To determine how many moles of air are inside the volleyball using the ideal gas law, we can follow these steps:
1. Given Constants and Conversion:
- The ideal gas constant, [tex]\( R \)[/tex], is [tex]\( 8.314 \frac{L \cdot kPa}{mol \cdot K} \)[/tex].
- The standard atmospheric pressure, [tex]\( P \)[/tex], is [tex]\( 101.3 \, kPa \)[/tex].
- The volume, [tex]\( V \)[/tex], is [tex]\( 5.27 \, L \)[/tex].
- The temperature in Celsius is [tex]\( 25^\circ C \)[/tex].
2. Convert Celsius to Kelvin:
- Using the conversion formula [tex]\( K = {^\circ}C + 273.15 \)[/tex]:
[tex]\[
T = 25 + 273.15 = 298.15 \, K
\][/tex]
3. Apply the Ideal Gas Law:
- The ideal gas law is given by [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 in Kelvin.
4. Rearrange the equation to solve for [tex]\( n \)[/tex]:
- [tex]\( n = \frac{PV}{RT} \)[/tex]
5. Substitute the known values into the equation:
- [tex]\( P = 101.3 \, kPa \)[/tex]
- [tex]\( V = 5.27 \, L \)[/tex]
- [tex]\( R = 8.314 \frac{L \cdot kPa}{mol \cdot K} \)[/tex]
- [tex]\( T = 298.15 \, K \)[/tex]
Substituting these values, we get:
[tex]\[
n = \frac{101.3 \, kPa \times 5.27 \, L}{8.314 \frac{L \cdot kPa}{mol \cdot K} \times 298.15 \, K}
\][/tex]
6. Calculate [tex]\( n \)[/tex]:
[tex]\[
n \approx 0.215 \, mol
\][/tex]
So, the number of moles of air inside the volleyball is approximately [tex]\( 0.215 \, mol \)[/tex].
This confirms that none of the given choices matches our calculated result. The choices provided:
- A. [tex]\(0.278 \, mol\)[/tex]
- B. [tex]\(3.37 \, mol\)[/tex]
- C. [tex]\(3.59 \, mol\)[/tex]
- D. [tex]\(25.2 \, mol\)[/tex]
- E. [tex]\(28.0 \, mol\)[/tex]
do not align with the correct calculation. Therefore, there is likely an issue with the provided options, as the correct number of moles of air inside the volleyball should indeed be around [tex]\( 0.215 \, mol \)[/tex].
