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Basic algorithm of thermodynamics (BAT)

Zhigang Suo's picture

I have just taught an undergraduate course on thermodyanmics and written up my lectures as a book. The book is posted as a google doc. If you cannot access the google doc, here is a PDF file of the book. The following paragraphs describe my approach. I hope to hear about your experience with thermodynamics. Please leave your comments below. Please help me hear from more people by forwarding this post. Thank you.

Thermodynamics for everyone

Thermodynamics should be a play for everyone, just as English, physics, chemistry, and calculus should. The logic of thermodynamics is expressed mostly in English, with a little physics, chemistry, and calculus. I’ll be careful in using English, and even more careful in using physics, chemistry, and calculus. 


English is a wonderful language, but is not invented for thermodynamics. I will mostly use nouns and verbs, and mostly avoid adjectives. Along with words, I will use pictures, videos, and equations.  A picture is worth a thousand words. A video is worth a thousand pictures. An equation is worth a thousand videos. For example, the equation f = ma has generated countless words, pictures, and videos. 


The facts of physics, chemistry, and calculus are easy to state, but applying them takes practice. You experience thermodynamics in daily life and you will do experiments in labs. 


A girl meets a boy. They fall in love. They live happily ever after. How many times have we heard this story? Yet we keep telling it. Each telling is as fresh as that by Homer or Shakespeare. It is a great story, has endless variations, and is fundamental to many other stories. It is a story to experience for a lifetime. 


So is thermodynamics. After this play, you will recognize the story of thermodynamics no matter who tells it. Aha, you will say, this is yet another play of thermodynamics! One day, you will tell your own story of thermodynamics. You cannot avoid telling the story so long as you stay in touch with Nature. Thermodynamics is a fundamental play of Nature. 


This book does not teach the history of thermodynamics. It is impractical to teach thermodynamics by tracing the steps (and missteps) of Carnot, Clausius, Boltzmann, and Gibbs, just as it is impractical to teach calculus by tracing the steps of Newton and Leibniz. A subject and its history are different things. Mixing them in an introductory book does injustice to both. 


This said, the history of thermodynamics is interesting, illuminating, and well-documented, full of dramas of triumph and despair. Nature works without science. It is humans who create science to understand Nature. To study science is to study Nature and humans. We celebrate past creators, and nurture future ones. We build a collective memory that helps humans survive, prosper, and be happy. 


Missteps of the creators leave scars on thermodynamics, some of which are unhealed to this day. Healing may expedite if we learn some history. 


I will place a few names and years in this book as landmarks. You can read the history of thermodynamics online, starting with this Wikipedia entry


But historians do not create history. You might as well dip into original works, many of which are available online in English. Even a cursory reading of the works by the creators will enhance your enjoyment of the play. Stars shine before street lamps pollute the sky.

Ignore the laws

An average person knows many facts of Nature. Energy transfers from a hot place to a cold place. Friction warms things. Ice melts in hands. Ink disperses. Perfume smells. Scientists know when to conserve energy, and when to minimize it. 


Yet, even many great scientists feel uncomfortable with the laws of thermodynamics. Trust our own eyes: the emperor wears no clothes. Feeling entropy through the second law is like blind men feeling an elephant. It is odd to teach entropy today without letting students know what entropy is.


I will not structure this play of thermodynamics around the zeroth, first, and second laws. It is often claimed that these laws define the three thermodynamic properties: temperature, internal energy, and entropy. This claim is false. I will mention them in passing, so you see how they mislead 


No one practices thermodynamics with these laws. They are sterile. They misrepresent Nature. They belong to history, not to a book of current practice. The situation is reminiscent of Chinese medicine. The medicine works, but the theory of medicine is faulty, made up before the facts of Nature came to light. 


I will focus on the basic algorithm of thermodynamics (BAT) that directs calculation and experiment, in a way that thermodynamics has been practiced since the time of Gibbs (1873). You will learn to run the BAT on everything thermodynamic. 

