How does the energy mass equivalence not contradict conservation of energy?

Question

As we all know energy and mass are equivalent. When I burn wood a small amount of mass is converted to energy. In theory, I could convert energy to mass. This is not practically possible, but, if I understand it correctly, mathematically they are equivalent ($E=mc^2$). Mass, however, can be expressed in different ways. I can have a kg of iron or a kg of gold and they have the same mass. Mathematically, they can be converted to the same energy and then back to the same mass. So, mathematically, a kg of gold can be converted to a kg of iron. But iron and gold are not the same, so I should need energy to make the transformation. So, where is this extra energy coming from?

Addendum: Let me explain more, there are other questions (e.g. Why can't I do this to get infinite energy?) in this forum that go like this: if I can have a space station and I shoot energy to the space station, and in the space station they convert the energy back to mass and drop it on Earth, and somehow we could use that kinetic energy, then when the mass is on earth, we convert it to energy again and keep doing it, therefore we have unlimited energy (the kinetic energy). The problem, in that reasoning, is that when we send the mass up the space station the wave is also affected by gravity (decrease frequency).

What is the issue in my description?

Answer 1

This is similar to the problem of going between two villages with a hill between:

      /\
__A__/  \__B__

A and B both have the same gravitational potential energy, but in order to get from A to B you have to expend energy.

Q1: Where does the energy come from to get over the hill?

A1: You have to supply it. You can't just spontaneously teleport from A to B. Similarly, the rest energy bound up in 1kg of iron won't spontaneously turn into 1kg of gold; you have to give it a "push" to nudge it out of its current stable configuration

Q2: Where does the energy go after you do get over the hill?

A2: In principle you could re-capture it on the way down the hill, by regenerative braking or some similar mechanism. In practice though there will be energy loss due to friction and other inefficiencies. Similarly, in principle one could recover the initial energy you had to use to "push" the 1kg of iron into a 1kg of gold configuration, but in practice there would be losses. In our current state of the art there would in fact be huge losses!

Answer 2

One has to be clear in physics on the frame of reference.

There is Newtonian mechanics, where mass is conserved as well as momentum and energy.

There is special relativity mechanics where mass is not conserved whereas energy and momentum are conserved. Mass is the length of four vector $(E,p_x,p_y,p_z)$, describing an object where $E$ is the energy and $p$ the three dimensional momentum vector. $$ \sqrt{P \cdot P} = \sqrt{E^2 - {\left( p c \right)}^2 } = m_0 c^2 $$

$m_0$ is the invariant or rest mass of the object.

As we all know energy and mass are equivalent

So the above statement is correct only when the particle is at rest, the invariant mass is the same as the relativistic mass of the confusing $E=mc^2 .$

What I am saying is that if you convert a kg of gold into energy, then you convert it back into mass, and that mass is a kg of iron, you have converted a kg of iron into gold. That should require energy, where did it come from

Your question answers itself, you cannot convert one object into another object except with interactions, and interactions require energy, so you cannot convert "a kg of iron into gold" (which alchemists tried for years). You could not do it with Newtonian mechanics because your mass-is-energy half-understood assumption needs special relativity. An individual iron nucleus scattered off an appropriate nucleus might produce an atom of gold, it would need energy to have the interaction, and a lot of extra radioactive debris would arise.

For the addendum:

and in the space station they convert the energy back to mass

The energy can be in the chemical energy of a bomb, or maybe petroleum oil. If you shoot pure energy (gamma rays) it will evaporate the station, and in any case photons cannot be "caught" as they always move with velocity $c .$

As explained above the laws of special relativity are such that an enormous amount of energy has to be spent to get a small amount of mass, as in accelerators. The problem is not in balancing gravitational energy, which is in the Newtonian frame, but that mass to energy and vice verso conversion obeys special relativity constraints.

Answer 3

You are probably thinking of nuclear reactions, where nuclei decay to other nuclei, all the while conserving other quantities you can learn about, like baryon number, or charge, and so on...

Energy/mass is also conserved there, and small small mass accounting discrepancies between parent and daughter particles are matched by kinetic energy, or photons, etc..., released.

