Stoichiometry and the energy of reactions
When one micron of silicon is oxidized, 2.27 microns of oxide grow. That number does not come from a measurement: it comes from the balanced equation, and it has to be taken into account when designing the gate thickness.
01The balanced equation is a recipe
A balanced chemical equation does not say what happens: it says in what proportion it happens. The coefficients are the recipe, and they are read in moles, not in grams or in volumes.
This particular reaction is the one that creates the gate oxide of every MOS transistor: the wafer goes into a furnace at 1000 °C with oxygen, and the silicon at the surface turns into SiO₂. Since the oxide occupies 2.27 times the volume of the silicon consumed, for each micron of oxide grown, 0.44 microns came out of the wafer. That number is pure stoichiometry, and it has to be taken into account when designing the final thickness.
1. Balance the equation. 2. Convert the data to moles, dividing by the molar mass. 3. Compare with the coefficients and find the limiting reactant. 4. Calculate products and leftovers in moles. 5. Go back to grams if needed. Skipping step 2 is the source of almost all errors.
02The limiting reactant
When the reactants are not in the exact proportion of the recipe, one runs out first. That is the limiting reactant, and it is the only one that decides how much product is obtained: there is some of the other left over, and what is left over does not react no matter how much of it is there.
Pick one of the reactions that appear in electronics manufacturing and enter the moles of each reactant. The calculation shows which one limits, how much product comes out, how much is left over and how much energy is released.
In industry, one almost never works in exact proportion. In oxidation a huge excess of oxygen is used so that the limiting reactant is always the silicon at the surface and the rate does not depend on the flow; in etching with hydrofluoric acid, the excess acid guarantees that the oxide is removed entirely. The excess is chosen, not suffered.
What stoichiometry calculates is the theoretical yield: the maximum possible. The actual yield is always lower, because of side reactions, equilibrium or handling losses. The ratio between the two is the percent yield, and in manufacturing processes it is also called yield: the fraction of the chips on a wafer that work.
03The energy of the reaction
Every reaction exchanges energy with its surroundings, because breaking bonds costs energy and forming them gives it back. If it gives back more than it costs, the reaction is exothermic and \( \Delta H \lt 0 \); if it is the other way around, endothermic.
That enthalpy is a state function —it depends on where you start and where you end, not on the path— is what makes it possible to add enthalpies. That is Hess's law, and it is the reason why a single table of enthalpies of formation is enough to calculate any reaction, even one that nobody has ever measured.
| Reaction | \( \Delta H \) | What it implies in practice |
|---|---|---|
| Si + O₂ → SiO₂ | −911 kJ/mol | Very exothermic: silicon oxidizes on its own in air, and that 2 nm native layer has to be removed before any process |
| SiHCl₃ + H₂ → Si + 3 HCl | +219 kJ/mol | Endothermic: that is why the Siemens reactor operates at 1100 °C and consumes the energy it does |
| Zn + 2 MnO₂ → ZnO + Mn₂O₃ | −270 kJ/mol | The energy of an alkaline battery. Its voltage comes from the free energy (about −290 kJ/mol), divided by the charge transferred |
The reaction \( \mathrm{2Al + Fe_2O_3 \rightarrow Al_2O_3 + 2Fe} \) releases 852 kJ per mole and reaches 2500 °C. It is the same chemistry that makes the aluminum of traces oxidize instantly in air: aluminum rips the oxygen away from almost anything. That is why soldering on aluminum needs an aggressive flux, and why aluminum oxide is such a stable dielectric in electrolytic capacitors.
04In the lab
React measured amounts of baking soda with vinegar in a container on a balance and record the loss of mass due to the CO₂ released. Repeat, varying the proportion until you identify experimentally the point at which the limiting reactant changes.
Use a polystyrene-cup calorimeter to measure the temperature rise when dissolving sodium hydroxide in water and when neutralizing it with hydrochloric acid. Calculate \( \Delta H \) per mole and compare with the tabulated value.
Experimentally determine the enthalpy of three related reactions —solid NaOH in water, solid NaOH with HCl, and NaOH in solution with HCl— and verify that the first plus the third give the second.
05Common mistakes
- Comparing grams with grams to find the limiting reactant. The comparison is done in moles divided by their coefficient.
- Balancing by changing the subscripts. That changes the substance; only the coefficients are touched.
- Calculating the product from the excess reactant. The limiting reactant decides it, always.
- Confusing theoretical yield with actual yield. One comes from the calculation; the other, from the balance.
- Forgetting the sign of \( \Delta H \). Negative is exothermic: the system gives off energy.
- Reversing products and reactants in Hess's law. It is products minus reactants, and when a reaction is reversed the sign changes.
06Self-assessment
How is the limiting reactant identified?
By dividing the moles of each reactant by its stoichiometric coefficient: the smallest quotient rules.
What does it mean for \( \Delta H \) to be negative?
That the reaction is exothermic: it releases energy to the surroundings.
What does Hess's law say?
That the enthalpy of a reaction does not depend on the path, so steps can be added to obtain an overall reaction.
What is the enthalpy of formation of gaseous O₂?
Zero: it is an element in its standard state.
Why does the silicon deposition reactor need heat?
Because the reaction is endothermic, with \( \Delta H \approx +219 \) kJ per mole of silicon deposited.
If one micron of silicon is oxidized, how much oxide grows?
About 2.27 microns, because SiO₂ occupies more volume than the silicon it comes from.
07Further reading
- Raymond Chang and Kenneth A. Goldsby. Chemistry. 12th ed. (Spanish edition, “Química”), McGraw-Hill, 2017. Chapters 3 and 6: stoichiometry, limiting reactant, yield and thermochemistry.
- Ralph H. Petrucci et al. General Chemistry. 11th ed. (Spanish edition, “Química general”), Pearson, 2017. Many worked problems on limiting reactant and Hess's law, with the detail of the units.
- Stephen A. Campbell. Fabrication Engineering at the Micro- and Nanoscale. 4th ed., Oxford University Press, 2013. Thermal oxidation of silicon, with the Deal–Grove model and the volume ratio of this unit.