An Introduction to Thermal Physics

Citation

Schroeder, D. V. (2021). An Introduction to Thermal Physics. Oxford University Press. (First published 2000 by Addison Wesley Longman.)

Chapter Summary

Eight chapters in three parts, followed by two appendices, annotated suggested reading and reference data: Part I sets out the first and second laws, and Parts II and III apply them with the tools of thermodynamics and of statistical mechanics respectively.

Chapter 1: Energy in Thermal Physics

A survey of thermal equilibrium, the ideal gas and a microscopic model of it, equipartition, heat and work (including compression work), heat capacities, latent heat and enthalpy. A final section on rates of processes (conduction, viscosity, diffusion) is optional.

Chapter 2: The Second Law

The second law is reached by counting microstates: two-state systems and the Einstein solid, then two interacting Einstein solids, whose multiplicity function becomes extremely sharp for large systems. The chapter handles "very large numbers" and Stirling's approximation, computes the multiplicity of an ideal gas, and defines entropy, including the entropy of mixing and reversible versus irreversible processes.

Chapter 3: Interactions and Implications

Temperature, pressure and chemical potential are defined as partial derivatives of the entropy, leading to the thermodynamic identity. The chapter relates entropy to heat and predicts heat capacities for a paramagnet, the Einstein solid and the ideal gas, all microcanonically (as the author's overview notes).

Chapter 4: Engines and Refrigerators

Efficiency limits for heat engines and refrigerators follow from the first and second laws, with the Carnot cycle as the ideal case. The chapter then treats real devices: internal combustion and steam engines, the throttling process, liquefaction of gases and ways of approaching absolute zero.

Chapter 5: Free Energy and Chemical Thermodynamics

One of the two longest chapters (about 70 pages, like Chapter 7). The Helmholtz and Gibbs free energies are introduced as available work and as a force toward equilibrium, then applied to fuel cells and batteries, phase transformations (the Clausius–Clapeyron relation, the van der Waals model), mixtures, dilute solutions and osmotic pressure, and chemical equilibrium.

Chapter 6: Boltzmann Statistics

The Boltzmann factor and partition function, applied to paramagnets and rotating molecules, followed by the equipartition theorem, the Maxwell speed distribution, the link between partition functions and free energy, and the ideal gas revisited.

Chapter 7: Quantum Statistics

The Gibbs factor leads to the boson and fermion distribution functions, which are applied to degenerate Fermi gases (with the density of states and the Sommerfeld expansion), blackbody radiation (from the Planck spectrum to the cosmic background radiation), the Debye theory of solids and Bose–Einstein condensation.

Chapter 8: Systems of Interacting Particles

Two independent sections go beyond ideal systems: weakly interacting gases via the cluster expansion and the second virial coefficient, and the Ising model of a ferromagnet, solved exactly in one dimension and studied with the mean field approximation and Monte Carlo simulation.

Appendices A–B: Elements of Quantum Mechanics; Mathematical Results

Appendix A summarises the quantum mechanics the text uses (wavefunctions, the particle in a box, the harmonic oscillator, angular momentum, many-particle systems); Appendix B collects Gaussian integrals, the gamma function, Stirling's approximation and the integrals of quantum statistics.

Key Concepts

Part I: Fundamentals (Chapters 1–3)

• Multiplicity: the number of microstates consistent with a macrostate; for large systems it is so sharply peaked that the most probable macrostate is effectively certain, the book's route to the second law.
• Entropy: defined statistically as the logarithm of the multiplicity, then connected to heat flow.
• Thermodynamic identity: the relation that ties energy changes to temperature, pressure and chemical potential, each defined as a derivative of entropy.

Part II: Thermodynamics (Chapters 4–5)

• Carnot cycle: the ideal cycle that reaches the maximum efficiency allowed by the second law between two temperatures.
• Coefficient of performance: a refrigerator's ratio of heat extracted to work supplied.
• Helmholtz and Gibbs free energies: the energy that must be supplied as work (or can be recovered) at constant temperature, or at constant temperature and pressure.
• Clausius–Clapeyron relation: the slope of a phase boundary in terms of latent heat and volume change.

Part III: Statistical Mechanics (Chapters 6–8)

• Boltzmann factor and partition function: the relative probability of a state of a system in contact with a reservoir, and the sum over states from which averages and free energy follow.
• Fermi–Dirac and Bose–Einstein distributions: the average occupancies of single-particle states for fermions and bosons.
• Ising model: a lattice of interacting spins used as a model of ferromagnetism.

