AP Environmental Science

## Learning Outcomes

- Define **half-life** and explain why radioactive decay follows an exponential pattern
- Calculate how much of a radioactive sample remains after a given time or number of half-lives
- Infer how many half-lives have passed from fractions such as 1/2, 1/4, and 1/8
- Explain how half-life affects nuclear waste storage and risk over short and long timescales
- Distinguish between short-term radioactivity and long-term persistence
- Apply half-life to radiometric dating using parent and daughter isotopes
- Avoid common exam mistakes about radioactive decay and nuclear waste

## AP Environmental Science Syllabus

For the AP Environmental Science, you are expected to understand half-life applications in nuclear power, with a focus on the following syllabus points:

- Know that **half-life** is the time required for half of the radioactive atoms in a sample to decay
- Understand that radioactive decay is **exponential**, not linear
- Calculate the amount of a radioactive substance remaining after a given number of years or half-lives
- Infer elapsed time from the fraction of parent isotope remaining
- Explain why used nuclear fuel does not have one single half-life
- Distinguish between shorter-lived fission products and longer-lived transuranic elements in nuclear waste
- Apply half-life to **radiometric dating**, including carbon-14 for recent once-living material and uranium systems for very old rocks
- Explain why nuclear waste management requires isolation over very different timescales

## Test Your Knowledge

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, look more closely at that area during your revision._

1. What is half-life, and why is it not the same as “time until a substance is gone”?
2. If 25% of the original radioactive parent isotope remains, how many half-lives have passed?
3. Why does used nuclear fuel not have a single half-life?
4. How does a short half-life differ from a long half-life in terms of hazard and persistence?
5. Why is carbon-14 useful for recent once-living material, but not for very old rocks?

## Introduction

Half-life is one of the most testable ideas in nuclear power because it connects science, math, and environmental decision-making. AP Environmental Science does not usually require advanced nuclear physics, but it does expect you to explain how radioactive materials change over time and why that matters for waste storage, accident risk, and dating materials from the past.

This topic matters because nuclear power is often presented as a trade-off. It produces low operational greenhouse gas emissions compared with fossil fuels, but it also creates radioactive waste that remains hazardous for very different lengths of time. Half-life helps you explain that trade-off clearly and quantitatively.

> **Key Term: half-life**
> The time required for half of the radioactive atoms in a sample to decay.

### How half-life works and why decay is exponential

Radioactive materials contain unstable nuclei. Over time, some of those nuclei decay and release energy as radiation. The key point is that the sample does not lose the same amount in each equal time period. Instead, it loses the same **fraction** each half-life.

That is why decay is exponential. Suppose a sample starts with 100 g of a radioisotope.

- After 1 half-life, 50 g remain
- After 2 half-lives, 25 g remain
- After 3 half-lives, 12.5 g remain
- After 4 half-lives, 6.25 g remain

Notice what changed. The sample did not lose 50 g every time. It lost half of what remained. This is the idea students often miss.

> **Key Term: radioactive decay**
> The process by which an unstable atomic nucleus loses energy by emitting radiation and changes into another nucleus or isotope.

A useful APES rule is:

- after 1 half-life: **1/2** remains
- after 2 half-lives: **1/4** remains
- after 3 half-lives: **1/8** remains
- after 4 half-lives: **1/16** remains

If you remember that pattern, many exam questions become much easier.

You may also see the decay formula:

$$
N(t)=N_0\left(\frac{1}{2}\right)^{t/T_{1/2}}
$$

Where:

- $N_0$ = original amount
- $N(t)$ = amount remaining after time $t$
- $T_{1/2}$ = half-life

For AP Environmental Science, the repeated-halving method is often enough, but the formula helps confirm the logic.

> **Test Tip:** If an exam question gives a half-life and asks for the amount left, first see whether the time is an exact multiple of the half-life. If it is, use repeated halving before reaching for a calculator.

Here is a worked example.

A sample has a half-life of 20 years and begins with 80 g. How much remains after 60 years?

Step 1: Find the number of half-lives.

$$
60 \div 20 = 3
$$

So 3 half-lives have passed.

Step 2: Halve the sample three times.

