In this post, we will see the book The Differential Equations of Thermodynamics by V. V. Sychev.

About the book

Thermodynamics, as is known, is constructed quite simply. Two of its main laws have been established experimentally, and by applying mathematical tools to them we can obtain the range of conclusions in which thermodynamics is so rich.

The mathematical tools of thermodynamics are simple but in

certain aspects at the same time quite sophisticated. Neglecting

some of these sophisticated “trifles” often results in crude mistakes, even in reputable works on thermodynamics.The restricted size of the usual textbooks on thermodynamics does not permit discussing more extensively these important questions concerning the mathematical tools. For this reason it was felt necessary to consider these problems in a special book, which though limited in size would at the same time go into details.

Naturally, the author does not aim at a presentation of thermo-

dynamics and its physical, chemical, and technical applications.

These have been sufficiently discussed in the existing textbooks and monographs. The purpose of this book is more modest—to deepen the reader’s knowledge of the mathematical tools of thermodynamics, to systematize them, and at the same time to emphasize questions that are often a source of error in thermodynamic calculations. The book is therefore designed to meet the needs of students and graduates majoring in thermal physics, physical engineering, and physico- technical specialities who already have a background in general thermodynamics. I hope that the book may also prove useful to scientists, engineers, and teachers specializing in thermodynamics.

The book was translated from the Russian by *Eugene Yankovsky *and second edition was published by Mir in 1983.

You can get the book here.

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**Contents**

**1. Recollections of Thermodynamics: A Survey Chapter**

1.1 Basic Concepts of Thermodynamics 9

1.2 The Equations of the First and Second Laws of Thermodynamics 12

**2. The Mathematical Tools of Thermodynamics**

2.1 Derivatives of Functions of Several Variables 16

2.2 Pfaffian Forms and Total Differentials 19

2.3 The Relationships Between Derivatives 26

2.4 The Legendre Transformation 28

2.5 The Discontinuities of Thermodynamic Functions 29

2.6 Jacobians 31

**3. Characteristic Functions and Their Properties**

3.1 The Basic Characteristic Functions 35

3.2 The Chemical Potential 47

3.3 The Massieu-Planck Functions 57

3.4 The Grand Potential and the Kramers Function 70

3.5 Other Characteristic Functions 75

**4. The Maxwell Equations**

4.1 Simple Systems 88

4.2 Complex Systems 91

4.3 Systems with Variable Amounts of Substance 94

**5. Simple Thermodynamic Systems**

5.1 Partial Derivatives of Thermodynamic Potentials 101

5.2 The Gibbs-Helmholtz Equations 108

5.3 Equations for Heat Capacities 115

5.4 Equations for Entropy 121

5.5 Other Important Partial Derivatives of Thermodynamic Functions 123

5.6 The Differential Equation of an Isentrope. The Laplace Equation 126

5.7 Basic Thermodynamic Equations for Flow Processes 128

**6. The Discontinuities of Thermodynamic Quantities on Boundary Curves**

6.1 Crossing Boundary Curves: Salient Points and Discontinuities of Thermodynamic Functions 136

6.2 Discontinuity Equations for Thermodynamic Functions on Boundary Curves 140

**7. Phase Transitions**

7.1 A General Survey 148

7.2 The Clausius-Clapeyron Equation and Its Analogs 151

7.3 The Phase Transition Equation at Different Pressures in the Phases

7.4 Isolines in a Two-Phase Region in the p, T Plane .

7.5 Discontinuity Equations for Thermodynamic Quantities on the Boundary of a Two-Phase Region 161

7.6 Heat Capacities on Boundary Curves 172

7.7 Thermodynamic Quantities in a Two-Phase Region 175

7.8 Equations Relating Thermodynamic Quantities on the Left and Right Boundary Curves 180

7.9 Equations for Second- Order Phase Transitions 190

**8. The Mathematics of the Critical Point**

8.1 Thermodynamic Relations for the Critical Point 197

8.2 Describing the Thermodynamic Properties of a Substance at the Critical Point 205

**9. Complex Thermodynamic Systems**

9.1 The Basic Thermodynamic Relations for Complex Systems 210

9.2 Systems in a Magnetic Field 211

9.3 Systems in an Electric Field 229

9.4 Systems in a Gravitational Field 234

9.5 Elastically Deformed Systems 235

9.6 Voltaic Systems 238

9.7 Two-Dimensional Surface Systems 204

9.8 Radiation in a Cavity as a Thermodynamic System 254

Notation Index 257

Subject Index

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