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 2 mg. what was the initial mass (in mg) of the sample? what is the mass 5 weeks after the start?

Explanation:

Step1: Calculate the number of half - lives in 24 days

The half - life of Palladium - 100 is $t_{1/2}=4$ days. The number of half - lives $n$ in $t = 24$ days is $n=\frac{t}{t_{1/2}}=\frac{24}{4}=6$.

Step2: Find the initial mass

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 $m = 2$ mg and $n = 6$. Rearranging the formula for $m_0$ gives $m_0=m\times2^n$. Substituting the values, we get $m_0=2\times2^6=2\times64 = 128$ mg.

Step3: Calculate the number of days in 5 weeks

Since 1 week has 7 days, 5 weeks have $t'=5\times7 = 35$ days. The number of half - lives $n'$ in 35 days is $n'=\frac{35}{4}=8.75$.

Step4: Find the mass after 5 weeks

Using the decay formula $m'=m_0(\frac{1}{2})^{n'}$, with $m_0 = 128$ mg and $n'=8.75$. So $m'=128\times(\frac{1}{2})^{8.75}=128\times\frac{1}{2^{8.75}}$.
$2^{8.75}=2^{8}\times2^{0.75}=256\times2^{\frac{3}{4}}=256\times\sqrt[4]{8}\approx256\times1.68179 = 430.54$.
$m'=\frac{128}{430.54}\approx0.297$ mg.

Answer:

Initial mass: 128 mg
Mass after 5 weeks: 0.297 mg