Question

Calcium carbide reacts with water to produce acetylene gas, \(\mathrm{C}_{2} \mathrm{H}_{2}\) $$\mathrm{CaC}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{Ca}(\mathrm{OH})_{2}(a q)+\mathrm{C}_{2} \mathrm{H}_{2}(g) $$ Calculate the volume (in liters) of acetylene produced at \(26^{\circ} \mathrm{C}\) and \(684 \mathrm{mmHg}\) from \(0.075 \mathrm{~mol} \mathrm{CaC}_{2}\) and excess \(\mathrm{H}_{2} \mathrm{O} .\)

Short answer

The volume of acetylene gas produced is approximately 2.05 liters.

Step-by-step solution

Identify the Given Data

We are given that 0.075 moles of \( \mathrm{CaC}_2 \) react, and that the conditions are \( 26^{\circ} \mathrm{C} \) and \( 684 \mathrm{mmHg} \). We need to find the volume of acetylene gas \( \mathrm{C}_2\mathrm{H}_2} \).

Use Ideal Gas Law

The ideal gas law \( PV = nRT \) relates pressure (\(P\)), volume (\(V\)), moles of gas (\(n\)), the ideal gas constant (\(R\)), and temperature (\(T\)). First, convert pressure to \( \mathrm{atm} \) and temperature to \( \mathrm{K} \).

Convert Pressure to Atmospheres

To convert 684 mmHg to atmospheres, use the conversion 1 atm = 760 mmHg: \[P = \frac{684 \, \mathrm{mmHg}}{760 \, \mathrm{mmHg/atm}} = 0.9 \, \mathrm{atm}\]

Convert Temperature to Kelvin

Convert temperature from Celsius to Kelvin using the formula: \[T = 26^{\circ} \mathrm{C} + 273.15 = 299.15 \, \mathrm{K}\]

Calculate the Volume of Gas Using Ideal Gas Law

Using the ideal gas law \( PV = nRT \), solve for \(V\), where \( n = 0.075 \) mol, \( R = 0.0821 \, \mathrm{L\cdot atm/mol\cdot K} \), \( P = 0.9 \, \mathrm{atm} \), and \( T = 299.15 \, \mathrm{K} \):\[V = \frac{nRT}{P} = \frac{0.075 \, \mathrm{mol} \times 0.0821 \, \mathrm{L\cdot atm/mol\cdot K} \times 299.15 \, \mathrm{K}}{0.9 \, \mathrm{atm}} \approx 2.05 \, \mathrm{L}\]

Key concepts

Gas Pressure Conversion

Understanding how to convert gas pressure is crucial when using the Ideal Gas Law in varying units. Most pressure values in chemistry problems are measured in millimeters of mercury (mmHg) or atmospheres (atm). Each unit has its situations where it’s most applicable. The conversion between these units follows this simple relationship: 1 atm is equivalent to 760 mmHg.

To convert from mmHg to atm, you divide the pressure in mmHg by 760. For example, if you have a pressure of 684 mmHg, to convert to atm, you use the formula:

• \[ P = \frac{684 \, \text{mmHg}}{760 \, \text{mmHg/atm}} = 0.9 \, \text{atm} \]

Such conversions help ensure accurate calculations in equations like the Ideal Gas Law where pressure has to be in atm.

Temperature Conversion

Temperature conversion is essential when applying the Ideal Gas Law, which requires temperature in Kelvin. The Kelvin scale is preferred in scientific calculations because it starts at absolute zero, offering a true zero point for thermodynamic calculations.

To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature. If the given temperature is 26°C, the conversion is straightforward:

• \[ T = 26^{\circ} \text{C} + 273.15 = 299.15 \text{ K} \]

This critical conversion facilitates the calculation of gas volumes under varying conditions.

Chemical Reaction Stoichiometry

Stoichiometry in chemical reactions helps us understand the proportions of reactants and products involved. In the reaction between calcium carbide and water:

\[ \text{CaC}_2(s) + 2 \text{H}_2\text{O}(l) \rightarrow \text{Ca}(\text{OH})_2(aq) + \text{C}_2\text{H}_2(g) \]

It shows us that 1 mole of calcium carbide produces 1 mole of acetylene gas. Since we have 0.075 moles of calcium carbide, it directly translates to 0.075 moles of acetylene gas being produced. This 1:1 stoichiometric ratio simplifies the calculation process, allowing us to directly apply these mole values into the Ideal Gas Law.

Gas Volume Calculation

To determine the volume of a gas under specific conditions, the Ideal Gas Law comes into play:

\[ PV = nRT \]

Where:

• \( P \) is the pressure in atm.
• \( V \) is the volume in liters.
• \( n \) is the amount of substance in moles.
• \( R \) is the universal gas constant \( 0.0821 \, \text{L} \cdot \text{atm/mol} \cdot \text{K} \).
• \( T \) is the temperature in Kelvin.

To find the volume of acetylene gas, rearrange the equation to solve for \( V \):

• \[ V = \frac{nRT}{P} \]

Plugging in the values:

• \[ V = \frac{0.075 \, \text{mol} \times 0.0821 \, \text{L} \cdot \text{atm/mol} \cdot \text{K} \times 299.15 \, \text{K}}{0.9 \, \text{atm}} \approx 2.05 \, \text{L} \]

This calculation tells us that under the given conditions, 0.075 moles of calcium carbide will produce approximately 2.05 liters of acetylene gas.