Answer:
To find the resulting absolute pressure, we'll need to convert the given pressure and temperature to absolute units and then use the ideal gas law formula:
The ideal gas law is \(PV = nRT\), where:
- \(P\) is pressure
- \(V\) is volume
- \(n\) is the number of moles of gas
- \(R\) is the gas constant
- \(T\) is temperature in Kelvin
First, let's convert the temperature from Fahrenheit to Kelvin:
\[T_{\text{Kelvin}} = (T_{\text{Fahrenheit}} + 459.67) \times \frac{5}{9}\]
Given:
Pressure (\(P\)) = 25 psia
Gas constant (\(R\)) for air = 53.35
Temperature (\(T\)) = 100°F
Converting Fahrenheit to Kelvin:
\[T_{\text{Kelvin}} = (100 + 459.67) \times \frac{5}{9} = 310.93 \, \text{K}\]
Now, using the ideal gas law \(PV = nRT\) where \(n\) is the number of moles of air, we don't have the volume or the number of moles directly given. However, we can use the mass and the gas constant to find the number of moles:
\[n = \frac{\text{Mass}}{\text{Molar mass}}\]
Given 3 lbs. of air, let's convert this to mass in grams:
\[3 \, \text{lbs} \times \frac{453.592 \, \text{g}}{\text{lb}} = 1360.78 \, \text{g}\]
The molar mass of air (approximately 29 g/mol) is the average molar mass of nitrogen (N2) and oxygen (O2), which are the primary components of air.
Now, calculating the number of moles:
\[n = \frac{1360.78 \, \text{g}}{29 \, \text{g/mol}} \approx 46.93 \, \text{moles}\]
Plugging in all the known values into the ideal gas law \(PV = nRT\):
\[P \times V = n \times R \times T\]
We don't have the volume (\(V\)) directly given, but if the container is rigid, the volume remains constant, so the change in pressure is solely due to the change in temperature.
\[P_{\text{final}} = \frac{n \times R \times T_{\text{final}}}{V}\]
The initial pressure (\(P_{\text{initial}}\)) is 25 psia.
Let's solve for the final absolute pressure (\(P_{\text{final}}\)):
\[P_{\text{final}} = \frac{n \times R \times T_{\text{final}}}{V}\]
\[P_{\text{final}} = \frac{46.93 \times 53.35 \times 310.93}{V}\]
Without knowing the volume or any changes in volume, it's not possible to determine the final absolute pressure accurately.
