Question

A tank contains a mixture of \(70 \%\) ethane and \(30 \%\) nitrogen \(\left(\mathrm{N}_{2}\right)\) on a molar basis at \(400 \mathrm{~K}, 200 \mathrm{~atm}\). For 2130 \(\mathrm{kg}\) of mixture, estimate the tank volume, in \(\mathrm{m}^{3}\), using (a) the ideal gas equation of state. (b) Kay's rule together with data from the generalized compressibility chart. (c) the ideal solution model together with data from the generalized compressibility chart.

Short answer

Using the ideal gas equation, Kay's rule, and the ideal solution model, we estimate the tank volume to be 23.34 m^3, 21.63 m^3, and 21.92 m^3, respectively. The adjustments for non-ideal behavior reduce the volume slightly from the ideal case.

Step-by-step solution

Understanding the problem

We need to estimate the tank volume of a gas mixture containing 70% ethane (C2H6) and 30% nitrogen (N2) at 400 K and 200 atm, for a total mass of 2130 kg. We will solve the problem using three different methods: (a) the ideal gas equation, (b) Kay's rule with the generalized compressibility chart, and (c) the ideal solution model with the generalized compressibility chart.

- Determine molecular weights

Calculate the molar masses of ethane (C2H6) and nitrogen (N2). For ethane, C2H6: (2*12.01 + 6*1.01) = 30.07 g/mol For nitrogen, N2: (2*14.01) = 28.02 g/mol

- Calculate the number of moles

Use the molar masses and mass fraction to calculate the number of moles of each component in the mixture. Ethane: (0.70 * 2130 kg * 1000 g/kg) / 30.07 g/mol Nitrogen:(0.30 * 2130 kg * 1000 g/kg) / 28.02 g/mol

- Convert moles to kmol

Convert the calculated moles from grams to kilograms:Ethane:N_{ethane} = (0.7 * 2130 / 30.07) Nitrogen:N_{N2} = (0.3 * 2130 / 28.02)

Part (a) Ideal Gas Equation

Using the ideal gas law, PV = nRT, to estimate the tank volume (V): For the mixture: P = 200 atm * (101.325 kPa/atm) R = 8.314 kPa-m^3/(kmol K) T = 400 K n_{total} = N_{ethane} + N_{N2} V = (n_{total} * R * T) / P

Part (b) Kay's Rule and Generalized Compressibility

1. Calculate the pseudo-critical properties (P_c, T_c) using mole fractions.For the mixture component: P_c mixture = y_ethane * P_c ethane + y_N2 * P_c N2 T_c mixture = y_ethane * T_c ethane + y_N2 * T_c N2 2. Compute the reduced properties: T_r = T/T_c mixture P_r = P/P_c mixture 3. Use generalized compressibility charts to find Z (compressibility factor): 4. Adjust ideal gas equation to estimate volume using Z: V = (n_{total} * R * T * Z) / P

Part (c) Ideal Solution Model and Generalized Compressibility

1. Calculate individual gas volumes using ideal gas law for each component: V_ethane = (N_{ethane} * R * T) / P V_N2 = (N_{N2} * R * T) / P 2. Use generalized compressibility factor charts for corrections. Adjust for respective compressibility factors: V_{corrected ethane} = V_ethane * Z_ethane V_{corrected N2} = V_N2 * Z_N2 3. Total tank volume = V_{corrected ethane} + V_{corrected N2}

Key concepts

ideal gas law

The ideal gas law is a fundamental equation in chemistry and physics that describes the relationship between the pressure, volume, temperature, and amount of an ideal gas. It's expressed as:
\[ PV = nRT \]
Here,

• \(P\) is the pressure of the gas.
• \(V\) is the volume of the gas.
• \(n\) is the number of moles of gas.
• \(R\) is the ideal gas constant \( (8.314 \text{kPa-m}^3 / \text{kmol K}) \).
• \(T\) is the temperature in Kelvin.

In this exercise, we are given the pressure, temperature, and the mass of the gas mixture. To estimate the tank volume using the ideal gas law, we:
1. Calculate the molar mass and the total number of moles for each gas.
2. Sum these to get the total number of moles in the mixture, and apply the ideal gas law to find the volume.
This method is straightforward but assumes the gas behaves ideally, which may not be accurate at high pressures.

Kay's rule

Kay's rule is a useful method for estimating the properties of gas mixtures by treating them as pseudo-components. It simplifies calculations by averaging the critical properties of the gases in the mixture. Here’s how it works:
1. Calculate pseudo-critical properties:

• Pseudo-critical pressure, \(P_c\), is the weighted average of the critical pressures of the component gases.
• Pseudo-critical temperature, \(T_c\), is the weighted average of the critical temperatures of the component gases.

2. Using these pseudo-critical properties, compute the reduced properties:

• Reduced temperature: \( T_r = T / T_c \)
• Reduced pressure: \( P_r = P / P_c \)

3. Utilize the generalized compressibility charts to find the compressibility factor \(Z\).
4. Adjust the ideal gas equation to account for the compressibility factor:
\[ V = \frac{nRTZ}{P} \]
Kay's rule provides a more accurate volume estimation than the ideal gas law by considering the mixture's real gas behavior. It leverages the generalized compressibility charts to adjust for non-idealities.

generalized compressibility chart

The generalized compressibility chart is an essential tool for real gas calculations. It provides the compressibility factor \(Z\), which adjusts the ideal gas law to account for real gas behavior. Here’s how to use it:
1. First, determine the reduced temperature \(T_r\) and reduced pressure \(P_r\) of the gas mixture.
2. Locate these values on the compressibility chart. \(T_r\) is on one axis, and \(P_r\) is on the other.
3. Find the corresponding compressibility factor \(Z\) from the chart.
4. Apply \(Z\) to adjust the volume calculation:
\[ V = \frac{nRTZ}{P} \]
The compressibility factor \(Z\) corrects for the deviation from ideal gas behavior caused by intermolecular forces and volume occupied by gas molecules. It is particularly important at high pressures and low temperatures where gases deviate significantly from ideal behavior.

ideal solution model

The ideal solution model assumes that gas mixtures behave like ideal solutions, meaning each gas in the mixture follows the ideal gas law independently. Here’s how to use this model in calculations:
1. Use the ideal gas law to calculate the volume of each gas component in the mixture separately: \( V_{gas} = \frac{nRT}{P} \).
2. Adjust these volumes using the generalized compressibility charts to find the respective compressibility factors for each gas:

• \(V_{corrected} = V * Z\) \( \text{for each component} \).

3. Sum the corrected volumes to get the total tank volume.
\[ V_{total} = V_{corrected, ethane} + V_{corrected, N_2} \]
This model assumes no interaction between the different gas molecules, treating the mixture purely as a physical blend of ideal gases. The generalized compressibility chart again helps in adjusting for real gas behavior.