The first and second laws of thermodynamics are logically equivalent

Abstract

In the September 2022 issue of The Physics Teacher (TPT), Richard Kaufman and Harvey Leff showed the interdependence of the first and second laws of thermodynamics. Here, I go further and use the facts that the first law implies the second law, and that the second law implies the first law. This two-way implication establishes the logical equivalence of the first and second laws. Although the laws are logically equivalent (when one is true, then the other must be true), this does not mean that they are the same. The equivalence provides for a deeper and richer understanding of the laws of thermodynamics for students, teachers, and physicists. Surely, if we can know that the first and second laws are equivalent, then we should know.

Keywords

First law second law equivalent thermodynamics

Background

History has borne out that the laws of thermodynamics are universally valid; we might wonder if there are any connections between them. A review of the literature has shown some instances where people have looked for linkages between the laws of thermodynamics, specifically the zeroth and second laws.^1–4 Linkages between the first and second laws were found in the recently published The Physics Teacher (TPT) paper,⁵ “Interdependence of the First and Second Laws of Thermodynamics.” That paper was an extension of the TPT paper⁶ “What if energy flowed from cold to hot? Counterfactual thought experiments” (see also Appendix A).

The present study is an extension of those papers. Points from those papers are provided next (for the interested reader, this is elaborated on in Appendix B, “The first and second laws of thermodynamics, and their usage,” and discusses (1) the mathematical form of the first law with its equilibrium macroscopic states, (2) the second law, and (3) further review for the usage of these laws in this article, etc.):

First law of thermodynamics (1st): The first law of thermodynamics is a generalized form of conservation of energy that is adopted for thermodynamic processes and is based on the existence of equilibrium states. The first law applies to processes that connect initial and final equilibrium states i and f, and conserves energy.

Clausius statement of the second law (2nd): Heat energy transfer can only occur spontaneously from hotter temperature to colder temperature regions.

The papers^5,6 show that if heat energy flowed spontaneously in the other direction (i.e., colder temperatures to hotter temperatures), then thermal equilibrium would not result.^a In that case, temperature variations within an object would not smooth out, and would not lead to a uniform temperature necessary for the equilibrium conditions of thermodynamics. Note that the papers also discuss that thermodynamics deals with equilibrium states for the macroscopic properties that are required for the thermodynamic laws, which is also discussed in other literature as indicated later in this article.

The main results about the universal laws stated in the article, “Interdependence of the First and Second Laws of Thermodynamics,” by Kaufman and Leff,⁵ are quoted here:

Main result 1: “The first law of thermodynamics requires the Clausius statement to assure equilibration to the initial and final states i and f.”

Main result 2: “The Clausius statement requires energy conservation from the first law to assure that all heat processes conserve energy throughout their duration.”

Logic and the equivalence

The present study uses simple concepts and terminology from logic. “Two propositions or statements are said to be equivalent when the truth of one implies the truth of the other.”⁷ In other words, when “if then” statements can be shown in both directions, then the statements are equivalent.^b

The main results 1 and 2 from Kaufman and Leff, which were quoted above, are implications about the universal laws. Main result 1 states that, “The first law of thermodynamics requires the Clausius statement to assure equilibration to the initial and final states i and f.” So, if the first law is true, then the second law must be true; the first law implies the second law. Main result 2 states that, “The Clausius statement requires energy conservation from the first law to assure that all heat processes conserve energy throughout their duration.” So, if the second law is true then the first law must be true; the second law implies the first law. Therefore, we have the following:

The first law implies the second law: $1^{s t} \Rightarrow 2^{n d}$

The second law implies the first law: $2^{n d} \Rightarrow 1^{s t}$

These two-way implications establish that: The first and second laws are logically equivalent.

This means that if one of these laws is true, then so is the other. In mathematical notation, the equivalence is $1^{s t} \Leftrightarrow 2^{n d}$ .

Points for teachers and students

In this section, I elaborate on points for teachers and students. This includes points about the equivalence of the first and second laws of thermodynamics and a deeper understanding of thermodynamics overall.

I begin by reiterating a point made by Kaufman and Leff⁵ that students should understand about the first law and equilibrium.

Students might be surprised that the first law requires the second law, because the first law is often mistakenly viewed to be only a statement of energy conservation. Commonly overlooked is the need for the Clausius statement to guarantee that initial and final equilibrium states can be reached. Such surprises during the learning process can lead to more intense thought, and perhaps provide a spark that leads to enhanced student understanding.

This is important to keep in mind for the following subsections.

