Question

The curie (Ci) is a commonly used unit for measuring nuclear radioactivity: 1 curie of radiation is equal to \(3.7 \times 10^{10}\) decay events per second (the number of decay events from \(1 \mathrm{~g}\) radium in \(1 \mathrm{~s}\) ). a. What mass of \(\mathrm{Na}_{2}{ }^{38} \mathrm{SO}_{4}\) has an activity of \(10.0 \mathrm{mCi}\) ? Sulfur38 has an atomic mass of \(38.0\) and a half-life of \(2.87 \mathrm{~h}\). b. How long does it take for \(99.99 \%\) of a sample of sulfur- 38 to decay?

Short answer

The mass of \(\mathrm{Na}_{2}{ }^{38} \mathrm{SO}_{4}\) with an activity of 10.0 mCi is approximately \(3.50 \times 10^{-9}~\mathrm{g}\). And it takes about \(6.42 \times 10^4~\mathrm{s}\) for 99.99% of a sample of sulfur-38 to decay.

Step-by-step solution

Part a: Calculate the mass of \(\mathrm{Na}_{2}{ }^{38} \mathrm{SO}_{4}\)

We need to find the mass of \(\mathrm{Na}_{2}{ }^{38} \mathrm{SO}_{4}\) that has an activity of 10.0 mCi. Step 1: Convert activity to decay events per second Given activity in mCi, we need to convert it to decay events per second: \(A = 10.0 \times 10^{-3}~\mathrm{Ci} \times 3.7 \times 10^{10}~\mathrm{decay/s/Ci} = 3.7 \times 10^8~\mathrm{decay/s}\) Step 2: Calculate the decay constant Now let's find the decay constant using the given half-life: \(\lambda = \frac{\ln{2}}{2.87~\mathrm{h}} \times \frac{3600~\mathrm{s}}{1~\mathrm{h}} = 6.68 \times 10^{-5}~\mathrm{s^{-1}}\) Step 3: Calculate the number of radioactive nuclei Using the activity formula, we can find the number of radioactive nuclei for \(\mathrm{Na}_{2}{ }^{38} \mathrm{SO}_{4}\): \(N = \frac{A}{\lambda} = \frac{3.7 \times 10^8~\mathrm{decay/s}}{6.68 \times 10^{-5}~\mathrm{s^{-1}}} = 5.54 \times 10^{12}\) Step 4: Calculate the mass of \(\mathrm{Na}_{2}{ }^{38} \mathrm{SO}_{4}\) Now let's find the mass of the sample using the number of radioactive nuclei: \(mass = \frac{N \times m_{S}}{N_{A}}\) Where \(m_{S}\) is the atomic mass of sulfur-38 (38.0 g/mol) and \(N_{A}\) is Avogadro's number (\(6.022 \times 10^{23}\) mol/g): \(mass = \frac{5.54 \times 10^{12}~\mathrm{nuclei} \times 38.0~\mathrm{g/mol}}{6.022 \times 10^{23}~\mathrm{mol^{-1}}}\) \(mass \approx 3.50 \times 10^{-9}~\mathrm{g}\) So, the mass of \(\mathrm{Na}_{2}{ }^{38} \mathrm{SO}_{4}\) with an activity of 10.0 mCi is about \(3.50 \times 10^{-9}~\mathrm{g}\).

Part b: Calculate the decay time for 99.99% of sulfur-38

Now, let's find how long it takes for 99.99% of a sample of sulfur-38 to decay. Step 1: Calculate the remaining percentage of radioactive nuclei 99.99% decay means only \(1 - 0.9999 = 0.0001\) (0.01%) of radioactive nuclei remain. Step 2: Use the decay formula to find the time Let \(N = 0.0001N_{0}\), then we have: \(0.0001N_{0} = N_{0} \times e^{-\lambda t}\) Step 3: Solve for time Divide both sides by \(N_{0}\) and take the natural logarithm of both sides to solve for time: \(t = \frac{-\ln{0.0001}}{\lambda}\) Using the decay constant found in part a: \(t = \frac{-\ln{0.0001}}{6.68 \times 10^{-5}~\mathrm{s^{-1}}}\) \(t \approx 6.42 \times 10^4~\mathrm{s}\) So, it takes about \(6.42 \times 10^4~\mathrm{s}\) for 99.99% of a sample of sulfur-38 to decay.

Key concepts

Decay Events Per Second

Understanding the frequency of decay events per second is crucial when studying the radioactivity of a substance. Radioactivity is the process by which an unstable atomic nucleus loses energy by emitting radiation. This is measured in terms of decay events per second, which indicates how many atoms in a sample disintegrate every second, producing radiation in the process. One of the standard units for this measurement is the curie (Ci), where 1 Ci is equivalent to an immense amount, precisely \(3.7 \times 10^{10}\) decay events per second. This unit provides a tangible number to represent the rate at which a substance undergoes nuclear decay.

As an example, let's take a closer look at an exercise involving the unit curie: when we're given the radioactivity of a substance in millicuries (mCi), like in our exercise scenario, it's necessary to convert this into decay events per second to further analyze the substance's properties. This conversion lays the groundwork for more complex calculations, such as determining the mass of a radioactive sample or predicting how long a radioactive isotope will take to reach a certain level of decay.

Half-Life Calculation

The half-life of a radioactive isotope is a pivotal concept in nuclear chemistry; it's the time it takes for half the atoms in a given sample to decay. This constant, unique to each isotope, allows scientists to calculate not only the rate of decay but also to predict the behavior of radioactive material over time.

When we calculate the half-life, we're often interested in how much time will pass before a certain percentage of the original sample remains. To perform such calculations, we need the decay constant (\(\text{defined as } \[ \lambda = \frac{\ln{2}}{t_{1/2}} \]\)), which is inversely proportional to the half-life. The decay constant is a signature to each radioactive isotope and plays a central role in the exponential decay formula used to determine the remaining mass or quantity of an isotope after a given time. For instance, in the given exercise, knowing the half-life of sulfur-38 allows us to find the decay constant and subsequently perform calculations related to the decay process.

Radioactive Decay

Radioactive decay is a stochastic (random) process at the level of single atoms, which results in an unstable nucleus releasing energy to attain a more stable configuration. This decay can occur in several ways, including alpha decay, beta decay, and gamma radiation, each involving the emission of different particles or energy. Its unpredictability on an atomic level nevertheless follows a pattern that can be described statistically for a large number of nuclei, and is characterized by the half-life, which we worked with in the exercise.

By using formulas derived from the principles of radioactive decay, we can quantify how a sample's activity declines over time. For example, the exercise demonstrates how to calculate the time necessary for a significant percentage of sulfur-38 to decay using its decay constant and the natural logarithm of the remaining proportion of the sample. Understanding the step-by-step computations in radioactive decay not only allows one to predict the behavior of radioactive samples but also holds importance in applications ranging from medical treatments with radioactive isotopes to determining the age of archaeological finds through radiocarbon dating.