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I tried to work this out. My thinking is as follows. If the same amount of energy is given to a low mass particle and a high mass particle, the low mass particle is faster. A particle with a mass of 1 will be twice as fast as a particle with a mass of 4 but the momentum of the particle with a mass of 1 will be twice that of the particle with mass 2, which seems to go against what is seen with regards to molar mass and specific impulse.

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    $\begingroup$ I think this question might do better on the Space Exploration SE site. $\endgroup$ – Fred Larson Jul 28 '14 at 16:32
  • $\begingroup$ Since specific impulse is per unit of mass. So you should not be comparing a low mass particle with a high mass particle, but a bunch of low mass particles with a smaller bunch of higher mass particles of the same total mass. And the molecular mass falls out of equation and only the total mass matters. Besides products of burning high molecular mass fuel (like kerosene) don't have high molecular mass themselves. $\endgroup$ – Jan Hudec Jul 28 '14 at 19:02
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Your question is slightly confusing. What you seem to ask is why, given that momentum is proportional to the energy, and speed increases only with the square of the energy input, a heavy rocket fuel is so much worse than a light fuel.

For a rocket, the speed of the burnt gases exiting the nozzle is important. Heavy molecules need more energy for the same acceleration, so the lighter the rocket fuel is, the higher are the nozzle speeds. Momentum is less important, because the mass of the fuel is on both sides of the equation: Heavy fuel in the rocket's tanks needs more energy for the same acceleration, and the chemical energy of the fuel is not proportional to its molecular mass.

Thrust is mass throughput times exit speed. For a given mass, thrust can be increased if the exit speed can be maximized. Since this mass must be accelerated, the total fuel mass needs to be as small as possible, not the number of fuel molecules (as your question implies). As you say, using lighter molecules helps to achieve this. The only problem is that lighter fuels need more volume, so a hydrogen rocket is bigger than one using alcohol (V-2) or Kerosene (Vostok).

To get the best specific impulse, the mixture in a hydrogen rocket engine is not stoichiometric, but uses only as much oxygen as required to heat up the gases up to their dissociation temperature. Adding more oxygen would set free more energy, but this would only be used up to ionize the combustion gases, not to heat them up more (Oberth effect).

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