Problem 28 - Algebra and Trigonometry

Question

Use the exponential decay model, \(A=A_{0} e^{k t},\) to solve Exercises \(28-31 .\) Round answers to one decimal place. The half-life of thorium- 229 is 7340 years. How long will it take for a sample of this substance to decay to \(20 \%\) of its original amount?

Step-by-Step Solution

Answer

It will take approximately 16050.1 years for a sample of thorium-229 to decay to \(20\%\) of its original amount.

Step 1: Identify the Given Values

The half-life of thorium-229 (\(t_{1/2}\)) is given as 7340 years and we need to find the time it will take to decay to \(20\%\) of its original amount, which means \(A/A_{0} = 0.2\). The decay constant (\(k\)) can be found using the formula for half-life, \(t_{1/2} = ln(2)/k\).

Step 2: Calculate the Decay Constant

Rearranging the half-life formula to solve for \(k\), we get \(k = ln(2)/t_{1/2}\). Substituting \(t_{1/2}\) = 7340, we calculate \(k\) to be approximately \(-9.4*10^{-5}\) per year to three significant figures. It's negative because it's a decay process.

Step 3: Calculate the Time to Decay to \(20\%\) of Original Amount

Now we substitute \(A/A_{0} = 0.2\), \(k = -9.4*10^{-5}\), and \(A = A_{0}e^{kt}\) in the decay model and solve for \(t\). Rearranging the equation for \(t\) gives \(t = ln(A/A_{0})/k\). Substituting the known values gives \(t\) approximately 16050.1 years.