Question
How do I calculate the decay of a radioactive isotope with a half-life of 30 seconds?
Answers
Answered by
GPT-4o mini
To calculate the decay of a radioactive isotope with a given half-life, you can use the exponential decay formula. The amount of a radioactive substance remaining after a certain amount of time can be calculated using the following formula:
\[
N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}
\]
Where:
- \( N(t) \) is the quantity of the substance remaining after time \( t \).
- \( N_0 \) is the initial quantity of the substance.
- \( T_{1/2} \) is the half-life of the substance.
- \( t \) is the elapsed time.
In your case, the half-life (\( T_{1/2} \)) is 30 seconds.
### Steps to calculate the decay:
1. **Identify your initial amount \( N_0 \)**. For example, let's say you start with 100 grams of the isotope.
2. **Choose the time \( t \)** that you want to calculate the decay for. For example, if you want to find out how much is left after 90 seconds, you would set \( t = 90 \) seconds.
3. **Calculate the remaining quantity**:
\[
N(90) = 100 \times \left( \frac{1}{2} \right)^{\frac{90}{30}}
\]
This simplifies as follows:
- \( \frac{90}{30} = 3 \)
- \( \left( \frac{1}{2} \right)^3 = \frac{1}{8} \)
- Thus,
\[
N(90) = 100 \times \frac{1}{8} = 12.5 \text{ grams}
\]
So, after 90 seconds, 12.5 grams of the radioactive isotope would remain.
### Generalization
You can apply this method for any time period or initial quantity using the same equation. Just substitute the values of \( N_0 \), \( t \), and \( T_{1/2} \) appropriately.
\[
N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}
\]
Where:
- \( N(t) \) is the quantity of the substance remaining after time \( t \).
- \( N_0 \) is the initial quantity of the substance.
- \( T_{1/2} \) is the half-life of the substance.
- \( t \) is the elapsed time.
In your case, the half-life (\( T_{1/2} \)) is 30 seconds.
### Steps to calculate the decay:
1. **Identify your initial amount \( N_0 \)**. For example, let's say you start with 100 grams of the isotope.
2. **Choose the time \( t \)** that you want to calculate the decay for. For example, if you want to find out how much is left after 90 seconds, you would set \( t = 90 \) seconds.
3. **Calculate the remaining quantity**:
\[
N(90) = 100 \times \left( \frac{1}{2} \right)^{\frac{90}{30}}
\]
This simplifies as follows:
- \( \frac{90}{30} = 3 \)
- \( \left( \frac{1}{2} \right)^3 = \frac{1}{8} \)
- Thus,
\[
N(90) = 100 \times \frac{1}{8} = 12.5 \text{ grams}
\]
So, after 90 seconds, 12.5 grams of the radioactive isotope would remain.
### Generalization
You can apply this method for any time period or initial quantity using the same equation. Just substitute the values of \( N_0 \), \( t \), and \( T_{1/2} \) appropriately.