Physical chemistry was a powerful, central tool in the discovery of ammonia synthesis. However, this does not mean the full potential of the theory was (or needed to be) employed to solve the problem. By studying the use of the theory in Haber’s and Nernst’s work, we are able to examine the role physical chemistry played in terms of theoretical and mathematical possibilities versus practical, industrial application. These considerations are based on the preceding text as well as the derivations in Appendices A and B, but are presented here with minimal mathematics. As before, a consideration of the appendices will improve understanding of the following discussion.

The key to the success of physical chemistry is contained in the use of the free energy, A, in Eq. B.1, as a quantitative measure of describing chemical reactions. Here, more background is provided to illustrate its role in Haber’s and Nernst’s work as well as the extent to which a precise expression of its value was needed. This viewpoint also frames ammonia synthesis in a new light: as part of the completion of classical thermodynamics (and, therefore, “classical” physical chemistry). What would later become known as the third law of thermodynamics is considered to have been the final step in this development (although it was not apparent at the time Nernst published it) (Suhling 1972). However, a theory itself is not complete without experimental substantiation. This complementary dynamic is what took place between Haber and Nernst throughout their exchange as ammonia research supplied meaningful data for the confirmation of Nernst’s heat theorem. The discrepancies Nernst observed between theory and experiment could have been based on behavior in nature that had not yet been discovered or due to an error in his assumptions or calculations. Haber and Nernst’s interaction showed, at least for the ammonia system, that the deviations were caused by uncertainties in thermal data and not by the physical/mathematical descriptions of the free energy.

The origin of these descriptions lies in research on the development of heat during chemical reactions. Depending on experimental conditions, the free energy is referred to as the Gibbs energy or the Helmholtz free energy, named for Josiah Willard Gibbs and Hermann von Helmholtz, who independently derived expressions and published them in 1878 (Gibbs 1878a,b) and 1882 (von Helmholtz 1882).Footnote 1 The “isolation” of the free energy as a fundamental value arose from attempts to assess chemical affinity through the heat released during a chemical reaction. In the first half of the 1800s, there were many investigations into this phenomenon (Nernst 1914), (Partington 1964, pp. 608–620, 684–699), (Suhling 1972), (Bartel 1989, pp. 71–73). By mid-century, quantitative attempts to determine the maximum work available from a chemical reaction of homogeneous substances (for example the electromotive force (E.M.F.) that could be expected from an electrochemical cell) had resulted only in the unsatisfactory assumption that the heat of reaction was equal to the maximum work (and a measure for the chemical affinity). That is, ΔU = ΔA, with the spontaneous chemical reaction generating the most heat. These ideas were (at least partially) supported by James Joule, Julius Thomsen, Marcelin Berthelot, Helmholtz and others and formulated in different ways under various names, including the Principle of Maximum Work. Despite its simplicity, the general validity of the claim was viewed with skepticism as specific chemical systems behaved in obvious contradiction. Some chemical reactions provided an E.M.F. while proceeding endothermically (ΔU was negative) and there were also indications the heat of reaction was temperature dependent. With the convergence of chemistry and energy science, it became evident that there were two kinds of energy involved in a chemical reaction. The quantities were, in general, not equal and their magnitude depended on reaction conditions. One was always in the form of heat and the other could be obtained partially or fully as electrical energy or as another form of work. A chemical reaction proceeded spontaneously in a way that minimized the latter, the free energy.

It is a temperature-dependent dynamic that describes a balance in nature:

$$\displaystyle \begin{aligned} \textit{Free}\ \textit{Energy} = \textit{Total}\ \textit{Energy} - \textit{Temperature} \times \textit{Entropy} {} \end{aligned} $$
(13.1)

