Question

Copper-64 is used to study tumors of the brain. It has a half-life of 12.8 hours. What is the activity (in Ci) of \(1.50 \mathrm{mg}\) \(\mathrm{Cu}-64 ?\)

Short answer

Answer: After following the step-by-step solution, the activity of the Copper-64 sample in Curies can be calculated as: Activity in Ci = \(\frac{\frac{\ln 2}{12.8} \times \frac{1.50}{64 \times 1000} \times 6.022 \times 10^{23}}{3.7 \times 10^{10}}\ \text{Ci}\)

Step-by-step solution

Determine the number of atoms of \(\mathrm{Cu}-64\)

Firstly, we need to find out how many atoms of \(\mathrm{Cu}-64\) are in the given sample of 1.50 mg. To do this, we can use the molar mass of \(\mathrm{Cu}-64\) and Avogadro's number. The molar mass of \(\mathrm{Cu}-64\) is 64 g/mol. Calculate the number of moles of \(\mathrm{Cu}-64\) in the sample and then use Avogadro's number (\(6.022 \times 10^{23}\) atoms/mol) to find the number of atoms. Number of moles of \(\mathrm{Cu}-64 = \frac{1.50 \ \text{mg}}{64\ \text{g/mol}} \times \frac{1\ \text{g}}{1000\ \text{mg}}\) Number of moles of \(\mathrm{Cu}-64 = \frac{1.50}{64 \times 1000} \ \text{mol}\) Number of atoms of \(\mathrm{Cu}-64 = \text{Number of moles of}\ \mathrm{Cu}-64 \times \text{Avogadro's number}\) Number of atoms of \(\mathrm{Cu}-64 = \frac{1.50}{64 \times 1000} \times 6.022 \times 10^{23} \ \text{atoms}\)

Calculate the activity of the sample

To find the activity, we need to calculate the decay constant using the half-life and then multiply it by the number of atoms in our sample. Decay constant (\(\lambda\)) = \(\frac{\ln 2}{\text{half-life}}\) \(\lambda = \frac{\ln 2}{12.8\ \text{hours}}\) Now, we can calculate the activity (A) in becquerels (Bq). Activity (A) = \(\lambda \times \text{Number of atoms}\) A = \(\frac{\ln 2}{12.8} \times \frac{1.50}{64 \times 1000} \times 6.022 \times 10^{23}\ \text{Bq}\) Finally, we need to convert the activity from Bq to Ci. There are \(3.7 \times 10^{10}\) Bq in 1 Ci. Activity in Ci = \(\frac{\text{Activity in Bq}}{3.7 \times 10^{10}\ \text{Bq/Ci}}\) Activity in Ci = \(\frac{\frac{\ln 2}{12.8} \times \frac{1.50}{64 \times 1000} \times 6.022 \times 10^{23}}{3.7 \times 10^{10}}\ \text{Ci}\) By calculating the above expression, we will find the activity of the \(\mathrm{Cu}-64\) sample in Curie (Ci).

Key concepts

Half-Life

The concept of half-life is crucial in understanding radioactive decay. It is the amount of time required for a quantity of a radioactive isotope to reduce to half its original amount. This concept is a key component in radioactivity calculations, as it helps determine how stable or unstable an isotope is.

For instance, Copper-64 has a half-life of 12.8 hours, meaning that every 12.8 hours, half of the sample will have decayed into other elements or isotopes. Normally, the longer the half-life, the slower the rate of decay. This information is used to determine the safety and effectiveness of radioactive materials used in medical, industrial, and scientific applications.

Avogadro's Number

Avogadro's number, approximately equal to \(6.022 \times 10^{23}\), is the number of atoms, ions, or molecules in one mole of a substance. It serves as a bridge between the macroscopic scale we observe and the microscopic scale of atoms and molecules.

When dealing with radioactive isotopes, Avogadro's number allows us to translate mass (like the 1.50 mg of Copper-64 in our exercise) into a count of individual atoms, which is essential for further radioactive decay calculations. This conversion is a fundamental step before we can calculate the sample's activity or decay rate.

Decay Constant

The decay constant, symbolized by \(\lambda\), is a probability factor that describes the likelihood of a single atom to decay per unit time. In essence, it gives us the natural logarithm's rate at which the radioactive atoms are expected to decay. Its value depends on the specific isotope and is directly related to its half-life.

The decay constant is used to find out how quickly a radioactive isotope decays, which is a crucial part of determining the activity of the isotope. By knowing the half-life, we can compute the decay constant using the formula \(\lambda = \frac{\ln 2}{\text{half-life}}\), which helps to understand the kinetics of decay for the radioactive isotope.

Activity of Radioactive Isotope

The activity of a radioactive isotope, often measured in becquerels (Bq) or curies (Ci), represents the number of decay events that occur per unit of time. It is a direct measurement of the isotope's radioactivity and depends on both the decay constant and the number of undecayed atoms present.

To calculate the activity, we multiply the decay constant by the number of atoms (calculated using Avogadro's number and the mass of the sample). For example, as shown in the solution steps, we can determine the activity of a given Copper-64 sample, which is effectively a measure of how many Copper-64 atoms are decaying each second. Understanding the activity is important for safe handling of radioactive materials and for applications where specific levels of radiation are required.