# Hydrogen Bomb Mass to Energy?

How much mass is converted to energy when a hydrogen bomb explodes? I remember an eighth grade chemistry class where, by going through the nuclear processes, my teacher estimated that roughly 2g of matter was converted in a fission bomb.This is a surprisingly small amount of mass! I have never seen the process involved in a fusion device.

• An interesting variant would be to ask how much of the energy came directly from hydrogen fusion. There's a pretty large fission bomb used to initiate the fusion, and I seem to recall some weird stuff described in "Dark Sun," by Richard Rhodes. – Carl Witthoft Sep 11 '14 at 14:43

$50\text{ Mt}$ TNT means that the energy is equivalent to $50 \, 000 \, 000\text{ t}$ of TNT, and $1\text{ t}$ of TNT is equivalent to $4184\text{ MJ}$.

So Tsar Bomba released $50 \, 000 \, 000 \times 4184 = 209 \, 200 \, 000 \, 000 \text{ MJ} = 2\cdot10^{11}\text{ MJ}$.

Now, given that $E=mc^2$, we have $m=\frac{2\cdot10^{17}}{299 \, 792 \, 458^2}=2.3\text{ kg}$ as said above.

For comparison, Little Boy did not convert more than $1\text{ g}$.

While it does take $2.33\:\mathrm{kg}$ of any mass to be converted to 50MT of energy using the calculations stated before ( $m = \frac{E}{c^2} = \frac{2.092 \cdot 10^{17}}{8.987 \cdot 10^{16}} = 2.33\:\mathrm{kg}$ ), be aware that just a small fraction of the total hydrogen in a thermonuclear device is converted to energy since most of the hydrogen is fused into helium. So do not conclude that the 50MT device contained only $2.33\:\mathrm{kg}$ hydrogen.

It takes 4 hydrogen atoms to fuse into one helium atom. This process is called the proton-proton chain reaction and is a multi-step reaction. If you start with 4 hydrogen atoms, each with a unified atomic mass unit (u) of 1.007825 u, then your starting fuel mass is $4 \cdot 1.007825\:\mathrm u = 4.03130\:\mathrm u$. When fused to helium, the product mass is 4.00260 u. The difference between the mass of the fuel (4 hydrogens) and the mass of the product (1 helium) is $4.0313\:\mathrm u - 4.0026\:\mathrm u = 0.0287\:\mathrm u$.

This difference between the mass of the fuel and the mass of the product is how much mass gets converted to energy in Einstein's equation. Therefore if you start with a fuel mass of 4.0313u of hydrogen (4 hydrogen atoms) and 0.02870 u gets released as energy, then only 0.712% of the hydrogen fuel gets converted to energy, not 100%. So even though a thermonuclear reaction gives off an impressive amount of energy, the proton-proton chain is less than 1% efficient.

Using the efficiency of 0.712%, we calculate that the amount of hydrogen fuel we need to start with is $\frac{2.33\:\mathrm{kg}}{0.00712} = 327 \: \mathrm{kg}$

So it takes 327kg of hydrogen to produce 50MT of energy by using a thermonuclear reaction ( Proton-Proton chain).

Of that 327kg, only 0.712% or 2.33kg is converted to energy.

Note:

Thermonuclear weapons use lithium deuteride as the bulk source of hydrogen. This substance is a solid salt at normal temperatures. When high energy neutrons from the primary bombard the lithium deuteride, a chain reaction starts first by creating tritium (2 protons, 1 neutron) then this tritium fuses with the deuterium (1 proton, 1 neutron) in the salt to form helium.

The most powerful hydrogen bomb ever exploded had a TNT equivalent of 50 Mt TNT, if I remember correctly, TNT energy equivalent is 4184 MJ/kg, that gives a mass loss of about 2.3 kg, if my calculations are correct.

Of course all other replies do not consider the fact that in a hydrogen bomb only 30% of yield comes from fusion, ie. hydrogen. The rest comes from fission as in the known sequence:

fission (plutonium) -> fusion (hydrogen) -> fission (depleted uranium in the case)

So we have to correct all of the calculations by multiplying them by, say 30% and then we'll get the correct numbers. We have to remember that even though less hydrogen is used even more depleted uranium will be used.