Context Independence

Two years after my first encounter with Robert Rosen’s work, I’m re-reading Essays on Life Itself. I’d like to think that I’m gaining more insight in my second reading, but have no way to prove it. Perhaps it’s the double-edged sword of gaining expertise. The more you know, the more your eyes are opened to what you don’t know. Questions beget more questions.

 

Today’s post surrounds a phrase Rosen uses: context independence. It shows up in chapter 2 (“Biological Challenges to Contemporary Paradigms”) in a section that questions the reduction of biology into physics. I’ll quote Rosen to set up the issue at stake. Subsequently, I will look at some examples at the introductory chemistry level.

 

Here’s Rosen: “… the complexity of organisms, in the conventional view, is interpreted as a measure of how special they are… Complexity is measured in a system by counting the number of distinguishable ingredients it contains and the interactions that constrain them, by how complicated it looks to us… Moreover, it is conventional to suppose a kind of gradient of complexity in the material world, and to suppose that we move in this gradient from generic simplicity to sparse, non-generic complexity by simple accretion, by a rote addition operation… Conversely, a reduction of one of these rare complex systems is merely an inverse rote operation of subtracting… Also presumably, these rote operations of adding and subtracting do not change the material basis of the systems themselves: simple systems are the same whether they are alone or whether they have been added to a larger one.”

 

Rosen then asserts: “This kind of context independence of simple systems is one central feature of scientific objectivity; its main corollary is that one must never pass to a larger system (i.e., a context) in trying to understand a given one but must only invoke simpler subsystems, specifically those that manifest a complete independence of context in their properties and behaviors.” Rosen thinks this is wrong. Instead, “complex systems are far more generic than simple, context-independent ones… analysis and synthesis are not simple rote operations, nor are they in any sense inverses of one another… context-independence with objectivity is itself far too special and cannot be [foundational].”

 

In my view, chemistry is fundamentally about explaining phenomena at a particular scale: the level of atoms and molecules. And a chemical reaction is about making and breaking chemical bonds between atoms and molecules. That’s a tiny scale, typically at the level of tenths of nanometers (or 10^-9 to 10^-10 meters). Most small molecules, and even medium-sized ones (from a student’s perspective), have sizes in this rage. Larger (macro)molecules and polymers with thousands of subunits might extend into the micron range, just barely visible by a light microscope.

 

Are there seemingly immutable context-independent features of chemistry? Setting aside nuclear reactions for now, my students would (correctly) say that elements do not change identity in a chemical reaction or interaction. Hydrogen remains hydrogen; it doesn’t change into helium. Oxygen remains oxygen; it doesn’t change into nitrogen. Since an element’s identity (a label, a name) is only dependent on the number of protons, and since the number of protons does not change in any chemical reactions (chemistry is about the movement and interaction of electrons!), that the atom identities are in some sense, context independent. The number of protons never changes – so that’s a context independent feature.

 

What else might be context independent? The number of neutrons probably, and anything else that involves the nucleus. So maybe anything involving the nucleus is context independent. What about the electrons? An atom can gain or lose electrons without changing its elemental identity – its name (but what does a name really mean?) – although for practical purposes, these are just the outer electrons or the ones furthest away from the nucleus. The inner electrons are held very strongly by the nucleus, and most chemical reactions (and interactions) that chemists care about under most conditions do not involve energies strong enough to extract these electrons.

 

The rightmost column of the periodic table consists of the noble gases, the only elements that exist as isolated atoms under standard conditions (room temperature and sea-level pressure). Under these conditions, one could surmise that individual atoms can be context independent. Therefore, properties of isolated atoms should also be context independent. Examples might be the ionization energies and electron affinities encountered by students in general chemistry. We can measure these properties for any element, not just for the noble gases, by vaporizing and breaking any chemical bonds between the atoms. Imagining an isolated atom, another property we might be able to measure is polarizability – the ability for the electron cloud to be distorted away from spherical symmetry. But would these properties change in non-isolated atoms? Probably. And that’s why we’re careful to define the ionization energy as referring to the atom, while the analogous work function refers to a bulk metal. The two values are different, but they do correlate. (Students often confuse ionization energy and work function because of their closely similar definitions.)

 

Sticking with pure elements for the moment, properties unique to each element can be found from their phase diagrams. The two properties most familiar to students are the melting point and the boiling point. Let’s be careful in our definition. These should be the normal melting point and the normal boiling point, i.e., at sea-level atmospheric pressure. To be even more specific, these temperatures are when the two phases are at equilibrium. Hence, the point in melting or boiling point. What we’ve done, in fact, is defined the context, and once we’ve done so, we can start to define properties. But now we’re straying into context dependence rather than independence.

 

What about the holy grail of what defines chemistry, the chemical bond? Students learn in general chemistry that there are broadly three kinds of chemical bonds (ionic, covalent, metallic) depending on the elements involved – whether they are metals or non-metals. On top of that there are several types of “intermolecular” forces: dispersion (temporary dipole, related to polarizability), permanent dipole (related to polarity), and hydrogen-bonding (a misnomer that adds to student confusion). It’s why I didn’t list atomic size as a property earlier. The actual measurement of atomic size on the same objective scale for all elements is tricky because of the different bonds and interactions between atoms when they get together. (Good luck trying to measure the size of an isolated atom other than a noble gas!)

 

You might hope that after defining the two elements in a chemical bond, that the bond lengths and bond strengths can be tabulated and used. They are! And they’re useful, which I try to impress upon my students. But I also tell them the caveats – many of these tabulated values are averages. The information on the H–H covalent bond probably does refer to the H₂ molecule but the information on the C–C covalent bond might, or might not, be accurate for the particular molecule you’re considering. The students see this starkly in their textbooks for the C=O bond where both the average value (~730 kJ/mol) and the very different specific value (~800 kJ/mol) in CO₂ are provided. And if you want to go wild thinking about this, how might these change when Ant-Man resizes?

 

The bugbear illustrating the difference between context dependence and context independence in G-Chem is electronegativity. Because the students learn about electronegativity trends across the periodic table, and because they are often exposed to the Pauling values, they think that electronegativity is a context independent property. This leads to them making spurious arguments invoking electronegativity in a variety of contexts. Every time I hear one of these spurious arguments, I ask the students for the definition of electronegativity and try to emphasize its context dependence. Electronegativity is an extremely useful context, and it is invoked throughout the curriculum: organic chemistry, inorganic chemistry, biochemistry, and even physical chemistry (although I hope that here students see the physical basis for context dependence). The attempt to replace Pauling values by the seemingly less context dependent Mulliken values doesn’t work as well. (I think the Allred-Rochow values are an improvement, but they require more work and context.)

 

Ultimately the issue of context independence is about defining two regimes: an inside and an outside separated by a boundary of some sort. In thermodynamic models, we refer to these as the system and the surroundings. But the cutoff has to be made somewhere if you’re trying to establish some independence of the system (inside) from its surroundings (outside). The key word here is ‘some’. The degree of independence will depend on how and where the boundary is placed. In small-molecule chemistry, I’d argue that a practical separation is the nucleus plus core electrons versus the outer electrons. Others might draw the boundaries differently depending on context. To return to Rosen, in a truly complex system, there is no largest model, and therefore any boundary you draw is inadequate and there is no context independence.