Calculate the volume of carbon dioxide at 20.0°C and 0.956 atm produced from the complete combustion of 7.50 kg of methane.

To determine the volume of \( \text{CO}_2 \) produced, we need to follow these steps:

1. **Write the balanced chemical equation for the combustion of methane:**

   \[
   \text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O}
   \]

   This equation tells us that 1 mole of methane (\( \text{CH}_4 \)) produces 1 mole of carbon dioxide (\( \text{CO}_2 \)).

2. **Calculate the moles of methane (\( \text{CH}_4 \)) given the mass:**

   The molar mass of methane (\( \text{CH}_4 \)) is:
   \[
   12.01 \, \text{g/mol} + (4 \times 1.008 \, \text{g/mol}) = 16.04 \, \text{g/mol}
   \]

   Convert 7.50 kg of methane to grams and then to moles:
   \[
   7.50 \, \text{kg} = 7500 \, \text{g}
   \]
   \[
   \text{Moles of } \text{CH}_4 = \frac{7500 \, \text{g}}{16.04 \, \text{g/mol}} \approx 467.67 \, \text{mol}
   \]

3. **Determine the moles of \( \text{CO}_2 \) produced:**

   From the balanced equation, 1 mole of \( \text{CH}_4 \) produces 1 mole of \( \text{CO}_2 \). Therefore:
   \[
   \text{Moles of } \text{CO}_2 = 467.67 \, \text{mol}
   \]

4. **Use the Ideal Gas Law to find the volume of \( \text{CO}_2 \) at the specified conditions:**

   The Ideal Gas Law is:
   \[
   PV = nRT
   \]
   where:
   - \( P = 0.956 \, \text{atm} \)
   - \( n = 467.67 \, \text{mol} \)
   - \( R = 0.0821 \, \text{L} \cdot \text{atm} / (\text{mol} \cdot \text{K}) \)
   - \( T = 20.0^\circ \text{C} = 293.15 \, \text{K} \)

   Rearrange to solve for \( V \):
   \[
   V = \frac{nRT}{P}
   \]

   Substitute the values:
   \[
   V = \frac{(467.67 \, \text{mol}) \times (0.0821 \, \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K}) \times 293.15 \, \text{K}}{0.956 \, \text{atm}}
   \]

   Calculating this gives:
   \[
   V \approx 11,323.6 \, \text{L}
   \]

**Answer:** The volume of carbon dioxide produced is approximately **11,323.6 liters**.

Would you like further details or clarification on any part of this solution?

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**Related Questions:**

1. How would the volume change if the temperature were increased to 50°C?
2. How would this calculation differ at a pressure of 1.5 atm?
3. What is the volume of water vapor produced under the same conditions?
4. How does the Ideal Gas Law account for non-ideal gas behavior?
5. How would the volume change if we started with 10 kg of methane?

**Tip:** Always check that the temperature in gas law calculations is converted to Kelvin.

Learn how to calculate the volume of carbon dioxide produced from the complete combustion of 7.50 kg of methane at 20°C and 0.956 atm. This solution covers stoichiometry, the Ideal Gas Law, and unit conversions, and is suitable for students in Grades 10-12.

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