to its final strength in a single addition and the heat measured. In the other, the same final dilution is reached in several separate additions, with the heat of each step measured and the steps added up.

Theory predicted

Nothing in particular was predicted, because there was no principle to predict from. Heat was widely thought to be a substance — caloric — squeezed out of bodies as they combined, and on that picture there is no obvious reason why the total should not depend on how the squeezing was done.

They measured

The sum of the steps matched the single step, as closely as the calorimeter could distinguish them. The same held for his neutralisations: the heat of a reaction was fixed by its starting and finishing states and by nothing else.

How sure could they be? The agreement is the entire result, so its quality is the whole question. Hess’s calorimetry was good to a per cent or so — crude beside a modern bomb, and far better than the difference he was looking for would have been if the totals genuinely depended on the route. He repeated the comparison on several systems before publishing.

Why it mattered

It makes the heat of a reaction a property of the two states rather than of the procedure — which means it can be added, subtracted, and obtained by arithmetic from other reactions. A quantity you cannot measure directly stops being unavailable and becomes a matter of finding two you can.

An engraved portrait of Germain Henri Hess in a high collar and dark coat, facing slightly left. — Born in Geneva, raised in Russia from the age of three, trained as a physician at Dorpat and appointed professor at the Technological Institute in St Petersburg. He wrote the first Russian chemistry textbook, and the law that carries his name was published when he was thirty-eight. He died at forty-eight, ten years before anyone could explain it.

The law of constant heat summation

The heat change of a reaction depends only on the initial and final states, and not on the route between them. So if a change can be broken into steps, the heats of the steps add up to the heat of the whole. And — this is where the value is — if you know all the steps but one, you know that one too. It is worth noticing how little this claims. It says nothing about whether a reaction will go, nothing about how fast, nothing about what the mechanism is. It says one thing: the accounts do not depend on the route. And that one thing turns a laboratory measurement into an algebra. You may add reactions like equations, reverse one and flip the sign of its heat, double one and double its heat — and every result you get out is as good as a measurement, because it is the sum of measurements.

The picture to carry is a hill. Walk from the car park to the summit and the height you gain is fixed by those two places. Straight up the face, round by the long ridge, or with a detour to a café halfway down — the difference in altitude comes out the same every time. It is a property of where you started and where you stopped. The distance you walked is not like that at all. The ridge is four times as long as the face. Enthalpy is the altitude. A quantity with this property is called a state function: it has a definite value in each state, so a change in it is a difference between two numbers, and differences do not remember paths. That is the whole of Hess’s law, and it is why the diagram below is drawn as levels rather than as a story. There is no time axis in it. There is only how high each state sits.

Two arrows are solid, because those are burns you can do in a calorimeter. One is dashed, because nobody can. Press Solve by difference and watch the dashed one acquire a number.

Loading the energy diagram…

The canonical case. Graphite burns cleanly to CO₂ and carbon monoxide burns cleanly to CO₂, so both of those are honest measurements. Burning carbon to carbon monoxide and stopping is not a reaction anyone can perform — you always get a mixture — so the arrow from the top level to the middle one has never been measured by anybody. Subtract the two measured legs and it appears: −110.53 kJ/mol, the standard enthalpy of formation of carbon monoxide, which is in every data table in chemistry and was never once obtained directly.

That is not a conjuring trick kept for one awkward compound. It is how the tables were built. Enthalpies of formation are mostly unmeasurable directly. Making methane by heating graphite with hydrogen is not something anyone does; the equilibrium is poor and the reaction is slow to the point of not happening. So nobody has ever measured the heat of forming methane from its elements. Enthalpies of combustion are almost all easy. Put a weighed sample in a steel bomb, fill it with oxygen at thirty atmospheres, fire it electrically, and measure the temperature rise of the water tank it is sitting in. Berthelot had this working by 1881, and heats of combustion were soon being published by the thousand. So you burn methane, you burn graphite, you burn hydrogen, and you subtract. Three measurements you can make, standing in for one you cannot. Open any data book at the table of standard enthalpies of formation and you are looking at the output of that subtraction, done many thousands of times.

Walk the direct route, then press Take the long way and walk it again. The running total is the height of the marker, and nothing else.

Loading the energy diagram…

The same cycle with the missing step filled in, so both routes are now available. Take either. The running total is literally the marker’s height on the enthalpy axis, so arriving at the same bottom level means arriving at the same number — which is what a state function is, drawn rather than asserted. Change the cycle and the point survives: graphite to diamond, or methane to its elements, both by the same subtraction.

