To truly understand Gibbs’s groundbreaking 3rd paper on thermodynamics “On the Equilibrium of Heterogeneous Substances” (1875-1878), one should first read his first two papers (1872, 1873) to gain proper background and context. [1]
Gibbs’ first two papers and the rise of graphical techniques
There’s opportunity at interfaces. If you want to find a fertile field of study, find an interface. Gibbs certainly did. In writing his Ph.D. thesis at Yale, Gibbs became proficient in the use of mathematical graphing techniques for scientific inquiry. In his travels and studies in Europe and his own study back in New Haven, Gibbs also became proficient in additional mathematics as well as in contemporary thermodynamics. Gibbs saw his opportunity at the interface between mathematics and thermodynamics, using the former to explain and bring new insights to the latter. Needless to say, for Gibbs, this field was indeed fertile and led to his first two publications and then his third.
The use of graphical techniques in thermodynamics arguably began when Watts created the indicator diagram to quantify the efficiency of his steam engine. This classic PV diagram, directly linked as it was to the quantity of work done, became the favored approach for many years thereafter and was highly effective for elucidating the steps in Carnot’s cycle. But things changed upon Gibbs’ discovery of Clausius in the early 1870s. Whether it occurred as a single eureka moment or a more gradual realization, I don’t know. But there had to be a moment when Gibbs looked at the result of Clausius’ Ninth Memoir (1865-66) [2],
dU = T dS – P dV [Eq. 1]
and realized that an opportunity was sitting there in front of him. As a young professor with an interest in finding his own niche of contribution, he pursued this equation, following unbreakable threads of cause-effect logic. Given that this equation is true, what must be the consequences? Deductive logic based on an induced hypothesis at its finest.
When Clausius developed this equation, his primary focus was on the mechanical theory of heat and its role in the relationships between heat, work, and internal energy. Entropy arrived towards the end of his research, and he certainly brought attention to it, but it wasn’t central to his efforts. His equation, published in 1866, sat relatively idle, largely because people, including Clausius himself, had a difficult time grasping what entropy was all about. But where others saw confusion, Gibbs saw a new beginning. He realized that regardless of what entropy actually meant, its sheer existence and features were what were important, for they enabled the creation of the above equation based purely on the thermodynamic properties of matter as governed by the 1st and 2nd Laws of Thermodynamics. It was a true equation of state, or more accurately, a derivative of such an equation, and embodied no reference to path dependency. And it was in this equation that Gibbs realized the existence of its parent equation based on the relationship U(S,V). In this way energy and entropy, two new and abstract properties, became central to Gibbs’ effort to shift focus from Carnot’s cycle to the properties of matter.
Moving beyond the PV indicator diagram: U(S,V)
In effect, Gibbs’ realization occurred by working backwards. With his mathematically trained mind, Gibbs saw Clausius’ fundamental equation as the result of a mathematical process that started with a parent equation, U(S,V), whose total differential
dU = (dU/dS)v dS + (dU/dV)S dV
led him to conclude, by comparison with [Eq. 1], that
T = (dU/dS)v
P = – (dU/dV)S
and thereby provided him an exact thermodynamic definition for temperature and pressure. Gibbs proposed that since the U(S,V) relationship for an closed system led to Clausius, and since Clausius is valid, then the function U(S,V) must exist. He wasn’t concerned with the exact nature of the function or what it looked like; that was for experiment to determine. He was solely concerned that mathematical logic proved its existence. And it was within U(S,V) that he saw a new set of thermodynamic coordinates on which to graph physical properties.
My favorite Gibb’s graph
Gibbs viewed the U(S,V) relationship as a single, curved surface in three-dimensional space for which every point on this surface complied with the U(S,V) equation and its differential [Eq. 1]. This equation is only valid once entropy is at a maximum, which occurs when all internal energy gradients have dissipated. For this reason, Gibbs referred to this curved plane as a “surface of dissipated energy” and illustrated it very clearly in his 2nd paper by assuming constant volume to reduce the graph from complicated 3D to easy-to-read 2D (see Figure 35.1 below from my book based on Gibb’s Fig. 3 in his 2nd paper. The clarity provided to me by this graph is why it became my favorite of Gibbs’s).