Postulates and facts 

I play down the laws of thermodynamics for another reason. The Euclidean geometry must have impressed the creators of thermodynamics. In Euclidean geometry, a few facts, labeled as postulates (i.e., laws), derive all other facts. 


Thermodynamics has never been practiced this way. A few facts do play special roles in setting up the BAT, but are insufficient to derive most other facts. For example, we will use the BAT to develop a theory of temperature and a theory of melting, but these theories do not predict this fact: ice melts at zero Celsius.


Euclidean geometry is a wrong model for thermodynamics, and is even a wrong model for practical use of geometry. In thermodynamics, numerous facts are significant, and cannot be derived from other facts. You will have to learn numerous facts individually.


Watch the Feynman Lecture on the relation between mathematics and physics. The discussion on Greek and Babylonian traditions of mathematics starts at 23:30.

Big data

Ours is the age of molecules and the age of data. Molecules generate big data of thermodynamics. The BAT guides us to measure, curate, and use the big data.


Thermodynamic data are measured in many ways, including thermometry, calorimetry, thermochemistry, and electrochemistry.


Steam tables list temperature, pressure, volume, energy, enthalpy, and entropy of a single species of molecules, H2O, in various states. Similar tables exist for numerous other pure substances, most notably for refrigerants. 


For each species of molecules, ideal gas tables list enthalpy and entropy as functions of temperature. These tables are used to analyze mixing and reaction of any number of ideal gases.


Solids and liquids that mix many species of molecules generate enormous amounts of data. Gathering these data remains an unfinished business, and has become a part of the Material Genome Initiative.

Logic, intuition, and application are distinct aspects of thermodynamics

Open a text of physics, chemistry, biology, environmental science, materials science, or food science, and you see large sections on thermodynamics. Entropy affects all natural phenomena. The ubiquity comes because entropy is a universal force of Nature. 


But the ubiquity also makes entropy difficult to learn. To apply entropy to an engine, for example, you need to know entropy and the engine. This historical development was hard in creating thermodynamics in the nineteenth century, and is even harder in learning thermodynamics today. Students today rarely have first-hand knowledge of engines, and many will never care about engines. I forgo the historical development, and do not use the operation of an engine to develop the logic of entropy.


This book will develop the logic of entropy from first principles, intuition of entropy from everyday experience, and application of entropy in many domains. 


I will keep the logic, intuition, and application in separate sections. The same logic and intuition can be taught to everyone, but application will be domain-specific. The situation is analogous to learning calculus. The same rules of differentiation and integration are taught to everyone, but different examples of application will be effective for engineers and economists. 


After the logic and intuition are in place, the engine will come as an example of application. You can choose to study it or not. The book contains many other examples of application, and will have more when I have time, so that the book will help the reader master thermodynamics in any domain. Multidisciplinary study is effective if we are disciplined.


To develop intuition, we will look at familiar phenomena:


  • Ideal gas

  • Ideal gas mixture. humidity, dew, frost

  • Change of phases. Ice, water, steam

  • Dispersion of ink

  • Dissipation of energy

  • Degradation of energy


These phenomena will remind you of empirical facts—things that you already know. Looking at these phenomena will prepare you for the logic and application, and should not overwhelm you with empirical facts. 


Thermodynamics makes ordinary ideas extraordinarily effective. The range of applications is enormous, richer than Nature itself, including natural phenomena and human inventions. Any one of the standard textbooks will have copious examples. This book will describe some:


  • Pure substance

  • Incompressible pure substance

  • Ideal gas

  • Ideal gas mixture

  • van der Waals model

  • Osmosis

  • Steady-flow devices (e.g., turbine, compressor, throttle, heat exchanger, nozzle, and diffuser)

  • Cycles

  • Power plants

  • Refrigerators

  • Internal combustion engines

  • Heating, ventilation, and air-conditioning

  • The ascent of sap

  • Chemical reactions

  • Fuel cell


All these applications, particularly engines, historically contributed to the development of thermodynamics. But all applications are incidental to the logic of thermodynamics. You can master thermodynamics without studying engines, but you do need to work through some applications. 


