Question

6.7: homework assignment
score: 2.5/10 answered: 2/10
question 3
the half - life of palladium - 100 is 4 days. after 16 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 (in mg) 5 weeks after the start?
you may enter the exact value or round to 4 decimal places.
question help: video
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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 \(t = 16\) days and \(T=4\) days, then \(n=\frac{16}{4}=4\).
The formula for radioactive decay is \(m = m_0(\frac{1}{2})^n\), where \(m\) is the final mass, \(m_0\) is the initial mass, and \(n\) is the number of half - lives. We know \(m = 7\) mg and \(n = 4\). Rearranging the formula for \(m_0\) gives \(m_0=m\times2^n\).
Substitute \(m = 7\) and \(n = 4\) into the formula: \(m_0=7\times2^4\).

$$m_0=7\times16 = 112$$

Step2: Calculate the number of half - lives for 5 weeks

Since 1 week has 7 days, 5 weeks have \(t=5\times7 = 35\) days. Using \(n=\frac{t}{T}\) with \(T = 4\) days, \(n=\frac{35}{4}=8.75\).
Using the formula \(m = m_0(\frac{1}{2})^n\), with \(m_0 = 112\) and \(n = 8.75\)

$$m=112\times(\frac{1}{2})^{8.75}=112\times2^{-8.75}$$

$$2^{-8.75}=2^{-(8 + 0.75)}=2^{-8}\times2^{-0.75}=\frac{1}{256}\times\frac{1}{\sqrt{2^{1.5}}}=\frac{1}{256}\times\frac{1}{2\sqrt{2}}$$

$$m = 112\times\frac{1}{256}\times\frac{1}{2\sqrt{2}}=\frac{112}{512\sqrt{2}}=\frac{7}{32\sqrt{2}}\approx\frac{7}{32\times1.4142}\approx0.1543$$

Answer:

The initial mass is \(112\) mg. The mass 5 weeks after the start is approximately \(0.1543\) mg.