Thermodynamics for the curious I: the laws

Photo by Adam Kring on Unsplash

Some time ago, while searching for random videos on the internet (Ehhh… Procrastinating), I came across the surprise that the university where I did my Ph.D. is uploading to their YouTube channel a playlist with the title: "física para curiosos" (physics for the curious). The playlist contains a series of talks about contemporary topics in physics but presented in a simple and informal language to a non-specialist audience, generally for non-scientists (or as they say: "for the curious"). Among the talks they already uploaded, I found one given by professor Guillermo Cabrera entitled: “A corrida pelo frio: uma breve história da supercondutividade”  (The race for cold: a brief history of superconductivity).  Immediately, I took my laptop, lay on my sofa with a comfortable pillow and my coffee, and started watching the talk, because if Professor Cabrera would give a talk about thermodynamics I must watch it!

–“Now, I'll tell you some stories, and you all know, as we get old, we start collecting stories, and I'm old enough to be able to do thermodynamics with the ones I already have…”– With this joke, Professor Cabrera began his talk which he divided into three parts: the first part, as the title suggests, was about the process of cooling the so-called permanent gases, starting by Oxygen and finishing with Helium having a boiling temperature of approximately -231.15 °C; the second part dealt with the conductivity of metals at these new very low temperatures recently reached and ended with the discovery of superconductivity. Finally, the third part – the longest one, probably ⅔ of the total presentation – was about the applications and the mathematical development of Superconductivity, which is a quantum phenomenon. Thus, a talk that was supposed to be about thermodynamics (according to my expectations: a brief history of superconductivity was even in the title, after all) ended up being about quantum mechanics… Oof (read as a sigh of disappointment), I wanted to clarify that I already made peace with quantum mechanics, and I embraced it as part of my current research. I even have friends who work with quantum physics, I’m a quantum-friendly guy! However, the main title was “a race for cold”, and during the talk, Cabrera told about the Joule-Thompson experiment, an experiment at constant enthalpy employed to liquefy gases, the inversion temperature, which justifies why Helium can only be liquefied after Hydrogen, and Hydrogen after Nickel, and Nickel after Oxygen. 

–“Ok, someone will probably ask about the Joule-Thompson experiment, or what is enthalpy? – I thought naively. When the talk came to an end and the section for questions was opened, all the questions were about quantum mechanics and other quantum phenomena not related to the talk. – “Why?” – I thought out loud (indeed, I shouted) – “Does this disrespect continue?” – I continued thinking aloud (shouting). – “Thermodynamics is the key point here, this phenomenon wouldn’t have been discovered without thermodynamics. The Joule-Thompson experiment is an outstanding thermodynamic experiment that shows all the power of thermodynamics. Does anyone know that Thompson is Lord Kelvin? And Kelvin is the name of the unit of temperature used in thermodynamics. Anyone interested in the word similar to energy but different, enthalpy. Why does this quantity appear in thermodynamics? And what is its utility? What is a thermodynamic potential? have I already said thermodynamics?” – more thoughts out loud (shouts!).  After a while, my neighbors came to ask me nicely to stop shouting, I mean, thinking out loud. 

I studied the Joule-Thompson experiment in my undergrad twice. First, in my regular course of thermodynamics and later, in the statistical mechanics one. However, I was just aware of how this experiment encapsulates the full breadth of the scope of thermodynamics, the third time when Professor Cabrera ended his review of thermodynamics by explaining it. The Argentinian writer Jorge Luis Borges once wrote “Cualquier destino, por largo y complicado que sea, consta en realidad de un solo momento: el momento en que el hombre sabe para siempre quién es” (Any destiny, however long and complicated it may be, actually consists of a single moment: the moment when a man knows forever more who he is), and I have to say that my moment was that class. At that moment, that day, when the review was over I had two certainties in my life: first, I knew what kind of scientist I wanted to be and second, I became aware of my incurable syndrome of loving thermodynamics. So, hereinafter, I will do my best to emulate that review class, but for a “curious” audience. I hope that some of you will grasp some of the excitement I felt.

