To determine the number of moles of air inside the volleyball, we need to use the Ideal Gas Law, which is given by the equation:
[tex]\[ PV = nRT \][/tex]
Where:
- [tex]\( P \)[/tex] is the pressure in pascals (Pa).
- [tex]\( V \)[/tex] is the volume in cubic meters (m³).
- [tex]\( n \)[/tex] is the number of moles.
- [tex]\( R \)[/tex] is the ideal gas constant (8.314 J/(mol·K)).
- [tex]\( T \)[/tex] is the temperature in Kelvin (K).
Let's follow the steps to solve this problem:
1. Convert temperature from Celsius to Kelvin:
The temperature in Kelvin [tex]\( T \)[/tex] is given by:
[tex]\[ T = T_{\text{Celsius}} + 273.15 \][/tex]
Substituting the given temperature:
[tex]\[ T = 24.6 + 273.15 = 297.75 \, K \][/tex]
2. Convert pressure from kilopascals to pascals:
The pressure in pascals [tex]\( P \)[/tex] is:
[tex]\[ P = 130.75 \, \text{kPa} \times 1000 \, \text{Pa/kPa} = 130750 \, \text{Pa} \][/tex]
3. Convert volume from liters to cubic meters:
The volume in cubic meters [tex]\( V \)[/tex] is:
[tex]\[ V = 5.27 \, \text{liters} \times 0.001 \, \text{m}^3/\text{liter} = 0.00527 \, \text{m}^3 \][/tex]
4. Rearrange the ideal gas law to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{PV}{RT} \][/tex]
Now, substitute the known values:
[tex]\[ n = \frac{130750 \, \text{Pa} \times 0.00527 \, \text{m}^3}{8.314 \, \text{J/(mol·K)} \times 297.75 \, \text{K}} \][/tex]
5. Calculate the number of moles [tex]\( n \)[/tex]:
[tex]\[ n \approx 0.28 \, \text{mol} \][/tex]
Thus, the number of moles of air inside the volleyball is approximately 0.28 mol.
So, the correct answer is:
A. 0.278 mol
