Question 14.F.1: Suppose that you are working for a company that prepares rad......
Suppose that you are working for a company that prepares radioisotopes for medical imaging. You receive an order from a researcher requesting 15 mg of { }^{45}Ti for an experiment. The customer’s laboratory is located roughly 1.5 hours from your production source. How would you determine what mass of the isotope to produce to fill this order?
Strategy The key issue here is that the isotope you produce will decay during the time that it is in transit to the research lab. Because the experiment requires a set mass of the isotope, you must consider the kinetics and produce more than is required so that there will be enough { }^{45}Ti left upon delivery. The most important piece of information to look up is the half-life of { }^{45}Ti. Once that value is known, we can use the integrated rate law for first-order kinetics to carry out the needed calculations.
Once we know the half-life, we can determine the rate constant, k:
k=\frac{0.693}{t_{1 / 2}}
Next we turn to the integrated rate law for a first-order reaction. In Equation 14.1, we wrote this as
N=N_0 e^{-k t}
Here we can use masses in place of the N and N_0 terms, giving us
m=m_0 e^{-k t}
Our goal is to find m_0, the initial mass of { }^{45}Ti needed. We have to make an assumption about the delivery time and the amount of time it would take to package the isotope for delivery. If we assume 30 minutes from production to getting the isotope aboard the delivery vehicle, we might use 120 minutes as our time.
\begin{gathered}m=m_0 e^{-k t} \\m_0=\frac{m}{e^{-k t}}=\frac{15\text{ mg} }{e^{-k(120\text{ min} )}}\end{gathered}