While looking for references to write this post I found a nice video where the lecturer starts his speech by saying: “the thermodynamic potentials are the icing on the cake of classical thermodynamics, it’s here that we will see the vastness of phenomena and processes that can be explained by means of them.” With this in mind, the best thing to do now would be to introduce our main characters a.k.a the potentials, but we need some context before, and when I say some context I mean some background in physics and mathematics. So, our first stop will be the definition of thermodynamics and its postulates. In fact, thermodynamics has four postulates, but the last one belongs to the realm of quantum mechanics, then I won’t mention it here, but I let this link for those interested (in the references therein, there’s one link to a video of Neil deGrasse Tyson explaining the impossibility of reaching absolute zero, btw).

Classical thermodynamics is the branch of physics (chemists will say that is a branch of chemistry, engineers will say it is a branch of engineering, but this is my blog and I’ll say that it is a branch of physics… thank you) interested in the study of energy and the way of changing it from one place to another and from one form to another. Here, the adjective classical plays a crucial role. In fact, it is important to make a distinction between classical thermodynamics and statistical thermodynamics (which I prefer to call statistical physics or statistical mechanics). The latter is based on the atomic structure of matter and makes its predictions through averages of the collective behavior of the atoms that make up the system. Classical thermodynamics, on the other hand, makes no assumptions about the microscopic world, it is a totally phenomenological theory, and therein lies its power and its great breadth. In the words of Albert Einstein:

“It is the only physical theory of universal content, which I am convinced, that within the framework of applicability of its basic concepts will never be overthrown.”

Due to its phenomenological nature quite often students regard classical thermodynamics as very strange, hard to grasp, or as an ex-friend of mine told me, boring (of course she isn’t my friend anymore). For these reasons, I think, generally (at least in physics), thermodynamics is taught by mixing both classical and statistical thermodynamics to gain some “feeling” with the theory. It won’t be my case here, I’ll avoid any reference to the microscopic world; otherwise, this post will become a book. Thus, once clear on this point it is time to enunciate the postulates (three of them as I said before).

For historical reasons, the postulates are enumerated from zero instead of one. In fact, the second postulate was the first one to be formulated, the first was the second, and years later scientists noticed that something was missing in the foundations of thermodynamics. This lack was filled with a new postulate, however, at that time the first and second postulates were already widely established in the scientific community and the new postulate was named the zero law. It is necessary to understand two concepts before stating the zero law: these are the concepts of temperature and heat. Fun fact number 1, the concept of temperature is a consequence of the second postulate, and heat is a consequence of the first, so we will have spoilers here! Unfortunately, these two terms are used in our daily lives with a different meaning than the one employed in thermodynamics. Thus, it is important to understand how these terms are used here. First, for the concept of temperature, I’ll follow one of my favorite books on thermodynamics “An introduction to thermal physics” by Daniel V. Schoeder, who defines the concept of temperature by saying: “I won’t be ready to tell you what temperature really is until chapter 3 – when the second postulate is stated –. For now, however, let’s start with a very naive definition:

Temperature is what you measure with a thermometer”  

Even though this definition may be unsatisfactory when it is presented for the first time it contains everything we need at this moment. I mean, there is a property of every system which can be measured, that's it! Now, why would we want to measure it? This is another thing that leads us to the concept of heat. In thermodynamics, heat is a manner of transporting energy between bodies when they are at different temperatures. Here, as the saying goes, the devil is in the details. In thermodynamics, heat is associated with a manner of transporting energy (please note that heat isn’t energy, but instead a mechanism to transport it) when bodies are in contact at different temperatures. At this point, we are not interested in the direction of this transport, I mean, who gives energy to whom. This is a problem of another postulate, try to guess which one (... The second). Now if heat is a manner of transporting energy when the bodies are at different temperatures (I will repeat this sentence as much as I can), what does happen when the bodies are at the same temperature? Well, if heat is a manner of transporting energy when bodies are at different temperatures (you see, I repeat it again) when both are at the same temperature there is no transport of energy as heat and we say the bodies are in thermal equilibrium! Now we should state the zero law. If two bodies are in thermal equilibrium, let's say body A and body B, then if we put one of these bodies, let’s say A, in contact with a new one C, and there is no transfer of energy as heat, it means that A and C are in thermal equilibrium. In such a case, the zero law affirms that C and B are also in thermal equilibrium. The zero law is about the transitivity of thermal equilibrium!!! If two bodies are in equilibrium and a third one is in equilibrium with one of them we can always affirm that all of them are in thermal equilibrium. This could sound obvious, but it isn’t. Imagine for example you have a rich friend, and this friend has another rich friend, if you are together talking about something, it is not necessarily true that you are rich, it could be, but it could also not be, then transitivity isn’t obvious. But, in thermodynamics thermal equilibrium is transitive and is stated by the zero law: 

