Question

A chemist wishing to do an experiment requiring \({ }^{47} \mathrm{Ca}^{2+}\) (halflife \(=4.5\) days) needs \(5.0 \mu \mathrm{g}\) of the nuclide. What mass of \({ }^{47} \mathrm{CaCO}_{3}\) must be ordered if it takes \(48 \mathrm{~h}\) for delivery from the supplier? Assume that the atomic mass of \({ }^{47} \mathrm{Ca}\) is \(47.0 .\)

Short answer

The chemist must order approximately 11.3 µg of CaCO3 to have 5.0 µg of Ca-47 for the experiment, considering the 48-hour delivery time.

Step-by-step solution

Convert the half-life and delivery time to the same unit of time

To be consistent with our calculations, we need to convert the half-life and the delivery time into the same unit of time, which is hours in this case. Half-life of Ca-47 = 4.5 days × 24 hours/day = 108 hours Delivery time = 48 hours

Calculate the decay constant

We will calculate the decay constant (λ) using the formula: \( \lambda = \frac{ln(2)}{t_{1/2}} \), where \(t_{1/2}\) is the half-life. Decay constant (λ) = \( \frac{ln(2)}{108} \approx 0.00642 \ \mathrm{h}^{-1}\)

Determine the initial mass of Ca-47 required

We will use the radioactive decay formula to find the initial mass of Ca-47 considering the 48-hour delivery time. \(m_{initial} = \frac{m_{final}}{e^{-\lambda t}} \), where \(m_{final}\) is the desired mass (5.0 µg), t is the delivery time (48 hours), and λ is the decay constant. Initial mass of Ca-47 required = \(\frac{5.0}{e^{-0.00642 \times 48}} \approx 5.31 \ \mu\mathrm{g}\)

Calculate the mass fraction of Ca-47 in CaCO3

To find the mass fraction of Ca-47 in CaCO3, we will divide the atomic mass of Ca-47 by the molecular mass of CaCO3. Atomic mass of Ca-47 = 47.0 Molecular mass of CaCO3 = 47.0 (Ca) + 12.0 (C) + 3 × 16.0 (O) = 100.0 Mass fraction of Ca-47 in CaCO3 = \(\frac{47.0}{100.0}\) = 0.47

Calculate the mass of CaCO3 needed

We will now calculate the mass of CaCO3 required using the mass fraction of Ca-47 and the initial mass of Ca-47 required. Mass of CaCO3 required = \(\frac{5.31 \ \mu g}{0.47}\) ≈ 11.3 µg So, the chemist must order 11.3 µg of CaCO3 to have 5.0 µg of Ca-47 for the experiment, considering the 48-hour delivery time.

Key concepts

Half-Life

The concept of half-life is crucial in understanding radioactive decay. It refers to the amount of time it takes for half of the atoms in a sample of a radioactive isotope to decay. This is an exponential decay process, meaning that in each successive half-life, half of the remaining radioactive atoms will decay. For example, if you start with a 100g sample and the half-life is 2 years, after 2 years you would have 50g remaining, after 4 years, 25g, and so on.

Knowing the half-life of a radioisotope helps chemists and physicists predict how long a sample will remain active or how long it might pose a hazard. Understanding this concept also allows for calculations similar to the textbook exercise, where it is necessary to account for the decay of an isotope over a certain period, such as the 48-hour delivery time for the Ca-47 isotope.

Decay Constant

The decay constant, denoted by the symbol \(\lambda\), represents the probability per unit time that an atom of a radioactive substance will decay. It is related to the half-life of the isotope and can be calculated using the formula \(\lambda = \frac{\ln(2)}{t_{1/2}}\), where \(t_{1/2}\) is the half-life of the substance. The decay constant is a cornerstone of the mathematical equations used to describe the rate of radioactive decay.

The decay constant is inherently linked to the stability of the nucleus; the larger the decay constant, the faster the isotope will decay. It provides an essential means to quantify and compare the stableness of different isotopes, facilitating radioisotope purification and other processes in nuclear chemistry.

Radioisotope Purification

Radioisotope purification is a technique used to isolate a single type of radioactive isotope from a mixture of isotopes. This process is particularly important because the presence of impurities can influence the behavior and measurement of radioactive samples in scientific experiments. The purification process may involve chemical separation techniques, distillation, or the use of centrifuges.

In our exercise, once the chemist receives the sample of Ca-47, they may further purify it to ensure that it is suitable for experiments. The precision in the initial calculation of the required mass, accounting for decay, allows for more efficient and effective purification, reducing waste and improving the accuracy of subsequent experiments.

Nuclear Chemistry

Nuclear chemistry is the study of changes that occur in atomic nuclei. This field delves into various reactions, including radioactive decay, nuclear fission, and fusion. A fundamental aspect of nuclear chemistry is understanding how elements transform into others, emitting radiation in the process. This not only relates to natural processes but also to the synthesis of new elements and isotopes in a laboratory setting.

For instance, the need to calculate the original mass of \(^{47}\mathrm{CaCO}_{3}\) for the chemist's experiment is directly tied to the concepts of nuclear chemistry. It involves predicting the changes that \(^{47}\mathrm{Ca}^{2+}\) will undergo during its decay process and ensuring that a sufficient amount is present for the experiment after accounting for radioactive decay.