Problem 58 - Chemistry The Molecular Science

Question

Acetylene can be made by reacting calcium carbide with water. $$\mathrm{CaC}_{2}(\mathrm{~s})+2 \mathrm{H}_{2} \mathrm{O}(\ell) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{2}(\mathrm{~g})+\mathrm{Ca}(\mathrm{OH})_{2}(\mathrm{aq})$$ Assume that you place \(2.65 \mathrm{~g} \mathrm{CaC}_{2}\) in excess water and collect the acetylene over water. The volume of the acetylene and water vapor is \(795 \mathrm{~mL}\) at \(25.0^{\circ} \mathrm{C}\) and a barometric pressure of \(735.2 \mathrm{mmHg}\). Calculate the percent yield of acetylene. The vapor pressure of water at \(25^{\circ} \mathrm{C}\) is \(23.8 \mathrm{mmHg}\).

Step-by-Step Solution

Answer

The percent yield of acetylene is approximately 73.36%.

Step 1: Determine Molar Masses

First, calculate the molar mass of calcium carbide (\( \mathrm{CaC}_2 \)) and acetylene (\( \mathrm{C}_2\mathrm{H}_2 \)). \( \mathrm{CaC}_2 \) molar mass: \( 40.08 + (2 \times 12.01) = 64.10 \ g/mol \). \( \mathrm{C}_2\mathrm{H}_2 \) molar mass: \( 2 \times 12.01 + 2 \times 1.008 = 26.04 \ g/mol \).

Step 2: Calculate Moles of \( \mathrm{CaC}_2 \)

Using the given mass of \( \mathrm{CaC}_2 \), calculate the moles using its molar mass: \[ \text{moles of } \mathrm{CaC}_2 = \frac{2.65 \ g}{64.10 \ g/mol} \approx 0.04134 \ mol \]

Step 3: Determine Theoretical Moles of \( \mathrm{C}_2\mathrm{H}_2 \)

The balanced equation shows a 1:1 mole ratio between \( \mathrm{CaC}_2 \) and \( \mathrm{C}_2\mathrm{H}_2 \), so the theoretical moles of \( \mathrm{C}_2\mathrm{H}_2 \) produced is the same as the moles of \( \mathrm{CaC}_2 \): \( 0.04134 \ mol \).

Step 4: Calculate Theoretical Volume of \( \mathrm{C}_2\mathrm{H}_2 \) at Standard Conditions

Use the ideal gas law to convert moles of \( \mathrm{C}_2\mathrm{H}_2 \) into volume at 298 K and 711.4 mmHg (735.2 mmHg minus the vapor pressure of water 23.8 mmHg). Use \( R = 0.0821 \ L \cdot atm/mol \cdot K \). Convert pressure to atm: \[ 711.4 \ mmHg \times \frac{1 \ atm}{760 \ mmHg} = 0.9355 \ atm \] \[ V = \frac{nRT}{P} = \frac{0.04134 \ times \ 0.0821 \ times \ 298}{0.9355} \approx 1.084 \ L \]

Step 5: Calculate Percent Yield

Convert the actual collected volume of gas to liters: \( 795 \ mL = 0.795 \ L \). Calculate the percent yield of the reaction: \[ \text{Percent Yield} = \left(\frac{0.795 \ L}{1.084 \ L}\right)\times 100 \approx 73.36\% \]

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry that helps us understand how gases behave under various conditions. It's expressed in the formula \( PV = nRT \), where:

• \( P \) is the pressure of the gas,
• \( V \) is the volume,
• \( n \) is the number of moles,
• \( R \) is the ideal gas constant (0.0821 L·atm/mol·K),
• \( T \) is the temperature in Kelvin.

To use the Ideal Gas Law in our percent yield problem, we need to adjust for the vapor pressure of water and use the adjusted pressure in our calculations. The actual pressure exerted by the acetylene is the total pressure minus the vapor pressure of the water. This means:\[P_{acetylene} = P_{total} - P_{water ext{ }vapor}\]Once we have the pressure of acetylene, the Ideal Gas Law allows us to calculate the volume of the gas that should be produced under standard conditions. This is crucial for determining how much acetylene we should expect to obtain from our theoretical calculations.

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the proportionate relationships among elements and compounds in chemical reactions. It relies on the understanding of balanced chemical equations, helping to calculate the amounts of reactants needed and products formed.In the given reaction:\[\mathrm{CaC}_{2}(s)+2 \mathrm{H}_{2}O(l) \rightarrow \mathrm{C}_{2} \mathrm{H}_{2}(g)+\mathrm{Ca}(\mathrm{OH})_{2}(aq)\]We see a 1:1 mole ratio between calcium carbide \(\mathrm{CaC}_{2}\) and acetylene \(\mathrm{C}_{2}\mathrm{H}_{2}\). This means that for every mole of \(\mathrm{CaC}_{2}\) used, one mole of \(\mathrm{C}_{2}\mathrm{H}_{2}\) is expected to be formed, assuming the reaction proceeds to completion.By knowing the molar mass of \(\mathrm{CaC}_{2}\), we can calculate the number of moles used in the reaction. This helps us predict the theoretical yield of the product \(\mathrm{C}_{2}\mathrm{H}_{2}\). This theoretical yield is essential for calculating percent yield, as it represents the maximum possible product we could obtain if the reaction were perfectly efficient with no losses.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid form at a given temperature. It's an important factor to consider when collecting gases over a liquid, as in this acetylene generation experiment.At \(25^{\circ}C\), the vapor pressure of water is \(23.8\ \mathrm{mmHg}\). This means that some of the pressure inside the collection container is due to water vapor, not just the acetylene gas. To find the pressure exerted solely by the acetylene, we must subtract this vapor pressure from the total pressure in the container:\[P_{acetylene} = P_{total} - P_{water ext{ }vapor}\]By doing this subtraction, we get the corrected pressure value that represents only the acetylene gas. This corrected pressure is used in our Ideal Gas Law calculation to ensure we determine the theoretical volume of acetylene accurately. Understanding and accounting for vapor pressure is key for precise gas measurements in chemical experiments.