Critical Analysis

Strengths

• A deliberately balanced plan. The preface positions the book in the middle of the "thermo-to-statmech continuum": Part I develops the laws from both microscopic and macroscopic viewpoints, and Parts II and III extend each side in turn. Harvey Gould's review in Physics Today (August 2000) judged the book more balanced between the two approaches than Ralph Baierlein's Thermal Physics, which he reviewed alongside it.
• Applications across the sciences. The preface aims to include as many applications as possible, and the contents bear this out, from fuel cells and nitrogen fixation to the cosmic background radiation; topics in the problems run from the adiabatic lapse rate to white dwarfs and Hawking radiation. Gould singled out the table of thermodynamic data, which students use for problems such as the diamond–graphite transition and the power of a lead-acid cell.
• Problems placed inside the text. Problems close almost every section, ranging from thought questions and order-of-magnitude estimates to extended projects, and some later text relies on results derived in them (the author names Problems 3.24–3.25 on the Einstein solid). Gould welcomed the problems that call for a computer or programmable calculator, which allow questions beyond closed-form answers.
• Accessible, flexible exposition. The style is informal and direct, and Gould judged the arguments easier for most undergraduates to follow than Baierlein's. The author's website adds a section-by-section dependency map, course plans and a recommended-problem list.

Limitations

• Too long for one semester, by the author's own account. The preface says the book outgrew its one-semester goal and that several sections must be omitted; the author's own course skips Sections 1.7, 4.3, 4.4, 5.4–5.6 and all of Chapter 8.
• Less sophisticated arguments, some uneven spots. Gould noted that the arguments are less sophisticated than Baierlein's (the other side of their accessibility), and he found the treatment of the virial expansion unnecessarily abstruse.
• Content fixed in 2000. The Oxford University Press issue reprints the original text with only page corrections, so the book reflects the field as of its first publication; the author also describes his historical remarks as deliberately oversimplified.
• Limited depth in advanced statistical mechanics. Interacting systems receive a single chapter of two sections, and the preface acknowledges that some topics are omitted or treated superficially, pointing readers to the annotated suggested reading.

Where It Fits in the Course

In this course list Schroeder sits beside Stephen and Katherine Blundell's Concepts in Thermal Physics; both are described as treating thermodynamics and statistical mechanics together at undergraduate level, so either can anchor a first course, and each offers a second account of any topic the other leaves unclear. Two shorter, gentler books suit different needs: Finn's Thermal Physics is described as a concise, mainly classical introduction to thermodynamics, and Don Lemons's A Student's Guide to Entropy is a short companion on a single concept; either can be read before or alongside Schroeder, particularly Lemons's guide next to Chapters 2–3.

Ralph Baierlein's Thermal Physics covers the same ground at a more demanding level. Gould's review compared the two directly: he found Baierlein's arguments more sophisticated and more focused on statistical mechanics, and recommended Schroeder as the course text with Baierlein as a source of supplementary material. Pathria and Beale's Statistical Mechanics, a graduate text on ensembles, phase transitions and the renormalization group, is the natural next step after Schroeder's Chapters 6–8 for students who want the theory in depth.

How to Use This Book

The book is written for physics undergraduates from the second to the fourth year, and its applications also suit students of chemistry, earth science and engineering. The preface asks for a year of calculus-based introductory physics and a year of calculus. Students whose introductory course had no thermal physics should spend extra time on Chapter 1; those who have not met quantum physics should consult Appendix A while reading Chapters 2, 6 and 7. Multivariable calculus is introduced gradually and is helpful but not essential as a corequisite.

A one-semester plan, following the author's course pages:

• Core for everyone: Part I (Chapters 1–3), apart from Section 1.7 on transport, which is optional; do not skip Section 2.5 on the ideal gas, whose results recur later. Then Sections 5.1–5.2 on free energy and Chapter 6 (Sections 6.3 and 6.4 can be dropped).
• Statistical-mechanics emphasis: go from Chapter 3 straight to Chapter 6, picking up Sections 5.1–5.2 before Section 6.5, then continue into Chapter 7 (the Sommerfeld expansion is optional) and, if time allows, Chapter 8.
• Thermodynamics emphasis: read straight through and stop at the end of Chapter 6; nothing later depends on Chapter 4, so it can also be dropped in a statistical-mechanics course.
• Two semesters: Chapters 1–5 in the first, the rest of the book in the second, with some of the harder problems in Chapters 7–8.

Section 7.4 on blackbody radiation can be read at any time after Section 6.2 (apart from its subsection on photons), and the two sections of Chapter 8 are independent of Chapter 7 and of each other.

Read with a pencil and treat the problems as part of the text: read every problem even if only a few are attempted, and work the problems the later chapters rely on (3.24–3.25 in particular). The website's recommended-problem list is a good filter, and its corrections page should be checked. The solution manual is restricted to instructors, so self-learners need to check results against limiting cases and estimates. When buying a used copy, check the edition: the author warns that the abridged international editions (roughly 330 pages instead of 422) omit Chapter 8, both appendices, the preface, the suggested reading and the index.