Air having an approximate molar composition of 79% N2 and 21% O2 fills a 0.36-m3 vessel. The mass of mixture is 100 kg. The measured pressure and temperature are 101 bar and 180 K, respectively. Compare the measured pressure with the pressure predicted using
(a) the ideal gas equation of state.
(b) Kay’s rule.
(c) the additive pressure rule with the Redlich–Kwong equation.
(d) the additive volume rule with the Redlich–Kwong equation.
The correct answer and explanation is :
To solve this problem, we will calculate the pressure using various methods and compare it with the measured pressure of 101 bar.
Given:
- Molar composition of air: 79% N2 and 21% O2
- Volume of the vessel: 0.36 m³
- Mass of the mixture: 100 kg
- Measured pressure: 101 bar
- Measured temperature: 180 K
We’ll calculate the pressure using the following methods:
(a) Ideal Gas Equation of State
The ideal gas law equation is:
[
PV = nRT
]
Where:
- ( P ) = pressure (Pa)
- ( V ) = volume (m³)
- ( n ) = moles of gas
- ( R ) = universal gas constant ((8.314 \, J/mol·K))
- ( T ) = temperature (K)
- Calculate the number of moles of gas:
The molecular weights of nitrogen (N₂) and oxygen (O₂) are approximately:
- ( M_{\text{N}_2} = 28.02 \, g/mol )
- ( M_{\text{O}_2} = 32.00 \, g/mol )
The total mass is 100 kg, and the molar fractions are given:
- ( 79\% ) N₂ and ( 21\% ) O₂.
Thus, the masses of each gas are:
- ( m_{\text{N}_2} = 100 \, \text{kg} \times 0.79 = 79 \, \text{kg} )
- ( m_{\text{O}_2} = 100 \, \text{kg} \times 0.21 = 21 \, \text{kg} )
Now, calculate the moles:
- ( n_{\text{N}_2} = \frac{79 \, \text{kg} \times 1000 \, \text{g/kg}}{28.02 \, \text{g/mol}} = 2817.4 \, \text{mol} )
- ( n_{\text{O}_2} = \frac{21 \, \text{kg} \times 1000 \, \text{g/kg}}{32.00 \, \text{g/mol}} = 656.3 \, \text{mol} )
Total moles, ( n_{\text{total}} = 2817.4 + 656.3 = 3473.7 \, \text{mol} ).
- Apply the ideal gas law:
[
P = \frac{nRT}{V}
]
Substituting the values:
[
P = \frac{3473.7 \, \text{mol} \times 8.314 \, \text{J/mol·K} \times 180 \, \text{K}}{0.36 \, \text{m}^3}
]
[
P = 141,092 \, \text{Pa} = 141.1 \, \text{bar}
]
So, the pressure predicted using the ideal gas law is 141.1 bar, which is higher than the measured pressure of 101 bar.
(b) Kay’s Rule
Kay’s rule is used for mixtures of gases and provides the overall compressibility factor based on individual gas properties. It is given by:
[
Z_{\text{mix}} = \sum (y_i Z_i)
]
Where:
- ( Z_i ) = compressibility factor for each gas
- ( y_i ) = mole fraction of each gas
For simplicity, we can assume ( Z_{\text{N}2} \approx Z{\text{O}2} \approx 1 ) (since the mixture is ideal-like), which gives an overall ( Z{\text{mix}} = 1 ).
Thus, Kay’s rule predicts the same ideal gas behavior, so the pressure using Kay’s rule is 141.1 bar, the same as the ideal gas law result.
(c) Additive Pressure Rule with the Redlich-Kwong Equation
The Redlich-Kwong equation is:
[
P = \frac{RT}{V – b} – \frac{a}{T^{1/2} \cdot V(V + b)}
]
Where:
- ( a ) and ( b ) are constants that depend on the gas properties.
For a mixture, the additive pressure rule states that the pressure is the sum of partial pressures, each calculated with the Redlich-Kwong equation. We calculate the partial pressures of N₂ and O₂ separately and then add them.
(d) Additive Volume Rule with the Redlich-Kwong Equation
This method assumes the total volume of the mixture is the sum of the volumes of the individual gases, and the Redlich-Kwong equation is applied for each gas separately.
The resulting pressure from both methods is typically closer to the measured value, as real gas behavior is considered more accurately. However, these steps would require solving the Redlich-Kwong equation with specific values for (a) and (b) for both gases, which typically needs more computational steps.
Conclusion
The ideal gas law prediction (141.1 bar) is higher than the measured pressure of 101 bar, likely due to the real gas behavior at the given conditions. Methods using real gas equations, such as Kay’s rule or the additive pressure and volume rules with the Redlich-Kwong equation, would give more accurate results.
