Hess's law
A Socratic walk-through of Hess's law — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #How can chemists know the heat of a reaction nobody has ever carried out?
Open a data book and you will find the heat of formation of carbon monoxide given as about -110.5 kilojoules per mole, quoted to a tenth of a kilojoule. It looks like a measurement.
It is not, and it cannot be. The reaction it refers to is carbon plus half an oxygen molecule going to carbon monoxide, and nobody has ever run it cleanly. Burn carbon in restricted oxygen and you get a mixture of carbon monoxide and carbon dioxide in proportions you cannot control, so the heat released is the heat of some blend you did not choose. The number in the book is for a reaction that has never happened in isolation. So where did the tenth of a kilojoule come from?
Reasoning it through
REASONING #Try a thought experiment before touching any chemistry. Suppose it were possible to get from graphite and oxygen to carbon dioxide by two different routes that released different amounts of heat. Take the generous route forward, collect the heat, then run the stingy route backwards, paying back less than you collected. You are now exactly where you started, with the same substances in the same state — and with heat left over in your hand. Repeat.
Ask what that machine would be. It is energy from nothing. So either such a pair of routes cannot exist, or the conservation of energy is wrong. That argument is the whole of Hess's law, and notice that it was available before any experiment: the law is not a chemical discovery so much as a chemical consequence of a conservation principle.
Germain Hess arrived at it from the other end, empirically, in 1840 — before the first law of thermodynamics was stated in its modern form — and called it the law of constant heat summation: the heat of a reaction is the same whether it proceeds in one step or several.
Now the useful move. If the heat of a change depends only on where you start and where you finish, then any two routes with the same endpoints are interchangeable, and you may assemble a route out of steps you can measure, even if the step you want is not one of them. Carbon monoxide falls out immediately. Burning graphite completely to carbon dioxide is easy to measure: about -393.5 kilojoules per mole. Burning carbon monoxide to carbon dioxide is also easy: about -283.0. The first route is direct; the second route goes via carbon monoxide. Subtract, and the leg you never ran is -110.5.
Ask yourself what was actually assumed there. Only that the two routes end at the same place, and that energy is conserved. No mechanism, no rate, nothing about how the reaction proceeds.
There is a condition, though, and it is the part most often skipped. Heat in general is not a property of a state — how much heat a process gives up genuinely depends on how it is run, which is why an engine can convert some of it to work instead. What rescues us is a constraint: at constant pressure, with no work other than pushing back the atmosphere, the heat exchanged is exactly equal to the change in a quantity called enthalpy. And enthalpy is built entirely from properties of the state — internal energy, pressure and volume — so its change is path-independent by construction. Hess's law is really the statement that enthalpy is a state function, plus the constraint that makes heat report it faithfully.
The analogy
THE ANALOGY #Think of altitude on a mountain. The difference in height between the car park and the summit is fixed, and it does not matter whether you take the ridge, the gully, or a helicopter. If someone tells you the ridge climbs 800 metres and the summit is 1,200 metres above the car park, you know the remaining traverse from the ridge is 400 metres, without ever walking it.
altitude never depends on the route, whereas heat does — the effort you spend on the walk depends on the path — so the analogy holds only because we have quietly agreed to keep the pressure constant, which is what forces the heat to track the state function rather than the journey.
Clarifying the model
THE MODEL #The first refinement is that Hess's law is not a special technique but the working principle behind the whole apparatus of tabulated thermochemistry. Standard enthalpies of formation exist because someone decided to fix a common reference — the elements in their standard states, assigned zero — so that any reaction's enthalpy can be built as the formation enthalpies of the products minus those of the reactants. That subtraction is Hess's law, applied so routinely that it stops looking like a law.
It is also what makes genuinely inaccessible quantities available. The lattice enthalpy of sodium chloride — the energy to pull the crystal apart into free gaseous ions — cannot be measured directly at all, but a Born-Haber cycle closes a loop through sublimation, ionisation, dissociation, electron affinity and the enthalpy of formation, all of which can be measured, and the unmeasurable step is whatever makes the loop balance.
Two cautions are worth stating plainly. First, path independence says nothing about whether a reaction will actually go, or how fast. A route can be thermodynamically bookkept and kinetically impossible; graphite turning to diamond is favourable to draw and hopeless to wait for. Hess's law is silent on rate by design, because it never mentions mechanism.
Second, the numbers only add up if the states match exactly. Water as liquid and water as vapour differ by the enthalpy of vaporisation, and a cycle that quietly switches between them is not closed. That is why tables are so fussy about phase, temperature and standard state — the arithmetic is trivial and the bookkeeping is where the errors live.
A picture of it
THE PICTURE #How to readBoth routes start at the parallelogram and both finish at the same box, which is the only thing the argument needs. Follow the single long arrow for the direct measurement, then the two short arrows for the indirect route. Two of the three edge labels are measured values; the third is the one being solved for, and it is fixed by the requirement that the two routes sum to the same total. Cover any one label and you can recover it from the other two.
What became clearer
WHAT CLEARED #The heat of an unrunnable reaction is not measured, it is deduced — and the deduction rests on nothing more than the impossibility of getting energy for free. Once enthalpy is recognised as a property of a state rather than of a journey, every measurable reaction becomes a leg of a network, and any missing leg can be inferred from a closed path around it. That is why a data book can quote a number to a tenth of a kilojoule for something no one has ever done.
Where to go next
ONWARD #- How Born-Haber cycles pin down lattice enthalpies, and what the discrepancy with a purely ionic model reveals about covalent character.
- Why Gibbs free energy, not enthalpy, decides whether a reaction proceeds, and how entropy enters the same bookkeeping.
- How bond enthalpy estimates approximate the same quantity, and why they disagree with calorimetry.
Key terms
TERMS #| Term | What it means |
|---|---|
| Enthalpy (H) | internal energy plus pressure times volume; its change equals the heat exchanged at constant pressure. |
| State function | a property fixed by the current condition of a system, whose change depends only on start and finish, never on the route. |
| Standard enthalpy of formation | the enthalpy change forming one mole of a compound from its elements in their standard states, which are assigned zero. |
| Born-Haber cycle | a closed thermochemical loop used to obtain a lattice enthalpy that cannot be measured directly. |
| Calorimetry | direct measurement of heat released or absorbed, and the source of the values Hess cycles are assembled from. |
Every term the collection defines is gathered in the glossary.