“If two thermodynamic systems are in thermal equilibrium with each other, and also separately in thermal equilibrium with a third system, then the three systems are in thermal equilibrium with each other.”

Quickly returning to Schroeder’s book, after stating the zero law, temperature could be defined as:

“Temperature is a tendency of an object to spontaneously give up energy to its surroundings. When two objects are in thermal contact, the one that tends to spontaneously lose energy is at the higher temperature”

Ok, the second part of this definition is a big spoiler. In fact, this is the second postulate, but we will see it very soon.

During the introduction of the zero law, I used the word energy without saying what energy is, and the reason for that is: energy is the central concept of the first postulate (that is in fact the second) and now is its moment to shine! Now, fun fact number 2, energy is a fundamental concept in physics which means we cannot define it in a more fundamental way, but this doesn’t mean we cannot relate it to another concept which allows us to grasp the idea. Then, we can simply start with a naive question: if heat is a manner of transporting energy when systems are at different temperatures (I wrote it again), what should I call the transport of energy when a difference in temperature is not involved? The answer to this question leads us to another main concept in physics,

any manner of transporting energy that is not due to the temperature difference between systems is called work.

Even though the concept of work used in physics is different from the one used in our daily lives, I think the idea of energy is grasped from this definition. We can say that energy is the capacity of a system to perform work. Thus, a more energetic system can perform more work, and bodies at higher temperatures possess more energy and, therefore, more capacity to perform work. Again it is important to clarify that work is not energy; instead, like heat, is a manner of transporting energy (when a difference in temperature is not involved). Although we do not define what energy is, I think the idea becomes clear when we introduce the notion of work. However, for those interested in a more formal definition, I recommend the lectures on physics by Richard Feynmann. Particularly, in chapter 4 where the concept of energy is explained. The importance of the concept of energy in physics is because energy is a quantity that is always the same independent of what we do, I mean, we cannot change the quantity of energy in the universe, we can only transport it from one form to another. Since there is no single experiment proving the contrary this is a fundamental law in physics: the energy of the universe is a constant, and this is exactly the first (second) postulate often called the first law:

“when energy passes into or out of a system (as work or heat), the system's internal energy changes in accordance with the law of conservation of energy.”

The first law brings us the first of our protagonists, the internal energy which is defined as the quantity of energy that a system possesses just by the simple fact of existing in the universe. Internal energy is a thermodynamic potential, but I’ll tell you more about it after stating the second law. Before moving on to the second postulate (that is in fact the third) I’d like to say one more thing about the first law. Thermodynamics was developed during the XIX century when the industrial revolution was happening, and at that time the interest was focused on thermal machines, and how to construct better and better machines to perform work in a more efficient way. Thus, another way to state the first law is saying that it is impossible to construct a machine that can produce work indefinitely without energy input, this is a perpetual motion machine of the first kind (the video of this link really worth it). So, we cannot construct a machine that after building can produce work indefinitely by itself, I mean without providing external energy to work, these perpetual machines are forbidden by the first law!

There’s a book written by the physical chemist Charles Percy Snow called “the two cultures” when he states that “not knowing the second law of thermodynamics is equivalent to never having read a work by Shakespeare” (here you can change Shakespeare for the most famous writer of your mother tongue i.e. Cervantes, Camões, Molière, Sienkiewicz, etc). In the book, the author compares intellectuals from science and literature (the two cultures) and criticizes how they are alienated from others meanwhile for an intellectual in literature is unacceptable that someone has never read Shakespeare, most of them are unable to enunciate the second (third) postulate of thermodynamics or its implications. I, sadly, belong to the other group, I have never read anything by Shakespeare (Hamlet is now in my P.A.L.). However, I’m ok if people don’t know the second (third) postulate of thermodynamics, mainly because if you’re reading this and you didn’t know it, you will know it now.