The part the usual statement gets slightly wrong

You will very often see Hess’s law stated as "the heat of a reaction is independent of the route". That is not quite true, and the gap is worth having straight, because it is the difference between two symbols that get used as if they were one. Heat is not a state function. It is not a property a substance has; it is a way energy moves across a boundary, and how much of it moves depends on what else you let the reaction do. Burn hydrogen in a flame and you get about 286 kJ per mole as heat. Run the identical change in a fuel cell and you get substantially less heat, because a large share leaves as electricity instead. Same reactants, same products, different heat. Enthalpy is a state function. ΔH is fixed by the two states, always. The heat equals ΔH only under a condition: constant pressure, and no work done other than pushing the atmosphere out of the way. Hess worked in open vessels at atmospheric pressure with no electrodes in them, so the condition held everywhere he looked and the exception did not exist for him. The distinction was drawn later, and the function itself did not even get its modern name until around 1909, when Kamerlingh Onnes called it enthalpy — from the Greek for "to warm within".

You might think

Hess’s law is a consequence of the first law of thermodynamics.

Actually

It is, and that is not how it was found. Look at the dates. Hess published in 1840. Mayer argued that heat and work are the same currency in 1842 and was largely ignored. Joule measured the exchange rate over the middle of that decade. Helmholtz generalised it into conservation of energy in 1847, and Clausius set out the first law in 1850. There was no first law in 1840 for Hess to deduce anything from — energy was not yet a thing that was conserved, and heat was still widely thought to be a fluid. Hess measured a regularity, repeated it on several systems, satisfied himself it was general, and published it. Ten years later the rest of physics arrived and explained why it could not have come out any other way. That order — a solid empirical regularity first, the principle that accounts for it afterwards — is much commoner in science than the tidy retelling suggests, and it is worth noticing when it happens, because a law found this way is believed for reasons that survive even if the explanation later changes.

Problem

The reaction that cannot be run

Two combustion enthalpies, both straightforward bomb-calorimeter measurements: C(graphite) + O₂(g) → CO₂(g) and CO(g) + ½O₂(g) → CO₂(g). What is ΔH for C(graphite) + ½O₂(g) → CO(g) — the reaction nobody can perform cleanly? Give it in kJ/mol, with its sign.

Burning graphite to CO₂: ΔH = −393.51 kJ/mol
Burning CO to CO₂: ΔH = −282.98 kJ/mol

When a combination takes place, the quantity of heat evolved is constant, whether the combination occurs directly or indirectly and in stages.
Germain Henri HessThermochemische Untersuchungen, Annalen der Physik und Chemie 50, 1840. Translated from the German; Hess was Swiss-born, Russian by upbringing, and published in Berlin.

1783Lavoisier and Laplace melt ice to measure heat, and state that decomposing a compound costs exactly what forming it gave out.
1840Hess dilutes sulphuric acid in one step and in several, gets the same total, and publishes the law of constant heat summation.
1842Mayer argues that heat and work are the same currency. Almost nobody reads him.
1845Joule’s paddle wheel puts a number on the exchange rate between them.
1847Helmholtz, Über die Erhaltung der Kraft: conservation of energy as a general principle.
1850Clausius states the first law of thermodynamics. Hess dies the same year, aged 48, without ever seeing why his law was true.
1881Berthelot’s bomb calorimeter. Heats of combustion begin to be published by the thousand, and the tables fill up.
1909Kamerlingh Onnes gives the quantity its modern name: enthalpy.

Where this leaves us

The heat books balance, and they balance so dependably that you can extract a measurement you were never able to take. Enthalpy is an altitude; routes do not matter; the tables are built out of subtraction. And that hands the nineteenth century a very tempting idea. If every reaction has a definite heat, and reactions that give out heat are running downhill, then surely the heat tells you whether a reaction will go at all. Downhill happens; uphill does not. Thomsen said so in 1854, Berthelot said so more loudly in 1867, and it works most of the time — the great majority of reactions that go spontaneously are exothermic. Most of the time is not a law, though. Go into a chemist’s shop and buy an instant cold pack. Squeeze it, and a thin membrane breaks, and ammonium nitrate meets water and dissolves. It takes 25.7 kJ per mole in from its surroundings. The pouch goes down to near freezing in your hand while it happens. And it happens. Immediately, completely, and without anything pushing it.