Radioactivity — alpha, beta, gamma and half-life (KCSE Physics Form 4)

TL;DR

Radioactivity is when unstable atomic nuclei spontaneously break down, releasing energy and particles. These emissions are mainly alpha, beta, and gamma radiation, each with distinct properties. Half-life describes the time it takes for half of a radioactive sample to decay.

1. The Mental Model

Imagine an unstable Jenga tower that occasionally sheds blocks on its own. Radioactivity is like that: unstable atoms spontaneously "shed" parts to become more stable. These "shed" parts are what we call radiation.

2. The Core Material

What is Radioactivity?

Radioactivity is the spontaneous disintegration (breakdown) of an unstable atomic nucleus, accompanied by the emission of radiation. This process happens naturally in certain elements, called radioisotopes. The nucleus changes into a more stable form, often becoming a different element.

Types of Radiation

There are three main types of radiation you need to know: alpha (α), beta (β), and gamma (γ). They differ in their composition, charge, mass, and penetrating power.

Alpha (α) Radiation

• Composition: Consists of two protons and two neutrons, identical to a helium nucleus ($^4_2$He).
• Charge: +2e (positive).
• Mass: Relatively heavy.
• Penetrating Power: Very low. Can be stopped by a sheet of paper or a few centimetres of air.
• Ionizing Power: Very high. Because of its large charge and mass, it strongly interacts with matter, knocking electrons off atoms and creating ions.
• Deflection: Deflected by electric and magnetic fields.

Beta (β) Radiation

• Composition: High-energy electron ($^0_{-1}$e) or positron ($^0_{+1}$e). In KCSE, we usually focus on electron emission.
• Charge: -1e (negative) for electron emission.
• Mass: Very small (much lighter than an alpha particle).
• Penetrating Power: Moderate. Can penetrate paper but is stopped by a few millimetres of aluminium.
• Ionizing Power: Moderate. Less than alpha but more than gamma.
• Deflection: Deflected by electric and magnetic fields, but in the opposite direction to alpha particles (if negative) and by a greater angle due to its smaller mass.

Gamma (γ) Radiation

• Composition: High-energy electromagnetic wave (photon). It's not a particle.
• Charge: No charge (neutral).
• Mass: No mass.
• Penetrating Power: Very high. Can pass through several centimetres of lead or metres of concrete.
• Ionizing Power: Very low. It interacts weakly with matter.
• Deflection: Not deflected by electric or magnetic fields because it has no charge.

Here's a summary table to help you compare them:

Half-Life ($T_{1/2}$)

The half-life of a radioactive isotope is the time it takes for half of the original radioactive nuclei in a sample to decay. It's a constant value for a given isotope and is not affected by temperature, pressure, or chemical reactions.

Key idea: After one half-life, half the sample remains. After two half-lives, half of the remaining half (i.e., a quarter of the original) remains, and so on.

Let $N_0$ be the initial number of radioactive nuclei (or initial mass/activity).
Let $N$ be the number of radioactive nuclei (or mass/activity) remaining after time $t$.
Let $n$ be the number of half-lives that have passed.

The relationship is:
$N = N_0 \times (\frac{1}{2})^n$

Where $n = \frac{t}{T_{1/2}}$

This means you can calculate how much radioactive material is left after a certain time, or how long it takes for a certain amount to decay.

Decay Process Flow

Here's how the decay process works over time, based on half-life:

graph TD
    A[Start with N₀ atoms] --> B{After 1 Half-Life};
    B --> C[N₀/2 atoms remain];
    C --> D{After 2 Half-Lives};
    D --> E[N₀/4 atoms remain];
    E --> F{After 3 Half-Lives};
    F --> G[N₀/8 atoms remain];
    G --> H[...and so on];

3. Worked Example

A radioactive isotope has a half-life of 20 minutes. If you start with a sample containing 800g of the isotope, calculate:
a) The mass of the isotope remaining after 60 minutes.
b) The time it takes for the mass of the isotope to reduce to 50g.

Solution:

a) Mass remaining after 60 minutes:
* Given: Initial mass ($N_0$) = 800g, Half-life ($T_{1/2}$) = 20 minutes, Total time ($t$) = 60 minutes.
* First, find the number of half-lives ($n$):
$n = \frac{t}{T_{1/2}} = \frac{60 \text{ minutes}}{20 \text{ minutes}} = 3$ half-lives.
* Now, use the formula $N = N_0 \times (\frac{1}{2})^n$:
$N = 800g \times (\frac{1}{2})^3$
$N = 800g \times (\frac{1}{8})$
$N = 100g$
So, 100g of the isotope remains after 60 minutes.

b) Time to reduce to 50g:
* Given: Initial mass ($N_0$) = 800g, Final mass ($N$) = 50g, Half-life ($T_{1/2}$) = 20 minutes.
* We need to find $n$ first using $N = N_0 \times (\frac{1}{2})^n$:
$50 = 800 \times (\frac{1}{2})^n$
Divide both sides by 800:
$\frac{50}{800} = (\frac{1}{2})^n$
$\frac{1}{16} = (\frac{1}{2})^n$
* We know that $2^4 = 16$, so $(\frac{1}{2})^4 = \frac{1}{16}$.
Therefore, $n = 4$ half-lives.
* Now, calculate the total time ($t$):
$t = n \times T_{1/2}$
$t = 4 \times 20 \text{ minutes}$
$t = 80 \text{ minutes}$
It takes 80 minutes for the mass of the isotope to reduce to 50g.

4. Key Takeaways

• Radioactivity is the spontaneous breakdown of unstable atomic nuclei, releasing energy and particles.
• Alpha particles are heavy, positively charged, low penetrating, and highly ionizing.
• Beta particles are light, negatively charged (electrons), moderately penetrating, and moderately ionizing.
• Gamma rays are uncharged electromagnetic waves, highly penetrating, and weakly ionizing.
• Half-life is the time for half of a radioactive sample to decay, and it's constant for a given isotope.
• The amount of radioactive material remaining after 'n' half-lives is $N_0 \times (1/2)^n$.

Common mistakes to avoid:
- Confusing the penetrating power with ionizing power for different radiations.
- Assuming half-life means the substance completely disappears after two half-lives.
- Forgetting that gamma radiation is an electromagnetic wave, not a particle.
- Not correctly calculating the number of half-lives passed before applying the formula.

5. Now Try It

A radioactive source has an initial activity of 1280 Bq (Becquerel, a unit of activity). After 45 days, its activity is found to be 160 Bq. Determine the half-life of this radioactive source. What success looks like: You should be able to show your steps clearly, calculate the number of half-lives, and then correctly determine the half-life in days.