Composition over Phase Change

The practice of separating substances by distillation relies on the fact that a mixture and its vapor typically differ in composition, while a pure substance has the same constitution as its vapor. In distillation, the vapor is collected and cooled, providing a liquid that is enriched in one component. This distillate can successively be redistilled, purifying the relevant component to whatever degree is required. Constancy of composition over phase change might seem to provide a criterion for distinguishing pure substances from mixtures, but an immediate problem concerns azeotropes, which are mixtures of different compounds (like water and ethanol, or water and hydrogen chloride) that, under certain conditions, can be distilled without affecting their composition. One might regard azeotropes as single substances (i.e., compounds); presumably they would be secondary compounds of their components, much as ammonium chloride (NH₄Cl) is a compound of ammonia and hydrogen chloride, but this is not how they are regarded by chemists (see Moore Reference Moore1963, 140). Arguing for a thermodynamic conception of substance, Ostwald (Reference Ostwald1904, 514–16) attempted to distinguish between azeotropes and compounds proper on the grounds that azeotrope composition varies with pressure. In fact, altering the pressure at which distillation occurs is one way to separate the components. In this respect azeotropes are more like solutions and other mixtures. The composition of a saturated solution of salt in water, for instance, depends on temperature. In contrast, the composition of most compounds is constant within the range of physical conditions in which they exist. But why does variable composition disqualify an azeotrope from being a compound? Why not regard each compositional ratio as a distinct substance, yielding a whole class of related compounds, each existing within its own (narrowly defined) range of physical conditions? Interestingly, during the first half of the nineteenth century there was an extended debate over whether compounds must have a composition that is fixed and independent of physical conditions, because a range of substances (Berthollides) display variable composition, and not all could be regarded as solutions or mixtures (see Needham Reference Needham2008, 67–68; Reference Needham2010c). Constancy of composition over phase transitions therefore seems to fail as a criterion for being a pure substance because it requires appeal to an independent and contentious criterion, one that, moreover, does not clearly disqualify azeotropes.

The Phase Rule

Gibbs’ spookily elegant and abstract equation relates the number of components (C) in a multiphase system to the number of phases (P) present in it, and its variance (F), which is the number of independent physical variables required to characterize the thermodynamic state of the system:

Does the phase rule provide a criterion of sameness and difference for substances? The difficulty is that “components” cannot simply be equated with chemical substances, and in some systems it is a matter of some delicacy how to map one onto the other so as to preserve the truth of the phase rule: Needham (Reference Needham2010a, Reference Needham2010b) discusses the three-component system of calcium carbonate (CaCO₃), carbon dioxide (CO₂), and calcium oxide (CaO). Perhaps the phase rule itself does not tell us how many substances are present in a system but only once a case-by-case matching of phase-rule components to chemical substances has been achieved. If so, then the phase rule itself does not offer a criterion for distinguishing substances and mixtures but only a series of relationships upon which we can impose a carefully constructed interpretation that does so. The question is whether or not such an interpretation amounts to a bookkeeping exercise rather than an autonomous conception of substance.

Entropy of Isothermal Mixing

Whenever samples of two distinct substances in thermal equilibrium are allowed to mix, there is an increase in entropy. In contrast, samples of the very same substance generate no entropy change on isothermal mixing. Since entropy change is described within classical thermodynamics without any reference to microscopic structure, such entropic relations seem to offer a macroscopic criterion for the individuation of chemical substances. In the next two sections I will critically examine this criterion.

Case 1. Isotopic Variants

Protium (¹H) and deuterium (²D) are isotopes of hydrogen, differing in their mass but alike in their nuclear charges. Although there were clear indications of its existence earlier, deuterium was first isolated in 1932, the separation using the differential rates at which ¹H₂O and ²D₂O are electrolyzed (see Farkas Reference Farkas1935, 115–17). Mixing protium and deuterium involves a positive entropy change, so the entropy criterion counts them as distinct substances, as it does the different isotopes of any element. The same applies to isotopic variants of compounds. Thus, although water is overwhelmingly composed of ¹H₂¹⁶O, it typically contains traces of other isotopic variants and so must be counted as a mixture of several substances on the entropic criterion. As Needham puts it, “Lavoisier thought he had settled the composition of water. But in 1932 Harold Urey showed that what Lavoisier took to be a single substance was a mixture” (Reference Needham2008, 73).

