How much iron would result of fusing a human body?

Question

While discussing nonsense plots, a friend and I ran into the question of what would happen to a human enclosed perfectly with an enormous amount of energy, Antimatter or fusion level energy. It was a hypothetical for a story, and I started trying to figure out in my head what would happen.

Initially, I thought the body would be burned down to ashes(Carbon), and a cremation leaves roughly 5lbs of ash. I looked a bit into fusion reactions and saw some sample questions showing the variance in mass of results and products being around ~0.2%. Am I correct in guessing that if you fused the body you'd get around 4-5lbs of iron due to conservation of mass if no particles could escape?

Answer 1

Short answer - almost all of it.

Your body is a mixture of chemical elements. By number, most of the nuclei are hydrogen, by mass its mostly oxygen. In cremation, most of this oxygen, along with most of the other combustible things like carbon, hydrogen ends up going up the chimney. Hence the low residual mass.

Energy (including rest mass energy) is conserved in a fusion process. If you only allow neutrinos and radiation to escape during the fusion process then only a very small proportion of the mass is lost from the system.

Let's characterise the fusion process as turning 56 baryons in the form of 56 protons (hydrogen nuclei) into a single iron nucleus. The mass difference is $(56\times 1.00784 - 55.93494)m_u$, where $m_u$ is an atomic mass unit. As a percentage of the total mass, this is only 0.9 per cent, and is an upper limit, because the mass difference would be lower for carbon, oxygen etc.

Thus, at most, only 0.9 per cent of the mass is lost, so a 100 kg human would be turned into somewhere between 99.1 and 100 kg of iron. There would need to be a small corrections for the fraction of your body that was formed by elements heavier than iron to begin with, and which would not fuse to become iron, but this is less than 0.1 per cent.

Answer 2

69.8 kg

That's assuming that your original human weighed 70 kg. If that doesn't sound like much got burnt up, consider that the energy released in this reaction is equivalent to that in 3 millions tonnes of TNT exploding: $14.4 \times 10^{15}J$.

The maths

In order to work this out, let's see what humans are made of. According to Wikipedia, 61% of a human is Oxygen, 23% is Carbon, 10% is Hydrogen and 6% is something else by mass. Let's ignore the "other" and, assuming that our human weighs 70kg, use these masses:

• Oxygen: 46kg
• Carbon: 17kg
• Hydrogen: 7kg

The reason that our unlucky human would end up as iron is that iron has the highest nuclear binding energy (per nucleon). You can think of this as the strength with which a nucleus is held together. This means that nuclei made of anything other that iron can release energy by becoming iron instead.

So let's work out how much energy our three elements would release if they fused into iron. To do this, we need the nuclear binding energy of Hydrogen, Oxygen and Carbon, and Iron. Hydrogen is easy: it's zero. That's because a hydrogen nucleus is already a single proton by itself: it's not bound to any nucleons. For the others, we can look at the difference between their mass and the mass of their components. For example, an iron-56 nucleus weighs 55.9 atomic masses and contains 30 neutrons + 26 protons. Wolfram Alpha informs us that iron therefore requires 479 MeV of energy to break it apart, 8.6 MeV per nucleon.

So it's:

• Hydrogen : 0 MeV / nucleon
• Oxygen: 7.5 MeV / nucleon
• Carbon: 6.6 MeV / nucleon
• Iron: 8.6 MeV / nucleon

The change in energy (and therefore the energy released) for each nucleon that started in a H/C/O atom and ended in an Fe atom is:

• Hydrogen : -8.6 MeV / nucleon
• Oxygen: -1.2 MeV / nucleon
• Carbon: -2.0 MeV / nucleon

All we need to do now is multiply by the number of nucleons for each. So, for carbon it's 17kg / 12.011 atomic mass units * 12 nucleon per atom* 2 Mev per nucleon = $3.3 \times 10^{15}J$.

Add these up and we get $3.3 + 5.3 + 5.8 \times 10^{15}J = 14.4 \times 10^{15}J$. That's the amount of energy released by fusing your person (enough to satisfy the whole of humanity's electricity needs for 2 hours). The amount of iron left over must be the mass of the person (70kg) - the energy given off. $E = mc^2$ so there must be 69.8 kg of iron left over.

The reason why your 5 lbs of ash is wrong is that fusion works by an entirely different method to combustion. If you burnt a human body then you would indeed expect around 5 lbs of ash (which, by the way, is mostly not carbon since the majority of the carbon ends up as $\mathrm{CO}_2$). However the rest of the body's mass would be in the gasses released which would then undergo fusion along with everything else.