Both sides of this surface had relevance regarding what is and is not possible. For each U-V point, assuming a single phase only for now, there exists but one S point for an equilibrated system. As this point is a maximum, then all S points larger than this for the given U-V point represent an impossible state, while all S points smaller than this represent a possible state with the caveat that all such points represent non-equilibrated states for which entropy is not yet at a maximum. They are legitimate states but contain energy gradients of some form or another making the combined entropy value less than maximum. In sum, the final structure of the graph is a solid filling up the (possible) non-equilibrated portion of the U-S-V coordinate system, the surface of which quantifies the equilibrated state for which Clausius’ equation applies, and then emptiness signifying an impossible world.
Gibbs was particularly interested in two aspects of his own graph, the equilibrated surface itself and the distance between this surface and the points inside the non-equilibrated solid structure, for this distance quantifies the energy available to do useful work during the equilibration process as the point moves from the solid interior to the surface.
Maximum Work, Free Energy, Available Energy
Classical thermodynamics generally deals with equilibrated systems for which entropy is at a maximum. But for many industrial systems such as power generation, the initial system isn’t equilibrated, for if it were, no power generation would be possible. For example, when we burn coal to generate steam to do work, the initial mixture of air and coal is clearly not equilibrated. So how do we approach such problems?
The ability to produce useful work starts with an energy gradient of some kind. A waterfall – mechanical. A steam engine – thermal. An internal combustion engine – chemical. The nature of the gradient doesn’t matter; instead what matters is the fact that the initial system is not at its most probable state, that the system will naturally move towards that most probable state (and so dissipate the energy gradient and increase entropy), and that this natural tendency can be taken advantage of to do useful work. As Callen stated, “The propensity of physical systems to increase their entropy can be channeled to deliver useful work.”[3]
Energy gradients in such forms as, for example, mechanical (P), thermal (T), chemical (µ), can exist either internal to a given system or between the system and the surroundings or both.[4] The problem to solve in any of these situations is this: how to calculate the maximum amount of work that can be generated by dissipating the gradients? According to Gibbs’ energy minimization theory, the maximum amount of energy that can be removed from a system occurs when fully reversible processes are employed, thus leaving system entropy unchanged (Figure 35.1).
The practical challenge in the above theory is this. How does one go about calculating the maximum work based on such an ideal but non-defined process? It sounds rather complicated, doesn’t it? Fortunately, Gibbs provided us a way out, an approach that doesn’t even depend on the process chosen but instead depends only on the initial and final states. I’ll go into the details of this approach in an upcoming post.
As Gibbs himself noted in Fig. 35.1, the change in energy resulting from the isentropic process A-to-C is often referred to as the “available energy,” later to also become known as the “free energy”; thermodynamics terminology and exact definitions have been somewhat confusing throughout history. It’s the difference between two energy states that quantifies the energy stored in the internal gradients and thus represents the “maximum work” that can possibly be generated by reversibly dissipating these gradients.
If the distance between the initial state A and Gibbs’s surface of dissipated energy is zero, meaning that the initial state is already on this surface and internally equilibrated (maximum entropy and no internal energy gradients), then A has zero capacity to generate useful work, unless, of course, a gradient exists between A on the surface of dissipated energy and the environment.
Learn more!!!
Learn more about Gibbs’s development of thermodynamics in Chapters 35-39 of my book, Block by Block – The Historical and Theoretical Foundations of Thermodynamics.
References
[1] Gibbs, J. Willard. 1993. The Scientific Papers of J. Willard Gibbs. Volume One Thermodynamics. Woodbridge, Conn: Ox Bow Press.
[2] Clausius, R. 1867. The Mechanical Theory of Heat: With Its Applications to the Steam-Engine and to the Physical Properties of Bodies. Edited by Hirst, T. Archer. London: John Van Voorst. Ninth Memoir Appendix (1866), p. 366. Clausius wrote the equation TdS = dU + dW from his standpoint of, where does the heat go? For fluids, it was well understood that dW equaled PdV, but Clausius never explicitly included this as such in his fundamental equation.
[3] Callen, Herbert B. 1985. Thermodynamics and an Introduction to Thermostatistics. 2nd ed. New York: Wiley. p. 103.
[4] Tester, Jefferson W., and Michael Modell. 1997. Thermodynamics and Its Applications. 3rd ed. Prentice-Hall International Series in the Physical and Chemical Engineering Sciences. Upper Saddle River, N.J: Prentice Hall PTR. See Section 14 for their approach to accounting for both.
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