Calculating Radioactive Decay

Radioactive Decay Basics

Radioactive decay is a random and spontaneous process where unstable atomic nuclei lose energy by emitting radiation. This process changes the nucleus, often transforming it into a different element or isotope.

There are three main types of decay: alpha, beta, and gamma. Each type involves different particles or energy being emitted, but all reduce the number of unstable nuclei in a sample over time.

Because decay happens randomly, it is impossible to predict exactly when a particular nucleus will decay, but we can calculate how many nuclei remain or how the activity changes over time using mathematical models.

Calculating Decay

Radioactive decay follows an exponential decrease in the number of unstable nuclei. The key equation used to calculate the remaining number of nuclei after a certain time is:

$$N = N_0 e^{-\lambda t}$$

• $N_0$ = initial number of unstable nuclei
• $N$ = number of unstable nuclei remaining after time $t$
• $\lambda$ = decay constant (a unique value for each radioactive isotope)
• $t$ = time elapsed

The decay constant $\lambda$ represents the probability per second that a nucleus will decay. A larger $\lambda$ means the substance decays faster.

Activity ($A$) is the rate at which the nuclei decay, measured in becquerels (Bq), where 1 Bq = 1 decay per second. Activity is directly proportional to the number of unstable nuclei remaining:

$$A = \lambda N$$

This means as the number of nuclei decreases, the activity also decreases exponentially over time.

For example, if you know the initial number of nuclei and the decay constant, you can calculate how many nuclei remain after a given time, or find the activity at any moment.

For instance, if a sample starts with 10,000 nuclei and the decay constant is 0.001 s$^{-1}$, after 1000 seconds the number remaining is:

$$N = 10,000 \times e^{-0.001 \times 1000} = 10,000 \times e^{-1} \approx 10,000 \times 0.368 = 3,680$$

So about 3,680 nuclei remain after 1000 seconds.

Activity and Count Rate

Activity measures how many nuclei decay per second in a radioactive sample. It is measured in becquerels (Bq), where 1 Bq equals one decay per second.

The count rate is the number of decays detected by a detector per second. It depends on:

• The activity of the source
• The distance between the source and detector (count rate decreases with distance)
• Any shielding between the source and detector (which reduces count rate)

When performing calculations involving count rate, it is important to remember that the count rate is proportional to the activity but can be lower due to these factors.

For example, if a source has an activity of 500 Bq, but the detector only records 250 counts per second because of distance and shielding, the count rate is 250 counts/s.

Using Decay Equations in Calculations

To calculate the remaining number of nuclei or activity after a certain time, follow these steps:

1. Identify the initial number of nuclei $N_0$ or initial activity $A_0$.
2. Use the decay constant $\lambda$ (given or calculated from half-life).
3. Use the time $t$ elapsed.
4. Calculate the remaining nuclei $N = N_0 e^{-\lambda t}$ or activity $A = A_0 e^{-\lambda t}$.

If you need to find the decay constant $\lambda$ and you know the half-life $t_{1/2}$, use:

$$\lambda = \frac{\ln 2}{t_{1/2}}$$

(Half-life is covered in another topic, but this formula is useful for calculations. Note: Half-life is the time taken for half the nuclei in a sample to decay.)

Example: A radioactive isotope has a decay constant of 0.0005 s$^{-1}$ and initially has 20,000 unstable nuclei. Calculate how many nuclei remain after 2000 seconds.

Using the formula:

$$N = 20,000 \times e^{-0.0005 \times 2000} = 20,000 \times e^{-1} \approx 20,000 \times 0.368 = 7,360$$

So, 7,360 nuclei remain after 2000 seconds.