Question

the half - life of palladium - 100 is 4 days. after 24 days a sample of palladium - 100 has been reduced to a mass of 7 mg.
what was the initial mass (in mg) of the sample?

what is the mass 8 weeks after the start?

Explanation:

Step1: Calculate the number of half - lives

The formula for the number of half - lives \(n=\frac{t}{T}\), where \(t\) is the time elapsed and \(T\) is the half - life. Given \(t = 24\) days and \(T=4\) days, then \(n=\frac{24}{4}=6\).
The mass formula is \(m = m_0(\frac{1}{2})^n\), where \(m\) is the final mass, \(m_0\) is the initial mass. We know \(m = 7\) mg and \(n = 6\). So, \(7=m_0(\frac{1}{2})^6\).

Step2: Solve for the initial mass \(m_0\)

From \(7=m_0(\frac{1}{2})^6\), we can rewrite it as \(m_0=7\times2^6\).
Since \(2^6 = 64\), then \(m_0=7\times64 = 448\) mg.

Step3: Calculate the number of half - lives for 8 weeks

8 weeks is \(8\times7=56\) days. Using \(n=\frac{t}{T}\), with \(t = 56\) days and \(T = 4\) days, we get \(n=\frac{56}{4}=14\).
Using the formula \(m = m_0(\frac{1}{2})^n\), with \(m_0 = 448\) mg and \(n = 14\), we have \(m=448\times(\frac{1}{2})^{14}\).
Since \((\frac{1}{2})^{14}=\frac{1}{16384}\), then \(m=\frac{448}{16384}=\frac{7}{256}\approx0.02734375\) mg.

Answer:

Initial mass: \(448\) mg
Mass after 8 weeks: \(\frac{7}{256}\approx0.02734375\) mg