# Nuclear thermal rocket specific impulse calculation uses 1 amu, is that wrong?

Following the parameters in the Wikipedia nuclear thermal rocket article, it seems to paint somewhat of a shaky world view. Consider these quotes:

Current (2010) 25,000 pound-thrust reference designs (NERVA-Derivative Rockets, or NDRs) are based on the Pewee, and have specific impulses of 925 seconds.[citation needed]

and

A solid-core nuclear thermal rocket's fuel elements are unlikely to spread over a wide area because the elements are designed to withstand very high temperatures (up to 3500K) and high pressures (up to 200 atm)

I seriously doubt any of these actual parameters are realistic, but that's not important, I just want to talk about them in an academic sense. I think I understand the general idea of how a temperature can be turned into a specific impulse:

$$\frac{3}{2} k T = \frac{1}{2} m v^2$$

$$I_{sp} = \frac{ \sqrt{ \frac{3 k T}{m} } }{g}$$

If I use the above temperature, I can reproduce their specific impulse. Like so, and Google gives 950 seconds. I'm sure there are some other factors that could easily reduce that by 25. But in order to get that, I had to plug in $m= 1 \text{ amu}$.

That clearly can't be right! A nuclear thermal rocket heats cryogenic Hydrogen to produce Hydrogen gas, a diatomic gas with the formula $H_2$, not $H_1$. The molecular weight of the diatomic gas is obviously $2 \text{ amu}$, and there is no way to get that specific impulse (or anywhere close) using that mass.

So what's going on here? Did NASA engineers of the 60s demonstrate that heating of the Hydrogen gas would disassociate the molecule, or did some high school kid blindly plug numbers into the equation without thinking?

Yes

they considered the disassosciation of hydrogen

According to this source :

Previous testing used a maximum temperature of 2,750° K, short of the 3000+° K design temperature for the NCPS. The NTREES facility is designed to test fuel elements and materials in hot flowing hydrogen, reaching pressures up to 1,000 pounds per square inch and temperatures of nearly 5,000° F (2,760° C) – conditions that simulate space-based nuclear propulsion systems to provide baseline data critical to the research team.

Under that conditions the hydrogen would disassociate into atomic hydrogen

external source

at 5,000°K about 95% of the molecules in a sample of hydrogen are dissociated into atoms

• Your first source gives a temperature of almost half your second source. Do you think this is because of the high pressure lowering the dissociation T? Sep 18, 2013 at 15:03
• @AlanSE yes the second source is conducted experiment in lab. The high temperature decrease the disassociation temperature of hydrogen
– Hash
Sep 19, 2013 at 6:49

Some of the hydrogen will be disassociated. For the reaction mass that is not dissociated, and passes through the engine in the form of diatomic hydrogen, in addition to the three translational degrees of freedom, heat energy is also put into the vibrational and rotational energy of the hydrogen molecule. So the energy stored is 6/2kT, not 3/2 kT as in the first formula you quote. (The vibrational and rotational energy is converted into exhaust kinetic energy as the hydrogen cools as it expands through the nozzle. Some of it, however, isn't converted into kinetic energy because there isn't time. This is known as "frozen flow loss". Frozen flow losses depend on details of the engine.)