1. Introduction
It is claimed by every authority that phenomenological (also referred to as classical) thermodynamic theory is independent of the nature of matter. The theory's validity does not depend on either the composition or the structure of the matter. Thermodynamics embodies two abstract principles—conservation of energy and maximum distribution of energy—but absolutely no assumptions about the nature of matter. Whether the universe is made up of atoms and molecules, or radiation, or of yet unimaginable forms, thermodynamic theory is unaffected.
One would expect a theory thus formulated to be of little use in studying the properties of matter since there are no references to matter in the theory. How can such a theory make inroads into chemistry, which is exclusively concerned with forms of matter? Yet classical thermodynamics happens to be the most prevalent theory in investigating structure, properties, and reactivity in chemistry, hyphenated chemistries, and material science.
There is no mystery here for practicing scientists. They know the procedures through which theory and experiments are connected to provide insight into the properties of matter. Hence the purpose of this article is not to argue that there is something missing in the nexus between theory and experiment. Instead I will detail how theory and experiment are connected in chemical thermodynamics. I believe that a careful examination of the serpentine path between thermodynamic theory and chemical practice provides philosophical insight into the complexities involved in relating theory to the empirical world.
Undoubtedly chemical equilibrium is the area that has benefited most from thermodynamic theory. The equilibrium constant for a gas phase reaction, Kp, and the Gibbs energy change between the products and the reactants, ΔGo, are related by
where R = universal gas constant and T = temperature. For a reaction
the equilibrium constant is
where the p's are the partial pressures at equilibrium, and the change in Gibbs energy is
(The superscript indicates a pressure of 1 bar. The symbols, μ, denote chemical potentials: molar Gibbs energy for the species in the vessel.)
Equation (1) appears frequently in the pages of textbooks and monographs. It has reached an iconic status in chemical thermodynamics. It can be said, with some minor reservations, that the right side of equation (1) is from experiments and the left side from theory. This makes equation (1) a good candidate for investigation into the sly relation between theory and experiment in chemistry, my sole aim in this article.
2. Chemical Equilibrium: Relevant History
It was Claude Louis Berthollet's observation of sodium carbonate by Lake Natron, during his tour of Egypt with Napoleon's army (in 1799), that is credited with starting the study of chemical equilibrium. Berthollet correctly reasoned that sodium carbonate, a soluble salt, is precipitated by the excess amounts of sodium chloride reacting with calcium carbonate. The extent of the reaction between sodium chloride and calcium carbonate,
is not only controlled by the “affinity” of the chemicals but also their masses.
After nearly 7 decades, Cato Guldberg and Peter Waage gave a quantitative account of Berthollet's observation and formulated the law of mass action. With reference to equation (1), the forward rate of reaction is determined by
and the backward rate by
where the k's are the rate constants (affinities, in older usage) and the concentrations of the chemicals are indicated by brackets. The rates are affected by rate constants and by the concentrations (or masses). Equilibrium is established when the two rates become equal. The equilibrium constant (here as a function of concentrations) then is simply the ratio of the two rate constants.
Since then the connection between the kinetic and equilibrium aspects of chemical reactions has been seen as weak. Hence it is unlikely that one finds kinetics and equilibrium discussed in the same section in physical chemistry textbooks or in reviews of recent advances. There seem to be two reasons for this. One is that the authors of phenomenological thermodynamics books are often obsessed with keeping that subject uncontaminated by any dynamic considerations. The other is the realization that reactions do not actually proceed in the way expressed in the rate equations (4) and (5). There are usually several intermediate elementary reactions, many beyond our ability to monitor experimentally. If the simple scheme shown in equations (4) and (5) is all there is to chemical change, we can proceed without the help of thermodynamics.
By ignoring the connection between rates and equilibrium, however, we miss an important idea regarding the nature of substances that is the subject of an ongoing discussion in philosophy (Ruthenberg and van Brakel Reference Anderson2008). Chemical species in a dynamic equilibrium continuously undergo transformation, even though at widely different rates. Atoms in one molecule become part of another molecule at a certain frequency. A movie of atoms and molecules does not show permanence of any species in dynamic equilibrium. Under these conditions it becomes difficult to give individual identity to any chemical species. Of course, there is no problem identifying individual species if equilibrium happens to be static, or if the clock can be stopped.
