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.
