Question

A chemist wishing to do an experiment requiring \(^{47} \mathrm{Ca}^{2+}\) (half- life \(=4.5\) 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 \(\mathrm{u} .\)

Short answer

In conclusion, the chemist must order 150.6 µg of \(^{47} \mathrm{CaCO}_{3}\) for the experiment.

Step-by-step solution

Calculate the amount of \(^{47} \mathrm{Ca}^{2+}\) needed at the time of delivery

We are given that the experiment requires 5.0 µg of the nuclide and the delivery takes 48 hours. Since the half-life of \(^{47} \mathrm{Ca}^{2+}\) is 4.5 days, we need to calculate how many half-lives have passed during this delivery period. First, convert the 48 hours delivery period into days: \[48 \;\text{hours} = 48\;\text{hours} \times \frac{1\;\text{day}}{24\;\text{hours}} = 2\;\text{days}\] Next, calculate the number of half-lives that passed during these 2 days: \[\frac{2\;\text{days}}{4.5\;\text{days/half-life}} = 0.4444\;\text{half-lives}\] Now, we calculate how much \(^{47} \mathrm{Ca}^{2+}\) is needed at the time of delivery: \[5.0\;\mu\text{g} \times 2^{0.4444} = 7.079\;\mu\text{g}\]

Calculate the moles of \(^{47} \mathrm{Ca}^{2+}\) required

Now that we know how much \(^{47} \mathrm{Ca}^{2+}\) is needed at the time of delivery, we can calculate the required moles by dividing the mass by the atomic mass of \(^{47} \mathrm{Ca}\): \[7.079\;\mu\text{g} \times \frac{1\;\text{mol}}{47.0\;\text{u}} = 1.506\;\text{µmol}\]

Calculate the moles of \(^{47} \mathrm{CaCO}_{3}\) needed

Since one mole of \(^{47} \mathrm{CaCO}_{3}\) contains one mole of \(^{47} \mathrm{Ca}^{2+}\), the moles of \(^{47} \mathrm{CaCO}_{3}\) needed will be equal to the moles of \(^{47} \mathrm{Ca}^{2+}\) required: \[1.506\;\text{µmol}\; \mathrm{Ca}^{2+} = 1.506\;\text{µmol}\;\mathrm{CaCO}_{3}\]

Determine the mass of \(^{47} \mathrm{CaCO}_{3}\) needed to be ordered

To determine the mass of \(^{47} \mathrm{CaCO}_{3}\) needed, multiply the moles by the molar mass of \(^{47} \mathrm{CaCO}_{3}\). The molar mass of \(^{47} \mathrm{CaCO}_{3}\) is approximately 47.0 u (for \(^{47} \mathrm{Ca}\)) + 12.0 u (for C) + 3 × 16.0 u (for 3O) = 100.0 u: \[1.506\;\text{µmol}\;\mathrm{CaCO}_{3} = 1.506\;\text{µmol} \times \frac{100.0\;\text{u}}{1\;\text{mol}} = 150.6\;\mu\text{g}\; \mathrm{CaCO}_{3}\] In conclusion, the chemist must order 150.6 µg of \(^{47} \mathrm{CaCO}_{3}\) for the experiment.

Key concepts

Half-life

Half-life is a crucial concept in the study of radioactive decay, as it denotes the time required for half of the radioactive isotopes in a sample to decay. To calculate how much of a radioactive substance remains after a certain period, you need to know its half-life. In our given problem, the half-life of \(^{47} \mathrm{Ca}^{2+} \)is 4.5 days. Understanding half-life allows us to predict how much of the substance will remain after two days of delivery time.
To find the decay factor, divide the elapsed time by the half-life: 2 days/4.5 days = 0.4444 half-lives. This gives an idea of how much decay has occurred.Using the decay formula,\[\text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^{\text{Number of half-lives}}.\]

This calculation helps determine how much of the substance will be available for the experiment at the time of delivery after accounting for decay.

Radioactive Isotopes

Radioactive isotopes, also known as radioisotopes, are variants of chemical elements that emit radiation as their unstable nuclei break down into a more stable form. Radioisotopes have numerous applications, including in medical, scientific, and industrial fields. In the scenario outlined in the problem, \(^{47} \mathrm{Ca}^{2+} \) is a radioactive isotope utilized for experiments.
Understanding the behavior of radioactive isotopes is crucial as it determines how you plan and execute experiments. Since they decay into other elements over time, you must factor in the decay process when determining the quantities required at specific times.In practical applications, it is essential to consider how the decay impacts the number of isotopes you start with. Key concepts like decay constant, activity, and decay series relate to how quickly an isotope will lose its radioactivity, helping chemists and researchers manage and plan their experiments effectively.

Molar Mass Calculations

Molar mass calculations are essential in converting between the mass of a substance and the amount of substance (moles). When working with isotopes like \(^{47} \mathrm{Ca}^{2+} \)or compounds like \(^{47} \mathrm{CaCO}_{3} \),it's important to know the mass of one mole of these substances, expressed in units of grams per mole (g/mol) or atomic mass units (u).To calculate the molar mass, sum up the atomic masses of all atoms present in the formula. For \(^{47} \mathrm{CaCO}_{3} \), the molar mass is calculated as follows:

• Calcium (Ca) = 47.0 u
• Carbon (C) = 12.0 u
• Oxygen (O) = 16.0 u each, with 3 atoms = 48.0 u

Therefore, the molar mass of \(^{47} \mathrm{CaCO}_{3} \) is 107.0 u. Molar mass calculations allow chemists to relate mass to moles, thereby making it easier to scale quantities for reactions or when ordering materials.
Using molar mass, you can convert between grams and moles – a key step in determining how much of a substance is needed.