The only way to convert a kg of gold to energy completely is to annihilate it with a kg of antigold, and then you'd have 2kg of energy and no baryon number, in principle available to completely go to photons, or other matter-antimatter matched products and kinetic energy. A small fraction of such notional changes might include a kg of iron and a kg of anti-iron, but with such freakishly infinitesimal probability, that it may well not be constructive to think about.

All such mutations happen spontaneously: no extra energy is required, or used, for the species changes, of course. Energy/mass is strictly conserved.

The takeaway is that not any matter may be completely converted to energy, nor can it arise out of energy: The number of nucleons/baryons, that is, protons or neutrons, has to remain constant, and this is what fails in your fantasy experiment. (Something similar holds for the electrons-leptons which may convert to neutrinos and back, but let us not complicate matters.)

Not even one atom of iron or gold can completely convert to energy.

Answer 4

I would like to add something the other answers do not address.

You are saying "In theory, I could convert energy to mass. This is not practically possible", but in reality there is something called pair production.

Pair production is the creation of a subatomic particle and its antiparticle from a neutral boson. Examples include creating an electron and a positron, a muon and an antimuon, or a proton and an antiproton. Pair production often refers specifically to a photon creating an electron–positron pair near a nucleus. For pair production to occur, the incoming energy of the photon must be above a threshold of at least the total rest mass energy of the two particles, and the situation must conserve both energy and momentum.[1]

https://en.wikipedia.org/wiki/Pair_production

So it is practically possible, and it does happen since we know the universe is fundamentally quantum mechanical. This is an example of converting energy (photons) into particles with rest mass.

Answer 5

You assert that the conversion from one form of a 1 kg mass to another form of a 1 kg mass will require energy input. Let's assume that is true. That does not violate conservation anymore than driving your car out of your driveway, stopping at the stop sign, accelerating forward down the road, accelerating backward to stop, changing speeds, and finally ending in a stop. The net mechanical work done on the car during that trip is zero!

But work was done on the car at various points, some of it negative, some of it positive. And what is work? It is the transfer of energy from one system to another. Braking, air resistance, tire flexing, internal combustion.

The energy to make each step happens is the work done in various steps along the way, and that work is done by agents we call forces. Forces do work. Forces cause (unless you hold to Hmhe's philosophy of cause and effect) energy to move from one system to another. And that is conservation: $$E_{\mathrm{after}}=E_{\mathrm{before}}+W.$$

We don't create or destroy energy, we merely transfer it from one form and system to another.

But you can't ignore the other constraints. Energy (while conserved, always!) is not the only concern.

Answer 6

But iron and gold are not the same, so I should need energy to make the transformation. So, where is this extra energy coming from?

Hot gold contains more energy than cold gold. Same is true of iron.

If a customer comes into a shop and asks for 1 kg of hot gold, the shopkeeper gives him a 1kg chunk of hot gold.

The customer would have gotten more gold if he had asked for 1kg of frozen gold. Then the shopkeeper would have given him a 1kg chunk of frozen gold, which is more gold than 1 kg of hot gold.

The customer can transform his 1kg chunk of hot gold into a bigger 1kg chunk of cold gold if he so wishes. Here bigger means more gold atoms.

Similarly the customer may buy a 1 kg bag of flour, and later transform that into a much smaller 1kg chunk of gold. Here smaller means smaller number of atoms.

Answer 7

The energy you require to convert a kg of gold into a kg or iron comes from whatever power plant you are using to power your nuclear transmutation device. For example, maybe your device is some sort of nuclear smasher that smashes nuclei using particle beams, then catches all the debris and somehow fuses them together again. In this case you are probably going to require a lot of electrical power to run the vacuum pumps, accelerator electronics, microwave plasma heater, etc.

The point about energy here is that if you run this device without any energy loss to the surroundings, and without heating up your equipment overall, then the energy you get back from the final fusion step will exactly balance all the energy you initially acquired from the power plant. If you feed this energy back into the grid then your total electricity bill at the end of the month will be zero.

A final comment: as others have pointed out, to fund your operation you might like to consider converting iron into gold.