Throwing Hot Stuff in a Lake

[Blood Legend]

So lets say one of my nut case players threw an atom of gold into a lake, and that this single gold atom was heated to approximately 7.2 trillion degrees. What happens?

Does the atmosphere ignite before it even gets to the water or is one atom that hot not much of a problem?

[mlangsdorf]

I don't even know how you throw a single atom.

Anyway, an atom of gold weighs 3.2x10e-13 kg, so it's not going to have very much thermal mass, since thermal mass is roughly proportional to specific heat times mass - which I don't know, but is usually in the range of 10 to 1x10e6 J/K/kg. The change in temperature multiplied by the thermal mass is the total energy transfer. The difference between ambient temperature (of the air, the water, or whatever) and the atom's starting temperature is roughly 7.2x10e12.

So: the upper bound for total energy transfer in cooling it to ambient temperature is mass * specific heat * delta T, or 3.2x10E-13 * 1x10e6 * 7.2x10e12, or about 2 MJ. That's the equivalent of a couple of lbs of TNT going off in theory, but there's a big difference between 2 MJ happening in a second and 2 MJ happening over a minute.

It's been a long time since Heat and Energy Transfer for my engineering degree, but generally the speed of heat transfer is proportional to the energy difference, so this thing is going to be shedding heat quickly. There's going to be a streak of fire across the sky, and probably steam and an explosion when the atom hits the water. It's going to be very hot, but there's only at most 2 MJ of total energy transfer, so it's not going to be the end of the world.

Be aware the heat transfer is going to start as soon as the atom gets hotter than the local air, and a lot of that energy is going to be dumped into the air very near the thrower's hand.

Also, 2 MJ is the upper bound (if I did the right math right) for energy, and 2 KJ or less is also probably - metals seem to have specific heat on the lower end. 2 KJ is the explosion of 1 gram of TNT, which wouldn't be very impressive at all.

[johndallman]

Originally Posted by mlangsdorf

I don't even know how you throw a single atom.

Electrostatically, I think. At that temperature it's going to be completely ionised.

[whswhs]

Originally Posted by mlangsdorf

Also, 2 MJ is the upper bound (if I did the right math right) for energy, and 2 KJ or less is also probably - metals seem to have specific heat on the lower end. 2 KJ is the explosion of 1 gram of TNT, which wouldn't be very impressive at all.

The specific heat of metals is what it is because the atoms are fixed in place in a rigid matrix, allowing only vibration of the nuclei and excitation of the electron gas. But with a single atom of gold you can't meaningfully speak of "fixed in place." I think you need to treat it as an atom of an ideal gas.

I'm going to assume that "a trillion degrees" means a trillion K; if it's 1,000,000,000,000°F, you would multiply that by 5/9 to get K. Each degree of freedom of the atom carries kT/2 joules of energy, where k is the Boltzmann constant. Since this is a single atom free to move in three dimensions, its total energy is 3kT/2. The Boltzmann constant is 1.38x10^(-23), and 3/2 times that is 2.07x10^(-23). So your energy is 2.07x10^(-11) J.

Water's heat capacity is 4200 J per kg K, so your heating effect is just about 5x10^(-15) kg K. If you look at the smallest cubic micrometer (a volume of water comparable to a single bacterium), the collision will raise its temperature about 5 K or 9°F. Not a lot. Atoms are really, really tiny.

Now, your atom of gold is probably going to be travelling at a significant fraction of the speed of light. You've given it a kinetic energy of about 86 million electron volts. That's a healthy output for an early particle accelerator, higher than you could get from a van de Graaf, but it's way lower than what's used in current high-energy physics. Still, you'd ionize a fair number of water molecules; the bonding energy of water is about 7.5 electron volts, so you'd maybe break up 10 million water molecules. But that's still an incredibly tiny amount of water. I don't think you could get x-rays out of the collision, though, because the bond energy of water just isn't high enough for x-ray emission. A bond energy of 7.5 eV gets you up into the extreme ultraviolet, so you might have a brief flash, though not at a frequency your eyes can detect.

Corrigendum: I missed the "7.2" when I was workiing this out. That gets you up to 36 K or 65°F for that bacterium. But on a macroscopic scale it's still nothing much. Disrupting 72 million water molecules is hardly noticeable. Though with a total kinetic energy of about 620 MeV, you're about as effective as an early synchrotron, I think.

[Blood Legend]

The atom's in a superscience bottle made mostly out of hard light and magnets. He can safely throw it from his hand because shenanigans. For the purposes of math and gameplay, this bottle has no mass and simply stops existing when it needs to. It's also Fahrenheit.

