# Energy cost of creating body parts from nothing but energy

### What amount of energy would you need to expend to create an arm out of nothing but energy?

Of course we have $$E = m c^2$$ telling us the energy any amount of mass represents. Given the average of a total human male arm of $$m_{\text{arm}} = 5.7\text{ kg}$$ that would equal $$E_{\text{Matter}} = 5.123 × 10^{17} \text{ J}$$ of energy just for the matter.

The question is whether accounting for the matter is sufficient
There is still the chemical energy depending on the molecular structure. A block of graphite and a block of diamond are both just carbon, but they have different chemical energies.

I think the chemical energy is ultimately irrelevant considering the sheer magnitude of $$E_{\text{Matter}} = 512.3 \text{ PJ}$$ which would equal harvesting the energy of $$7115.3\text{ kg}$$ of Uranium-235 in a nuclear reactor.

An upper bounds calculation regarding the chemical energy of carbon as an example
Comparing the enthalpy of pure carbon (gas) and graphite (in German) with we get an energy difference $$\Delta{E}$$: $$H_{\text{carbon}} = 718.9 \frac{\text{kJ}}{\text{mol}}$$ $$H_{\text{graphite}} = 0 \frac{\text{kJ}}{\text{mol}}$$ $$n_{\text{C;arm}} = {m_{\text{arm}}}/{M_{C}} = 5.7\text{ kg} / 12 \frac{\text{g}}{\text{mol}} = 475 \text{ mol}$$
$$\Delta{E} = (H_{\text{carbon}} - H_{\text{graphite}} ) * n_{\text{C;arm}} = 341.4775 \text{ MJ}$$

If my calculation is even remotely correct then we are about 9 magnitudes below the energy required for just the matter. So it is easily a negligible amount of energy compared to the creation of matter.

Is there an important effect or principle that I did not account for? In my opinion what ever is to happen chemically is entirely negligible regarding the amount of energy required.

To clarify:
It is to be assumed that every atom is created in the spot it is needed and the liquid and solid matter does not need to be sorted from a blob of unsorted matter.

I am not as well versed in chemistry as in physics, but to my knowledge the numbers should be massively different (as they are in my calculation) as you do not need an extra nuclear reactor to fuel basic chemical reactions to form biochemical molecules and structures.

• I'm guessing water would be a good basis for calculation, since the body is 90% water. – Neil Jul 19 '18 at 12:49
• This got me thinking about how energy-dense gems are in Steven Universe. – The Square-Cube Law Jul 19 '18 at 12:49
• The only realistically relevant difference from E=mc^2 would be the amount of energy the process that recreates the limb requires to operate. E=mc^2 is how much energy is put into the product, but how much energy is needed by the machine/biology to coordinate the whole process? Probably a lot. – Tyler S. Loeper Jul 19 '18 at 14:08
• The chemical energy difference will be so small compared to the energy in mass it will not even show up on your measurement. It is like calculating the displacement of a cargo ship and worrying about how much buoyancy the paint adds. – John Jul 19 '18 at 14:22
• @TylerS.Loeper that is the efficiency of the magic, not the chemical energy in the arm. – John Jul 31 '18 at 21:56

This may be a very basic answer but I found this link with some calorific values :

Bone and meat calorific values

This is an approximation but for this mixture of meat and bone (the composition percentage is not specified) they measured a gross calorific value of 19.69 MJ.kg-1.

This leads to approximatively 112 MJ for your 5.7 kg arm.

From what I recall, the calorific value is the energy gained by burning the material which is pretty similar to separate the molecules into single atoms or at least into some very smaller molecules. If so, this is the opposite of the energy needed to creates molecules from single atoms.

But this measures were made on a powder so what is the energy needed to transform a raw arm into some sort of powder ? From this "funny" article, we learn that breaking a bone need between 375 J to 1 KJ. Even if we are taking 1 KJ to be sure, crushing this arm 100000 times will only consume 100 MJ.

It leads to a total of approximatively 200 MJ and this value is far far smaller than 5.123×10^17 J.

So if your question is whether accounting for the matter is sufficient, the answer would be yes as the rest is totally negligeable : Agregation energy to form an arm from atoms is light years away from the energy required to "create matter".

• The calorific value is not the same as the amount of energy contained within. Even when you burn a piece of paper, ash and gas is leftover. Both are still atoms, and both have plenty of energy left. The formula you should use is e=mc^2. – Neil Jul 19 '18 at 14:11
• @Neil Nah, Freedomjail is on the right track. He's got E=mc^2, but is trying to figure out the bond energy in the molecular structure. So burning reconfigures the molecular structure to a lower energy state, but doesn't completely atomize the matter and break every chemical bond. However that's a really smart way to come up with a lower bound/rough order of magnitude estimate for the energy in the molecular bonds. It's close enough that you can compare it against the E=mc^2 for creating the matter itself and see that the bond energy is negligible by comparison. – MParm Jul 19 '18 at 14:32