To find the decay constant for elementium-224, we follow these steps:
1. Understand the relationship between half-life and decay constant:
The decay constant ([tex]\(k\)[/tex]) can be found using the equation:
[tex]\[
k = \frac{\ln(2)}{t_{\frac{1}{2}}}
\][/tex]
Where:
- [tex]\(\ln(2)\)[/tex] is the natural logarithm of 2, which is approximately 0.693.
- [tex]\(t_{\frac{1}{2}}\)[/tex] is the half-life of the substance.
2. Given data:
The half-life ([tex]\(t_{\frac{1}{2}}\)[/tex]) of elementium-224 is 21.99 hours.
3. Substitute the values into the formula:
[tex]\[
k = \frac{\ln(2)}{21.99}
\][/tex]
4. Calculate the decay constant:
First, calculate [tex]\(\ln(2)\)[/tex]:
[tex]\[
\ln(2) \approx 0.693
\][/tex]
Now substitute [tex]\(\ln(2)\)[/tex] and the half-life into the formula:
[tex]\[
k = \frac{0.693}{21.99}
\][/tex]
5. Perform the division:
[tex]\[
k \approx 0.031526
\][/tex]
6. Round the result to 4 decimal places:
[tex]\[
k \approx 0.0315
\][/tex]
Therefore, the decay constant for elementium-224, rounded to four decimal places, is [tex]\(0.0315\)[/tex].