- Start: 80 g
- After 20 years: 40 g
- After 40 years: 20 g
- After 60 years: 10 g

Answer: **10 g remain**

A second type of question works backward.

A sample has 25% of its original parent isotope remaining. How many half-lives have passed?

25% = 1/4, which means **2 half-lives**.

This is not difficult once you connect percentages to simple fractions.

Common mistakes:

- Thinking one half-life means the sample is gone
- Treating decay as linear
- Forgetting to convert years into number of half-lives first

### Half-life and nuclear waste management

Half-life becomes environmentally important when you apply it to nuclear waste. Nuclear fission in reactors creates radioactive by-products. Some decay quickly and give off intense radiation or heat in the short term. Others decay very slowly and stay dangerous for thousands of years or longer.

This is why waste management is difficult.

> **Key Term: used nuclear fuel**
> Fuel removed from a reactor after use that remains highly radioactive and must be stored safely.

A major exam distinction is that used nuclear fuel does **not** have one half-life. It is a mixture of many radionuclides, each with a different decay rate.

> **Key Term: radionuclide**
> A radioactive isotope that undergoes decay and emits radiation.

That mixture changes over time. Early on, shorter-lived isotopes dominate the hazard. Much later, longer-lived isotopes become the main concern.

Important examples:

- **Iodine-131**: about 8 days
- **Cesium-137**: about 30 years
- **Strontium-90**: about 29–30 years
- **Plutonium-239**: about 24,000 years
- **Uranium-235**: about 700 million years
- **Uranium-238**: about 4.5 billion years

Why do shorter and longer half-lives matter in different ways?

A shorter half-life usually means a faster decay rate. That often means greater short-term activity and more immediate radiation output. But it also means the material declines much more quickly.

A longer half-life usually means lower decay rate at any given moment, but far greater persistence. The material stays in the environment much longer.

This is a classic APES trade-off:

- **short half-life**: more intense short-term hazard
- **long half-life**: longer-term storage problem

For example, cesium-137 and strontium-90 are major concerns in the first decades after fuel is removed from a reactor because they contribute strongly to heat and radioactivity. But after about 1,000 years, long-lived transuranic elements such as plutonium-239 become more significant in the remaining hazard.

> **Key Term: transuranic element**
> An element with an atomic number greater than uranium, often formed in reactors and important in long-term radioactive waste risk.

This is why waste storage needs different strategies over different timescales. Near-term storage must manage heat and strong radiation. Long-term disposal must isolate waste from groundwater, the environments, and people for very long periods.

Here is a worked example using cesium-137.

A storage container has 160 g of cesium-137 with a half-life of 30 years. How much remains after 120 years?

Step 1: Find half-lives.

$$
120 \div 30 = 4
$$

Step 2: Halve four times.

- 160 g
- 80 g
- 40 g
- 20 g
- 10 g

Answer: **10 g remain**

That still is not zero. This is a common exam trap. Even after four half-lives, material remains.

Now compare with plutonium-239.

If 80 g of plutonium-239 has a half-life of about 24,000 years:

- after 24,000 years: 40 g remain
- after 48,000 years: 20 g remain

That is the environmental problem in one glance. The radioactivity persists across timescales far beyond ordinary human planning.

> **Test Tip:** In waste questions, separate **immediate hazard** from **persistence**. Students often write vaguely that “radioactive waste is dangerous.” Better answers explain which isotopes matter first, and which matter later.

Another key distinction: **used nuclear fuel** is one kind of radioactive waste, but not a synonym for all radioactive waste. AP questions may also refer to low-level and high-level radioactive waste. High-level waste remains dangerous much longer and needs much stricter isolation.

### Radiometric dating: using half-life to estimate age

Half-life is also useful far beyond power plants. It allows scientists to estimate the age of materials by comparing the amount of radioactive parent isotope left to the amount of daughter product formed.

> **Key Term: parent isotope**
> The original radioactive isotope that decays over time into another substance.
>
> **Key Term: daughter isotope**
> The isotope or element produced when a parent isotope decays.

This works because decay rates are predictable. If the half-life is known, and you know how much parent remains, you can estimate elapsed time.

> **Key Term: radiometric dating**
> A method for estimating age by measuring the relative amounts of radioactive parent isotopes and their daughter products.

