The title of this article refers to a fun example that pertains to the relationship between mass and energy. But before we get to that, some definition is required. For the sake of general readership, I will try to keep things simple.

To start, we need to get into some physics. In particular, we need to look to special relativity and the principle of mass-energy equivalence. This principle tells us that energy has mass. To phrase it slightly differently, anything that has mass can be said to have an equivalent amount of energy. Thus, to the question of whether energy has mass, the answer is ‘yes’. This was a key lesson offered to us by Einstein’s famous equation,

    \[ E=mc^2 \]

In that the fundamental quantities of mass and energy are directly related to one another, for the purposes of the present discussion we can re-write the above formula in a slightly more nuanced way

    \[E_0=m_0c^2\]

. This means a particles rest mass has a rest energy. One will note how the rest energy is multiplied by

    \[c^2] which is the speed of light. This equation helps us explain, among other things,  a very important and certainly nuanced question in particle physics. *** Let's think of the Large Electron-Positron (LEP) collider at CERN. One of Einstein's predictions concerned how mass <em>increases</em> without limit as the velocity of an object, or in this case a particle, approaches the speed of light. The implication is that, if we imagine a particle that has been projected, say, in the horizontal direction, travelling near the speed of light, whatever force we apply to this particle as it is travelling will have very little effect on its velocity. Take a strip of paper and crumple it into a tiny ball. Pretend that this little ball of paper is our particle, moving in a horizontal direction nigh to the speed of light. Now take the tip of a pen, and as you move this particle in a horizontal direction, apply onto the particle - in other words, imagine the tip of the pen as some constant force being applied to the particle. What happens is that, as our particle continues to approach the speed of light, as its velocity gets closer and closer to the speed of light, acceleration becomes more difficult. This is because there is an increase in inertia, and inertia relates to mass. And so, it can be said in simple terms that with this increase in inertia there is also an increase in mass.  The logical implication, if we keep things very handy wavy, is that an increase in mass also relates to an increase in velocity. Hence, \[m=\gamma m_0\]

Or we can say,

    \[m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}\]

In the example of our imagined particle, it is travelling so fast that it has gained mass. Thus, the constant force that we have applied onto it struggles to induce any sort of acceleration  onto it.

Think of it this way, in the  Large Electron-Positron collider particles are moving so fast that they can experience as much as 100 000 times an increase in mass. This extra, or added mass, comes from the energy as a result of work done on the particle to achieve its acceleration.

Hence, if E=mc^2, what is a very cool thing to realise is not so much how energy has been converted to mass. In other words, one might be inclined to describe an energy-mass transfer (or vice versa), but they shouldn’t mistake this as a conversion of energy to mass. As noted above, the more accurate description is that energy has mass (there is no conversion).

For our particle, it has gained so much kinetic energy (KE) that its mass has increased, say for the sake of this example, almost 100 000 times its original value. This is a dramatic increase, to say the least. But it is not uncommon, at least when thought about in the context of today’s massive particle colliders.

This begs the question, why is there a need in our modern particle colliders to project particles at such speeds? The simple abbreviated answer is that, if we were to smash our particle into another, all of this energy has to go somewhere. This is just the law of the conservation of energy speaking. Thus, in the context of a collider, like LEP, much of that energy transfers to the creation of new particles. This is also why there is a tremendous push in particle physics to build even bigger and more powerful colliders that can project particles with even greater energy, with the aim of potentially discovering new and possibly even more fundamental particles.

***

In the future, we can discuss these points in a more rigourous way and also explore the maths.

But for the sake of keeping the present engagement short, let’s end with a wonderful thought. There is a very textbook, yet evocative example, that one will often read when it comes to learning how mass and energy relate with one another. Typically, the example consists of something very practical, like a glass of water. The question that follows is: how much energy is stored in this glass of water? The maths is quite simple.

Let’s assume we have poured ourselves a 500ml glass of water. We know that 1 milliliter of water (ml) equals 0.0010 kilograms (kg). We can therefore determine that our glass of water has 0.50 kg of mass. The question is: how much energy is stored in this glass? We think about this by simply substituting our values into Einstern’s equation:

 

energy and mass

That is an incredible amount of energy. To put it into perspective, 4.5x10^{16} J= 12.5 TWh. To offer further context, in 2014 the demand for electricity in the UK was approximately 301.7TWh. That is a lot of energy in just 500ml of water.

***

There is potentially another way of arriving at this conclusion, if we were to consider constructing a simple theoretical model. It would look something like this:

We could start by noting that there are approximately 167.28×10^{23} H2O molecules in 500 ml of water. We know that each water molecule has two hydrogen atoms. Therefore, we can simply multiply the number of molecules in our glass by two: 167.28×10^{23} \cdot 2 = 3.345x10^{25}.

Now, let’s take as an assumption that in the process of nuclear fusion (it would also require nuclear fission, but I am leaving that out), 4 hydrogen atoms combine to make 1 helium atom.

There is also only ~0.7% conversion of initial mass to energy (that is a lot of potential fusion energy lost!).

So we can start by taking the number of molecules and divide by four, which gives the number of possible reactions.

    \[ R_{poss}=\frac{3.345x10^{25}}{4}=8.364x10^{24}\]

Now we must find, in our nuclear fusion reaction, how much energy would be released. We know that the energy released from one reaction would be approximately 4.3575 x 10^{-12} J per reaction. So, to get a sense of total energy potential, we could multiply R_poss by the amount of energy released in a single reaction,

    \[E = 8.364x10^{24} reactions \cdot 4.3575x10^{-12}Jr^{-1}= 3.644x10^{13} Joules\]

This number is much lower than previously estimated, due to the loss of energy in the fusion process.

Convert from joules to KWh, and we get, rounding our answer, 10100000 kWh of potential fusion energy. Now if we convert from kilo-watt hours to mega-watt hours, we get 10100 MWh. We can then finally convert this to tera-watt hours, which gives us 1.01x10^{-2}.

In 2014, demand for electricity in the UK was approximately 301.7TWh.

So, on my estimating, a glass of water would contribute \frac{1.01x10^{-2}}{301.7} \cdot 100\% = 3.345x10^{-5} of the total need in the UK in 2014.

This was roughly calculated based on a 0.7% mass to energy conversion.  The earlier calculation assumed ~100% conversion, which would only be possible if we were able to cleanly annihilate matter with anti-matter. But if nuclear fusion/ fission were to improve over the coming decades, and this conversion were to increase from, say, 0.7% to 5%, then already we’re looking at a tremendous amount of energy captured from just a 500ml of water. Imagine, then, a 20% conversion of mass to energy or, dare I say, 50%!

 

Sources

Molar mass of water: http://en.wikipedia.org/wiki/Properties_of_water

Energy released from single fusion reaction: https://en.wikipedia.org/wiki/Nuclear_fusion

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