# 11.8 Examples of Mass-Energy Conversion

It is appropriate to think of matter as "frozen energy." This is a startling realization because mass seems to be tangible, "stuff" you can hold, while energy seems to be intangible, something that you can't grasp, like radiation. In Einstein’s simple and famous equation:

E = mc^{2}

E stands for energy and m stands for mass. Energy can be created from mass using the conversion factor c. The conversion factor is a universal constant, the speed of light. Since the speed of light is an enormous number, 300,000 kilometers per second, a tiny amount of matter is equivalent to an phenomenal amount of energy.

Mushroom cloud from Castle Bravo, the first US hydrogen bomb explosion on Bikini Atoll. Click here for original source URL.

To calculate the conversion of mass into energy, scientists must use a consistent set of units. When units are in the international metric system (or the SI system, from Système Internationale, because it was developed in France), mass is in kilograms and energy is in Joules. The speed of light is 3 x 10^{8} meters per second. Let's consider the total energy consumption per year of all the people in the United States. It is about 8 x 10^{18} Joules (this is equivalent to each person having ten lights turned on at the same time, since we are a rather wasteful society). The equivalent mass is given by moving the factor c^{2} to the other side of the equation, so that m = E / c^{2}. The result is m = (8 x 10^{18} Joules) / (3 x 10^{8} m/s)^{2} = 88 Joules s^{2} / m^{2}. Since a unit of 1 Joules s^{2}/m^{2} is equal to 1 kilogram, the answer is 88 kilograms.

The mass-energy of one kilogram of material is about 10^{17} Joules. What does this enormous number mean? It means that if we could liberate the entire energy content of the mass corresponding to a large person, we could power the country for a year. In a fusion process we only get about 0.1% of the energy out, so we would need 1000 times the mass to get the required amount of energy. That larger mass is still only about 88 x 1000 kg or about 10 tons, compared to the 2.2 billion metric tons of chemical fuels that we actually burn to keep our homes lit and our factories working every year. Nuclear energy sources are about ten million times more efficient than chemical energy sources.

Apollo 11 Saturn V rocket. Click here for original source URL.

The Saturn V rocket carried astronauts to the Moon. A rocket the size of a ten-story building lifted a payload about the size of a minivan free of the Earth's gravity. Over 90% of the mass of the rocket consisted of the highly volatile fuel needed to accelerate the payload. The energy source was chemical energy. If on the other hand the energy source had been pure mass-energy, only 10 grams of fuel would have been needed to get to the Moon. NASA could have dispensed with a ten-story building full of chemical fuel in favor of a pocketful of nuclear fuel!

The contrast between mass-energy and chemical energy can be made clear with the simple example of a hamburger. A quarter-pound burger has a nutritional value of about 250 Calories, which is 250 x 4186 = 10^{6} Joules. This is the amount of chemical energy your body can extract from the burger. If we apply the equation above with a mass of 0.15 kg, the mass-energy of the meat is about 10^{16} Joules. Thus, we only extract one part in 10^{10} (or a ten-billionth) of the possible energy in a hamburger by chemical means. If your body could extract mass-energy, you could live for your entire life off 1/30 of a gram (1/1000 of an ounce )of food!

Now imagine something very small, such as the period at the end of a sentence in a printed book. The ink in that period weighs roughly 1/100,000 gram or 10 micrograms. What is the equivalent amount of energy using Einstein's handy equation? In the previous example, the annual United States energy consumption was about 8 x 10^{18} Joules, which is roughly 10^{19} Joules. We can take a shortcut by noticing that 1/100,000 gram is about a ten billionth (10^{-10}) of the amount of mass in the first example. So the energy release from the dot of ink is about 10^{19} / 10^{10} = 10^{9} Joules. The unit of energy consumption is called a Watt, after the famous Scottish scientist and engineer. A Watt is a Joule per second, so a light bulb might consume 100 Joules per second. If the typical house consumes energy at a rate of 10 kilowatts, or 10,000 Joules per second, then 10^{9} Joules will last 10^{9} / 10^{4} seconds. If only we could harness it, the mass-energy in the dot at the end of this sentence could run a family home for a day.

Finally, an equation goes in both directions. We have been talking about a tiny amount of mass being equivalent to a huge amount of energy. But energy has an equivalent mass as well. A photon is a "particle" of light and by Einstein's equation it has a tiny mass. This got Einstein thinking. If everything with mass feels the force of gravity, then photons should feel gravity too. If gravity makes low mass objects curve in their trajectory near a massive object, then a massive object should bend the trajectory of a photon. If gravity causes objects to lose energy as they escape from the influence of a massive object, then photons should lose energy leaving a massive object. These insights led Einstein to his greatest achievement: the general theory of relativity.

James Watt. Click here for original source URL.