The Ideal Gas Law is expressed with the formula:
[tex]\[ PV = nRT \][/tex]
where:
- [tex]\( P \)[/tex] is the pressure of the gas,
- [tex]\( V \)[/tex] is the volume of the gas,
- [tex]\( n \)[/tex] is the number of moles,
- [tex]\( R \)[/tex] is the gas constant, and
- [tex]\( T \)[/tex] is the temperature of the gas.
Given Data:
- Volume ([tex]\( V \)[/tex]) = 0.0225 m³
- Temperature ([tex]\( T \)[/tex]) = 275 K
- Pressure ([tex]\( P \)[/tex]) = 105 N/m²
- Gas constant ([tex]\( R \)[/tex]) = 8.314 J/(mol·K)
To find the number of moles ([tex]\( n \)[/tex]), we need to rearrange the Ideal Gas Law to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{PV}{RT} \][/tex]
Step-by-Step Solution:
1. Identify known quantities:
[tex]\[
V = 0.0225 \text{ m}^3 \\
T = 275 \text{ K} \\
P = 105 \text{ N/m}^2 \\
R = 8.314 \text{ J/(mol·K)}
\][/tex]
2. Substitute these values into the equation:
[tex]\[
n = \frac{105 \text{ N/m}^2 \times 0.0225 \text{ m}^3}{8.314 \text{ J/(mol·K)} \times 275 \text{ K}}
\][/tex]
3. Calculate the numerator:
[tex]\[
105 \text{ N/m}^2 \times 0.0225 \text{ m}^3 = 2.3625 \text{ N·m} = 2.3625 \text{ J} \quad (\text{since } 1 \text{ N·m} = 1 \text{ J})
\][/tex]
4. Calculate the denominator:
[tex]\[
8.314 \text{ J/(mol·K)} \times 275 \text{ K} = 2286.35 \text{ J/mol}
\][/tex]
5. Divide the numerator by the denominator:
[tex]\[
n = \frac{2.3625 \text{ J}}{2286.35 \text{ J/mol}} = 0.001033306361668161 \text{ mol}
\][/tex]
So, the number of moles ([tex]\( n \)[/tex]) of the gas is approximately [tex]\( 0.001033 \)[/tex] mol.
