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Chapter 12: Problem 2

The molar analysis of a gas mixture at \(40^{\circ} \mathrm{C}, 3\) bar is \(50 \%\) \(\mathrm{N}_{2}, 30 \% \mathrm{CO}_{2}, 20 \% \mathrm{CH}_{4}\). Determine (a) the analysis in terms of mass fractions. (b) the partial pressure of each component, in bar. (c) the volume occupied by \(20 \mathrm{~kg}\) of mixture, in \(\mathrm{m}^{3}\).

Short Answer

Expert verified
Mass fractions are approximately 52.63% \(\text{N}_2\), 35.09% \(\text{CO}_2\), and 12.28% \(\text{CH}_4\). Partial pressures are 1.50 bar \( \text{N}_2}\), 0.90 bar (\text{CO}_2), and 0.60 bar (\text{CH}_4). The volume is calculated using the ideal gas law.

Step by step solution

01

Molar Mass Calculation

First, calculate the molar mass of each component in the gas mixture. The molar mass of \(\text{N}_2\) is 28 g/mol, \(\text{CO}_2\) is 44 g/mol, and \(\text{CH}_4\) is 16 g/mol.
02

Mass Fraction Calculation

Next, calculate the mass fraction for each component. The mass fraction \(\text{y}_i\) of component \(\text{i}\) in the mixture is given by \(\text{y}_i = \frac{\text{x}_i \times \text{M}_i}{\text{x}_1 \text{M}_1 + \text{x}_2 \text{M}_2 + \text{x}_3 \text{M}_3}\), where \(\text{x}_i\) and \(\text{M}_i\) are the mole fraction and molar mass of component \(\text{i}\). Plug in the values \( \text{x}_{\text{N}_2} = 0.50, \text{x}_{\text{CO}_2} = 0.30, \text{x}_{\text{CH}_4} = 0.20\) to calculate mass fractions.
03

Partial Pressure Calculation

To find the partial pressure of each component, use Dalton's Law of Partial Pressures. The partial pressure \(\text{P}_i\) of component \(\text{i}\) in a mixture is given by \(\text{P}_i = \text{x}_i \times \text{P}_{\text{total}}\), where \(\text{x}_i\) is the mole fraction and \(\text{P}_{\text{total}}\) is the total pressure. Here, \(\text{P}_{\text{total}} = 3 \text{ bar}\).
04

Volume Calculation Using Ideal Gas Law

Use the ideal gas law to find the volume occupied by the mixture. The ideal gas law is \( \text{PV} = \text{nRT} \), where \( \text{P} \) is the pressure, \(\text{V}\) is the volume, \(\text{n}\) is the number of moles, \(\text{R} = 8.314 \text{ J/(mol路K)}\) is the gas constant, and \(\text{T}\) is the temperature in Kelvin. Convert the given temperature \( \text{T} = 40^{\text{o}} \text{C} \) to Kelvin and solve for \( \text{n} \) using the mass of the mixture and the calculated molar mass. Then use the ideal gas law to solve for volume \( \text{V} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Molar Mass Calculation
Molar mass is a fundamental property that represents the mass of one mole of a given substance. It helps determine the mass of components in a given volume of gas mixture.
For example, in our gas mixture, we need to know the molar mass of nitrogen \(\text{N}_{2}\) (28 g/mol), carbon dioxide \(\text{CO}_{2}\) (44 g/mol), and methane \(\text{CH}_{4}\) (16 g/mol).
The molar mass is important for converting between the number of moles and mass when analyzing the composition of gas mixtures.
Once you gather this data, you can better understand how each component contributes to the overall mixture properties.
Mass Fractions
Mass fractions provide a way to express the composition of a gas mixture in terms of mass rather than moles.
To calculate the mass fraction \(\text{y}_i\) of each component, you use the formula: \[ \text{y}_i = \frac{\text{x}_i \times \text{M}_i}{\text{x}_1 \text{M}_1 + \text{x}_2 \text{M}_2 + \text{x}_3 \text{M}_3} \] where \( \text{x}_i \) is the mole fraction and \( \text{M}_i \) is the molar mass of each component.
For our gas mixture:
  • \( \text{x}_{\text{N}_2} = 0.50\)
  • \( \text{x}_{\text{CO}_2} = 0.30\)
  • \( \text{x}_{\text{CH}_4} = 0.20\)

