Verified Answer :
- Determine the moles of CO₂ using Avogadro's number: \(n = \frac{8.1 \times 10^{25}}{6.022 \times 10^{23}} \approx 134.51 \, \text{mol}\).
- Calculate the mass in kilograms using molar mass: \(m = 134.51 \, \text{mol} \times 0.04401 \, \text{kg/mol} \approx 5.9196 \, \text{kg}\).
- Compute volume in cubic meters using density: \(V = \frac{5.9196 \, \text{kg}}{1.98 \, \text{kg/m}^3} \approx 2.9897 \, \text{m}^3\).
- Convert to liters: \(V = 2.9897 \, \text{m}^3 \times 1000 \approx \boxed{2989.7} \, \text{L}\).
Explanation
- Problem Analysis
To determine the volume of CO₂ produced, we first analyze the given data:
- The total number of CO₂ molecules is \(N = 8.1 \times 10^{25}\).
- Avogadro's constant is \(N_A \approx 6.022 \times 10^{23} \, \text{molecules/mol}\).
- The molar mass of CO₂ (\(M\)) is calculated as the sum of atomic masses: \(12.01 + 2 \times 16.00 = 44.01 \, \text{g/mol}\), which is \(0.04401 \, \text{kg/mol}\).
- The density of CO₂ is given as \(\rho = 1.98 \, \text{kg/m}^3\).
- We aim to find the volume in liters (L), noting that \(1 \, \text{m}^3 = 1000 \, \text{L}\).
- Calculating Moles and Mass
First, we find the number of moles of CO₂ (\(n\)) using the relationship \(n = \frac{N}{N_A}\):
\[n = \frac{8.1 \times 10^{25} \, \text{molecules}}{6.022 \times 10^{23} \, \text{molecules/mol}} \approx 134.51 \, \text{mol}.\]
Next, we determine the total mass (\(m\)) of this amount of CO₂:
\[m = n \times M = 134.51 \, \text{mol} \times 0.04401 \, \text{kg/mol} \approx 5.9196 \, \text{kg}.\]
- Calculating Volume in Liters
Now we calculate the volume (\(V\)) in cubic meters using the formula \(V = \frac{m}{\rho}\):
\[V = \frac{5.9196 \, \text{kg}}{1.98 \, \text{kg/m}^3} \approx 2.9897 \, \text{m}^3.\]
Finally, we convert the volume from cubic meters to liters by multiplying by \(1000\):
\[V_L = 2.9897 \, \text{m}^3 \times 1000 \, \text{L/m}^3 = 2989.7 \, \text{L}.\]
Rounding the result to a reasonable precision, the total volume is approximately \(\boxed{2989.7}\) L.
Examples
Understanding the volume of gases produced in chemical reactions is crucial for industrial safety and process control. For instance, in carbon capture technologies, engineers calculate the volume of CO₂ emitted from combustion to determine the capacity needed for storage or conversion systems. By using density and molar relationships, they ensure that environmental impact assessments are accurate and that infrastructure, such as containment tanks or pipes, is appropriately sized for the volume of gas being managed.