Question

complete the table shown to the right for the half - life of a certain radioactive substance. 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 rate. We can solve for \(k\) by re - arranging the formula: \(k = \frac{\ln2}{T}\).

Step2: Substitute the value of \(T\)

Given \(T = 2095\) years. Substitute \(T\) into the formula \(k=\frac{\ln2}{T}\). So \(k=\frac{\ln2}{2095}\).

Step3: Calculate the value of \(k\)

We know that \(\ln2\approx0.693147\). Then \(k=\frac{0.693147}{2095}\).

$$k=\frac{0.693147}{2095}\approx0.000331$$

Answer:

\(0.000331\)