Question

Air having an approximate molar composition of \(79 \%\) \(\mathrm{N}_{2}\) and \(21 \% \mathrm{O}_{2}\) fills a \(0.36-\mathrm{m}^{3}\) vessel. The mass of mixture is \(100 \mathrm{~kg}\). The measured pressure and temperature are 101 bar and \(180 \mathrm{~K}\), 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.

Short answer

Measured pressure is 101 bar. Use ideal gas law, Kay's Rule, and Redlich-Kwong equations to predict pressure and compare.

Step-by-step solution

Ideal Gas Equation

Use the ideal gas equation of state: \[ PV = nRT \] Where:\( P \) = pressure in bars (convert to Pascals)\( V \) = volume in \( m^3 \)\( n \) = moles of gas\( R \) = universal gas constant \(8.314 \frac{J}{mol \, K} \)\( T \) = temperature in KelvinFirst, calculate the number of moles, \( n \), using the molar mass of air:\[ n = \frac{100 \text{ kg}}{0.79 \times 28.014 + 0.21 \times 31.998} \]Solve for \(n\) and plug all values into the ideal gas equation to find the pressure.

Kay's Rule

Kay’s rule involves calculating pseudocritical properties based on composition. First, find critical properties for pure \(N_2\) and \(O_2\):\[ T_c (N_2) = 126.2 \text{ K}, \, P_c (N_2) = 33.94 \text{ bar} \]\[ T_c (O_2) = 154.6 \text{ K}, \, P_c (O_2) = 50.43 \text{ bar} \]Calculate pseudocritical properties:\[ T_c = 0.79 \times 126.2 + 0.21 \times 154.6 \]\[ P_c = 0.79 \times 33.94 + 0.21 \times 50.43 \]Use these values in the compressibility factor equation. Then, use the compressibility factor to modify the ideal gas equation:\[ P = \frac{{ZRnT}}{V} \]

Additive Pressure Rule with Redlich-Kwong

Use the Redlich-Kwong equation for individual gases:\[ P = \frac{RT}{V_m - b} - \frac{a}{\sqrt{T}V_m(V_m + b)} \]where:\[ a_{N_2} = 4.18, \, a_{O_2} = 5.48 \]\[ b_{N_2} = 0.037, \, b_{O_2} = 0.0318 \]Calculate mixture parameters:\[ a_m = 0.79^2 a_{N_2} + 0.21^2 a_{O_2} \]\[ b_m = 0.79 b_{N_2} + 0.21 b_{O_2} \]Use these in Redlich-Kwong to find the pressure for the mixture.

Additive Volume Rule with Redlich-Kwong

For the additive volume rule, again use individual Redlich-Kwong for \( N_2 \) and \( O_2 \). Adjust volumes:\[ V_{m,N_2} = 0.79 V \]\[ V_{m,O_2} = 0.21 V \] Calculate pressures for each and sum:\[ P = P_{N_2} + P_{O_2} \]Use the mixture Redlich-Kwong to find pressures for each species.

Key concepts

Ideal gas law

The Ideal Gas Law is a fundamental equation in thermodynamics used to describe the behavior of an ideal gas. It relates pressure (P), volume (V), temperature (T), and the number of moles (n) of a gas using the equation: \( PV = nRT \). The gas constant (R) has a value of 8.314 J/(mol·K). To apply this law:

• Calculate the number of moles using: \( n = \frac{mass}{molar \, mass} \)
• Convert pressure to Pascals if needed (1 bar = 100,000 Pa)
• Plug the values into the Ideal Gas Law to solve for the desired variable.

In our exercise, we simplify by first finding the moles of the gas mixture using its mass and molar mass. Then, we apply the Ideal Gas Law to estimate the pressure, providing a theoretical understanding before checking against actual values.

Kay's rule

Kay's rule helps us address real gases, especially mixtures, by calculating pseudocritical properties. This involves using the critical properties of individual gas components to find equivalent properties for the mix. First, find the critical temperature (T_c) and critical pressure (P_c) for each component:

• T_c(N2) = 126.2 K, P_c(N2) = 33.94 bar
• T_c(O2) = 154.6 K, P_c(O2) = 50.43 bar

Next, calculate pseudocritical properties:

• T_c_mix = 0.79 * 126.2 + 0.21 * 154.6 K
• P_c_mix = 0.79 * 33.94 + 0.21 * 50.43 bar

These pseudocritical properties help determine a compressibility factor (Z), adjusting the Ideal Gas Law to accommodate real gas behavior: \( P = \frac{ZRnT}{V} \). This provides a more accurate pressure prediction for the gas mixture.

Redlich-Kwong equation

The Redlich-Kwong equation improves predictions for gas mixtures by refining the van der Waals equation. It is represented by: \[ P = \frac{RT}{V_m - b} - \frac{a}{\root{2}{T} \times V_m(V_m + b)} \]where a and b are substance-specific constants. For our gas components, we use:

• a(N2) = 4.18 and a(O2) = 5.48
• b(N2) = 0.037 and b(O2) = 0.0318

Mixture parameters are calculated as:

• a_m = (0.79^2 * a_N2) + (0.21^2 * a_O2)
• b_m = (0.79 * b_N2) + (0.21 * b_O2)

These values allow inserting into the Redlich-Kwong equation to predict the mixture's pressure, accounting for interactions between molecules. This provides more accurate estimates compared to the ideal gas law.

Thermodynamic equations of state

Thermodynamic equations of state, like the Ideal Gas Law and Redlich-Kwong equation, describe relationships between pressure, volume, and temperature in substances. They help predict gas behavior under various conditions. Some key concepts include:

• Equation of state: Mathematical models relating state variables (such as P, V, and T).
• Ideal Gas Law: Assumes gases consist of non-interacting particles.
• Real Gas Laws: Include interactions between particles, e.g., the Redlich-Kwong equation.

Equations of state are essential for engineers and scientists to design systems involving gases like engines or chemical reactors. Each model's complexity balances accuracy and computational requirements, with simple models preferred for ease, while more detailed ones offer better precision.