Problem 16 - Analytical Chemistry

Question

\({ }^{60} \mathrm{Co}\) is a long-lived isotope \(\left(t_{1 / 2}=5.3 \mathrm{yr}\right)\) frequently used as a radiotracer. The activity in a 5.00 -mL sample of a solution of \({ }^{60} \mathrm{Co}\) is \(2.1 \times 10^{7}\) disintegrations/sec. What is the molar concentration of \({ }^{60} \mathrm{Co}\) in the sample?

Step-by-Step Solution

Answer

The molar concentration of { }^{60} \mathrm{Co} is \(4.11 \times 10^{-6} \text{ mol/L}\).

Step 1: Understanding the relationship

First, we need to understand the relationship between disintegration rate (activity) and the number of radioactive nuclei. The activity (A) of a sample is related to the number of radioactive nuclei (N) and the decay constant (λ) by the formula: \[ A = \lambda N \].

Step 2: Determine the decay constant

The decay constant (λ) can be calculated using the half-life ( t_{1/2}), which is given by the formula: \[ \lambda = \frac{0.693}{t_{1/2}} \]. Plugging in the half-life of { }^{60} \mathrm{Co}, which is 5.3 years (converted to seconds since activity is per second), we find:\[ \lambda = \frac{0.693}{5.3 \times 365 \times 24 \times 3600} \, \text{s}^{-1} \].

Step 3: Calculate the number of radioactive nuclei

Next, calculate the number of radioactive nuclei (N) using the activity provided and the decay constant computed. Rearrange the formula for activity:\[ N = \frac{A}{\lambda} \]. Substitute the value of A (2.1 \times 10^{7} \, \text{s}^{-1}) and \(\lambda\) calculated in Step 2.

Step 4: Find the moles of { }^{60} \mathrm{Co}

Convert the number of radioactive nuclei (N) found in Step 3 into moles. Remember, 1 mole is equal to Avogadro's number, \(6.022 \times 10^{23}\) nuclei/mole. So,\[ \text{moles of } { }^{60} \mathrm{Co} = \frac{N}{6.022 \times 10^{23}} \].

Step 5: Determine the molar concentration

The molar concentration of { }^{60} \mathrm{Co} is found by dividing the moles of { }^{60} \mathrm{Co} by the volume of the solution in liters (5.00 mL = 0.005 L). \[ \text{Molar concentration} = \frac{\text{moles of } { }^{60} \mathrm{Co}}{0.005} \].

Key Concepts

Radiotracer

Radiotracers are radioactive isotopes used in various scientific fields to track the movement and chemical processes in a system. In medicine, they help in imaging and diagnosing by being introduced into the body, where their progress can be visualized. In this exercise, Co^{60} stands out as the radiotracer. It is often chosen because it has a half-life of 5.3 years, meaning it remains active for a significant but not excessive amount of time. This property makes it effective for experiments that require monitoring over weeks, months, and even years. These isotopes have a distinct disintegration rate useful for various calculations and examinations, including determining quantity and concentration in a solution.

Half-life Calculation

The half-life of a radioactive isotope is the time taken for its activity to reduce to half its original value. Knowing an isotope's half-life is crucial as it helps you understand how long the isotope will remain active. For ^{60}Co, the half-life is given as 5.3 years. Calculating the decay constant from this primary attribute involves the formula \( \lambda = \frac{0.693}{t_{1/2}} \). This formula helps convert the half-life into a rate at which the nuclei will decay per unit time. By converting 5.3 years into seconds before plugging it into the formula, we adjust for the fact that activity measurements are often in disintegrations per second.

Molar Concentration Calculation

Molar concentration measures the number of moles of a substance per liter of solution. To find this value for ^{60}Co, first calculate how many moles are present by dividing the number of radioactive nuclei by Avogadro's number (6.022 \times 10^{23}). This yields how many moles of ^{60}Co are present in the given volume of the solution. Divide this by the volume of the solution in liters to find molarity. For a solution volume of 5.00 mL (0.005 L), molarity is computed as \( \text{Molar concentration} = \frac{\text{moles of } { }^{60} \text{Co}}{0.005} \), giving a precise concentration.

Decay Constant

The decay constant (\lambda) is a probability rate of decay per unit time for a radioactive sample. It provides a direct link to the half-life of a particle and hence its decay dynamics. The formula to find the decay constant using half-life is \( \lambda = \frac{0.693}{t_{1/2}} \), representing the natural logarithm approximation for half-time decay. Converting the half-life from years to the matching unit for disintegration rate (seconds) ensures that the decay constant accurately fits into the larger process of calculating the number of remaining radioactive atoms and their activity.

Disintegration Rate

The disintegration rate or activity of a radioactive sample is denoted by A and measures the number of decays occurring per second. It's given initially in the problem as 2.1 \times 10^7 \text{s}^{-1}. Calculating the actual number of radioactive nuclei requires rearranging the formula \( A = \lambda N \), or \( N = \frac{A}{\lambda} \). Knowing this activity helps pinpoint how much of the radioactive substance remains active at a given point. The disintegration rate is central to solving problems related to radiotracers as it helps determine concentration precisely and efficiently.