# How much pressure would be needed to contain a 1 gigaton nuclear bomb explosion within a sphere of one meter radius?

How much pressure would be needed to contain the largest human exploitable nuclear bomb within a sphere of one radius?

Also would it be possible to create a magnetic field that controlled some substrate material like a thick wall of sand so it ended up like an underground explosion?

• Given that 1kg of uranium 235 completely fissioned is equivalent to 17,500kg of TNT explosive, to produce a '1 megaton' explosion (equivalent to 1 billion tons TNT) you will need approximately 57,143kg of uranium, which at 18,000kg/m^3 would need at least 3m^3 so would take up 3/4 of the volume inside a sphere of 1m radius.
– theo
Dec 18 '14 at 7:01
• Send this to what-if.xkcd.com immediately! Dec 18 '14 at 13:27
• I doubt you could assemble 57+ tonnes of U-235 in one place without a critical assembly forming before you got finished. Jun 11 '16 at 8:41
• @theo I expect bombs of this size would have a substantial fusion component. Jan 1 '17 at 11:34

## 1 Answer

You might get an order of magnitude estimate as follows.

We make the rough assumption that everything ends up in its vessel as a monoatomic ideal gas - actually it will be a plasma, with a thermal energy per mole of $\frac{3}{2}\,R\,T_{final}$, where $T_{final}$ is the thermodynamic temperature of the plasma.

Neglecting heats of vaporisation (we assume that only a small amount of the energy released is taken up vapourising everything), we thus equate the energy released $\Delta\,H = 4\times 10^{18}{\rm J}$ to the total thermal energy of the plasma, so that, from the ideal gas equation:

$$\Delta\,H=\frac{3}{2}\,\mu\,R\,T_{final} =\frac{3}{2} P\,V$$

and we know $V = 4\,\pi\,r^2$ with $r=1{\rm m}$, whence I get $P\sim2\times 10^{17}{\rm Pa}$, or about $10^{12}$ atmospheres!

The pressure at the centre of the Earth is six orders of magnitude smaller than this, so if we were to contain the explosion in a vessel at the centre of the Earth, the volume would fleetingly rise to of the order of six orders of magnitude bigger than our sphere, i.e. it would fleetingly open a hole up of about $1{\rm km}$ radius at the centre of the Earth. It would swiftly shrink again as the heat leaked off into the Earth. I don't know much about seismology, but I'm guessing that such a disturbance at the Earth's centre might even be quite a noticeable at the surface everywhere on Earth. If you work out the amplitude of the spherical wave at the ground arising from a disturbance with $1{\rm km}$ amplitude at $1{\rm km}$ from the centre of the Earth, its pretty minuscule, but sensitive instruments might detect it.

• I wasn't even considering seismologically what could happen, I was just pipe dreaming about some sort of nuclear containment defense device when I asked this. But I always wondered about those tsunamis. Dec 18 '14 at 19:38
• @vajra78 I was just trying to relate my calculated pressure to something real and known. There is an intriguing theory (not mainstream, I'm not sure how much weight it has nowadays) that there is a nuclear reactor at the Earth's centre, made of uranium and akin to the Proterozoic Natural Reactor in the Gabon. This kind of calculation shows nicely how it might work. If the fissile material were crushed too hard, a nuclear explosion begins, but the Earth is so big that as soon as the energy of the explosion expanded the reactants .. Dec 19 '14 at 0:00
• @vajra78 .. to any significant degree, the reaction would be quenched. The pressure would end up holding the reactants at exactly the right density to be near critical. Dec 19 '14 at 0:01
• Unless of course the pressure vessel turns out to be an excellent neutron reflector. Nov 24 '15 at 23:15