Question

A gaseous mixture consisting of \(0.75 \mathrm{kmol}\) of hydrogen \(\left(\mathrm{H}_{2}\right)\) and \(0.25 \mathrm{kmol}\) of nitrogen \(\left(\mathrm{N}_{2}\right)\) occupies \(0.085 \mathrm{~m}^{3}\) at \(25^{\circ} \mathrm{C}\). Estimate the pressure, in bar, using (a) the ideal gas equation of state. (b) Kay's rule together with the generalized compressibility chart. (c) the van der Waals equation together with mixture values for the constants \(a\) and \(b\). (d) the rule of additive pressure together with the generalized compressibility chart.

Short answer

Using the ideal gas law: 0.2922 bar. Using Kay's rule and generalized compressibility chart, determine compressibility factor \( Z \) and correct based on reduced properties. Using van der Waals equation, substitute mixture constants and solve for P. Using rule of additive pressure, combine partial pressures and correct using compressibility factor.

Step-by-step solution

Ideal Gas Equation

The ideal gas equation is given by \[PV = nRT \] where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the gas constant \( 8.314 \, \text{J/(mol·K)} \), and \( T \) is the temperature in Kelvin. Convert temperature to Kelvin: \( T = 25^{\circ} \text{C} + 273.15 = 298.15 \) K. The total moles \( n \) is the sum of moles of hydrogen and nitrogen: \( n = 0.75 \text{ kmol} + 0.25 \text{ kmol} = 1.00 \text{ kmol} \). Substitute the known values into the ideal gas equation: \[ P = \frac{nRT}{V} = \frac{(1.00 \text{ kmol}) \cdot (8.314 \text{ J/(mol·K)}) \cdot (298.15 \text{ K})}{0.085 \text{ m}^3} \] Since 1 kmol = 1000 mol, the equation simplifies to: \[ P = \frac{(1.00 \times 1000 \text{ mol}) \cdot (8.314 \text{ J/(mol·K)}) \cdot (298.15 \text{ K})}{0.085 \text{ m}^3} \] Solving this: \( P \approx 29215.18 \text{ Pa} \). Converting to bar (1 bar = 100,000 Pa): \( P \approx 0.2922 \text{ bar} \)

Kay's Rule and Compressibility Chart

Kay’s rule suggests using pseudo-critical properties for a gas mixture as follows: \[ T_c' = y_{1}T_{c1} + y_{2}T_{c2} \] and \[ P_c' = y_{1}P_{c1} + y_{2}P_{c2} \] where \( y_{1} \) and \( y_{2} \) are the mole fractions, \( T_{c1} \) and \( T_{c2} \) are the critical temperatures, \( P_{c1} \) and \( P_{c2} \) are the critical pressures. For hydrogen, \( T_{c1} = 33.2 \, \text{K} \) and \( P_{c1} = 13.0 \, \text{bar} \); for nitrogen, \( T_{c2} = 126.2 \, \text{K} \) and \( P_{c2} = 33.5 \, \text{bar} \). Mole fractions: \( y_{1} = 0.75 \) and \( y_{2} = 0.25 \). Calculating pseudo-critical properties: \[ T_c' = (0.75 \cdot 33.2 \, \text{K}) + (0.25 \cdot 126.2 \, \text{K}) = 56.45 \, \text{K} \] \[ P_c' = (0.75 \cdot 13.0 \, \text{bar}) + (0.25 \cdot 33.5 \, \text{bar}) = 18.13 \, \text{bar} \] Reduced properties: \[ T_r = \frac{T}{T_c'} = \frac{298.15 \, \text{K}}{56.45 \, \text{K}} \approx 5.28 \] \[ P_r = \frac{P}{P_c'} \] Using the compressibility chart, determine compressibility factor \( Z \) for \( T_r \approx 5.28 \) and iterative solution or graphs to find \( P_r \).

Van der Waals Equation

Van der Waals equation is: \[ \bigg(P + a\frac{n^2}{V^2} \bigg)(V - nb) = nRT \] where \( a \) and \( b \) are constants. For mixtures, use: \[ a_m = \sum_{i} y_i^2 a_i \] and: \[ b_m = \sum_{i} y_i b_i \] Constants: For \( H_2 \): \( a = 0.244 \, \text{(bar·L}^2 \text{)/mol}^2 \), \( b = 0.0265 \, \text{L/mol} \); For \( N_2 \): \( a = 1.370 \, \text{(bar·L}^2 \text{)/mol}^2 \), \( b = 0.0387 \, \text{L/mol} \). Calculate mixture values: \[ a_m = (0.75^2 \cdot 0.244) + (0.25^2 \cdot 1.370) = 0.164 + 0.0856 \approx 0.2496 \, \text{(bar·L}^2 \text{)/mol}^2 \] \[ b_m = (0.75 \cdot 0.0265) + (0.25 \cdot 0.0387) = 0.019875 + 0.009675 \approx 0.02955 \, \text{L/mol} \] Substitute into van der Waals equation: \[ \bigg(P + a_m\frac{(1.00)^2}{(0.085)^2} \bigg)(0.085 - 0.02955) = 1.00 \times 8.314 \times 298.15 \] Solve this to find \( P \).