Thermodynamics deals with equilibrium states and macroscopic properties for its laws

It should be emphasized to students that thermodynamics deals with equilibrium states, as stated by Çengel and Boles⁸:

Thermodynamics deals with equilibrium states. The word equilibrium implies a state of balance. In an equilibrium state there are no unbalanced potentials (or driving forces) within the system. (Emphasis in original)

They also state that (Çengel and Boles, p. 15⁸):

When a process proceeds in such a manner that the system remains infinitesimally close to an equilibrium state at all times, it is called a quasi-static, or quasi-equilibrium, process. A quasi-equilibrium process can be viewed as a sufficiently slow process which allows the system to adjust itself internally so that properties in one part of the system do not change any faster than those at other parts. (Emphasis in original)

Thermodynamic equilibrium exists when mechanical, chemical, and thermal equilibrium occurs throughout a system (Zemansky and Dittman, p. 30⁷ and Cengal and Boles, pp. 13-14⁸). “Thermodynamics does not attempt to deal with any problem involving the rate at which a process takes place” (Zemansky and Dittman, p. 30⁷).

Consider a mixer that stirs a liquid in a sealed container, like James Prescott Joule's experiments. Before the mixer is turned on, the temperature of the liquid is measured with a thermometer. Then the mixer stirs the liquid for a period of time. While the mixer is running, the liquid is not at a homogeneous temperature. We must wait until after the mixer is turned off for equilibrium to be established in order to measure the final temperature of the liquid. This demonstrates how equilibrium states are required in thermodynamics.

It should be further emphasized to students that thermodynamics deals with measurable macroscopic properties. For example, Zemansky and Dittman, p. 4⁷ state:

…a macroscopic description of a system involves the specification of a few fundamental measurable properties of a system. Thermodynamics, then, is the branch of natural science that deals with the macroscopic properties or characteristics of nature and always includes the macroscopic coordinate of temperature for every system. The presence of temperature distinguishes thermodynamics from other macroscopic branches of science, such as geometrical optics, mechanics, or electricity and magnetism. (Emphasis in original)

Contrast this with what they say about microscopic characteristics (Zemansky and Dittman, p. 5⁷):

Statistical mechanics … is the branch of natural science that deals with the microscopic characteristics of nature…. In short, a microscopic description of a system involves various assumptions about the internal structure of the system and then calculations of system-wide characteristics. (Emphasis in original)

It is outside of formal thermodynamics where microscopic characteristics, non-equilibrium, or non-quasi-static processes may be investigated or addressed. Zemansky and Dittman further state (Zemansky and Dittman, p. 6⁷):

Throughout its history, the study of thermodynamics has always looked for general laws, relationships, and procedures for understanding macroscopic temperature-dependent phenomena.

Accordingly, the general laws of thermodynamics use measurable macroscopic properties such as state variables for temperature T, pressure P, volume V, internal energy U, and entropy S. On its website, NASA recognizes the importance of the laws of thermodynamics with the macroscopic properties of thermodynamics.⁹

…thermodynamics deals only with the large scale response of a system which we can observe and measure in experiments…. There are three principal laws of thermodynamics … Each law leads to the definition of thermodynamic properties which help us to understand and predict the operation of a physical system.

Of note, if there were a macroscopic state in which no cycle or process occurred, then energy must be conserved, and equilibrium must be maintained for the state. Therefore, the first and second laws must be true to maintain a macroscopic state. If heat energy flowed from colder to hotter, then any microscopic fluctuation in the macroscopic state would result in non-equilibrium.

Thermodynamic laws are universal laws

People seek to understand how the universe works and have prized universal truths, such as thermodynamic laws, that are true everywhere and for all times. Observations that have limited applicability, scope, duration, or experimental evidence do not rise to the level of such “laws.” That is, if it were ever observed that a law failed in any part of the universe at any time, then the law would not be a universal law of nature. Accordingly, students should understand that thermodynamic laws are universal in their application.

Usage of logic and equivalence

When two statements imply each another, they are equivalent. This is well-established and irrefutable by mathematical logic and is shown in truth tables. It is important for students to understand what equivalence means and what it does not. Equivalent statements mean that when one statement is true, then the other must be true. Similarly, when one statement is false, then the other statement is false. Equivalent statements cannot have one statement be true and the other false; they are both true or false together.

A key point is that when statements are equivalent, it does not mean that they are equal (i.e., they are not the same statements).

