Question
What is half-life and how do I calculate the remaining amount of a substance after 270 days if its half-life is 90 days?
Answers
GPT-4o mini
Half-life is the time required for a quantity to reduce to half its initial value. This concept is commonly used in fields such as physics, chemistry, and biology, particularly in relation to radioactive decay or pharmacokinetics.
To calculate the remaining amount of a substance after a certain period of time, given its half-life, you can use the following steps:
1. **Determine the number of half-lives that have passed**: Divide the total time by the half-life.
2. **Calculate the remaining amount**: Use the formula for exponential decay based on the number of half-lives that have elapsed.
In your case:
- The half-life of the substance is 90 days.
- The total time elapsed is 270 days.
**Step 1**: Calculate the number of half-lives:
\[
\text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{270 \text{ days}}{90 \text{ days}} = 3
\]
This means that 3 half-lives have passed.
**Step 2**: Calculate the remaining amount using the formula:
\[
\text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^{\text{Number of half-lives}}
\]
Let’s assume the initial amount is \(N_0\).
\[
\text{Remaining amount} = N_0 \times \left(\frac{1}{2}\right)^3 = N_0 \times \frac{1}{8}
\]
Therefore, after 270 days (3 half-lives), the remaining amount of the substance will be \( \frac{1}{8} \) (or 12.5%) of the initial amount. You would just replace \(N_0\) with the actual initial quantity to get the specific remaining amount.
To calculate the remaining amount of a substance after a certain period of time, given its half-life, you can use the following steps:
1. **Determine the number of half-lives that have passed**: Divide the total time by the half-life.
2. **Calculate the remaining amount**: Use the formula for exponential decay based on the number of half-lives that have elapsed.
In your case:
- The half-life of the substance is 90 days.
- The total time elapsed is 270 days.
**Step 1**: Calculate the number of half-lives:
\[
\text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{270 \text{ days}}{90 \text{ days}} = 3
\]
This means that 3 half-lives have passed.
**Step 2**: Calculate the remaining amount using the formula:
\[
\text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^{\text{Number of half-lives}}
\]
Let’s assume the initial amount is \(N_0\).
\[
\text{Remaining amount} = N_0 \times \left(\frac{1}{2}\right)^3 = N_0 \times \frac{1}{8}
\]
Therefore, after 270 days (3 half-lives), the remaining amount of the substance will be \( \frac{1}{8} \) (or 12.5%) of the initial amount. You would just replace \(N_0\) with the actual initial quantity to get the specific remaining amount.