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Chapter 8: Problem 19

If isotopic distribution of \(\mathrm{C}^{12}\) and \(\mathrm{C}^{14}\) is \(98.0 \%\) and \(2.0 \%\), respectively, then the number of \(\mathrm{C}^{14}\) atoms in \(12 \mathrm{~g}\) of carbon is (a) \(1.032 \times 10^{22}\) (b) \(1.20 \times 10^{22}\) (c) \(5.88 \times 10^{23}\) (d) \(6.02 \times 10^{23}\)

Short Answer

Expert verified
Answer: (b) \(1.20 \times 10^{22}\)

Step by step solution

01

Calculate the ratio of C-14 atoms to the total number of atoms

According to the isotopic distribution, 2% of the total carbon atoms in the sample are C-14. Hence, the ratio of C-14 atoms to the total number of carbon atoms is \(2/100 = 1/50\).
02

Calculate the number of moles of carbon in the sample

To find the number of moles of carbon, we will use the formula: Number of Moles = \(\frac{\text{Mass of the sample}}{\text{Molar mass of the element}}\) The molar mass of carbon is 12 g/mol. Therefore, the number of moles in 12 g of carbon is: Number of Moles = \(\frac{12 \text{ g}}{12 \text{ g/mol}} = 1 \text{ mol}\)
03

Calculate the total number of atoms in the sample

Since there are 1 mole of carbon atoms in the sample, we can find the total number of carbon atoms using Avogadro's number (\(N_A = 6.022 × 10^{23}\) atoms/mol). Total number of carbon atoms = Number of Moles × \(N_A\) = \(1 \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} = 6.022 \times 10^{23} \text{ atoms}\)
04

Calculate the number of C-14 atoms in the sample

We know that the ratio of C-14 atoms to the total number of carbon atoms is 1/50. Applying this ratio to the total number of atoms in the sample, we can find the number of C-14 atoms: Number of C-14 atoms = \(\frac{1}{50} \times 6.022 \times 10^{23} \text{ atoms} = 1.204 \times 10^{22} \text{ atoms}\)
05

Compare the calculated value with the options

The calculated number of C-14 atoms in the sample is \(1.204 \times 10^{22}\) atoms, which is closest to option (b) \(1.20 \times 10^{22}\). Therefore, the correct answer is: (b) \(1.20 \times 10^{22}\)

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