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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Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Chapter 12 on Thermodynamic Property Relations emphasizes the fundamental role of partial differentials in thermodynamics. It covers the difference between partial and total differentials, the derivation and application of exact differentials, and key relations such as Maxwell relations and the Clapeyron equation. These tools enable the analysis of changes in internal energy, enthalpy, entropy, and specific heats, offering powerful methods to evaluate both ideal and real gas systems in various thermodynamic processes.

Learning Objectives

1

Differentiate between partial (?) and total (d) differentials in thermodynamic contexts.

2

Explain the concept of exact differentials and how they lead to relations such as the Maxwell relations.

3

Derive and apply the Clapeyron equation and related expressions for changes in internal energy, enthalpy, and entropy.

4

Analyze the role of specific heats (cv and cp) in thermodynamic processes and their derivations through partial differentials.

5

Evaluate the practical applications of partial differential relations in understanding properties of both ideal and real gases.

Key Concepts

CONCEPT

DEFINITION

Partial Differential (∂)

A derivative where one or more variables are held constant; used to express changes in a function with several variables.

Total Differential (d)

The overall derivative of a function taking into account the simultaneous variations of all its variables.

Exact Differential

A differential that corresponds to a state function, ensuring that its integral is path-independent; the foundation for deriving Maxwell relations.

Maxwell Relations

A set of equations derived from the equality of mixed second partial derivatives of thermodynamic potentials, used to relate measurable and non-measurable properties.

Clapeyron Equation

An expression that relates the changes in pressure and temperature during a phase transition to the latent heat and volume change, derived using partial differential relations.

Internal Energy (U)

The total energy contained within a system, changes in which can be expressed using partial derivatives with respect to entropy and volume.

Enthalpy (H)

A thermodynamic potential defined as H = U + PV, used to describe processes occurring at constant pressure.

Entropy (S)

A measure of the disorder or randomness in a system, whose changes are central to second law analyses in thermodynamics.

Specific Heats (cv and cp)

Quantities that represent the amount of heat required to raise the temperature of a system at constant volume (cv) or constant pressure (cp), derived using partial differentials.

Example Problems

Example 1

What is the difference between partial differentials and ordinary differentials?

Example 2

Consider the function $z(x, y)$. Plot a differential surface on $x-y-z$ coordinates and indicate $\partial x, d x, \partial y, d y,(\partial z)_{x}$ $(\partial z)_{y},$ and $d z$

Example 3

Consider a function $z(x, y)$ and its partial derivative $(\partial z / \partial y)_{x}$. Under what conditions is this partial derivative equal to the total derivative $d z / d y ?$

Example 4

Consider a function $z(x, y)$ and its partial derivative $(\partial z / \partial y)_{x}$ If this partial derivative is equal to zero for all values of $x$, what does it indicate?

Example 5

Consider a function $z(x, y)$ and its partial derivative $(\partial z / \partial y)_{x}$. Can this partial derivative still be a function of $x ?$
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Step-by-Step Explanations

QUESTION

How do you derive a Maxwell relation from a given thermodynamic potential?

STEP-BY-STEP ANSWER:

Step 1: Identify a thermodynamic potential (e.g., the Helmholtz free energy A) expressed as a function of temperature (T) and volume (V).
Step 2: Write the differential form of the potential, such as dA = -S dT - P dV, where S and P are the entropy and pressure respectively.
Step 3: Recognize that since A is a state function, its second mixed partial derivatives must be equal; thus, ∂²A/∂T∂V = ∂²A/∂V∂T.
Step 4: Set up the equality using the chain rule, equating the partial derivative of -S with respect to V to the partial derivative of -P with respect to T.
Step 5: Simplify to obtain the Maxwell relation: (∂S/∂V)_T = (∂P/∂T)_V.
Final Answer: The derived Maxwell relation is (∂S/∂V)_T = (∂P/∂T)_V, which links entropy and pressure changes with respect to volume and temperature.

Derivation of a Maxwell Relation

QUESTION

How is the Clapeyron equation derived using partial differentials and used to relate phase transitions?

STEP-BY-STEP ANSWER:

Step 1: Begin with the equality of the chemical potentials for two phases in equilibrium.
Step 2: Express changes in the Gibbs free energy (or chemical potential) in terms of pressure and temperature differentials.
Step 3: Differentiate the equilibrium condition and rearrange to isolate the dp/dT term.
Step 4: Introduce the latent heat (L) and the specific volume change (Δv) during the phase transition.
Step 5: Obtain the Clapeyron equation: dp/dT = L/(TΔv).
Final Answer: The Clapeyron equation, dp/dT = L/(TΔv), provides a relation between the slope of the phase boundary, latent heat, and volume change.

Application of the Clapeyron Equation

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Common Mistakes

  • Failing to clearly distinguish between partial (?) and total (d) differentials, leading to incorrect interpretations of thermodynamic equations.
  • Assuming all differentials are exact without verifying the criteria required for state functions.
  • Neglecting the importance of sign conventions when deriving Maxwell relations, which may cause errors in subsequent calculations.
  • Overlooking the limitations of ideal approximations, particularly when applying derived expressions to real gases without appropriate corrections.