To solve for the number of moles of air in the pressure cooker, we will use the Ideal Gas Law, which is given by the equation:
[tex]\[PV = nRT\][/tex]
where:
- [tex]\(P\)[/tex] is the pressure,
- [tex]\(V\)[/tex] is the volume,
- [tex]\(n\)[/tex] is the number of moles,
- [tex]\(R\)[/tex] is the ideal gas constant,
- [tex]\(T\)[/tex] is the temperature.
Given the following values:
- Pressure [tex]\(P = 2024 \, \text{kPa}\)[/tex],
- Volume [tex]\(V = 5.68 \, \text{liters}\)[/tex],
- Temperature [tex]\(T = 394 \, \text{K}\)[/tex],
- Ideal Gas Constant [tex]\(R = 8.314 \, \frac{\text{L} \cdot \text{kPa}}{\text{mol} \cdot \text{K}}\)[/tex],
we need to solve for the number of moles, [tex]\( n \)[/tex].
1. Rearrange the Ideal Gas Law to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{PV}{RT} \][/tex]
2. Substitute the given values into the equation:
[tex]\[ n = \frac{(2024 \, \text{kPa}) \cdot (5.68 \, \text{liters})}{(8.314 \, \frac{\text{L} \cdot \text{kPa}}{\text{mol} \cdot \text{K}}) \cdot (394 \, \text{K})} \][/tex]
3. Simplify the expression by performing the multiplication and division:
[tex]\[ n = \frac{2024 \times 5.68}{8.314 \times 394} \][/tex]
4. The exact calculation leads to:
[tex]\[ n \approx 3.5095594367765703 \, \text{moles} \][/tex]
Thus, the number of moles of air in the pressure cooker is approximately [tex]\(3.51 \, \text{moles}\)[/tex] when rounded to two decimal places.