3. Thermodynamics of Gibbs: A Brief Account
Current investigations of chemical equilibrium start with J. W. Gibbs's formulation of thermodynamic theory. François Messieu and Gibbs independently showed that an economical and reliable theory of equilibrium thermodynamics can be built by minimizing characteristic functions (called potentials). The minimization is with respect to nonequilibrium states for a particular set of values for environmental variables. Almost simultaneously these two remarkable scientists liberated thermodynamics from imaginary cycles and impractical processes. It is only Gibbs's formulation of the theory, however, that finds applications in chemistry since he alone introduced chemical composition into it.
The two functions that play pivotal roles in determining the conditions for chemical equilibrium are the Gibbs energy, G, and the chemical potential, μ. Gibbs energy at a given temperature and pressure, and at any chemical composition, is a minimum with respect to all possible nonequilibrium states at the same temperature and pressure. In the differential form, the Gibbs energy is given by the expression
where the μ's are the chemical potentials (S = entropy, T = temperature, V = volume, P = pressure, n = number of moles).
Since dG = 0 at equilibrium, it follows at once that
when temperature and pressure are fixed at a particular set of values. This simple equation is the sole basis for studying heterogeneous and homogenous chemical equilibria of all varieties. If we have a single chemical in two phases (say a and b), the above summation reduces to
Since the chemical potentials are not zero this equation is true iff
(This is a reminder that chemical equilibrium is dynamic; atoms and groups of atoms or molecules are continuously exchanged between products and reactants.) It follows from equations (8) and (9) that the condition for equilibrium is
Equation (10) is equally true for two chemicals in equilibrium in a single phase. Very generally
is the law of chemical equilibrium, which also includes phase equilibrium.
In a letter to the American Academy of Arts and Sciences (1881), Gibbs wrote: “The leading idea which I followed in my paper on the Equilibrium of Heterogeneous Substances was to develop the rôles of energy and entropy in the theory of thermo-dynamics equilibrium. By means of these quantities the general condition of equilibrium is easily expressed, and by applying this to the various cases we are led at once to special conditions which characterize them” (quoted in Reference RukeyserRukeyser 1942, 267).
Equation (11) is the general condition for equilibrium and it follows directly from the first and second laws of thermodynamics without approximations or provisos. The equation embodies the logical rigor and simplicity of a well-structured deductive theory. In order to go beyond this stage we have to know the chemical potential of every species as a function of pressure, temperature, and the number of moles of all species in equilibrium. Maxwell's relations (Euler's test for exactness) give the temperature and pressure dependence of chemical potential, as shown by the following equation:
The quantities si and 
are the molar entropy and volume of the ith species. For ideal solutions—mixtures of two or more chemicals that retain their “private properties” in solution—these quantities can be defined. Equation (12) holds for nonideal solutions as well with a slight extension of the concept of private entropy and volume. This, however, is as far as thermodynamics takes us. Considering that molecular composition did not enter into the formulation of the lofty theory of thermodynamics, it should not surprise us that it cannot take us beyond this point.
4. Extending and Amending: Chemical Reaction Equilibrium
An equation that allows us to go beyond the point where phenomenological thermodynamics leaves us is
where Xi = the mole fraction of the ith species. The last term in this equation is obtained from either statistical theory for entropy of mixing or from Raoult's law, which holds for dilute solutions. In both cases ideal behavior is assumed.
As equation (2) points out, we need the chemical potential as a function of partial pressure. This is obtained for an ideal gas by integrating equation (13),
In this equation, 
is the chemical potential of the gas at 1 bar pressure. There is no way of knowing the absolute value of the chemical potential. It can be given a value only with reference to what in thermodynamic literature is called the standard state. Equations (1), (2), and (3) follow from the above equation.
5. Tenuous Connection to Thermodynamics
In the above derivation we used the ideal gas equation of state. Equations of state are not part of thermodynamic theory. They do not contradict the laws of thermodynamics, but they cannot be derived from the laws of thermodynamics. An equation of state is a mathematical expression that connects the variables of the state; it is not derived, but deduced by assuming a model for intermolecular forces. Thus the molecular hypothesis, which is not in the original theory, gets incorporated into thermodynamics.