The situation is similar in calculus. The calculation of the orbits of planets historically contributed to the development of calculus, but is incidental to the logic of calculus. You can master calculus without studying the orbits of planets, but you do need to work through some applications.


The logic of thermodynamics parallels that of probability. We will need a few basic ideas of probability, but a course on probability is not a prerequisite for learning thermodynamics. The logic of thermodynamics requires six ideas: isolated system, sample space, property, state, process, and equilibrium. 


  • An isolated system is a part of the world that does not interact with the rest of the world.

  • An isolated systems flips—rapidly and ceaselessly—to a set of quantum states. In the language of probability, the isolated system is an experiment, each quantum state is a sample point, and the set of all quantum states of the isolated system is the sample space.

  • An internal variable x is a function that maps the sample space to a set X, such as a set of values of energy, values of volumes, numbers of H2O molecules, and numbers of electrons. An internal variable is called a random variable in probability, and is called a (thermodynamic) property in thermodynamics.

  • When an internal variable is fixed at a value x in a set X, the isolated system flips among the quantum states in a subset of the sample space. The number of quantum states in this subset is a function of the internal variable, Ω(x). A subset of the sample space is called an event in probability, and is called a (thermodynamic) state in thermodynamics.

  • A process of the isolated system corresponds to a sequence of values of the internal variable. 

  • After the system is isolated for a long time, and after the internal variable is allowed to vary for a long time, the system flips to every quantum state in the sample space with equal probability. The isolated system is said to have reached (thermodynamic) equilibrium.

Basic algorithm of thermodynamics (BAT) 

The logic leads to the basic algorithm of thermodynamics (BAT):


  1. Construct an isolated system with an internal variable x. 

  2. Find the number of quantum states in a subset as a function of the internal variable, Ω(x). 

  3. Change x to keep Ω(x) constant for a reversible process, or maximize Ω(x) for equilibrium.

  4. Change x to increase Ω(x) for an irreversible process.


Define the subset entropy of the isolated system by S(x) = log Ω(x). Entropy is a dimensionless, absolute, extensive, thermodynamic property. Keeping Ω(x) constant is equivalent to keeping S(x) constant. Increasing Ω(x) is equivalent to increasing S(x). 


When a phenomenon requires an isolated system with multiple internal variables, each being a function from the sample space to a distinct set, the BAT runs just the same. 

Pattern of application

Every thermodynamic phenomenon runs on the BAT. This book will run the BAT on many phenomena. Every application of thermodynamics displays the same pattern as follows. 


Describe a phenomenon. Examples include the function of a throttle, the ascent of sap, the processes in a power plant, the molecules in a reaction, and the function of a fuel cell.


Run the BAT. Identify an isolated system with internal variables. Go through the other steps.


Use data. This book describes data of four kinds: pure substances, incompressible pure substances, ideal gases, and ideal gas mixtures.


Give predictions. Examples include condition of equilibrium, direction of change, and thermodynamic efficiency.

Concrete examples 

A logical place to start the book is the basic algorithm of thermodynamics. However, this logical starting point is removed from everyday experience. 


Thermodynamics is an abstract subject, but has numerous concrete examples. We will begin with five: 


  • ideal gas

  • relative humidity

  • incompressible pure substance

  • water and steam

  • ice, water, and steam


These concrete examples motivate the logic of thermodynamics, sharpen our intuition, and underlie numerous applications. We will return to these concrete examples throughout the book.


If you know these concrete examples, please jump to the section on entropy, and return to the concrete examples when the need arises.

If you have studied thermodynamics before, and cannot wait to watch the end of this play, please jump to the summary of this play of thermodynamics.

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