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Chapter 19: Problem 35

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

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

Step by step solution

01

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: \[5.0\;\mu\text{g} \times 2^{0.4444} = 7.079\;\mu\text{g}\]
02

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 \(^{47} \mathrm{Ca}\): \[7.079\;\mu\text{g} \times \frac{1\;\text{mol}}{47.0\;\text{u}} = 1.506\;\text{µmol}\]
03

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: \[1.506\;\text{µmol}\; \mathrm{Ca}^{2+} = 1.506\;\text{µmol}\;\mathrm{CaCO}_{3}\]
04

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.

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Most popular questions from this chapter

Which do you think would be the greater health hazard: the release of a radioactive nuclide of Sr or a radioactive nuclide of Xe into the environment? Assume the amount of radioactivity is the same in each case. Explain your answer on the basis of the chemical properties of Sr and Xe. Why are the chemical properties of a radioactive substance important in assessing its potential health hazards?

Calculate the binding energy in J/nucleon for carbon-12 (atomic mass \(=12.0000\) u) and uranium-235 (atomic mass \(=\) 235.0439 u). The atomic mass of \(_{1}^{1} \mathrm{H}\) is 1.00782 \(\mathrm{u}\) and the mass of a neutron is 1.00866 u. The most stable nucleus known is \(^{56}\) Fe $(\text { see Exercise } 50)\( . Would the binding energy per nucleon for \)^{56} \mathrm{Fe}$ be larger or smaller than that of \(^{12} \mathrm{C}\) or \(^{235} \mathrm{U}\) ? Explain.

Technetium- 99 has been used as a radiographic agent in bone scans $(43 \mathrm{Tc} \text { is absorbed by bones). If } 43 \mathrm{Tc} \text { has a half-life of }\( 6.0 hours, what fraction of an administered dose of \)100 . \mu \mathrm{g}\( 43 \)\mathrm{Tc}$ remains in a patient's body after 2.0 days?

What are transuranium elements and how are they synthesized?

Which of the following statement(s) is (are) true? a. A radioactive nuclide that decays from \(2.00 \times 10^{21}\) atoms to $5.0 \times 10^{20}$ atoms in 16 minutes has a half-life of 8.0 minutes. b. Nuclides with large \(Z\) values are observed to be \(\alpha\) -particle producers. c. As \(Z\) increases, nuclides need a greater proton-to-neutron ratio for stability. d. Those light nuclides that have twice as many neutrons as protons are expected to be stable.
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