The fusion of thermodynamics with molecular models is done routinely in chemistry. Chemists see no problem in doing so, no epistemological shock. Those who keep theory and experiment in separate compartments and look for a direct relation between the two, however, should be uneasy. It would appear that in deriving equation (1), the empirical and theoretical became inseparable. Without the fundamental laws of thermodynamics we would not have the general law for equilibrium. Without the experimental input through an equation of state we do not gain anything useful from the general law.
There is a logical inconsistency in using the ideal gas law for deriving the expression for the equilibrium constant. Molecules in an ideal gas neither attract nor repel each other. They are hard spheres that feel an infinite force of repulsion only during a collision. How can we justify their reacting to form products? The simple answer is, we cannot.
The inadequacy of the theory based on ideal gases is made clear by experiments. In the ideal gas approximation, the equilibrium constant is not a function of total pressure. Yet in practice we find that pressure has a marked effect on the value of the equilibrium constant.
Before examining why the ideal gas equation is still widely used in equilibrium studies, let us examine how the theory is modified (improved?) by incorporating the nonideality. A natural extension of the above theory is to use a more appropriate equation of state in place of the ideal gas equation. That will modify equation (14) and give a different formula for the equilibrium constant. Moreover, the formula will be different for different equations of state.
In practice, the form of equation (14) is preserved by defining another function called the fugacity, f. Fugacity is not a quantity that can be directly measured or deduced from a fundamental theory. However, it can be deduced from the difference between the molar volumes of the “real” gas and the ideal gas. Hence it is a correction term added to the ideal gas volume. Instead of incorporating nonideality into equation (13), we incorporate it into the variable that goes into the function. There is no saving of labor, since the volume of the real gas has to be determined experimentally or by using a more suitable equation of state. The only advantage is that this ruse allows us to preserve the simple form of equation (13).
Currently we use fugacities in the equilibrium constant expression instead of partial pressures. All this trouble improves the match between the calculated and measured equilibrium constants, but still misses the mark for accurate prediction. The reason for this has to do with the nature of nonideality. The nonideality of a gas when it is the only component is different from its nonideality in a mixture. The story gets even messier now because to account for the extra nonideality in a mixture various ad hoc hypotheses are used.
Equilibrium constants are also measured in liquid solvents. Here the natural variable is the mole fraction of each species. However, nonideality is even more pronounced now, and the corrections to keep the equilibrium constant a true constant become even more convoluted with very limited success.
6. Imperfect Theory
I have given a picture of the imperfect and inelegant science that is widely used in chemistry. Can this practice be justified? It cannot, if we insist that theory and experiment have to be two separate layers without commingling.
Why do chemists continue to use an imperfect theory? An unkind answer is that they don't know any better. I will, however, argue that that answer is wrong, that the imperfect theory plays a crucial role in chemical research, and that a more rigorous theory will not help chemical practice. Consider the gas phase equilibrium. Gas phase reactions are notoriously slow, unless a catalyst mediates the reaction. Unlike biological catalysts (enzymes), man-made catalysts are not specific. So while they speed up the reaction, they also catalyze further reactions of reactants and products. Many processes in nature—atmospheric reactions, for example—are series of coupled reactions. Thus prediction of the exact value for any chemical equilibrium serves no purpose. It cannot be tested if a catalyst is used, or if the reaction is coupled to other reactions. However, an experimentalist needs guidance in designing the apparatus for manufacture of chemicals, or for simple laboratory preparation. It is foolish, impractical, and expensive to plan an experiment without at least a vague idea of what is expected. Thus a theory that gives a ballpark estimate is needed. Since the aim is to manufacture or prepare (and certainly not to test the goodness of the theory), a relatively simple theory that gives a range of values is as good as one that gives an exact number.
There is yet another consideration. Thermodynamic and kinetic requirements do not always match. In the manufacture of ammonia, thermodynamic theory predicts improved yields as temperature is lowered; kinetic theory, on the other hand, predicts increased speed of reaction as temperature is raised. For a successful operation of the plants, optimization has to find a balance between the two requirements. Hence the actual value of K is less useful than how it changes with temperature.
