Ray Patrick

Starships, Nukes, and SI

by Ray Patrick (other posts)

A minor spat erupted on an Ars Technica article about the upcoming Starship launch on Saturday. User WickWick was downvoted to oblivion for his comment suggesting the use of pure SI in calculation. (A totally sensible point!)

The discussion was about another user comparing the energy released by Starship to that released by the Hiroshima bomb. When attempting to compare them quantitatively, user malachykidd used a promiscuous mixture of SI, “millions and billions,” and “kilotons of TNT.”

I think this comparison is a neat physics problem which brings together a lot of concepts from mechanics and chemistry. I also think WickWick was correct, but you knew that already. Because I think it’s interesting, I will recreate the calculation here (in sensible units), adding detail where I think necessary. I will also limit all calculated figures to three significant digits.

Question: How does the energy released by launching SpaceX’s Starship (whose Super Heavy first stage consumes 800 tons of methane) compare to the energy released by the Hiroshima bomb (15 kilotons of TNT)?

The mythical “ton of TNT” is actually nothing more than the nice, round number of 1 billion (106) calories. One calorie (the quantity of energy required to heat a gram of water by 1 degree Celsius) is 4.184 J. Therefore, a “ton of TNT” is 4184 MJ.

“Little Boy,” the nuclear bomb used on Hiroshima, had a 15 kt yield. In other words, it released 62.8 TJ of energy.

Super Heavy’s methane tank holds 800 Mg of methane.

Methane is CH4, so its molar weight is the sum of the molar weights of one carbon atom (12.011 g/mol) and four hydrogen atoms (1.0078 g/mol each). This turns out to be about 16 g/mol. Therefore, the tank contains (800 × 106 g)/(16 g/mol) = 49.9 Mmol of methane.

The combustion of a stoichiometric mixture of methane and oxygen proceeds according to this formula:

CH4 + 2O2 → CO2 + 2H2O

This is a balanced equation. Therefore, by Hess’s law, the enthalpy change for the formation of the reactants in the process of methane-oxygen combustion is the same as the enthalpy of formation of methane.

In other words, methane-oxygen combustion has enthalpy equal to the (negative) enthalpy of formation of methane, plus twice that of oxygen, plus that of carbon dioxide, plus twice that of water. (The standard state of oxygen is the oxygen molecule in its gas state; therefore the enthalpy of formation of gaseous oxygen is zero by definition.)

Here are the enthalpies of formation for the substances under consideration. (By convention, the enthalpy of exothermic reactions has a negative sign.)

Substance Enthalpy of formation (kJ/mol)
Methane (CH4) -74.6
Carbon dioxide (CO2) -394
Water (H2O) -286
Oxygen (O2) 0

Using the above values, the energy released per mole of methane in methane-oxygen combustion must be:

74.6 kJ/mol + 0 kJ/mol + (-394 kJ/mol) + 2(-286 kJ/mol) = -891.4 kJ/mol.

So, when combusted with oxygen, methane releases 891 kJ/mol (891 GJ/Mmol). Therefore, combusting all the methane in the SH tank would produce (891 GJ/Mmol * 49.9 Mmol) = 44.5 TJ of energy.

Recall that Little Boy released 62.8 TJ of energy. Therefore, a Starship launch releases 44.5/62.8 TJ/TJ or 70.9% of the energy released over Hiroshima.

Wrong Question

Of course, the original point of comparing Starship to Little Boy was to use fear-based imagery and cause the hearer to imagine Starship gouging out a huge canyon where once had been protected wetlands. Using energy to make this point is disingenuous and shows the author’s complete unfamiliarity with even basic mechanics, as power (energy released per unit time) is the relevant metric here.

Little Boy released all 62.8 TJ of its explosive energy in about 1 microsecond, so that its power output was, at bare minimum, 6280 EW. Starship, on the other hand, consumes all but 20 tons of its methane (in other words, releases 97.5% of its total energy) over approximately six minutes. This is (43.4 TJ)/(360 s) = 121 GW.

Little Boy’s power output was 6.28 × 1021 W, while Starship’s will only be about 1.21 × 1011 W. This means that Little Boy emitted about 5 × 1010 (50 billion) times more power than Starship will. Ultimately, comparing a Starship launch to the Hiroshima bombing is not valid.

Topics: metric technology