Question from my dad

So, my dad asked the following question the other night.

"If we have 1 kilogram of water, what is the maximum amount of energy we could collect from it, assuming Einstein is right and the equation E=mc^2 is correct?"

First, we have to assume some things that are not exactly correct.

1) We could actually "collect" that 'E' above in some fashion.
2) We can turn 100% of the energy into useful work.

So, E=mc^2:
in words:
Kinetic energy is given by the mass of an object times its velocity (we saw this when talking about the small asteroid impacting in Sudan a topic back). Let's assume that we can actually recover 100% of the energy of 1 kilogram of water moving at the speed of light. Ignore the fact that we cannot accelerate a massive particle to the speed of light. The reality of the equation above is that the kinetic energy of 1 kilogram of water (anything, in truth) moving at the speed of light is given by the above equation, not that we can get that much energy from a kilogram of something moving at that speed.

The speed of light is approximately 3x10^8 m/s.

So, the energy involved in 1 kilogram of something moving at the speed of light:

E = 1*(3x10^8)^2 = 9*10^16 Joules (that basically the same as 1x10^17 joules).

Okay. Let's assume we can convert all of that (impossible) kinetic energy into usable energy, say electricity. How long could we power the United States?

According to wikipedia, the US uses 105 exajoules in one year (2005). That's 10^20 joules. So, 1 kilogram of water would provide 1/1000 of the energy used in the US in one year, assuming you could capture and use 100% of the kinetic energy of 1 kilogram moving at the speed of light (ignore that the Lorentz equations require inifinite energy to accelerate anything with mass TO the speed of light).

How much energy do we get from oil? 4*10^19 joules, or about 40% of our total usage. So, if we could capture the energy of 400-500 kg moving at the speed of light and turn 100% of it into energy, we could cease using any oil and continue to power the US at its current consumption rate.

Let's get a little more realistic now.

How much energy can we get from 1 kg of fuel in a nuclear reactor? According to this site, we can get about 1.7x10^11 joules of usable energy in a year. That's about 1 billion times less energy than the US uses. We're currently getting about 8% of our energy from nuclear. That implies that we're using about .08*10^9 = 80,000,000 kg of nuclear fuel today.

However, wikipedia indicates that a large reactor can generate about 10^15 joules from 1 kg of fuel. So, I have an error bar of four orders of magnitude on the energy density of nuclear fuels. That means, if these two sources are at all reliable, we're using something between 8 thousand and 80 million kg of nuclear fuel to provide us with 8% of our energy.

Obviously, this is much better than the mass used in the chemical energy generation (fossil fuel burning). 1 kg of coal will produce about 5.2*10^7 joules of energy. We use about 2.5*10^19 joules of coal-produced energy in a year, so we're using something on the order of 4*10^11 kg of coal a year. We get about 6.2*10^7 joules of energy from one kg of oil. Since we use about 4*10^19 joules of energy (per year) from oil, we're using about 6.5*10^11 kg of oil per year. We use about 2.5*10^19 joules of gasoline per year. We get about 2.6*10^7 joules per kg of gasoline, so we're using about 9.6^11 kg of gasoline per year.

In total, we're using 9.6*10^11 + 4*10^11 + 6.5*10^11 kg = 2*10^12 kg of fossil fuels per year. Compared with up to 8*10^7 kg of nuclear fuel per year.

Even if we decided to go 100% nuclear, with rounding up, we'd be using at most 100 times less material to go fully nuclear. Remember, we're mining both materials out of the ground. Of course, there are issues with making nuclear safe, but there are as many pressing issues with making fossil fuels safe.

[preaching]
The WHO estimates 2.4 million people die each year due to pollution. Let's (unfairly, maybe) lump all of that pollution into that generated by the burning of fossil fuels. Deaths due to nuclear plant operation? About 1000 per year, including all of the disasters and the very slight increases in cancer risk to the public (did you know that coal-fired plants release more radioactive materials than nuclear plants do?).

Here's a nice chart that shows the normalized deaths due to electricity generation:

(Electricity generation accounts for about 40% of total primary energy)

Fuel	Immediate fatalities 1970-92	Who?	Normalised to deaths per TWy* electricity
Coal	6400	workers	342
Natural gas	1200	workers & public	85
Hydro	4000	public	883
Nuclear	31	workers	8

Now, add the deaths due to pollution from burning fossil fuels and nuclear power plant-caused deaths are way, way, way below the noise. That's ignoring the increased safety of newer nuclear plants, and that chart doesn't have the deaths due to pollution from burning coal. If you include the deaths due to pollution, that normalized deaths increases to something like 8000 for coal and natural gas.
Imagine we could stop using coal and natural gas and replace them with nuclear, and assume we use the higher number of kg quoted above, just to give us the increase in material we'd need as a proxy for the increase in number of plants we'd need. Assuming a linear increase in the number of normalized deaths with number of plants, we'd have about 800 deaths per TWy for nuclear. Zero for coal+natural gas. That's a net decrease (assuming 8,000 deaths due to coal, natural gas, and pollution) of about 7000 deaths per TWy. The world uses about 16 TW per year. That would mean about 112,000 fewer deaths per year. Considering most of those deaths are due to pollution, we'd also be raising the quality of life of many, many people.

Why isn't the decrease about 2.4 million if all of the deaths due to pollution are due to coal- and natural gas-fired power plants? Because we burn gasoline too, and wood for heating, etc. Also, the above approximations are very poor and linear usually doesn't work well. Given all of that, it's safe to say that a 100% decrease in the use of fossil fuel-fired power plants would make life a lot better for a lot of people, even if we add in the risks due to nuclear fuel.

Of course, if we used less energy, generated a higher % of it from things like solar, created less trash, etc., we'd be even better off.
[/preaching]