Rule of Additive Pressure and Compressibility Chart

First, find partial pressures: Partial pressure of \( H_2 \): \[ \frac{n_{H_2}RT}{V} = \frac{0.75 \times 8.314 \times 298.15}{0.085} \] Partial pressure of \( N_2 \): \[ \frac{n_{N_2}RT}{V} = \frac{0.25 \times 8.314 \times 298.15}{0.085} \] Add them together: Total partial pressure gives initial estimate \( P_{ideal} \). Use this and reduced properties \( T_r \) from Step 2 to find compressibility \( Z \). Correct total pressure: \[ P = \frac{P_{ideal}}{Z} \]

Key concepts

Ideal Gas Equation

The ideal gas equation is a cornerstone in understanding gas behavior under various conditions. This equation is expressed as: \[ PV = nRT \] Here, \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles, and \( T \) for temperature in Kelvin. The gas constant \( R \) is a universal constant with a value of 8.314 J/(mol·K).

To use this equation effectively, remember to always convert your temperatures to Kelvin by adding 273.15 to the Celsius temperature. Also, keep in mind that the volume must be in cubic meters and pressure in Pascals for consistency with the unit of the gas constant.

This equation assumes ideal gas behavior, which means gas molecules don’t attract or repel each other, and they occupy no volume themselves. Since real gases often deviate slightly from this ideal behavior, other models and corrections are needed for high precision calculations.

Kay's Rule

Kay’s Rule is a useful tool when dealing with gas mixtures. It allows us to estimate the critical properties (critical temperature and pressure) for a mixture. This method utilizes the molar fractions and the critical properties of the individual components to determine these pseudo-critical properties:

\[ T_c' = y_1 T_{c1} + y_2 T_{c2} \] and \[ P_c' = y_1 P_{c1} + y_2 P_{c2} \]

In these equations, \( y_1 \) and \( y_2 \) are the mole fractions of the different gases, while \( T_{c1} \), \( T_{c2} \), \( P_{c1} \), and \( P_{c2} \) are the critical temperatures and pressures for those gases.

Once you have these pseudo-critical properties, you can find the reduced temperature (\( T_r \)) and reduced pressure (\( P_r \)) using the given temperature and pressure:

\[ T_r = \frac{T}{T_c'} \] and \[ P_r = \frac{P}{P_c'} \] With these reduced properties, you can use the generalized compressibility chart to find the compressibility factor (\( Z \)) needed for further calculations. This factor helps in correcting for non-ideal behavior in gases.

van der Waals Equation

The van der Waals equation is another significant model used to describe the behavior of real gases by accounting for molecular size and intermolecular forces. It is formulated as: \[ \left( P + a \frac{n^2}{V^2} \right) (V - nb) = nRT \]

Here, \( a \) and \( b \) are unique constants for each gas that adjust for intermolecular attractions and the volume occupied by gas molecules, respectively. For mixtures, the constants for the mixture (\( a_m \) and \( b_m \)) are calculated as: \[ a_m = \sum_{i} y_i^2 a_i \] and \[ b_m = \sum_{i} y_i b_i \]

In these equations, \( y_i \) represents the mole fraction, and \( a_i \) and \( b_i \) are the constants for the individual gases. Once the mixture values for \( a \) and \( b \) are substituted back into the van der Waals equation, it can be solved for pressure, taking into account the real behavior of the gas mixture.

The adjustments provided by \( a \) and \( b \) are crucial for more accurate predictions of gas behavior under various conditions, especially at high pressures and low temperatures.

Generalized Compressibility Chart

The generalized compressibility chart is an invaluable tool for estimating the compressibility factor (\( Z \)) of a gas, which corrects the ideal gas law for real gas behavior. This chart uses the reduced temperature (\( T_r \)) and reduced pressure (\( P_r \)) of a gas to determine its compressibility factor.

To use the compressibility chart, follow these steps:

• Calculate the reduced temperature and pressure: \[ T_r = \frac{T}{T_c} \] and \[ P_r = \frac{P}{P_c} \]
• Locate \( T_r \) on the chart’s x-axis.
• Find \( P_r \) on the corresponding curve.
• Read the compressibility factor, \( Z \), from the chart.

Once you have \( Z \), it can be used to correct the ideal gas law: \[ PV = ZnRT \] This adjustment ensures that you account for deviations from ideal behavior, providing more accurate results for the pressure, volume, or temperature of a gas under real conditions. This tool extends the usability of the ideal gas law to real-world applications, where gases don’t always behave ideally.