Mass-Energy Equivalence

So let's talk about mass energy equivalence. This is a consequence of Einstein's theory of special relativity. Now I'm not going to go into all the background as to why it's true but we'll talk a little bit about what it means.

So it means that mass is concentrated energy so whenever you have mass, it means you've got lots and lots of energy just kind of sitting there. How much energy? It's given by Einstein's famous relation e=mc squared. m is the mass which we'll put in kilograms, and c is the speed of light in a vacuum which is approximately 3 times 10 to the 8 meters per second. 300 million in SI units. And I'm going to square it? No wonder we're saying that mass is concentrated energy. This number is huge. So for example, a 20 gram marble, just little marble, 20 grams, contains the same amount of energy as is released in the explosion of a 500,000 ton hydrogen bomb. A little marble like that. So why aren't we afraid of marbles?

Well, this energy is really really really difficult to release, very difficult to release. So if you've got a marble sitting there, there's almost no way that you could release all that energy. The only way that there is to release all of that mass energy is through something called matter-antimatter annihilation. So we've got matter coming in, antimatter coming in, all the energy is released form both of them. Matter and antimatter both have mass but there's not a lot of antimatter around, very little. So we do have antielectrons, so let's just look at this process. I've got a positron, which is an antielectron and an electron, they come together and annihilate and just produce energy. This gamma means photons which basically just means energy for our purposes. So how much energy is released? Well, the mass of an electron or a positron is 9.11 times 10 to the -31 kilograms. So the energy that's released 2 mc squared because I got it from the electron and from the positron. And that turns out to be 1.64 times 10 to the -13 joules. It's not very much, but that was two electrons. Well, an electron and a positron. Think about it. If you've got a mol of these suckers, that's a lot of energy. Turns out to be about 10 to the 10 joules for one mol of electron positron pairs.

Alright. So what does this actually mean? We're not going to do this matter antimatter conversion, because there's not enough antimatter around. It turns out that almost every single release of energy actually can be understood in terms of this energy mass equivalence. So let's look at chemical energy. Alright. So, water is a molecule. It's formed by taking two hydrogen atoms and an oxygen atom and making two bonds. Bond between oxygen and hydrogen, bond between oxygen and the other hydrogen. These bonds cost energy, how much energy? 918 kilojoules per mol. So that means that there's about 1.5 times 10 to the -18 joules of bond energy per molecule. Now what we can do is we can use this to indicate that the mass of a water molecule is slightly less than the mass of hydrogen times 2, plus the mass of oxygen. The difference is associated with the mass of the bond energy. So how do we get that? Well, we're going to say, okay. The change in mass is the energy, is given by the change in mass times c squared because e=mc squared. But we're not going to use the whole mass because we're not destroying the hydrogens and the oxygen, we're just destroying the bonds. So you can think about it like the mass of the bond itself, okay? And it's just associated with that energy.

Now, we don't talk bout that in Chemistry because this change in mass divided by the mass is like 10 to the -10. That means in order to even see the difference in mass between water and two hydrogens and an oxygen. We would need to write out 10 digits in the mass and the only difference would be out there in the tenth digit. Nobody cares about that. So that's why we don't talk about it that way, we just talk about the energies. Alright. So this is the way that we usually see Einstein's relation, energy is equal to a change in mass times c squared. What's the difference in mass between what did we start with, what did we end with, multiply that by c squared and then you got the energy released. Alright.

In nuclear reactions it's a bit of a larger situation. We've got for example the alpha decay of uranium 238. So uranium 238 goes to thorium 234 and helium 4. Now if you add up the mass if this helium nucleus and this thorium nucleus, you get a smaller mass than the mass of this uranium nucleus. Smaller by how much? Well again, it's not by a lot. But the change in mass over the mass, so this is the relative mass, turns out to be about 2 times 10 to the -5. Well, it's still small but it's 5 orders of magnitude bigger than the chemical energy released. So that's one way that you can understand nuclear reactions releasing so much more energy than chemical reactions because the fraction of the mass that's released in terms of energy is 100,000 times greater. Alright. So, what about other processes that we could use to release all of this untouched energy? Well, neutron stars and black holes are probably our best bet for releasing the largest amount of this mass energy other than matter antimatter annihilation. It turns out that with a neutron star you can get relative releases of energy of order 7 percent. So that's 0.07 versus 0.00002, alright? So that's a huge amount of energy and with some types of rotating black holes, you can get it up to almost half, 42 percent.

So that's Einstein's energy mass relationship and it's really really really simple. All you need to do is multiply by the speed of light squared. That's it.