To complete the table for the radioactive substance Polonium (Po-198), let's walk through the steps to determine the decay rate given the half-life.
Given:
- The half-life (T) for Polonium (Po-198) is 1.8 minutes.
We are asked to find the decay rate (k) based on this half-life.
The relationship between a radioactive substance's decay rate and its half-life is given by the formula:
[tex]\[ k = \frac{\ln(2)}{T} \][/tex]
1. First, we need to calculate the natural logarithm of 2, which is a constant:
[tex]\[ \ln(2) \approx 0.693 \][/tex]
2. Using the half-life (T = 1.8 minutes), we substitute into the formula:
[tex]\[ k = \frac{0.693}{1.8} \][/tex]
3. Performing the division to find the decay rate (k):
[tex]\[ k \approx 0.3850817669777474 \][/tex]
4. Finally, we round the decay rate to four decimal places:
[tex]\[ k \approx 0.3851 \][/tex]
So, the completed table will be:
\begin{tabular}{|c|c|c|}
\hline
\begin{tabular}{c}
Radioactive \\
Substance
\end{tabular} & Decay Rate, k & Half-Life T \\
\hline
\begin{tabular}{c}
a) Polonium \\
(Po-198)
\end{tabular} & [tex]\(0.3851\)[/tex] per min & 1.8 mins \\
\hline
\end{tabular}
