To determine the number of moles of air in an empty water bottle at [tex]\(15^\circ \text{C}\)[/tex] and standard pressure, we will use the ideal gas law, which is represented by the equation:
[tex]\[ PV = nRT \][/tex]
where:
- [tex]\( P \)[/tex] is the pressure in atmospheres.
- [tex]\( V \)[/tex] is the volume in liters.
- [tex]\( n \)[/tex] is the number of moles of gas.
- [tex]\( R \)[/tex] is the ideal gas constant, [tex]\(0.0821 \frac{L \cdot atm}{mol \cdot K}\)[/tex].
- [tex]\( T \)[/tex] is the temperature in Kelvin.
### Step-by-Step Solution:
1. Convert the temperature from Celsius to Kelvin:
[tex]\[ T = 15^\circ \text{C} + 273.15 = 288.15 \, \text{K} \][/tex]
2. Use the ideal gas law equation:
[tex]\[ n = \frac{PV}{RT} \][/tex]
Here:
- [tex]\( P = 1 \, \text{atm} \)[/tex] (standard atmospheric pressure).
- [tex]\( V = 0.500 \, \text{L} \)[/tex] (volume of the bottle).
- [tex]\( R = 0.0821 \frac{L \cdot atm}{mol \cdot K} \)[/tex].
- [tex]\( T = 288.15 \, \text{K} \)[/tex].
3. Substitute the known values into the ideal gas law equation:
[tex]\[ n = \frac{(1 \, \text{atm}) \times (0.500 \, \text{L})}{(0.0821 \frac{L \cdot atm}{mol \cdot K}) \times (288.15 \, \text{K})} \][/tex]
4. Calculate the number of moles [tex]\( n \)[/tex]:
[tex]\[ n \approx 0.0211 \, \text{moles} \][/tex]
Therefore, the water bottle contains 0.0211 moles of air.
