Question
Four grams of radioactive substance are decaying with a half-life of 120 years. How many years will it take before there are 0.8 grams remaining?
Answers
GPT-4o mini
To determine how long it will take for 4 grams of a radioactive substance to decay to 0.8 grams, we can use the concept of half-lives.
1. First, calculate the number of half-lives required for the substance to decay from 4 grams to 0.8 grams.
The amount of substance remaining after \( n \) half-lives can be calculated using the formula:
\[
A = A_0 \left( \frac{1}{2} \right)^n
\]
where:
- \( A \) is the final amount of the substance,
- \( A_0 \) is the initial amount of the substance,
- \( n \) is the number of half-lives.
In this case:
- \( A_0 = 4 \) grams,
- \( A = 0.8 \) grams.
Substituting the values into the formula, we have:
\[
0.8 = 4 \left( \frac{1}{2} \right)^n
\]
2. Simplify the equation:
\[
\frac{0.8}{4} = \left( \frac{1}{2} \right)^n
\]
\[
0.2 = \left( \frac{1}{2} \right)^n
\]
3. Now, we convert 0.2 to a fraction:
\[
0.2 = \frac{2}{10} = \frac{1}{5}
\]
4. We need to express \( \frac{1}{5} \) in a way that relates to \( \left( \frac{1}{2} \right)^n \).
Remember that \( \frac{1}{2} = 2^{-1} \), so we can rewrite:
\[
\left( \frac{1}{2} \right)^n = 2^{-n}
\]
The next step involves taking the logarithm of both sides:
\[
\log(0.2) = n \cdot \log(0.5)
\]
From this, we can isolate \( n \):
\[
n = \frac{\log(0.2)}{\log(0.5)}
\]
5. Computing the values using a calculator:
- \( \log(0.2) \approx -0.699 \)
- \( \log(0.5) \approx -0.301 \)
Now calculate:
\[
n \approx \frac{-0.699}{-0.301} \approx 2.32
\]
This means it will take approximately 2.32 half-lives to get from 4 grams to 0.8 grams.
6. Multiply the number of half-lives by the duration of each half-life:
\[
\text{Total time} = n \times \text{half-life} = 2.32 \times 120 \, \text{years} \approx 278.4 \, \text{years}
\]
Thus, it will take approximately **278.4 years** for the amount of the radioactive substance to decay from 4 grams to 0.8 grams.
1. First, calculate the number of half-lives required for the substance to decay from 4 grams to 0.8 grams.
The amount of substance remaining after \( n \) half-lives can be calculated using the formula:
\[
A = A_0 \left( \frac{1}{2} \right)^n
\]
where:
- \( A \) is the final amount of the substance,
- \( A_0 \) is the initial amount of the substance,
- \( n \) is the number of half-lives.
In this case:
- \( A_0 = 4 \) grams,
- \( A = 0.8 \) grams.
Substituting the values into the formula, we have:
\[
0.8 = 4 \left( \frac{1}{2} \right)^n
\]
2. Simplify the equation:
\[
\frac{0.8}{4} = \left( \frac{1}{2} \right)^n
\]
\[
0.2 = \left( \frac{1}{2} \right)^n
\]
3. Now, we convert 0.2 to a fraction:
\[
0.2 = \frac{2}{10} = \frac{1}{5}
\]
4. We need to express \( \frac{1}{5} \) in a way that relates to \( \left( \frac{1}{2} \right)^n \).
Remember that \( \frac{1}{2} = 2^{-1} \), so we can rewrite:
\[
\left( \frac{1}{2} \right)^n = 2^{-n}
\]
The next step involves taking the logarithm of both sides:
\[
\log(0.2) = n \cdot \log(0.5)
\]
From this, we can isolate \( n \):
\[
n = \frac{\log(0.2)}{\log(0.5)}
\]
5. Computing the values using a calculator:
- \( \log(0.2) \approx -0.699 \)
- \( \log(0.5) \approx -0.301 \)
Now calculate:
\[
n \approx \frac{-0.699}{-0.301} \approx 2.32
\]
This means it will take approximately 2.32 half-lives to get from 4 grams to 0.8 grams.
6. Multiply the number of half-lives by the duration of each half-life:
\[
\text{Total time} = n \times \text{half-life} = 2.32 \times 120 \, \text{years} \approx 278.4 \, \text{years}
\]
Thus, it will take approximately **278.4 years** for the amount of the radioactive substance to decay from 4 grams to 0.8 grams.