We are not saying that $[X]$ is equal to $[Y]$ . Since $[X]$ and [ $Y]$ represent two different statements, they cannot be the same. What we are saying is, they always produce the same truth value, regardless of the truth values of the underlying propositional variables. That is why we write $[X] \equiv [Y]$ instead of $[X] = [Y]$ .¹⁰ (Emphasis in original)

This point about equivalent statements not being equal is worthy of repeating to students since this article does not show that the first law is the second law. That is, this article has shown that the first and second laws are equivalent, yet they are not the same. The laws say different things and are used by students and engineers as part of first- or second-law calculations.

Equivalence and its use in physics

Equivalence is used in physics, beyond what has already been shown in this article. For example, in addition to the Clausius statement (CS) of the second law, there is the Kelvin–Planck statement (KPS) of the second law, as discussed in the book by Zemansky and Dittman (Zemansky and Dittman, p. 153⁷):

It is impossible to construct an engine that, operating in a cycle, will produce no effect other than the extraction of heat [energy] from a reservoir and the performance of an equivalent amount of work.

The book uses the same logic for equivalence to show an equivalence of the KPS and CS (Zemansky and Dittman, p. 156⁷):

Two propositions or statements are said to be equivalent when the truth of one implies the truth of the second, and the truth of the second implies the truth of the first.

The proof is known from many sources, including many physics and thermodynamics textbooks. By using a hypothesized engine and refrigerator, it is demonstrated that $C S \Rightarrow K P S$ , and $K P S \Rightarrow C S$ , so the Clausius and Kelvin–Planck statements of the second law are equivalent, $K P S \Leftrightarrow C S$ (Zemansky and Dittman, pp. 156–158⁷ and Çengel and Boles, p. 254⁸). The Principle of Entropy Increase ( $Δ S_{u n i v e r s e} \geq 0$ ) is “entirely equivalent to” these other forms of the second law.¹¹ This is another example of the use of equivalence in physics, so it is important that students understand the concept.

Experiments and the thermodynamic laws

This article shows that the first and second laws of thermodynamics are equivalent. Since they are universal laws, they must both apply everywhere, if either one of them is true. In other words, we could not have a universe where the first law was true, and the second law was false (or vice versa). That is, an experiment in which a law holds part of the time would not be universal. If a student does an experiment where the first law is shown to be true, then this automatically shows that the second law is true (and vice versa). Students may not have realized that they are showing that both laws are true whenever they show that one is true.

Consilience to relate concepts

Consilience is a term that has been used for the linking of ideas that are thought to be from different disciplines. When very different ideas are brought together, they are said to have a high consilience. Although this article sticks to the discipline of thermodynamics, there is a high consilience between the first and second laws, which were previously viewed to be non-equivalent. While there was really no doubt that both laws were true in our universe, the equivalence shows that they must be true together (and not true independent of each other). It is hoped that this deeper connection between the laws provides for a deeper understanding of them.

Conclusion

There is much that can be taught, learned, and appreciated about the laws of thermodynamics—especially in the course of showing that the first and second laws are equivalent.

Footnotes

Acknowledgements

I thank the anonymous reviewers for their thoughts and suggestions on improving this article. I also thank my wife, Kara Kaufman, for proofreading this article.

Declaration of conflicting interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Richard Kaufman

Data availability statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Notes

Appendix A: Further suggested linkages between the laws

There is a paper that suggests a linkage between the laws, especially that a violation of the second law would violate the first law, as indicated in part of the abstract¹²:

In this work, we bring to light evidences [sic] to prove the absolute validity of the second law as a fundamental law of physics. For this purpose, we propose a short revisit of the history of the discovery of the second law in order to highlight the connection between the second law and the first law of energy conservation. We then demonstrate that the perpetual motion machine of the second kind also violate the first law of thermodynamics, albeit indirectly, contrary to the common belief. This result confirms the second law is an inviolable fundamental law of physics, just like the law of energy conservation. Denying one of these conjoined twin laws is to deny the other. Any presumed violation of the second law, even a probabilistic one, inevitably violates the laws of energy and mass conservation, and undermines all fundamental laws of physics and chemistry.

The article discusses entropy, statistical entropy, Maxwell's demon, thermodynamic laws, perpetual motion devices, etc. In the conclusion section, the article states:

We have then proved that, the perpetual motion machine of the second kind, regarded most textbooks […] as a process violating the second law but not the first law, also necessarily violate the first law of energy conservation, albeit indirectly. This result confirms the unbreakable link between the two laws.

