Alright, let's solve this problem step by step. We need to determine the number of moles of gas in the cylinder using the ideal gas law equation:
[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 of the gas,
- [tex]\( R \)[/tex] is the ideal gas constant, and
- [tex]\( T \)[/tex] is the temperature of the gas.
Given the following:
- Volume ([tex]\( V \)[/tex]) = 11.8 liters
- Pressure ([tex]\( P \)[/tex]) = 43.2 psi
- Temperature ([tex]\( T \)[/tex]) = 25 °C
1. Convert pressure from psi to atm:
The pressure is given in psi, and we need it in atmospheres (atm). The conversion factor is:
[tex]\[ 1 \, \text{atm} = 14.696 \, \text{psi} \][/tex]
So, to convert the pressure:
[tex]\[ \text{Pressure in atm} = \frac{43.2 \, \text{psi}}{14.696 \, \text{psi/atm}} \][/tex]
This conversion gives us:
[tex]\[ \text{Pressure in atm} = 2.9395753946652152 \, \text{atm} \][/tex]
2. Convert temperature from Celsius to Kelvin:
The temperature is given in Celsius, and we need it in Kelvin. The conversion formula is:
[tex]\[ T(K) = T(°C) + 273.15 \][/tex]
So, to convert the temperature:
[tex]\[ T(K) = 25 + 273.15 \][/tex]
This conversion gives us:
[tex]\[ T(K) = 298.15 \, \text{K} \][/tex]
3. Solve for the number of moles ([tex]\( n \)[/tex]) using the ideal gas law:
Using the gas constant [tex]\( R \)[/tex] for units of [tex]\(\frac{\text{L} \cdot \text{atm}}{\text{K} \cdot \text{mol}}\)[/tex]:
[tex]\[ R = 0.0821 \, \frac{\text{L} \cdot \text{atm}}{\text{K} \cdot \text{mol}} \][/tex]
Substitute the known values into the ideal gas law equation:
[tex]\[ n = \frac{PV}{RT} \][/tex]
Using the calculated values:
[tex]\[ n = \frac{2.9395753946652152 \, \text{atm} \times 11.8 \, \text{L}}{0.0821 \, \frac{\text{L} \cdot \text{atm}}{\text{K} \cdot \text{mol}} \times 298.15 \, \text{K}} \][/tex]
By performing the calculation, we get:
[tex]\[ n = 1.4170613079091077 \, \text{moles} \][/tex]
Therefore, the number of moles of gas in the cylinder is approximately 1.4170613079091077 moles.
