Problem 79 - Chemistry and Chemical Reactivity

Question

Acetylene can be made by allowing calcium carbide to react with water: $$ \mathrm{CaC}_{2}(\mathrm{s})+2 \mathrm{H}_{2} \mathrm{O}(\ell) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{2}(\mathrm{g})+\mathrm{Ca}(\mathrm{OH})_{2}(\mathrm{s}) $$ Suppose you react \(2.65 \mathrm{g}\) of \(\mathrm{CaC}_{2}\) with excess water. If you collect the acetylene and find that the gas has a volume of \(795 \mathrm{mL}\) at \(25.2^{\circ} \mathrm{C}\) with a pressure of \(735.2 \mathrm{mm} \mathrm{Hg},\) what is the percent yield of acetylene?

Step-by-Step Solution

Answer

The percent yield of acetylene is approximately 77%.

Step 1: Calculate Moles of Reactant

First, we need to calculate the moles of calcium carbide (\( \text{CaC}_2 \)) used. The molar mass of \( \text{CaC}_2 \) is approximately \( 64.10 \text{ g/mol} \). Use the formula \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \).\[\text{moles of } \text{CaC}_2 = \frac{2.65\text{ g}}{64.10\text{ g/mol}} \approx 0.0413 \text{ mol}\]

Step 2: Determine Theoretical Yield

Based on the balanced equation, 1 mole of \( \text{CaC}_2 \) produces 1 mole of \( \text{C}_2\text{H}_2 \). Thus, the theoretical moles of \( \text{C}_2\text{H}_2 \) is the same as the moles of \( \text{CaC}_2 \). So, the theoretical moles of \( \text{C}_2\text{H}_2 \) is also approximately \( 0.0413 \text{ mol} \).

Step 3: Use Ideal Gas Law to Find Actual Moles

To find the actual number of moles of \( \text{C}_2\text{H}_2 \) collected, we use the ideal gas law \( PV = nRT \). Convert the volume to liters (\( 795 \text{ mL} = 0.795 \text{ L} \)), temperature to Kelvin (\( 25.2^{\circ} \text{C} = 298.35 \text{ K} \)), and pressure to atmospheres (\( 735.2 \text{ mmHg} = 0.967 \text{ atm} \)). Then, use \( R = 0.0821 \text{ L atm/mol K} \).\[ n = \frac{PV}{RT} = \frac{0.967 \text{ atm} \times 0.795 \text{ L}}{0.0821 \text{ L atm/mol K} \times 298.35 \text{ K}} \approx 0.0318 \text{ mol} \]

Step 4: Calculate the Percent Yield

Percent yield is calculated using the formula \( \text{Percent yield} = \frac{\text{actual moles}}{\text{theoretical moles}} \times 100 \% \).\[ \text{Percent yield} = \frac{0.0318 \text{ mol}}{0.0413 \text{ mol}} \times 100 \% \approx 76.99\% \]

Step 5: Final Result

Therefore, the percent yield of acetylene is approximately \( 77\% \).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation used in chemistry to relate the pressure, volume, temperature, and number of moles of a gas. This equation is expressed as \(PV = nRT\), where \(P\) is the pressure of the gas, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant (typically \(0.0821\, \text{L atm/mol K}\)), and \(T\) is the temperature in Kelvin.

In the exercise, the Ideal Gas Law helps determine the actual moles of acetylene gas produced from its measured volume, temperature, and pressure.

• Pressure needs to be converted from mmHg to atmospheres since the ideal gas constant \(R\) is in \(\text{atm}\).
• Volume must be in liters, hence the conversion from milliliters.
• Temperature is converted to Kelvin to match the units used in the ideal gas constant.

By rearranging the equation to solve for \(n\), the formula is transformed into \(n = \frac{PV}{RT}\). This allows us to calculate the actual amount of gas in moles, facilitating a comparison with the theoretical moles from stoichiometry.

Chemical Reactions

Chemical reactions are processes in which substances, known as reactants, are transformed into different substances, called products. Understanding these reactions requires knowledge of balanced equations, where the number of atoms for each element is conserved.

In the given exercise, the chemical reaction involves calcium carbide \(\text{CaC}_2\) reacting with water \(\text{H}_2\text{O}\) to produce acetylene \(\text{C}_2\text{H}_2\) and calcium hydroxide \(\text{Ca(OH)}_2\). The balanced chemical equation is:

\[\mathrm{CaC}_{2}(\mathrm{s})+2 \mathrm{H}_{2} \mathrm{O}(\ell) \longrightarrow \mathrm{C}_{2}\mathrm{H}_{2}(\mathrm{g})+\mathrm{Ca}(\mathrm{OH})_{2}(\mathrm{s})\]

This equation shows that one mole of calcium carbide reacts with two moles of water to produce one mole of acetylene, a direct mole-to-mole conversion commonly used in stoichiometry. Knowing the balanced equation is crucial for calculating the theoretical yield of a product from the quantity of reactants used.

Percent Yield

Percent yield is a measure used in chemistry to determine the efficiency of a chemical reaction. It indicates how much of the theoretical amount of product was actually obtained.

To calculate percent yield, use the formula:
\[\text{Percent yield} = \frac{\text{actual yield}}{\text{theoretical yield}} \times 100\%\]

In the exercise, the actual yield is the amount of acetylene gas calculated using the Ideal Gas Law. The theoretical yield is determined by stoichiometry, which predicts the amount of acetylene formed assuming complete reaction of the given amount of calcium carbide.

• A percent yield of less than 100% indicates that the reaction was not fully efficient, possibly due to side reactions, incomplete reactions, or practical losses.
• A percent yield over 100% is rare and usually indicates measurement errors.

Percent yield calculations help laboratories optimize conditions for maximum production of a desired product.

Mole Calculations

Mole calculations are an essential part of stoichiometry, helping chemists determine how many molecules, atoms, or ions are involved in a chemical reaction.
Calculating moles involves using the molar mass of a substance, which is the mass of one mole of that substance usually in grams/mole.

In this specific exercise, we began by calculating the moles of the reactant, calcium carbide \(\text{CaC}_2\), using its mass and molar mass.

• The formula used is \(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\).
• This provides the starting point for determining how much product can theoretically be formed.

This calculation is crucial because it defines the scale of the reaction and helps predict how much of each product will be generated based on the stoichiometry of the balanced chemical equation. Accurate mole calculations ensure that chemical reactions are understood quantitatively, allowing for precise predictions and analyses.