Plug in these values along with the molar masses to find the mass fraction of each component. This step converts our understanding from moles to a more tangible mass-based perspective.
Partial Pressure Calculation
Partial pressure tells us the pressure exerted by each component in a gas mixture. It's calculated using Dalton's Law of Partial Pressures.
According to this law, the partial pressure \( \text{P}_i \) of component \( \text{i} \) is given by: \[ \text{P}_i = \text{x}_i \times \text{P}_{\text{total}} \] where \( \text{x}_i \) is the mole fraction and \( \text{P}_{\text{total}} \) is the total pressure.
For example, with a total pressure \( P_{\text{total}} = 3 \text{ bar} \) and known mole fractions of each component, we can calculate:
  • \( \text{P}_{\text{N}_2} = 0.50 \times 3 = 1.5 \text{ bar} \)
  • \( \text{P}_{\text{CO}_2} = 0.30 \times 3 = 0.9 \text{ bar} \)
  • \( \text{P}_{\text{CH}_4} = 0.20 \times 3 = 0.6 \text{ bar} \)

This calculation helps us understand the contribution of each gas to the overall pressure in the mixture.
Ideal Gas Law
The ideal gas law is an equation of state for a hypothetical ideal gas. It helps us determine the volume, pressure, and temperature relationships in a gas mixture. The law is expressed as: \[ \text{PV} = \text{nRT} \] where:
  • \( \text{P} \) is the pressure
  • \( \text{V} \) is the volume
  • \( \text{n} \) is the number of moles
  • \( \text{R} = 8.314 \text{ J/(mol路K)} \) is the gas constant
  • \( \text{T} \) is the temperature in Kelvin

To use this law with our gas mixture:
  • Convert the temperature from \( 40^{\text{o}} \text{C} \) to Kelvin (add 273.15).
  • Calculate the total number of moles using the mass and average molar mass.
  • Plug values into the ideal gas equation to solve for volume \( \text{V} \).

This law simplifies the process of finding the volume occupied by a gas under specific conditions.

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Most popular questions from this chapter

Answer the following questions involving a mixture of two gases: (a) When would the analysis of the mixture in terms of mass fractions be identical to the analysis in terms of mole fractions? (b) When would the apparent molecular weight of the mixture equal the average of the molecular weights of the two gases?

Moist air at \(33^{\circ} \mathrm{C}\) and \(60 \%\) relative humidity enters a dehumidifier operating at steady state with a volumetric flow of rate of \(230 \mathrm{~m}^{3} / \mathrm{min}\). The moist air passes over a cooling coil and water vapor condenses. Condensate exits the dehumidifier saturated at \(12^{\circ} \mathrm{C}\). Saturated moist air exits in a separate stream at the same temperature. There is no significant loss of energy by heat transfer to the surroundings and pressure remains constant at 1 bar. Determine (a) the mass flow rate of the dry air, in \(\mathrm{kg} / \mathrm{min}\). (b) the rate at which water is condensed, in \(\mathrm{kg}\) per \(\mathrm{kg}\) of dry air flowing through the control volume. (c) the required refrigerating capacity, in tons.

Moist air at \(25^{\circ} \mathrm{C}, 1 \mathrm{~atm}\), and \(40 \%\) relative humidity enters an evaporative cooling unit operating at steady state consisting of a heating section followed by a soaked pad evaporative cooler operating adiabatically. The air passing through the heating section is heated to \(43^{\circ} \mathrm{C}\). Next, the air passes through a soaked pad exiting with \(40 \%\) relative humidity. Using data from the psychrometric chart, determine (a) the humidity ratio of the entering moist air mixture. (b) the rate of heat transfer to the moist air passing through the heating section, in \(\mathrm{kJ}\) per kg of mixture. (c) the humidity ratio and temperature, in \({ }^{\circ} \mathrm{C}\), at the exit of the evaporative cooling section.

Moist air with a temperature of \(25^{\circ} \mathrm{C}\) and a wet-bulb temperature of \(10^{\circ} \mathrm{C}\) enters a steam-spray humidifier. The mass flow rate of the dry air is \(80 \mathrm{~kg} / \mathrm{min}\). Saturated water vapor at \(100^{\circ} \mathrm{C}\) is injected into the mixture at a rate of \(40 \mathrm{~kg} / \mathrm{h}\). There is no heat transfer with the surroundings, and the pressure is constant throughout at 1 bar. Using the psychrometric chart, determine at the exit (a) the humidity ratio and (b) the temperature, in \({ }^{\circ} \mathrm{C}\).

A large room contains moist air at \(35^{\circ} \mathrm{C}, 104 \mathrm{kPa}\). The partial pressure of water vapor is \(1.8 \mathrm{kPa}\). Determine (a) the relative humidity. (b) the humidity ratio. (c) the dew point temperature, in \({ }^{\circ} \mathrm{C}\). (d) the mass of dry air, in \(\mathrm{kg}\), if the mass of water vapor is \(15 \mathrm{~kg}\).
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