Kerosene burning in oxygen reaches about 3700 K. Hydrogen burning in oxygen, at the mixture ratios engines actually use, reaches only 3000–3600 K. Yet the hydrogen engine's exhaust leaves 30% faster. How can a cooler flame make a faster jet?
Two ingredients: heat and lightness
Lesson 2 gave us the exhaust velocity of an ideal nozzle. Strip it to its essentials and one ratio remains:
The chamber temperature is how much energy each kilogram of gas holds. The molar mass is how heavy the molecules carrying it are. A propellant is good if it releases a lot of heat and leaves light molecules to carry it away.
Why the square root of ? Because temperature is the average kinetic energy of the molecules, . At the same temperature, a hydrogen molecule (mass 2) jiggles three times faster than a water molecule (mass 18). A nozzle's job is to turn that random jiggling into an ordered jet, and the jet can never go faster than the molecules could.
Burning at the right mixture
The Mixture ratio rOxidizer-to-fuel mass flow ratio, O/F. Engines run fuel-rich of stoichiometric because the lighter exhaust gives more Isp. is the rocket engineer's version of the equivalence ratio. For hydrogen and oxygen the stoichiometric ratio, where every hydrogen atom finds an oxygen, is 7.94. Combustion there is hottest. So why does no hydrogen engine run at 7.94?
Follow the purple dashed line. Adding fuel beyond stoichiometric leaves unburned hydrogen in the exhaust. It carries no chemical energy away — it has not burned — but it is extraordinarily light, and it drags the average molar mass down faster than it drags the temperature down. For a while keeps rising. The Isp peak for hydrogen sits near –5, far rich of stoichiometric; for kerosene the effect is smaller but still there, because the unburned fuel turns into light CO and H₂ rather than heavy CO₂.
Real engines don't even run at the Isp peak. The RS-25 runs at : slightly less Isp, but the extra oxygen is dense, so the hydrogen tank — the biggest and lightest-built structure on the vehicle — gets smaller. Engine designers optimize the vehicle, not the engine.
Dissociation: the ceiling on flame temperature
If you simply divide the heat of combustion by the heat capacity of the products, oxygen flames should reach 4500 K and more. Real ones stop around 3500–3900 K. The reason is DissociationAt chamber temperatures H₂O and CO₂ partly break into H, O and OH, storing energy that caps the flame temperature.: at those temperatures, water and carbon dioxide partly break apart into H, O, OH and CO. Breaking bonds absorbs energy, so the harder you try to heat the gas, the more of the extra energy is soaked up by pulling molecules apart.
Two consequences matter for rockets:
- Pressure helps. Dissociation increases the number of molecules; high pressure pushes back against it (Le Chatelier). Higher chamber pressure means a slightly hotter flame — one of many reasons engines have crept from 20 bar to 300 bar.
- Some energy can come back. As the gas expands and cools in the nozzle, the fragments recombine, releasing their stored energy as extra velocity — if the chemistry is fast enough.
Sutton calls the two limits Shifting equilibriumAssuming the exhaust keeps re-reacting (recombining) as it cools in the nozzle, returning dissociation energy as velocity. Frozen flow assumes it cannot. (composition keeps up with the falling temperature) and frozen flow (composition fixed at the throat). The truth lies in between, and the gap — a few percent of Isp — is one of the "real nozzle" losses of Lesson 2.
How the tables are made
The numbers in Sutton's performance tables come from programs like NASA's CEA, which find the chemical equilibrium composition by minimizing the Gibbs free energy of dozens of species at fixed pressure and enthalpy, then expand that mixture isentropically through the nozzle. The model inside this course is a much simpler cousin — product composition from bookkeeping rules, flame temperature from an energy balance, dissociation as a soft ceiling — but it is calibrated to land close to the tables:
| Propellant (O/F) | (K) | (m/s) | Isp (s)* | |
|---|---|---|---|---|
| LOX / LH₂ (4.0) | ~3000 | ~10 | ~2390 | ~390 |
| LOX / CH₄ (3.2) | ~3550 | ~20 | ~1840 | ~310 |
| LOX / RP-1 (2.6) | ~3700 | ~22 | ~1800 | ~300 |
| NTO / MMH (2.2) | ~3400 | ~21.5 | ~1720 | ~289 |
Going deeper
The energy balance behind the flame temperature
The adiabatic flame temperature comes from conservation of enthalpy: the enthalpy of the reactants (including their heats of formation, which for liquid propellants include vaporization) equals the enthalpy of the products at the flame temperature,
The catch is that the product composition itself depends on through dissociation, so the problem is solved iteratively: guess , find the equilibrium composition, check the energy balance, adjust. The same equilibrium solution also gives the effective and , and from them and .
- Exhaust velocity scales as : heat and lightness both matter.
- The mixture ratio for peak Isp is fuel-rich of stoichiometric, because excess fuel makes the exhaust lighter faster than it makes it cooler. Real engines trade a little Isp for denser propellant.
- Dissociation caps flame temperatures near 3500–3900 K; higher chamber pressure pushes the cap up slightly.
- Recombination in the nozzle returns some energy: shifting equilibrium is the upper limit, frozen flow the lower.
- Performance tables come from Gibbs-energy minimization plus isentropic expansion.
Next: how to feed these propellants into a chamber at hundreds of kilograms per second. The anatomy of a liquid rocket engine.