Mass Energy Conversion

Einstein's Mass-Energy Conversion Equation

So let's take a closer look at that equation, shall we?

$$E=mc^2$$

Where:

• E is energy,
• m is mass
• c is the speed of light.

The key point here is that all objects have an intrinsic amount of energy stored within them. The speed of light is a pretty large number (approximately 3x10⁸ m/s), so even a small particle can have a lot of energy stored within it.

To understand what I mean, let's look at an example.

Mass-Energy Conversion example

Let's say we have a cute tuxedo cat that is 3.63 kg (about 8 pounds). How much energy does this cat contain? Well, let’s plug it into our formula:

$$E=mc^2$$

$$E=(3.63\,kg)(3x10^8\frac{m}{s})^2$$

$$E=(3.63\,kg)(9x10^{16}\frac{m^2}{s^2})$$

$$E=3.267x10^17\frac{kg*m^2}{s^2}$$

$$1\,Joule(J)=1\frac{kg*m^2}{s^2}$$

$$E=3.267x10^17\,J$$

For reference, an atomic bomb releases about 1.5·10¹³ joules, so this is about 22,000 times stronger than that.

While you may now be side-eyeing your furry friend, it isn't actually a ticking time bomb. In reality, it's pretty hard to convert that mass into energy, which is why nuclear weapons are used in warfare instead of cats (or equally heavy objects).

Mass-Energy Conversion Reaction

One of the ways mass is converted to energy is through radioactive decay.

We can use the mass-energy conversion equation to calculate the energy emitted due to the loss of mass through particle emission.

For example, let's calculate the energy loss due to this reaction:

Fig.1-Radioactive decay of cesium

Here we see the decay of a cesium (Cs) atom. It converts one of its neutrons (n) into a proton (p+) and an electron (e-), which is ejected. Since the species is gaining a proton, it becomes barium (Ba).

First, we need to calculate the change in mass. We are going from a radioactive cesium-137 sample (mass 136.907 g/mol) to a neutral barium-137 sample (136.906 g/mol). So the change in mass is:

$$\Delta m=m_{product}-m_{reactant}$$

$$\Delta m=(136.906\frac{g}{mol})-(136.907\frac{g}{mol})$$

$$\Delta m=-0.001\frac{g}{mol}$$

If we assume that there is 1 mol of the sample, there is a -0.001 g or -1x10^-6 kg change in mass.

Now we can plug this into our mass-energy conversion formula:

$$\Delta E=\Delta m*c^2$$

$$\Delta E=(-1x10^{-6}\,kg)(3x10^8\frac{m}{s})^2$$

$$\Delta E=-9x10^{10}\,J$$

The amount of energy released here is much, much greater than that of a standard chemical reaction.

Mass-Energy Conversion in the Sun

Have you ever wondered how the sun produces energy? The answer is nuclear fusion.

Fig.2-Solar nuclear fusion

Mass Energy conversion in Atomic Bomb

Atomic bombs work due to a different process called nuclear fission.

The way an atomic bomb works is through a chain fission reaction:

1. A free neutron strikes the nucleus of a radioactive element (ex: uranium).
2. The strike knocks off a few neutrons from the radioactive nucleus.
3. These now free neutrons strike other nuclei, releasing more energy/neutrons.

This chain reaction kicks off almost instantaneously, which is why so much energy is released.

While the bombs themselves are massive, the mass change is a lot smaller. For example, one atomic bomb that weighed about 1.86x10⁷ kilograms only converted 0.9 grams of it into energy.

While that may seem small in theory, let's calculate the energy released.

For reference, that would be like if you set off over 22,000 tons of TNT.