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\begin{tabular}{|c|c|} \hline Ideal Gas Law & [tex]$PV = nRT$[/tex] \\ \hline Ideal Gas Constant & \begin{tabular}{l} [tex]$R = 8.314 \frac{L \cdot kPa}{mol \cdot K}$[/tex] \\ or \\ [tex]$R = 0.0821 \frac{L \cdot atm}{mol \cdot K}$[/tex] \end{tabular} \\ \hline Standard Atmospheric Pressure & [tex]$1 atm = 101.3 kPa$[/tex] \\ \hline Temperature Conversion & [tex]$T (K) = T (^{\circ}C) + 273.15$[/tex] \\ \hline \end{tabular}
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An empty water bottle is full of air at [tex]$15^{\circ}C$[/tex] and standard pressure. The volume of the bottle is 0.500 liters. How many moles of air are in the bottle?
The water bottle contains [tex]$\square$[/tex] mole of air.
Certainly! Let's solve the given problem step-by-step using the ideal gas law equation, [tex]\( PV = nRT \)[/tex].
1. Given values: - Pressure [tex]\( P \)[/tex]: 1 atm (standard atmospheric pressure) - Volume [tex]\( V \)[/tex]: 0.500 liters - Temperature [tex]\( T \)[/tex]: 15°C - Gas constant [tex]\( R \)[/tex]: 0.0821 L atm / mol K
2. Convert the temperature from Celsius to Kelvin: [tex]\[ T_{K} = T_{C} + 273.15 \][/tex] [tex]\[ T_{K} = 15 + 273.15 = 288.15 \, \text{K} \][/tex]
3. Rearrange the ideal gas law to solve for the number of moles [tex]\( n \)[/tex]: [tex]\[ n = \frac{PV}{RT} \][/tex]
4. Substitute the known values into the equation: [tex]\[ n = \frac{(1 \, \text{atm}) \times (0.500 \, \text{L})}{(0.0821 \, \text{L atm / mol K}) \times (288.15 \, \text{K})} \][/tex]
5. Calculate the number of moles: [tex]\[ n = \frac{0.500}{0.0821 \times 288.15} \][/tex] [tex]\[ n \approx 0.021 \][/tex]
The water bottle contains [tex]\( \boxed{0.021} \)[/tex] mole of air.
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