Question

complete the table shown to the right for the half - life of a certain radioactive substance. half - life 12.9 days decay rate, k k = □ (round to six decimal places as needed.)

Explanation:

Step1: Use the half - life formula

The formula for the half - life \(T\) of a radioactive substance is \(T=\frac{\ln2}{k}\), where \(T\) is the half - life and \(k\) is the decay constant.
We are given \(T = 13.9\) days. Rearranging the formula for \(k\), we get \(k=\frac{\ln2}{T}\).

Step2: Substitute the value of \(T\)

Substitute \(T = 13.9\) into the formula \(k=\frac{\ln2}{T}\).
Since \(\ln2\approx0.693147\), then \(k=\frac{0.693147}{13.9}\).

Step3: Calculate the value of \(k\)

\(k=\frac{0.693147}{13.9}\approx0.049867\)

Answer:

\(k\approx0.049867\)