Book cover for Thermodynamics: An Engineering Approach

Thermodynamics: An Engineering Approach

Yunus A. Cengel, Michael A. Boles

ISBN #9781259822674

9th Edition

2,694 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Chapter 13 on Gas Mixtures provides a comprehensive analysis of how gas mixtures behave under both ideal and real conditions. It emphasizes the utility of Dalton’s and Amagat’s laws in understanding the additive nature of partial pressures and volumes. For real-gas mixtures, the chapter introduces corrections using compressibility factors and Kay’s rule to account for non-idealities. Additionally, extensive properties are derived through summing individual contributions, with significant focus on mixing irreversibilities, entropy generation, and the minimum work required for separating mixtures—a critical aspect in both theoretical studies and practical process design.

Learning Objectives

1

Differentiate between ideal-gas and real-gas mixture behavior and identify the conditions under which each model is applicable.

2

Apply Dalton's and Amagat's laws to calculate partial pressures and volumes in gas mixtures.

3

Utilize correction methods such as compressibility factors and Kay’s rule for analyzing real-gas mixtures.

4

Derive extensive properties of gas mixtures by summing contributions from individual components.

5

Assess the impacts of mixing irreversibilities, entropy generation, and separation work on process design and performance evaluation.

Key Concepts

CONCEPT

DEFINITION

Ideal-Gas Mixture

A gas mixture in which individual gases are assumed to behave ideally, allowing direct relationships between mole fractions and properties like partial pressures and volumes.

Real-Gas Mixture

A gas mixture that requires corrections for intermolecular forces and non-ideal interactions, often using compressibility factors and Kay’s rule.

Dalton's Law

A principle stating that the total pressure of a gas mixture is the sum of the partial pressures of its individual components.

Amagat’s Law

A principle asserting that the total volume of a gas mixture is the sum of the volumes occupied by each individual gas, assuming constant pressure and temperature.

Compressibility Factor (Z)

A correction factor used to account for deviations from ideal gas behavior in real-gas mixtures.

Kay’s Rule

A method used to estimate the pseudo-critical properties of a gas mixture by weighing the contribution of each component, aiding in the correction of real-gas behavior.

Extensive Mixture Properties

Properties that depend on the amount or extent of the system, derived by summing the contributions from each component in the gas mixture.

Chemical Potential

A thermodynamic property that indicates the energy change when an additional amount of substance is introduced, important for understanding separation processes and work requirements.

Mixing Irreversibilities

Phenomena associated with entropy generation during the mixing of gases, which can affect the efficiency of separation and process design.

Minimum Work of Separation

The theoretical lower limit of energy required to separate a gas mixture into its components, taking into account factors like chemical potential and mixing irreversibilities.

Example Problems

Example 1

What are mass and mole fractions?

Example 2

Consider a mixture of several gases of identical masses. Will all the mass fractions be identical? How about the mole fractions?

Example 3

The sum of the mole fractions for an ideal-gas mixture is equal to $1 .$ Is this also true for a real-gas mixture?

Example 4

Somebody claims that the mass and mole fractions for a mixture of $\mathrm{CO}_{2}$ and $\mathrm{N}_{2} \mathrm{O}$ gases are identical. Is this true? Why?

Example 5

Consider a mixture of two gases. Can the apparent molar mass of this mixture be determined by simply taking the arithmetic average of the molar masses of the individual gases? When will this be the case?
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Step-by-Step Explanations

QUESTION

How do you calculate the partial pressure of a component in an ideal-gas mixture using Dalton's law?

STEP-BY-STEP ANSWER:

Step 1: Determine the mole fraction (x_i) of the component of interest in the gas mixture.
Step 2: Identify the total pressure (P_total) of the gas mixture.
Step 3: Calculate the partial pressure using the formula P_i = x_i * P_total.
Final Answer: The partial pressure P_i is obtained as the product of the component's mole fraction and the total pressure.

Application of Dalton's Law in Ideal-Gas Mixtures

QUESTION

How do you adjust the calculated properties of a gas mixture for real-gas behavior?

STEP-BY-STEP ANSWER:

Step 1: Calculate the ideal gas property (e.g., pressure or volume) using standard methods.
Step 2: Determine the compressibility factor (Z) for the mixture at the given conditions.
Step 3: Apply the correction by multiplying the ideal gas calculation by the compressibility factor.
Final Answer: The corrected property is obtained by adjusting the ideal gas value with the compressibility factor Z.

Calculating Real-Gas Corrections Using the Compressibility Factor

QUESTION

How do you relate chemical potential differences to the minimum work of separation in a gas mixture?

STEP-BY-STEP ANSWER:

Step 1: Identify the chemical potential of each component in the gas mixture.
Step 2: Determine the differences in chemical potential that are responsible for the driving force of separation.
Step 3: Use these differences, along with considerations of mixing irreversibilities and entropy generation, to calculate the minimum work required for separation.
Final Answer: The minimum work of separation is derived from the sum of the individual contributions of chemical potential differences across the components.

Estimating Separation Work via Chemical Potential

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Common Mistakes

  • Assuming ideal gas behavior for all gas mixtures without accounting for real-gas corrections when necessary.
  • Confusing the application of Dalton's law (which deals with pressures) with Amagat’s law (which deals with volumes).
  • Neglecting the impact of compressibility factors and Kay’s rule when analyzing high-pressure or non-ideal gas mixtures.
  • Overlooking the significance of mixing irreversibilities and entropy generation in the energy analysis of separation processes.
  • Misinterpreting the derivation of extensive mixture properties by failing to correctly sum the contributions of individual gases.