In 1923, following the discovery of isotopy among heavier elements (but before the discovery of deuterium itself), the International Union of Pure and Applied Chemistry (IUPAC) defined elements—and so the extensions of their names—in terms of nuclear charge (see Aston et al. Reference Aston, Baxter, Brauner, Debierne, Leduc, Richards, Soddy and Urbain1923). Hence, under standard chemical nomenclature protium and deuterium both fall within the extension of “hydrogen.” Needham (Reference Needham2008, 66) describes the adoption by IUPAC of the coarser-grained classification as a “convention” preserving the periodic table of the elements, and the unity of explanation that comes from regarding isotopes as chemically interchangeable, but this seems too thin a description of the decision. There was a principled discussion among chemists as to the similarities and differences between isotopes (see van der Vet Reference Van der Vet1979; Kragh Reference Kragh2000), and recognition that, as used by earlier chemists such as Lavoisier and Mendeleev, the extension of element names like “hydrogen,” “oxygen,” and “chlorine” encompassed heterogeneous populations of atoms that had a range of atomic weights but were alike in their nuclear charge (see Hendry Reference Hendry2006). It therefore seems more accurate to say that what IUPAC decided was to continue this earlier usage while making explicit the determinable property that had always governed the identity of elements. The finer-grained classification offered by the entropy criterion, and the possibility of separating isotopic variants, does show that there are genuine differences among the atomic populations that characterize each element, but this is to recognize a criterion of sameness and difference that was not part of the intellectual context that informed earlier thinking about the elements.

Case 2. Spin Isomers

Ortho-hydrogen (o-H₂) and para-hydrogen (p-H₂) are spin isomers, molecules that differ in respect of the alignment of their nuclear spins: in o-H₂ the spins are parallel, while in p-H₂ they are antiparallel. At standard temperature and pressure, o-H₂ and p-H₂ coexist in a ratio of approximately 3 : 1, while at absolute zero, the equilibrium distribution is 100% p-H₂. However, the activation energy for the transition between the two forms is significant, so liquid hydrogen can be “locked” into a nonequilibrium proportion of the higher-energy ortho form. Since the transition from o-H₂ to p-H₂ is exothermic, the spontaneous (although slow) transition to the low-temperature equilibrium gives rise to heating and the loss of significant amounts of liquid H₂ by evaporation. This can be addressed by catalyzing the transition on cooling.

How does the entropy criterion apply to o-H₂ and p-H₂? Since they display a positive entropy of mixing, it seems that they should be counted as distinct substances. Now it is true that the two species interconvert, and one of them, o-H₂, cannot be obtained pure, but this is true of many distinct substances that exist in chemical equilibrium with each other. If one wished to regard o-H₂ and p-H₂ instead as two forms of the same substance, differentiated by their physical states, some principled reason for doing so will need to be identified, one that is consistent with the motivation of the entropy criterion to provide a macroscopic perspective on substance identity. I will consider candidate reasons in the next section, when I discuss the hard cases together.

Case 3. Identical Atoms in Orthogonal Quantum States

Consider one mole each of two populations of atoms that are identical except that they are in different, though degenerate, quantum states and ψ. If (i.e., α is the “overlap” between and ψ), the entropy of mixing will be:Footnote ⁴

Note that the entropy change on mixing appears to depend continuously on α,Footnote ⁵ and

If the criterion of sameness of substance is that the entropy of mixing is zero, then populations of atoms in orthogonal quantum states, like the different streams of silver atoms in a Stern-Gerlach experiment, are different substances. This, of course, is counterintuitive: surely the recombination of the two Stern-Gerlach streams is a case of mixing the same substance (silver) in different physical states. The question once again is, what principled reasons are there for saying that, drawing only on macroscopic information?

Case 4. Nonisothermal Mixing of Identical Gases

If two samples of hydrogen at different temperatures are allowed to mix, there is also an increase in the overall entropy. They are not, of course, counted as separate substances by the entropy criterion, but that is only because the criterion is restricted explicitly to isothermal mixing. It is perfectly natural, however, to say that the two samples are just the same substance in different physical states.

How do these cases ground an objection to the entropic criterion? Suppose we admit case 1 as an instance of the mixing of different substances. In contrast, case 4 clearly involves mixing the same substance in different states. But cases 2 and 3 are more like case 4 than they are case 1. Entropy of mixing seems to track distinctions between different states rather than different substances and so should be rejected as a criterion of sameness and difference of chemical substance. In the next section I will consider three responses to these cases.

