To determine the moles of air contained in a 147-L cylinder at a pressure of 1.17 atm and a temperature of 341 K, we can use the ideal gas law, which is stated as:
[tex]\[ PV = nRT \][/tex]
where:
- [tex]\( P \)[/tex] is the pressure in atmospheres (atm)
- [tex]\( V \)[/tex] is the volume in liters (L)
- [tex]\( n \)[/tex] is the number of moles of the gas
- [tex]\( R \)[/tex] is the ideal gas constant ([tex]\(0.0821 \, \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K} \)[/tex])
- [tex]\( T \)[/tex] is the temperature in Kelvin (K)
The goal is to solve for [tex]\( n \)[/tex], the number of moles.
Step-by-step solution:
1. Identify the given values:
- Volume [tex]\( V = 147 \, \text{L} \)[/tex]
- Pressure [tex]\( P = 1.17 \, \text{atm} \)[/tex]
- Temperature [tex]\( T = 341 \, \text{K} \)[/tex]
- Ideal gas constant [tex]\( R = 0.0821 \, \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K} \)[/tex]
2. Rearrange the ideal gas law to solve for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{PV}{RT}
\][/tex]
3. Substitute the known values into the equation:
[tex]\[
n = \frac{(1.17 \, \text{atm}) \times (147 \, \text{L})}{(0.0821 \, \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K}) \times (341 \, \text{K})}
\][/tex]
4. Calculate the numerator:
[tex]\[
(1.17 \, \text{atm}) \times (147 \, \text{L}) = 171.99 \, \text{atm} \cdot \text{L}
\][/tex]
5. Calculate the denominator:
[tex]\[
(0.0821 \, \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K}) \times (341 \, \text{K}) = 27.9961 \, \text{L} \cdot \text{atm} / \text{mol}
\][/tex]
6. Divide the numerator by the denominator to find [tex]\( n \)[/tex]:
[tex]\[
n = \frac{171.99 \, \text{atm} \cdot \text{L}}{27.9961 \, \text{L} \cdot \text{atm} / \text{mol}} \approx 6.14 \, \text{mol}
\][/tex]
Therefore, the number of moles of air contained in the 147-L cylinder at a pressure of 1.17 atm and a temperature of 341 K is approximately [tex]\( 6.14 \)[/tex] moles.
