Questions: 7. The half-life of 99Tc is 6.00 hours. If it takes exactly 12.00 hours for the manufacturer to deliver a 99Tc sample to a hospital, how much must be shipped in order for the hospital to receive 10.0 mg ?

Solution Steps

The problem involves radioactive decay, specifically the decay of technetium-99 (\(^{99}\text{Tc}\)), which has a half-life of 6.00 hours. We need to determine the initial amount of \(^{99}\text{Tc}\) that must be shipped so that 10.0 mg is received by the hospital after 12.00 hours.

The decay of a radioactive substance can be described by the formula:

\[ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]

where:

• \(N(t)\) is the remaining quantity after time \(t\),
• \(N_0\) is the initial quantity,
• \(T_{1/2}\) is the half-life of the substance,
• \(t\) is the elapsed time.

We know:

• \(N(t) = 10.0 \, \text{mg}\) (the amount received by the hospital),
• \(t = 12.00 \, \text{hours}\),
• \(T_{1/2} = 6.00 \, \text{hours}\).

Substitute these values into the formula:

\[ 10.0 = N_0 \left(\frac{1}{2}\right)^{\frac{12.00}{6.00}} \]

Calculate the exponent:

\[ \left(\frac{1}{2}\right)^{\frac{12.00}{6.00}} = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

Substitute back into the equation:

\[ 10.0 = N_0 \times \frac{1}{4} \]

Solve for \(N_0\):

\[ N_0 = 10.0 \times 4 = 40.0 \, \text{mg} \]

Final Answer