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I just watched this video, and around 75 seconds they say

if king Midas touched a duck, and all of its atoms gained the necessary protons/neutrons/electrons to become gold, the duck would be 33x more massive and 20x as dense. The gold atoms are too close, and repel each other resulting in a explosion with the energy of half a ton of TNT

Why does this happen? is it electrons repelling each other? Is there some other force besides electromagnetic? What calculation is involved in coming to the ~2 Gigajoules of energy being released?

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    $\begingroup$ Not sure how they rationalized that since the duck is everywhere less dense than gold, not seeing how the gold atoms would be too close together, infact it seems they'd be too far apart. If anything it sounds more like the energy required to do that would instead lead to the area becoming extremely cold to produce the energy needed to make those extra particles. $\endgroup$ – Triatticus Mar 10 at 19:46
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    $\begingroup$ @Triatticus I think the notion is that if all the duck's atoms suddenly turned into gold, you'd wind up with an object that has a density higher than gold, and when you've packed more gold into a volume than the natural density of gold allows, bad things happen. You'll note they say the duck is 33x as massive, but only 20x as dense - that means the volume must increase by about 1.5x, and it does so very rapidly (if the volume stayed the same, it would be 33x as dense). $\endgroup$ – Nuclear Hoagie Mar 10 at 20:01
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    $\begingroup$ I'm not sure that the original story specifies that the number of atoms was conserved when Midas turned any given object to gold. I haven't read Ovid in the original Latin though, and Ovid probably heard it from somebody else, so who knows for sure? $\endgroup$ – Solomon Slow Mar 11 at 1:04
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The electromagnetic force (of the fundamental forces) does predominate, arising from Coulomb repulsion of the electrons and nucleus and the Pauli exclusion principle. If I wanted to estimate the energy $U$ released, I'd look to the strain energy. This is $$U=\int \sigma V\,d\varepsilon=\int K(\varepsilon)V\varepsilon\,d\varepsilon,$$ where $\sigma$ is the pressure required to obtain the equivalent compression, V is the volume, and $K(\varepsilon)$ is the bulk modulus (i.e., the volumetric stiffness) as a function of the compressive volumetric strain $\varepsilon$. Here, we're simply integrating the infinitesimal compression work done while keeping open the possibility that the bulk modulus varies with the strain, as the latter is very large.

(If the strain were small enough that the bulk modulus and volume could be considered constant, which is the typical treatment in linear elasticity, then we could move $K$ and $V$ out of the integral and obtain simply $KV\varepsilon^2/2$.)

Since bulk compression is generally recoverable, we could make the assumption that immediately after the transformation, the system releases this strain energy immediately in a variety of phenomena that all ultimately dissipate into thermal energy.

The bulk modulus of gold is about 150 GPa. Assuming that a 1 kg duck is mostly water (volume 0.001 m³), we have 60 moles of water or 180 moles (36 kg) of gold, as it's specified that each atom is changed to gold. This mass of gold (with density 19,000 kg/m³) would normally occupy 0.002 m³, so the volumetric strain is 0.5. Considering the bulk modulus to be strain-independent as a first pass, and approximating $V$ as the average of the initial and final volumes, we calculate $U=(150\,\mathrm{GPa})(0.0015\,\mathrm{m}^3)(0.5)^2/2=28\,\mathrm{MJ}$ of energy. Since a ton of TNT is reported to release 4 GJ, this energy corresponds to about 0.007 tons of TNT.

This is much less than the reported value of 0.5 tons of TNT, but note that we can expect the bulk modulus to increase sharply with increasing compression; this is a consequence of the pair potential having a much-higher-then-quadratic dependence on the interatomic spacing as we compress a material mightily. (Otherwise, the pressure needed to compress a material into nothingness would be exactly $K$.) That increase in the bulk modulus results in much more releasable strain energy stored in the material.

Please let me know if you see a calculation error. I'll look for the compression dependence of the bulk modulus of gold or similar metals.

Edit: Found a very useful figure from E. Güler and M. Güler, "Geometry optimization calculations for the elasticity of gold at high pressure," Advances in Materials Science and Engineering (2013):

enter image description here

Note that assuming a constant bulk modulus of 150 GPa would imply that 75 GPa is required to obtain a volumetric strain of -0.5. The initial slope is heading in that direction, but the actual value exceeds 1 TPa for the reasons discussed above. This corresponds to more stored energy and a much better fit with the quoted value. This makes me very curious as to what calculations they performed to obtain their estimate.

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There are ways to add more protons and neutrons to some atoms. For certain isotopes of hydrogen, if you slam two nuclei together hard enough they form a He nucleus. They also liberate a lot of energy. If you do this to a lot of nuclei all at once, this is a Hydrogen bomb. The energy of half a ton of TNT is small potatoes compared to other energy released.

For heavier nuclei than lead, you have to add a lot of energy to get more protons and neutrons to stick. You could not do it without providing about as much energy as a Hydrogen bomb releases.

Not all combinations work. Something like carbon or oxygen (because a duck contains a lot of these) and a target like gold (because that is what you want to make) might or might not be possible to convert.

A magic touch wouldn't do it. Typically you have to slam particles together at temperatures like the core of the Sun or hotter. It might take a chain of these kinds of reactions. As you can imagine, there are practical difficulties in addition to theoretical ones.

And however he did it, what King Midas wound up with wouldn't even slightly resemble a duck.

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