where Total Energy may denote either the internal energy or the enthalpy. At high temperatures the (Temperature ×Entropy) term dominates, and the free energy is minimized through reactions maximizing entropy. At low temperatures, this term becomes small and reactions occur that minimize the Total Energy. At non-extreme temperatures, nature strikes a balance between the total energy of the system and the product of temperature and entropy to minimize the free energy at equilibrium (Müller 2007, pp. 148–149). But ammonia has one more trick up its sleeve, which was the “mystery” that plagued researchers during the nineteenth century (see Part I, Chap. 4). In a mixture of NH3, N2, and H2 at 300 K and 1 bar, the equilibrium conditions favor a nearly full conversion to ammonia as the decrease in total energy during the NH3 production outweighs the decrease in entropy (the latter on its own would hinder the exothermic ammonia generation). However, no reaction occurs due to the potential barrier from nitrogen bonds (as well as from hydrogenation and blocked active catalyst sites) (Vojvodic et al. 2014). If the temperature is raised in order to overcome the energy barrier, the Temperature ×Entropy term also becomes large. The change in this value must then be countered by increasing the pressure, which in turn lowers the entropy. The catalyst facilitates the reaction because it reduces the temperature required to overcome the energy barrier. It also enables the use of lower pressure to still achieve an adequate ammonia yield (Müller and Weiss 2005, pp. 63–71), (Müller 2007, pp. 156–159), (Müller and Müller 2009, pp. 269–272). Thus, ammonia synthesis is described as a “balancing act.” This was the crucial knowledge that had been established by physical chemists, and was well-known to Fritz Haber and Walther Nernst as they began their work on ammonia synthesis (Haber and van Oordt 1905b, p. 343).

While Helmholtz’ definition of the free energy supplied decisive information about the behavior of chemical systems, it only allowed for a partial determination of the defined quantity. The differential equation still needed to be integrated, a mathematical operation that necessitates the addition of an unknown constant to the final equation (Appendix B, Eqs. B.1 and B.2). The change in chemical equilibrium with temperature could be calculated but the absolute value could not. By the time Haber and Nernst took up the problem of ammonia synthesis, the determination of this constant (and, thus, of chemical equilibrium) had become a central challenge in physical chemistry. “The problem,” wrote Lothar Suhling, “which faced the few chemists in the nineteenth century who were dealing with the application of thermodynamical principles on chemical equilibrium, was the old question of quantitative consideration of chemical affinities (Suhling 1972).” At the time of Haber’s breakthrough, there were two methods of solving this problem. Usually the chemical equilibrium (or alternatively the E.M.F) was measured directly, and the data point was used to determine the constant of integration (Nernst 1914, 1921). This method solved the problem in principle, but left the resulting values of chemical equilibrium subject to experimental error on the order of magnitude of any subsequent measurements. It defeated much of the purpose of the application of theory.

Walther Nernst’s goal was to find a way around this empirical determination of the constant, and with the help of a new assumption about nature, he succeeded. His theory was published in 1906 (Nernst 1906), and to be sure, the chemical industry recognized the value of such a theoretical tool: chemical yields could be determined without having to carry out the (possibly costly) reaction itself (Haberditzl 1960; Suhling 1972).

Nernst’s solution stated that the equilibrium of chemical systems depended on a specific behavior at temperatures approaching absolute zero. He had questioned the universal applicability of the principle of maximum work (ΔU = ΔA) since his investigations of chemical equilibrium and heats of reaction in the 1890s in Göttingen. The concept could not be universally applied and Nernst was convinced the resolution of the problem would be a profound discovery (Haberditzl 1960; Nernst 1914, 1921), (Bartel 1989, pp. 21–39). “A hidden law of nature,” he wrote in the first edition of his Theoretische Chemie (Theoretical Chemistry) in 1893, “underlies the ‘principle of maximum work’ whose clarification is of highest importance (Nernst 1893, p. 543).”Footnote 2 He was not alone in his suspicion. Others such as Le Chatelier and Fritz Haber himself (Haber 1905b, pp. 41, 50, 62–64) were considering the physical meaning of the integration constant. Nernst openly admitted that Le Chatelier had already formulated the problem clearly in 1888 (Chatelier 1888), but it was left to Nernst to present the answer. It would later became known as the third law of thermodynamics, though at first, it was only a postulate based on a hunch he had had after considering experimental data (Nernst 1906, pp. 5–7), (Haberditzl 1960, p. 408), (Farber 1966, pp. 185–186), (Hermann 1979, pp. 131–132), (Suhling 1993), (Müller 2007, pp. 170–172).