Nonideality is an even bigger problem in liquid solutions. Thermodynamic theory, however, finds a very different application now. From equilibrium constant measurements at different temperatures, chemists deduce energy and entropy changes in chemical transformation. A Quantitative estimate of bond energy in ATP came from such studies. Gibbs energy, enthalpy, and entropy are related to ln K. Hence the thermodynamic functions are not sensitive to small fluctuations in K values.
7. General Remarks
There are two layers in the structure of the physical sciences: (1) the theoretical, where every significant point is connected to another by mathematical expressions, and (2) the empirical, where data are collected under controlled experimental conditions. I do not claim that these layers are entirely independent of each other. Experimental work is at the least directed by some theoretical considerations. However abstract the theory gets, its aim has to be to provide some explanation of what scientists consider to be important facts.
The connection between the two layers is significant, particularly in some areas of chemistry. In actual practice these two layers are connected through intermediate layers where a series of further assumptions are made that do not fit into either the purely theoretical or the manifestly experimental layers.
In phenomenological thermodynamics, the theoretical layer is the Gibbs version of the theory, which is a set of relations on mathematical surfaces whose physical foundations are energy and entropy. The empirical layer is the experimental investigation of chemical and phase equilibrium. I have sketched the intermediate layers above.
I believe it is fair to say that most philosophical investigations are not concerned with the intermediate layers. This is understandable considering the complexities we face in these layers. Besides, the clear picture of science that emerges in the theoretical and empirical layers is missing in the middle. Yet without some sort of understanding of the intricacies in the middle layer, we get a very naive picture of the theory-experiment nexus.
It is comfortable to assume that once the fundamental laws are formulated in rigorous theoretical terms, all that is left is to apply them to different cases, in a straightforward manner. We get this impression not only from famous scientists but also from popular media, as evinced by the recent claims for the theoretical importance of the Hadron Collider experiments, which, it is stated, will provide a theory of everything. According to this view, the difficulties we meet in the intermediate layers call for no further ingenuity. The impression one gets is of a relay race. Once fundamental laws are passed down to the next runner, all he has to do is to carry them to the next runner or to the destination.
In an influential article, Philip Anderson objects to this view: “The main fallacy in this kind of thinking is that ‘reductionist’ hypothesis does not by any means imply a ‘constructionist’ one: The ability to reduce everything to simple fundamental laws does not imply the ability to start with those laws and reconstruct the universe” (Reference Rukeyser1972, 393). In Anderson's view, as I understand it, reduction has been achieved if the theoretical layer provides an explanation for the entities in the empirical layer, even a very qualitative one with several provisos. In that sense everything is either reduced or will be reduced to fundamental laws of physics. Anderson's use of the word “reduction” does not match the current usage in philosophy of science.
The more significant part of his statement is about the failure of the constructionist hypothesis, the idea that fundamental laws can give a complete explanation for all observations without further ado. The constructionist hypothesis, in my analysis, implies ignoring all the intermediate layers mentioned above.
Anderson goes on to say, “The constructionist hypothesis breaks down when confronted with the twin difficulties of scale and complexity. The behavior of large and complex aggregates of elementary particles, it turns out, is not to be understood in terms of a simple extrapolation of the properties of a few particles. Instead, at each level of complexity entirely new properties appear, and the understanding of the new behaviors requires research which I think is as fundamental in its nature as any other” (Reference Rukeyser1972, 393).
I find his views refreshing. Progress in science is not just the so-called fundamental laws codifying everything known. Chemistry cannot be understood without some input from the fundamental laws found in thermodynamics, quantum theory, and kinetic-statistical theories, whose origins are in physics, and yet chemistry is not a simple application of these laws.
The complexity of chemistry, and the difficulty of explaining it simply from fundamental laws, arises because chemistry is a multibody discipline with very large degrees of freedom for each chemical. Fundamental laws, by contrast, depend only on very few degrees of freedom. It is well known that a two-body problem can be solved in physics by making it into a one-body problem (through the concept of reduced mass) and the one-body problem can be solved only if radiation is ignored. Thus chemistry, a collection of multibody problems, cannot be quantified by mere application of a few principles in physics, which were developed for systems with the fewest degrees of freedom. Quantitative explanation in chemistry depends on extending the fundamental theory to systems with far more degrees of freedom. This calls for intermediate layers between fundamental theory, with minimum degrees of freedom, and the empirical world, with enormous degrees of freedom.