A common APES example is **carbon-14**.

Carbon-14 has a half-life of about **5,730 years** and decays to nitrogen-14. It is useful for dating relatively recent once-living material, such as wood, bone, or cloth, up to about 50,000 years.

Why only recent once-living material?

Because carbon-14 enters living things while they are alive. Once the organism dies, no new carbon-14 enters, so the existing carbon-14 simply decays. That makes the “clock” useful.

Here is a worked example.

A wooden sample has 12.5% of its original carbon-14 remaining. How old is it?

Step 1: Convert 12.5% to a fraction.

12.5% = 1/8

Step 2: Count half-lives.

- 1/2 after 1 half-life
- 1/4 after 2 half-lives
- 1/8 after 3 half-lives

So 3 half-lives have passed.

Step 3: Multiply by the half-life.

$$
3 \times 5,730 = 17,190 \text{ years}
$$

Answer: **about 17,190 years old**

Now compare that with **uranium-238**, which decays ultimately to **lead-206** and has a half-life of about **4.5 billion years**.

This makes uranium systems useful for dating very old rocks, not recent organic remains.

Why not use carbon-14 for ancient rocks?

- Carbon-14 is for **once-living material**
- Its half-life is too short for very old geologic timescales
- Old rocks usually do not preserve the carbon-based signal needed

Why not use uranium-238 for recent organic remains?

- It is found in rocks, not as the standard dating system for recent once-living material
- Its half-life is so long that it is poor for fine resolution on short timescales

This is a very common AP distinction.

A good answer also knows the assumptions behind radiometric dating:

- decay rate stays constant
- sample remained a closed system
- contamination is absent or accounted for
- parent/daughter relationships are understood

In AP Environmental Science, you usually will not need deep geochemistry, but you should be able to explain why different isotopes fit different dating problems.

### Common APES mistakes and how to avoid them

Students lose marks on half-life applications more from bad reasoning than from hard math. The main errors repeat each year.

First mistake: saying a substance is “used up” after one half-life. It is not. Half remains.

Second mistake: forgetting decay is exponential. Equal times do not remove equal amounts.

Third mistake: treating used fuel as if it has one half-life. It does not. It is a mix.

Fourth mistake: mixing up hazard and persistence. Short half-life does not always mean a bigger long-term problem; it usually means greater short-term activity.

Fifth mistake: using the wrong isotope for dating. Carbon-14 is for relatively recent organic material. Uranium systems are for very old rocks.

A stronger AP answer usually does three things:

- defines the half-life idea clearly
- applies it numerically or conceptually
- connects it to environmental consequences

For example, instead of writing:

“Radioactive waste lasts a long time.”

A stronger answer would say:

“Used nuclear fuel contains isotopes with very different half-lives. Shorter-lived fission products such as cesium-137 and strontium-90 create significant near-term heat and radiation, while longer-lived transuranic elements such as plutonium-239 create long-term storage concerns over thousands of years.”

That is more precise, more applied, and more likely to earn credit.

## Key Point Checklist

_This article has covered the following key knowledge points:_

- Half-life is the time required for half of the radioactive atoms in a sample to decay
- Radioactive decay follows an exponential pattern, not a linear one
- After 1 half-life, 1/2 remains; after 2, 1/4 remains; after 3, 1/8 remains
- One half-life does not mean a radioactive substance is gone
- Shorter half-life often means greater short-term activity, while longer half-life means greater persistence
- Used nuclear fuel does not have one single half-life because it contains many radionuclides
- Cesium-137 and strontium-90 are major shorter-term waste concerns, while plutonium-239 is a major long-term concern
- Half-life helps explain why radioactive waste requires storage and isolation over long periods
- Radiometric dating uses parent and daughter isotopes to estimate age
- Carbon-14 is useful for recent once-living material, while uranium-238 systems are useful for very old rocks
- APES questions often test both half-life calculations and environmental implications
- Strong exam answers distinguish clearly between immediate hazard and long-term persistence

## Key Terms and Concepts

- half-life
- radioactive decay
- used nuclear fuel
- radionuclide
- transuranic element
- parent isotope
- daughter isotope
- radiometric dating


Nuclear Power - Half-life applications