Therefore, after 1906, both a theoretical and a (theory-based) experimental solution were available to determine chemical equilibria. How does this situation help us identify the aspects of physicochemical theory that were necessary for Haber’s breakthrough?

If we consider Haber’s and Nernst’s results in Figs. 11.7, 11.8, 11.9, 11.10, and 11.11 and compare them to the supplied modern theoretical values, we see that Haber’s 1905 results were highly accurate considering there was no information available about the equilibrium between ammonia, nitrogen, and hydrogen at the time. By 1907, Haber’s accuracy would not (and did not need to) be further improved. He achieved his results not by using Nernst’s theory, but rather his own approximations for chemical equilibrium (Appendix A) combined with an experimental measurement. Complete knowledge of the theory of physical chemistry was, therefore, not needed to achieve the breakthrough. To be exact, Haber’s approximations were only accurate within a narrow temperature range surrounding the experimental data point (the deviation can be seen in the graphs). However, this local accuracy puts the current discussion into stark relief. The approximations were not perfect enough for a full description of the system, but they were powerful enough to achieve the intended objective: industrial upscaling. If, however, the goal had been to achieve full knowledge and control of the chemical system under arbitrary conditions, it would have been another matter.

Stepping back to consider the state of knowledge of the free energy before theories such as Helmholtz’ appeared, there was not enough information to allow for a controlled synthesis of ammonia under any conditions. I use the word “controlled” because it would, of course, have been possible (although unlikely) for someone to stumble upon the correct conditions. As illustrated above, equations like Helmholtz’ fostered the idea of the balancing act (Eq. 13.1) and, thus, a precise way in which conditions must to be changed in order to achieve new equilibria. Without this knowledge, even if someone had serendipitously achieved a synthetic production of ammonia under some set of conditions, it would not have been possible to extrapolate the suitable conditions needed for industrial synthesis (especially considering that these conditions were not technologically attainable in the nineteenth century).

Thermodynamic derivations of the free energy, such as those provided by Helmholtz or Gibbs, combined with Haber’s approximations and measurements (including his results on the effect and correct role of the catalyst) were the critical pieces of the physicochemical puzzle needed to achieve the breakthrough of ammonia synthesis. The importance of these combinations of knowledge is underscored by the failure of the late empirical approaches, like Edgar Perman’s. Even basic physicochemical principles, such as Le Chatelier’s principle, were not critical to Haber’s initial experiments, although they certainly influenced his thinking and were critical to the subsequent upscaling. Nernst’s theory provided a common starting point for the exchange of knowledge between Haber and Nernst as well as a solid theoretical framework for chemical synthesis and reassurance for industry. While such a framework that included all physicochemical principles was not required for industrialization, it would inevitably garner great interest as it was indispensable to securing scientific knowledge at a fundamental level.

It is important to note that this circumstance, while determined by the particular skills of those involved, was also shaped by the state of research at the time. A stronger contribution to the breakthrough from theory was not an impossibility. In 1906, Max Planck formulated an expression for entropy based on the probability interpretation of Ludwig Boltzmann and others that specified a zero-point for S at T = 0. From this reference, it was possible to derive Nernst’s heat theorem and provided the third law of thermodynamics with some theoretical underpinning. However, Planck’s approach did not convince everyone, among them Albert Einstein, and the correct physical starting point to arrive at his expression of entropy was debated for some time (Planck 1906, pp. 129–137), (Grandy 1987, pp. 16, 74), (Kox 2006) (Fig. 13.1). Had such a rigorous mathematical formulation allowed for an earlier substantiation of Nernst’s theorem, his theoretical determination of ammonia equilibrium—one that would have relied on the full theory of physical chemistry—may have been considered complementary to Haber’s initial measurements. At the time though, Nernst’s theory contained too much conjecture. Later, when the matter was settled and his theorem had proven correct and applicable the impact was clear: the field of classical thermodynamics was complete.

Fig. 13.1
figure 1

From left: Walther Nernst, Albert Einstein, Max Planck, Robert Millikan, and Max von Laue in 1931, probably at the home of the latter. Source: Archive of the Max Planck Society, Berlin-Dahlem, III. Abt. Rep. 57, NL Bosch; Akz 46/95, Picture Number III/2

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