To determine the number of moles of air inside the volleyball, we will use the Ideal Gas Law, represented by the equation:
[tex]\[ PV = nRT \][/tex]
where:
- [tex]\( P \)[/tex] is the pressure in kilopascals (kPa),
- [tex]\( V \)[/tex] is the volume in liters (L),
- [tex]\( n \)[/tex] is the number of moles,
- [tex]\( R \)[/tex] is the ideal gas constant in units of [tex]\( \text{L·kPa/(mol·K)} \)[/tex],
- [tex]\( T \)[/tex] is the temperature in Kelvin (K).
Given data:
- Temperature in Celsius: [tex]\(24.6^\circ \text{C}\)[/tex]
- Pressure in kilopascals: [tex]\(130.75 \text{ kPa}\)[/tex]
- Volume in liters: [tex]\(5.27 \text{ L}\)[/tex]
- Ideal gas constant: [tex]\(R = 8.314 \text{ L·kPa/(mol·K)}\)[/tex]
First, we need to convert the temperature from Celsius to Kelvin:
[tex]\[ T = 24.6 + 273.15 = 297.75 \text{ K} \][/tex]
Next, we rearrange the ideal gas law equation to solve for [tex]\( n \)[/tex] (number of moles):
[tex]\[ n = \frac{PV}{RT} \][/tex]
Substituting the given values into the equation:
[tex]\[ n = \frac{(130.75 \text{ kPa}) \times (5.27 \text{ L})}{(8.314 \text{ L·kPa/(mol·K)}) \times (297.75 \text{ K})} \][/tex]
From the calculation, we obtain:
[tex]\[ n \approx 0.2783495492918886 \text{ mol} \][/tex]
Rounding this to three significant figures, we get:
[tex]\[ n \approx 0.278 \text{ mol} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{0.278 \text{ mol}} \][/tex]
So, the correct option is:
A. 0.278 mol