5.1. Bullet Biting

Taking inspiration from Needham's acceptance that isotopic variants are different substances (see above), one might simply reject the intuitions that cases 2 and 3 involve the mixing of samples of the same substance in different physical states, embracing the entropic criterion's verdict that cases 1–3 are all cases of substantial difference.Footnote ⁶ There are two objections to this stance. First, one would lose the sense that chemical substances are continuants whose physical state may change. A difference in any physical quantity apart from temperature now seems to constitute a difference of substance, if mixing samples with different values of this quantity gives rise to a positive entropy change. It is simply unclear what relevance such a criterion for the sameness and difference of substances has for chemistry, or what light it could shed upon chemical thinking. Second, the exception for temperature is unmotivated. Why, apart from fiat, does a difference in temperature not constitute a difference of substance?

5.2. Equilibrium

One important difference between cases 1 and 4 is that in case 4, the entropy change arises from the relaxation of thermal disequilibrium, while in case 1 it arises from the relaxation of chemical disequilibrium. One might then seek to defend the entropy criterion by saying that two kinds of matter constitute different substances only if, when mixed isothermally, there is an overall entropy change, and this entropy change arises from the relaxation of chemical disequilibrium. But what is chemical equilibrium? In discussing equilibrium generally, textbooks of thermodynamics tend to appeal to the idea of a system lacking a disposition to undergo change. But what is “change”? Presumably, it is when the state at some time is replaced by a different state at some later time. Put this way, the notion of a change already seems to presuppose some idea of sameness and difference. Consider Zemansky, who first introduces mechanical equilibrium as the kind of state in which there is no “unbalanced force in the interior of a system” (Zemansky Reference Zemansky1957, 24). He then characterizes chemical equilibrium as follows: “When a system in mechanical equilibrium does not tend to undergo a spontaneous change of internal structure, such as a chemical reaction, or a transfer of matter from one part of the system to another, such as diffusion or solution, however slow, then it is said to be in a state of chemical equilibrium. A system not in chemical equilibrium undergoes a change of state that, in some cases, is exceedingly slow. The change ceases when chemical equilibrium is reached” (Reference Zemansky1957, 24).

Now apply this to the mixing of gases. Suppose we have two identical samples of gas at the same temperature and pressure in two vessels linked by a valve. In fact, some diffusion tends to occur when the valve is opened: some of the gas that was in one vessel will end up in the other vessel. Does this mean that the system overall is not at chemical equilibrium at the instant the valve is opened? No, because the diffusion of sample A into vessel B—which is of course balanced by some of sample B moving into vessel A—does not count as a “spontaneous change of internal structure” because the two gases are identical. Now suppose we have samples of two different gases (again at the same temperature and pressure) in two vessels linked by a valve. At the instant the valve is opened, before any mixing has occurred, the system is not in chemical equilibrium because the diffusion of the two samples through the valve does constitute a “spontaneous change of internal structure” in the system. But that change consists precisely in the difference between the two gases. The only difference between the two cases is that in the first the substances being mixed are identical, while in the second they are different. The notion of chemical equilibrium already seems to presuppose a criterion of sameness and difference of substance. Hence, it cannot be used to explicate that notion in a noncircular way.

5.3. Mechanisms of Relaxation

Consider another important difference between case 1 (isotopic variants) and case 4 (nonisothermal mixing). This requires relaxing the ideality assumption, that is, the assumption that the gases do not interact on mixing (recall that relaxing this assumption does not affect the “step change” underlying Gibbs’ paradox and the entropic criterion). In case 1 the nonequilibrium state just prior to mixing relaxes into equilibrium just by intermingling the two samples. None of the protium becomes deuterium, or vice versa, or changes in any other way with regard to the properties that differentiate the two samples. In case 4, in contrast, the system reaches thermal equilibrium via two processes: the hot gas and the cool gas intermingle, but the two bodies of matter also exchange energy, the hot stuff becoming cooler, the cool stuff warmer. This is a salient difference if we wish to regard substances as continuants that are capable of surviving changes to their (nonessential) properties. How does this apply to case 2? In this respect the mixing of o-H₂ and p-H₂ is closer to nonisothermal mixing: suppose we had pure samples of p-H₂ and (per impossibile) o-H₂, and allowed them to mix. Not only would the mingling occur but some of the p-H₂ would become o-H₂ (or vice versa, depending on the temperature). The same applies to case 3: as the two streams of silver atoms mix, they will not simply be juxtaposed but will interact in ways that alter their spin states. Hence, one might add the following modification to the entropy criterion: two bodies of matter are different substances if, and only if, they undergo a positive entropy change on isothermal mixing and the two-sample system approaches equilibrium via a mechanism in which there is no change in the individual samples in respect of the properties that differentiate them. With this proviso, the entropy criterion seems defensible, but mechanisms of relaxation seem to be a paradigmatically microstructural consideration, so adding the proviso means that the entropy criterion can no longer be part of a purely macroscopic conception of the sameness and difference of substance.