Introduction to Nuclear Equations and Half-Life
Welcome! In this chapter, we explore the "maths" of the nucleus. Atoms aren't always stable; sometimes they want to change to become more balanced. When they do this, they spit out radiation and turn into something new. We use nuclear equations to keep track of these changes and half-life to predict how long the process takes. Don't worry if it sounds like a lot of numbers—once you see the patterns, it’s as simple as basic addition and subtraction!
1. Radioactive Decay and Equations
When an unstable nucleus decays, it emits radiation. This changes the number of protons and/or neutrons in the nucleus. We show this using a nuclear equation, which looks a bit like a chemical equation.
The Golden Rule: In any nuclear equation, the Total Mass Number (top number) and the Total Atomic Number (bottom number) must be the same on both sides of the arrow.
Alpha (\(\alpha\)) Decay
An alpha particle consists of 2 protons and 2 neutrons (a Helium nucleus). When a nucleus emits an alpha particle:
- The mass number decreases by 4.
- The atomic number decreases by 2.
Example: \({}^{238}_{92}\text{U} \rightarrow {}^{234}_{90}\text{Th} + {}^{4}_{2}\alpha\)
Beta-minus (\(\beta^-\)) Decay
A neutron in the nucleus turns into a proton and an electron. The electron is shot out of the nucleus as a beta-minus particle.
- The mass number stays the same (because a neutron left but a proton replaced it).
- The atomic number increases by 1 (because there is one extra proton).
Example: \({}^{14}_{6}\text{C} \rightarrow {}^{14}_{7}\text{N} + {}^{0}_{-1}\beta^-\)
Beta-plus (\(\beta^+\)) Decay (Positron emission)
A proton turns into a neutron and a positron. The positron is shot out.
- The mass number stays the same.
- The atomic number decreases by 1 (because a proton was lost).
Example: \({}^{13}_{7}\text{N} \rightarrow {}^{13}_{6}\text{C} + {}^{0}_{+1}\beta^+\)
Gamma (\(\gamma\)) Radiation
Gamma rays are high-energy electromagnetic waves. They are just energy being "shaken off" by a nucleus.
- The mass number and atomic number both stay exactly the same.
Neutron (\(n\)) Radiation
Sometimes a nucleus just spits out a neutron.
- The mass number decreases by 1.
- The atomic number stays the same.
Quick Review: Remember that Beta-minus adds a proton (bottom number +1), while Beta-plus removes a proton (bottom number -1).
2. The Random Nature of Decay
Radioactive decay is random. This means we cannot predict exactly which nucleus will decay next, or exactly when a specific one will decay.
The Dice Analogy: Imagine you have 1,000 dice. If you roll them all, you can predict that about 1/6th will land on a '6', but you can't say which specific die will be a '6'. Radioactive atoms are just like those dice!
Activity and the Becquerel
Even though decay is random, we can measure how many decays happen every second for a large group of atoms. This is called Activity.
- Activity is the rate at which a source of unstable nuclei decays.
- It is measured in Becquerels (Bq).
- \(1 \text{ Bq} = 1 \text{ decay per second}\).
3. Half-Life
Because the activity of a source decreases over time (as there are fewer unstable atoms left to decay), we use half-life to describe how long it takes for a substance to disappear.
Definition: The half-life is the time taken for:
1. The number of radioactive nuclei in a sample to halve.
OR
2. The activity (count-rate) of a sample to halve.
Did you know? Half-lives can range from fractions of a second to billions of years, depending on the isotope!
Half-Life Calculations (Step-by-Step)
You might be asked to find the remaining activity after a certain amount of time. Use this simple method:
Example: A source has an activity of \(800 \text{ Bq}\). Its half-life is 2 hours. What is the activity after 6 hours?
1. Find the number of half-lives: \(6 \text{ hours} \div 2 \text{ hours} = 3 \text{ half-lives}\).
2. Halve the activity 3 times:
- Start: \(800 \text{ Bq}\)
- After 1 half-life: \(400 \text{ Bq}\)
- After 2 half-lives: \(200 \text{ Bq}\)
- After 3 half-lives: \(100 \text{ Bq}\)
Final Answer: \(100 \text{ Bq}\)
Finding Half-Life from a Graph
If you are given a graph of Activity vs. Time:
1. Look at the starting activity on the y-axis (e.g., \(100 \text{ Bq}\)).
2. Find exactly half of that value (\(50 \text{ Bq}\)).
3. Draw a horizontal line from that value to the curve, then a vertical line down to the x-axis (Time).
4. The value on the x-axis is the half-life.
Common Mistakes to Avoid
- Mistake: Thinking "half-life" means the substance is gone after two half-lives.
- Correction: No! After one half-life, you have 50%. After two, you have 25%. After three, you have 12.5%. It never quite reaches zero!
- Mistake: Forgetting to subtract 1 in Beta-plus equations.
- Correction: In \(\beta^+\), a proton turns into a neutron. Protons are the "identity" of the atom, so the atomic number must go down by 1.
Key Takeaways
1. Nuclear Equations: Always make sure the top numbers add up and the bottom numbers add up on both sides of the arrow.
2. Activity: Measured in Becquerels (Bq), representing decays per second.
3. Randomness: You can't predict an individual decay, but you can predict the pattern of a large group.
4. Half-Life: The constant time it takes for activity or the number of nuclei to halve.