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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 (\(N_A = 6.022 脳 10^{23}\) atoms/mol). Total number of carbon atoms = Number of Moles 脳 \(N_A\) = \(1 \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} = 6.022 \times 10^{23} \text{ atoms}\)
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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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Carbon Isotopes
Carbon is an element that can exist in different forms called isotopes. An isotope is a variant of an element that has the same number of protons but a different number of neutrons in its nucleus. This results in different atomic masses. The most common isotopes of carbon are Carbon-12 (\( \mathrm{C^{12}} \)) and Carbon-14 (\( \mathrm{C^{14}} \)).
\( \mathrm{C^{12}} \) is the stable and most abundant form, making up about 98% of natural carbon. It contains 6 protons and 6 neutrons. Meanwhile, \( \mathrm{C^{14}} \) is radioactive and makes up about 2% of natural carbon. It has 6 protons and 8 neutrons. This additional mass is why \( \mathrm{C^{14}} \) is heavier.
  • Isotopic Distribution: The percentage of each isotope in a sample, like 98% \( \mathrm{C^{12}} \) and 2% \( \mathrm{C^{14}} \), helps determine their relative abundance in a given sample.
  • Applications of Isotopes: Carbon isotopes are crucial in fields like radiocarbon dating, which uses the decay of \( \mathrm{C^{14}} \) to estimate the age of organic materials.
Moles Calculation
To understand chemical reactions and the amount of substances involved, we use the concept of "moles." A mole is a standard scientific unit for measuring large quantities of very small entities like atoms, molecules, or other specified particles. It is based on the number of atoms in exactly 12 grams of \( \mathrm{C^{12}} \), which is Avogadro's number.
  • Meaning: One mole of any substance contains the same number of entities as there are atoms in 12 grams of \( \mathrm{C^{12}} \).
  • Calculation: To find the number of moles, divide the mass of the sample by the molar mass of the element. For example, 12 grams of carbon corresponds to \( 1 \) mole because the molar mass of carbon is 12 g/mol.
The concept of a mole allows scientists to interconvert between mass and number of atoms, crucial for stoichiometry in chemical equations.
Avogadro's Number
Avogadro's number, named after the Italian scientist Amedeo Avogadro, is a fundamental constant used to describe the number of atoms, ions, or molecules in one mole of a substance. It is approximately \( 6.022 \times 10^{23} \) entities per mole.
  • Importance: Avogadro's number acts as a bridge between atoms, which are microscopic, and grams, which are macroscopic and measurable.
  • Application: It is used to precisely quantify the number of particles in a given mole, facilitating accurate calculations in chemistry.
For example, when we say there is \( 1 \) mole of carbon atoms, we mean there are \( 6.022 \times 10^{23} \) carbon atoms. Recognizing how Avogadro's number plays into calculations is key to understanding other chemistry concepts like molecular weight and composition.

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