## Table of Contents

## Radioactive Decay

**Main Article**: Radioactivity & Radioactive Decay

Radioactive decay is a fundamental process by which unstable atomic nuclei lose energy by emitting radiation. This process decreases the number of radioactive atoms in a substance over time, following an exponential decay pattern. Central to this concept is the half-life, a characteristic time that provides insight into the stability and decay rate of radioactive elements.

## Exponential Nature of Radioactive Decay

The exponential decrease in the number of radioactive atoms over time means that the quantity of radioactive material diminishes to half its initial amount over a period known as the **half-life**. This period is unique to each radioactive element, allowing scientists to predict the decay pattern of substances accurately.

## Defining Half-Life

The half-life of a radioactive isotope is defined as the duration required for half of the original unstable radioactive nuclei present in a sample to undergo decay. This definition is crucial for understanding how different substances behave over time and how their radioactive properties can be harnessed or mitigated.

### Variability Among Elements

Radioactive elements exhibit a wide range of decay rates, leading to significantly different half-life periods. For instance:

**Uranium-238**, a heavy element used in nuclear reactors and geological dating, has a half-life of over 400 million years.**Strontium-90**, a byproduct of nuclear fission with applications in medical and industrial fields, decays much faster with a half-life of 28 years.**Oxygen-18**, often used in paleoclimatology studies, has an extremely short half-life of less than 0.01 seconds.

### Practical Application: Sodium-24

An example of half-life in action can be observed with **Sodium-24 (Na-24)**, which has a half-life of 15 hours. By using a Geiger-Müller (GM) tube and a counter, scientists can measure the decay of Na-24 through the counts per second at various intervals. Starting with 20 million undecayed nuclei, after one half-life, only 10 million nuclei would remain.

### Stability Indicator

The half-life also serves as an indicator of a radioactive element’s stability. Elements with longer half-lives are considered more stable, as they decay at a slower rate compared to those with shorter half-lives.

## Measuring Radioactive Decay: The Decay Curve

A decay curve is a graphical representation of the decrease in activity (number of disintegrations per second) of a radioactive sample over time. By measuring the activity at various intervals, typically with a GM tube and ratemeter, scientists can plot this decay, which follows a predictable pattern.

### Understanding Decay Curve Dynamics

The decay curve demonstrates that the activity of a radioactive substance decreases by a consistent fraction in equal time intervals. For instance, if the activity of a substance falls from 80 to 40 disintegrations per second in 10 minutes, it will decrease from 40 to 20 in the next 10 minutes, and so on, indicating a half-life of 10 minutes. This pattern underscores the exponential nature of radioactive decay.

### Visualizing Decay: An Example

Consider a hypothetical element with an initial activity of 80 disintegrations per second. Over successive 10-minute intervals, its activity decreases as follows:

- After 10 minutes: 40 disintegrations per second
- After 20 minutes: 20 disintegrations per second
- After 30 minutes: 10 disintegrations per second

This pattern, where the activity halves every 10 minutes, exemplifies the element’s half-life.

## Mathematical Representation of Half-Life

The relationship between the initial and final quantity of a radioactive element and the number of half-lives elapsed can be expressed mathematically as:

$$\frac{N_{\text{final}}}{N_{\text{initial}}} = \left( \frac{1}{2} \right)^{n}$$

, where

$N_{\text{final}}$ is the number of remaining radioactive element

$N_{\text{initial}}$ is the number of initial radioactive element

$n$ is the number of half-life

### Calculating the Number of Half-Lives

To determine the number of half-lives elapsed given the initial and final quantities of a radioactive element, the formula can be transformed using logarithms:

$$n \text{log} \left( \frac{1}{2} \right) = \frac{N_{\text{final}}}{N_{\text{initial}}}$$

This equation is instrumental in calculating either the elapsed time given the half-life of the substance or the half-life itself if the time and quantities of the radioactive material are known.

## Experiment to Determine the Half-Life of Thoron

### Experimental Setup

The half-life of thoron, an alpha-emitting gas, can be determined through a simple yet effective experiment. The process begins by transferring thoron from a bottle to a flask by squeezing the bottle several times. After sealing the flask with a Geiger-Müller (GM) tube to measure the radiation, the experiment observes the decay of thoron’s radioactivity.