So how much gold would he need to supermelt to get a sizeable yet ridiculous explosion, something that'll shower water on the beach and kill plenty of d̶i̶v̶e̶r̶s̶ fish?

[Anaraxes]

Originally Posted by mlangsdorf

2 MJ. That's the equivalent of a couple of lbs of TNT going off in theory

I'm used to seeing a TNT equivalent defined as 4 MJ / kg. If so, more like half a kilo of TNT.

It's been a long time since Heat and Energy Transfer for my engineering degree, but generally the speed of heat transfer is proportional to the energy difference

For a single atom, isn't the mechanism (conduction, convection, radiation) going to dominate? It's not like one atom of anything can convect heat unless you just count the fact that it moves and keeps its energy, and "conduction" is really "bumping into another atom", so densities have to be such that atoms can actually bump. If the gold atom has ionized away all its electrons, and has no molecular bonds to vibrate or rotate, I'm not sure how it can radiate infrared or otherwise. I don't know how the macro-scale thermodynamics apply to such tiny quantities of mass and singular particles.

[whswhs]

Originally Posted by Anaraxes

I'm used to seeing a TNT equivalent defined as 4 MJ / kg. If so, more like half a kilo of TNT.

For a single atom, isn't the mechanism (conduction, convection, radiation) going to dominate? It's not like one atom of anything can convect heat unless you just count the fact that it moves and keeps its energy, and "conduction" is really "bumping into another atom", so densities have to be such that atoms can actually bump. If the gold atom has ionized away all its electrons, and has no molecular bonds to vibrate or rotate, I'm not sure how it can radiate infrared or otherwise. I don't know how the macro-scale thermodynamics apply to such tiny quantities of mass and singular particles.

A molecule has three degrees of freedom per atom; for example, a diatomic molecule has three forms of translational motion, two of rotational, and one of vibrational. A single atom, whether it has electrons or not, is normally capable only of translation. So "temperature" translates directly into "speed of motion."

On the other hand, if the gold atom is fully ionized, its electric field will be rather strong and will affect electrons in water molecules at a fairly large distance, molecularly speaking. And in liquid water, all those molecules are "in contact," that is, at distances where hydrogen bonding forces are effective.

[whswhs]

Originally Posted by Blood Legend

So how much gold would he need to supermelt to get a sizeable yet ridiculous explosion, something that'll shower water on the beach and kill plenty of d̶i̶v̶e̶r̶s̶ fish?

Ah, working it backward. That's a sounder method.

Let's pretend you want the equivalent of a kilogram of TNT. That has a radius of potentially lethal injury of about 10 meters. The energy is 1000 kcal, or about 4.2 megajoules.

If your gold is at 7.2 trillion degrees F, or 4 trillion K, you can figure it's acting as a gas, so again we can use the formula 3kT/2. Since one atom has about 2 x 10^(-11) J, and you need 4.2 x 10^6 J, you need about 2.1 x 10^17 atoms. Since a mole of gold is 6.023 x 10^23 atoms, that's about 3.5 x 10^(-7) moles. Gold's atomic weight is 197, so you need about 7 x 10^(-5) grams, or about 0.07 milligrams. That's about 0.0035 cubic millimeters, or a speck 1/150 of an inch on a side, if it were solid. If you have really good eyes you might be able to see that much gold.

What you're actually going to be doing is heating the water to 100°C, and then vaporizing it. The energy needed is on the order of 2500 joules per liter, so you'll be vaporizing around 1680 liters of water, or well over a ton. Or you might get a smaller volume of superheated steam.

If you want a much smaller explosion, divide the mass of gold and the volume of water vaporized by 1000, and the radius of lethality by 10.

[Anthony]

What you'll actually get is a radiation cascade in a random direction. However, energy estimates are off, possibly because an atom of gold does not weigh anywhere close to 3.2e-13 kg, it's about 3.3e-25 kg.

1K = 8.6173324(78)×10&−5 eV per particle, at a temperature of 4e+12K, works out to a per particle energy of 3.4e+8 eV/particle, which is a modest energy cosmic ray.

[whswhs]

Originally Posted by Anthony

What you'll actually get is a radiation cascade in a random direction. However, energy estimates are off, possibly because an atom of gold does not weigh anywhere close to 3.2e-13 kg, it's about 3.3e-25 kg.

1K = 8.6173324(78)×10&−5 eV per particle, at a temperature of 4e+12K, works out to a per particle energy of 3.4e+8 eV/particle, which is a modest energy cosmic ray.

I don't think any error in calculation can be a result of error on mass, because mass doesn't enter into the formula for converting temperature to energy. It's just E = 3kT/2.