Problem 20 - Chemistry The Science in Context

Question

The following quantities of trace gases were found in a 1.0 mL sample of air. Calculate the number of moles of each compound in the sample. a. \(1.4 \times 10^{13}\) molecules of \(\mathrm{H}_{2}(g)\) b. \(1.5 \times 10^{14}\) atoms of \(\mathrm{He}(g)\) c. \(7.7 \times 10^{12}\) molecules of \(\mathrm{N}_{2} \mathrm{O}(g)\) d. \(3.0 \times 10^{12}\) molecules of \(\mathrm{CO}(g)\)

Step-by-Step Solution

Answer

Question: Calculate the number of moles of H2(g), He(g), N2O(g), and CO(g) present in a 1.0 mL sample of air containing 1.4 × 10^13 molecules of H2(g), 1.5 × 10^14 atoms of He(g), 7.7 × 10^12 molecules of N2O(g), and 3.0 × 10^12 molecules of CO(g). Answer: In the 1.0 mL sample of air, there are: - 2.32 × 10^-11 moles of H2(g) - 2.49 × 10^-10 moles of He(g) - 1.28 × 10^-11 moles of N2O(g) - 4.98 × 10^-12 moles of CO(g)

Step 1: Calculate the number of moles of H2(g)

Given that there are \(1.4 \times 10^{13}\) molecules of \(\mathrm{H}_{2}(g)\), we can find the number of moles by dividing this by the Avogadro's constant: \(\text{moles of H2} = \frac{1.4\times 10^{13}\; \text{molecules}}{6.022 \times 10^{23}\; \text{molecules/mol}}\) \(\text{moles of H2} = 2.32 \times 10^{-11}\) moles

Step 2: Calculate the number of moles of He(g)

Given that there are \(1.5 \times 10^{14}\) atoms of \(\mathrm{He}(g)\), we can find the number of moles by dividing this by the Avogadro's constant: \(\text{moles of He} = \frac{1.5\times 10^{14}\; \text{atoms}}{6.022 \times 10^{23}\; \text{atoms/mol}}\) \(\text{moles of He} = 2.49 \times 10^{-10}\) moles

Step 3: Calculate the number of moles of N2O(g)

Given that there are \(7.7 \times 10^{12}\) molecules of \(\mathrm{N}_{2} \mathrm{O}(g)\), we can find the number of moles by dividing this by the Avogadro's constant: \(\text{moles of N2O} = \frac{7.7\times 10^{12}\; \text{molecules}}{6.022 \times 10^{23}\; \text{molecules/mol}}\) \(\text{moles of N2O} = 1.28 \times 10^{-11}\) moles

Step 4: Calculate the number of moles of CO(g)

Given that there are \(3.0 \times 10^{12}\) molecules of \(\mathrm{CO}(g)\), we can find the number of moles by dividing this by the Avogadro's constant: \(\text{moles of CO} = \frac{3.0\times 10^{12}\; \text{molecules}}{6.022 \times 10^{23}\; \text{molecules/mol}}\) \(\text{moles of CO} = 4.98 \times 10^{-12}\) moles After completing all the steps, we have the number of moles for each compound in the sample: - H2(g): \(2.32 \times 10^{-11}\) moles - He(g): \(2.49 \times 10^{-10}\) moles - N2O(g): \(1.28 \times 10^{-11}\) moles - CO(g): \(4.98 \times 10^{-12}\) moles

Key Concepts

Avogadro's constant

Avogadro's constant is a fundamental component in the field of chemistry. It represents the number of entities, such as atoms or molecules, in one mole of a substance. This constant is numerically defined as approximately \(6.022 \times 10^{23}\) entities per mole.

Avogadro's constant allows chemists to convert between the number of particles and the amount of substance in moles. For example, if you have \(6.022 \times 10^{23}\) molecules of water, it equates to 1 mole of water. This conversion is vital because chemical reactions are usually measured in moles, not in individual atoms or molecules.

Understanding Avogadro's constant also helps in calculating various physical properties and behaviors of substances in gaseous states, where measuring individual gas molecules would be impractical.

Molecular Quantities

Molecular quantities refer to the actual number of molecules or atoms present in a given sample. To move from molecular quantities to moles, we utilize Avogadro's constant.

Say you have a sample containing \(7.7 \times 10^{12}\) molecules of a compound like \(\mathrm{N}_2\mathrm{O}\). To find out how many moles this represents, you divide the molecular quantity by Avogadro's constant:

\[ \text{moles of } \mathrm{N}_2\mathrm{O} = \frac{7.7 \times 10^{12}}{6.022 \times 10^{23}} \]

This results in a very small number of moles because \(6.022 \times 10^{23}\) is such a large number. This conversion plays a fundamental role in laboratory settings, where precise measurements are required for chemical reactions.

Chemical Compounds

Chemical compounds are substances formed when two or more different types of atoms bond together. The air we breathe, for instance, contains trace amounts of compounds such as \(\mathrm{CO}\), \(\mathrm{H}_2\), \(\mathrm{N}_2\mathrm{O}\), and \(\mathrm{He}\).

These compounds exhibit different properties despite being in minute quantities. Each compound is characterized by a unique chemical formula that displays the types and numbers of atoms present.

For example:

• \(\mathrm{CO}\): Composed of one carbon and one oxygen atom.
• \(\mathrm{H}_2\): Composed of two hydrogen atoms.
• \(\mathrm{N}_2\mathrm{O}\): Composed of two nitrogen atoms and one oxygen atom.

By understanding chemical compounds, scientists can predict reactions, interactions, and behavior in various environments.

Trace Gases Concentration

Trace gases are present in very small concentrations in the atmosphere but can have significant effects on environmental and biological processes. Their concentration is often expressed as parts per million (ppm) or parts per billion (ppb), indicating how many parts of the gas exist per million or billion parts of air, respectively.

Determining the concentration of these gases, such as \(\mathrm{CO}\), \(\mathrm{H}_2\), \(\mathrm{He}\) and \(\mathrm{N}_2\mathrm{O}\), involves calculating the amount of substance (moles) in a given volume of air.

This is achieved through mole calculations, using principles such as Avogadro's constant to derive how many moles are present within a specified sample volume. Despite their low concentrations, trace gases are critical players in atmospheric chemistry, influencing aspects such as climate and air quality.