Nuclear Energy

You’ll be able to

Explain how nuclear fission generates electricity in a reactor
Weigh the benefits and risks of nuclear power, including waste and accidents
Calculate how much of a radioactive isotope remains after several half-lives

How a nuclear reactor works

Nuclear power runs on fission: the nucleus of a heavy atom — usually uranium-235 — is split by a neutron, releasing a large amount of heat and more neutrons that split further atoms in a controlled chain reaction. The heat boils water into steam, which spins a turbine to generate electricity — the same final step as a fossil-fuel plant, but with a nuclear heat source. Control rods absorb neutrons to regulate the reaction, and a coolant carries away heat. A key advantage: fission produces no CO₂ or air pollutants during operation.

The benefits and the risks

Nuclear power’s appeal is a large, reliable supply of electricity with no greenhouse gas emissions and a very high energy density (a small amount of uranium yields enormous energy). Its risks are serious: radioactive waste stays dangerous for thousands of years and has no permanent U.S. disposal site; uranium mining damages land; and a meltdown — overheating that breaches the reactor — can release radiation. Chernobyl (1986) resulted from a flawed design and operator error; Fukushima (2011) followed an earthquake and tsunami that knocked out cooling. Thermal pollution from warm coolant water discharged to rivers is another concern.

Radioactive decay and half-life

Radioactive waste loses its radioactivity through decay, measured by half-life — the time for half of a radioactive sample to decay. After one half-life, half remains; after two, a quarter; after three, an eighth. Isotopes in nuclear waste have half-lives ranging from years to tens of thousands of years, which is why the waste must be isolated for so long. A useful rule of thumb is that a sample is considered essentially safe after about 10 half-lives, when less than 0.1% of the original radioactivity remains.

Radioactive decay

amount remaining = initial amount × (1/2)ⁿ, where n = time / half-life

n is the number of half-lives that have elapsed. Each half-life halves the remaining amount: after n half-lives, the fraction left is (1/2)ⁿ.

Worked example

A sample of radioactive waste contains 80 grams of an isotope with a half-life of 30 years. How much of the isotope remains after 90 years?

1.Find the number of half-lives elapsed: n = time / half-life = 90 years / 30 years = 3.
2.Each half-life halves the amount, so the fraction remaining is (1/2)³ = 1/8.
3.Multiply the initial amount by that fraction: 80 g × 1/8.
4.Compute: 80 / 8 = 10 g.

Answer: 10 grams remain after 90 years (3 half-lives). You can also step it down: 80 → 40 (30 yr) → 20 (60 yr) → 10 (90 yr).

Checkpoint

A radioactive isotope has a half-life of 25 years. If you start with 100 grams, how much remains after 100 years?

Watch out

A common exam trap is to give the total time and expect you to divide by the half-life first. You cannot just halve once — count how many half-lives fit in the elapsed time (n = time ÷ half-life), then apply (1/2)ⁿ. Forgetting to find n is the top half-life error.

Checkpoint

Compared with a coal-fired power plant, an operating nuclear plant’s biggest ENVIRONMENTAL advantage is that it:

On the exam

For half-life problems, always compute n = elapsed time ÷ half-life first, then multiply the starting amount by (1/2)ⁿ. Stepping the halving down one period at a time (80 → 40 → 20 → 10) is a reliable check against calculator slips.

Answer the 2 checkpoints as you read.