Similarly, as with the zero law, we need to introduce two concepts in order to state the second (third) postulate: these are the concepts of spontaneous and irreversible processes. So, we say that a process is spontaneous when it happens without having to be driven by doing work of some kind. Here, “spontaneous” is associated with the tendency of something to occur naturally. Irreversible, on the other hand, is related to the direction of the process; a process is called irreversible when it can only occur spontaneously in one direction. A daily life example of this is a hot mug of coffee left in a room for a while: we noticed that the coffee is spontaneously becoming colder and colder until its temperature reaches that of the room in which it is placed. We never see the contrary happen, where we leave the mug with coffee in a room, and after a while we find the coffee hotter than before. The process can only happen in one direction. If we analyze this example carefully in the context of the previous laws, nothing prevents the mug of coffee from absorbing heat from the environment and becoming hotter. There’s no such violation at all. Therefore, there should exist a fundamental postulate different from the previous two ones that forbids this to happen, since there is no experiment that proves the contrary! Thus, based on all the available evidence scientists  conclude that:

“Heat always moves from hotter objects to colder objects (or «downhill»), unless energy in some form is supplied to reverse the direction of heat flow.”

This means that there is a direction in which the processes occur naturally, and in order to change this direction we need to do work. Since this result is postulated, not proven, one must have sufficient background and experimental testing to accept it with an easy heart, and as if that were not enough, the second (third) postulate offers several equivalent formulations, from which everyone may choose his own depending on his taste and ability to grasp. Having chosen one, the equivalent statement should be proven from the accepted one. Here, I’ll give two more equivalent statements of the second law, that sometimes are useful for understanding and accepting it. In the language of thermal machines, the second (third) postulate can be stated as: it is impossible to construct a machine that can transform all the heat absorbed in work, this is a perpetual machine of the second kind. Fun fact number 3, we always have to lose: when a thermal machine is constructed, part of the energy must be lost as heat. Then if the first law tells us about the quantity of energy, the second tells us about its quality. Sadly, now I cannot remember where I read this, but literally, to summarize the first and second laws in a single sentence:

“We can’t win (first law), we can’t tie (second law).” 

The last statement is, I think, the most widely known, and generally the one that is most scary because it introduces a property with many faces, and by many faces, I mean many interpretations. The statement is:

“There exists a quantity called entropy and all irreversible processes in isolated systems that lead to equilibrium are connected with an increase of it, until the entropy assumes its maximum when equilibrium is reached”

Before continuing, to see what I’m talking about when I say entropy has many faces, check out these short videos where entropy is explained: Video 1, Video 2, Video 3, Video 4, Video 5. I want to say all these videos are amazing, but except for the last one, all of them explain entropy with reference to the microscopic world; this is because they are in the realm of statistical thermodynamics. However, here we are interested in classical thermodynamics; therefore, entropy won’t be a measure of disorder, of the way energy spreads, of the lack of information about our system, or anything other than a measure of the irreversibility of a process. So, it is worth it to analyze in more detail the last statement. We can go by parts: “There exists a quantity called entropy” then as a consequence of the second law we have a new quantity called entropy and its value gives us information about the irreversibility of a process. At this point, it is not important how to compute it, but the second law also provides a way to do it. “All irreversible processes in isolated systems that lead to equilibrium are connected with an increase of it” The important part here is the word isolated: in an isolated system the entropy of a system increases until the equilibrium is reached. In fact, the thermodynamic potentials appear to answer the question: what happens when the system is not isolated anymore? Btw, probably someone could think at this moment that entropy is our second character, i.e. another thermodynamic potential, but in fact, entropy and energy represent the same thermodynamic potential. Thus, we still only have one of our main characters. “Until the entropy assumes its maximum when equilibrium is reached” this means that when entropy doesn’t change anymore the spontaneous irreversible process is over and the system reaches an equilibrium state. The important part here is the entropy doesn’t change because as we will see, in thermodynamics we are always interested in how things change. Therefore, our next stop will be in the mathematics of changes, and how we can measure changes using math.