The article appears to state one of the implications stated in the present study, namely that the second law implies the first law. Although it does refer to a linkage between the laws, it does not explicitly state a two-way implication—or state that the laws are equivalent, which is stated in the present study. The papers do not contradict each other, and, in fact, they both support any stated (or inferred) implication.

Appendix B: The first and second laws of thermodynamics,and their usage

In order to maintain the flow of the paper up to this point, logical equivalence of the laws was developed first. However, at the suggestion of an anonymous reviewer, I elaborate here on the laws discussed earlier, for the interested reader, without introducing anything not already covered by Kaufman and Leff.^5,6 The first law of thermodynamics can be expressed in a mathematical form⁵:

The first law applies to processes that connect initial and final equilibrium states, i and f. Energy can be transferred by work W and/or heat Q processes, with a change $Δ U$ in the stored internal energy…. The first law of thermodynamics, $Δ U = U_{f} - U_{i} = Q + W$ , accounts for these changes. Notably, the first law contains three related ideas: (1) the existence of the internal energy function U, (2) the principle of conservation of energy, and (3) the definition of heat as energy in transit by virtue of a temperature difference. (Emphasis in original) [citing Serway and Jewett,¹³ Fermi,¹⁴ and Zemansky and Dittman, p. 79⁷]

Therefore, the first law requires internal energy U and equilibrium states. Equilibrium states are only possible when heat energy flows spontaneously from hotter to colder (i.e., the Clausius statement of the second law must be true). Otherwise, spontaneous colder-to-hotter energy flow makes hotter objects hotter and colder objects colder. Such a colder-to-hotter spontaneous energy flow would cause temperature nonuniformity within an object, rather than the temperature uniformity necessary for the equilibrium conditions of thermodynamics. Therefore, the equilibrium states required by the first law relies on the second law to be true.

“Our everyday experience is that when hot and cold objects can exchange energy, their temperatures approach one another….”⁵ The Clausius statement details this spontaneous energy flow from a hotter temperature to a colder temperature. However, temperature is actually a macroscopic property for an equilibrium state⁶:

Thermodynamics variables such as (T, V, N) [temperature, volume, and number of atoms] characterize equilibrium for many macroscopic thermodynamic systems. Such equilibrium states have zero net energy flows at every part of their boundaries, and will persist until there is a change in one or more thermodynamic variables….

It can be noted that the “Clausius' definition of entropy $d S = d Q / T$ cannot be used in the C2H [colder-to-hotter energy flow] world because temperature cannot be defined.”⁶ Accordingly, both temperature and entropy could not be defined in a C2H world.

Next, I elaborate on the fact that entropy is a state variable that requires equilibrium and is used in the second law. Recall that entropy was used in a subsection of this article, “Equivalence and its use in physics,” to discuss an equivalent form of the second law (i.e., the Principle of Entropy Increase, $Δ S_{u n i v e r s e} \geq 0$ ). Moroz¹⁵ states, “It is the second law of thermodynamics that from a formal point of view allows us to introduce a macroscopic function, entropy S.” A NASA website¹⁶ also says, “The second law states that there exists a useful state variable called entropy.” Another NASA website adds⁹:

…thermodynamics deals only with the large scale response of a system which we can observe and measure in experiments…. The details of the process of reaching thermal equilibrium are described in the first and second laws of thermodynamics. (Emphasis added.)

Zemansky and Dittman further discuss entropy using initial and final equilibrium states in the following passage (Zemansky and Dittman, pp. 190–191⁷):

The existence of an entropy function S is deduced in the same way as that of the internal-energy function U, that is, by showing that a certain quantity is independent of choice of reversible processes connecting the initial equilibrium state with the final equilibrium state. Both U and S are state functions, which means that the difference of either function evaluated at the final and initial equilibrium states is independent of the path connecting the two states. (Emphasis added)

As noted in a subsection of this article, “Equivalence and its use in physics,” if there were a macroscopic state in which no cycle or process occurred, then energy must be conserved, and equilibrium must be maintained for the state. Therefore, the first and second laws must be true to maintain a macroscopic state. The measurements used by students or engineers rely upon macroscopic properties of initial and final equilibrium states in order to apply the first or second laws. As another anonymous reviewer pointed out, the mathematical form of the first law is used for calculations of a process or cycle, and it also ensures that energy is conserved; the second law is used to determine whether or not a process or cycle is possible.

Finally, I note that during a spontaneous heat energy flow process in our world, energy must be conserved if it is to be completely transferred from a hotter temperature to a colder temperature. This is why Kaufman and Leff⁵ state that “…the Clausius statement requires the first law to assure that energy is conserved throughout heat processes.”

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