To determine the decay rate [tex]\( k \)[/tex] for a radioactive substance given its half-life, we can use the formula for the decay constant in exponential decay processes:
[tex]\[ k = \frac{\ln(2)}{T_{\frac{1}{2}}} \][/tex]
where:
- [tex]\( \ln(2) \)[/tex] is the natural logarithm of 2, which is a constant approximately equal to 0.693.
- [tex]\( T_{\frac{1}{2}} \)[/tex] is the half-life of the substance in years.
Given the half-life [tex]\( T_{\frac{1}{2}} = 3552 \)[/tex] years, we substitute this value into the formula:
[tex]\[ k = \frac{\ln(2)}{3552} \][/tex]
[tex]\[ k = \frac{0.693}{3552} \][/tex]
By performing this division, we find:
[tex]\[ k \approx 0.000195016197 \][/tex]
Rounding this result to six decimal places, we get:
[tex]\[ k \approx 0.000195 \][/tex]
Therefore, the decay rate [tex]\( k \)[/tex] for the half-life of 3552 years is:
[tex]\[ k = 0.000195 \][/tex]
So, the completed table will be:
\begin{tabular}{|l|l|}
\hline
Half-Life & Decay Rate, [tex]\( k \)[/tex] \\
\hline
3552 years & 0.000195 \\
\hline
\end{tabular}
