Question

the half - life of palladium - 100 is 4 days. after 20 days a sample of palladium - 100 has been reduced to a mass of 1 mg. what was the initial mass (in mg) of the sample? what is the mass (in mg) 8 weeks after the start? you may enter the exact value or round to 4 decimal places.

Explanation:

Step1: Calculate the number of half - lives for 20 days

The formula for the number of half - lives \(n=\frac{t}{T}\), where \(t = 20\) days and \(T = 4\) days.
\(n=\frac{20}{4}=5\)

Step2: Use the formula for radioactive decay \(m = m_0(\frac{1}{2})^n\) to find the initial mass \(m_0\)

We know \(m = 1\) mg and \(n = 5\). Rearranging the formula \(m_0=m\times2^n\)
\(m_0=1\times2^5=32\) mg

Step3: Convert 8 weeks to days

Since 1 week = 7 days, 8 weeks \(t=8\times7 = 56\) days

Step4: Calculate the number of half - lives for 56 days

\(n=\frac{56}{4}=14\)

Step5: Use the formula \(m = m_0(\frac{1}{2})^n\) to find the mass after 8 weeks

We know \(m_0 = 32\) mg and \(n = 14\)
\(m=32\times(\frac{1}{2})^{14}\)
\(m=\frac{32}{16384}=\frac{1}{512}\approx0.001953125\approx0.0020\) mg

Answer:

The initial mass of the sample is \(32\) mg.
The mass 8 weeks after the start is approximately \(0.0020\) mg.