### Measurement and Observation

Upon reaching the maximum activity level, the count-rate is recorded every 15 seconds for the first 2 minutes, followed by every 60 seconds for a few minutes longer. The GM tube remains in place for at least an hour to ensure the thorough measurement of thoron’s decay. Additionally, background radiation is measured separately to correct the count-rates obtained during the thoron decay experiment.

### Analysis and Conclusion

By subtracting the average background count-rate from each reading and plotting the corrected count-rate against time, researchers can estimate thoron’s half-life, which is approximately 52 seconds. This experiment not only provides a direct method to determine the half-life of a radioactive element but also illustrates the random nature of radioactive decay.

### Random Nature of Radioactive Decay

During the thoron experiment, the variability in count-rate highlights the spontaneous and random nature of radioactive decay. Despite the predictability of the half-life concept, the decay of individual atoms occurs without any external stimulus and cannot be precisely predicted for specific atoms. This randomness underscores the inherent uncertainty in the timing of radioactive emissions, emphasizing that radioactive decay is a probabilistic process.

## Worked Examples

### Example 1: Understanding Half-Life

If a sample of Uranium-238 has an initial quantity of 1000 atoms, how many atoms would remain undecayed after 800 million years, given that the half-life of Uranium-238 is approximately 400 million years?

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Using the formula $\frac{N_{\text{final}}}{N_{\text{initial}}} = \left( \frac{1}{2} \right)^n$, where $n$ is the number of half-lives elapsed, we find $n = \frac{800 \text{ million years}}{400 \text{ million years}} = 2$. Therefore, the fraction of remaining atoms is $\left( \frac{1}{2} \right)^2 = \frac{1}{4}$. Thus, $1000 \times \frac{1}{4} = 250$ atoms of Uranium-238 would remain undecayed after 800 million years.

### Example 2: Calculating the Number of Half-Lives

A scientist starts with a sample of 160 atoms of Sodium-24. After 45 hours, only 10 atoms remain. How many half-lives have passed during this period?

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First, calculate the fraction of Sodium-24 that remains: $\frac{N_{\text{final}}}{N_{\text{initial}}} = \frac{10}{160} = \frac{1}{16}$. To find the number of half-lives ($n$), use the formula $\frac{1}{2^n} = \frac{1}{16}$. Solving for $n$ gives $2^n = 16$, thus $n = 4$. Therefore, 4 half-lives have passed during the 45 hours.

### Example 3: Decay Curve Interpretation

A radioactive substance has a half-life of 10 minutes. If the initial activity is 160 disintegrations per second, what will the activity be after 30 minutes?

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After 30 minutes, or 3 half-lives, the activity of the substance would decrease to $160 \times \left( \frac{1}{2} \right)^3 = 160 \times \frac{1}{8} = 20$ disintegrations per second.

### Example 4: Determining Half-Life from Experimental Data

In an experiment to determine the half-life of thoron, a researcher records a corrected count-rate of 1024 disintegrations per minute at the start. After 4 minutes, the count-rate drops to 64 disintegrations per minute. What is the half-life of thoron?

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The count-rate decreases by a factor of $\frac{64}{1024} = \frac{1}{16}$, which corresponds to $2^4$ (since $\frac{1}{2^4} = \frac{1}{16}$). This indicates 4 half-lives have passed in 4 minutes, so the half-life of thoron is $\frac{4 \text{ minutes}}{4} = 1$ minute.

### Example 5: Practical Application and Stability Indicator

If Strontium-90, with a half-life of 28 years, is used in a medical application requiring a minimal activity level of 25% of its initial activity, for how many years can it be effectively used before falling below this threshold?

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To fall to 25% of its initial activity, Strontium-90 would need to go through 2 half-lives ($\left( \frac{1}{2} \right)^2 = \frac{1}{4}$). Since each half-life is 28 years, it can be effectively used for $2 \times 28 = 56$ years before falling below the required activity level.