System States and Parameters

I’m having an epiphany! While lying down in bed. It helps to have had more sleep the night before. It helps that I’m having a lighter teaching semester with two small classes. It helps that I spent part of yesterday in interdisciplinary conversations with colleagues. It helps that I’m sitting in on a biochemistry course this semester (in preparation to teach it in the future) right after teaching my G-Chem II course where we’re now covering some applications of thermodynamics to solutions and mixtures. And now onward to the nerdy epiphany.

 

In the first week of class, one of my students stopped by office hours to ask me about state functions. I had just introduced the tripartite model of the thermodynamic universe consisting of the chemical system, the thermal surroundings (modeled by an insulated water bath), and the mechanical surroundings (modeled by a piston and shaft) – see the picture below. What we care about is measuring the energy changes in the system that result from a chemical process which typically involves making and breaking chemical bonds.

 

The internal energy of the system is a state function. It only depends on the chemical ‘state’ of the system – which atoms are bonded to each other. The system is thermodynamically closed, i.e., it is cleanly separated from the environment in that no atoms can enter or leave the system (although energy can be transferred between system and surroundings). The word tripping up this student was ‘function’ because it made him immediately think of functions in a math class. Since this is G-Chem and not P-Chem (where we delve into the mathematical functions), I convinced him to substitute the word ‘function’ with ‘property’ and he was satisfied. I didn’t spend time thinking about other appropriate words because the student had other questions for us to discuss. Should these instead be called state ‘variables’ or state ‘parameters’?

 

In thermodynamics, language and terminology can be tricky. For example, ‘heat’ is not a noun. And ‘work’ (in the piston and shaft model) is of a specific type (PV-work) that confuses students later when we introduce the definition of free energy. The word ‘function’ is often used in biochemistry in the context of structure-function relationships that relate (but don’t exactly map) to the seeming genotype-phenotype divide in biology. So for this post, let’s stick to the word ‘variable’ for now to describe system states. Most students would associate this word with an algebraic quantity that can change, and at the G-Chem level this works just fine. Thus, the internal energy of a chemical system is a state variable.

 

The internal energy of a system can change due to external variable changes such as changes in pressure and temperature of the environment. Pushing the piston down and compressing the system represents an increase in pressure. Hotter water in the thermal bath represents an increase in temperature. Assuming that the chemical system is composed of a gas (easiest for measuring changes due to P and T), we can use an equation to calculate V, the volume of the system, as P and T vary. For an ideal gas, the chemical identity of the gas does not matter; you just need to know how many gas molecules are present. Students happily use PV = nRT in their calculations.

 

Here’s part one of my epiphany. If P and T represent environmental forces that act on the system, then V is the phenotype of the system, and furthermore there is no specific genotype of the system beyond the tenets of an ideal gas: A large number of independent tiny particles moving randomly with only elastic collisions and lots of space to move in. You don’t need any other specifics if the gas behaves ideally. There is no ideal organism in biology. But there could be some abstract entity that we call ‘organism’ – a living being. We’re vague on the phenotype other than to list generic behaviors: it eats, it poops, it grows when food is plentiful, it dies for lack of food or if something else eats or injures it.

 

Here’s part two of my epiphany. In a real gas, the ‘equation of state’ PV = nRT fails to capture all the details and more complicated equations are needed. A simple one that we teach students is the van der Waals equation: (P + an²/V²)(V – nb) = nRT. This equation of state has two parameters that are dependent on the identity of the gas: a is related to the strength of the intermolecular forces and b is related to molecular size. I say ‘related’ because a and b are empirically-derived parameters from experiment and don’t exactly correspond to intermolecular forces and size respectively. In P-Chem we illuminate these relationships with mathematical models that allow us to approximate calculating a and b from first principles. But none of these models capture everything. With the van der Waals equation, you can calculate V, the system phenotype, taking into account a and b, the genotype of the system, given environmental variables P and T.

 

Here’s part three of my epiphany. There are more complicated equations that do better than the van der Waals equation in matching experimental results for the ‘behavior’ of gases, but these have more parameters. They have a larger genotype! In P-Chem my students learn the virial (not viral) equation, its advantages and disadvantages, and how to truncate the Taylor series expansion. We’re also using mathematical functions and so the language of state ‘functions’ is appropriate in P-Chem. And now for the kicker. To get everything precisely ‘correct’ mathematically requires an infinite number of parameters. So there is no largest model that truly describes the system. We truncate the model, i.e., we simplify it, for practical reasons.

 

How did this weird situation come about? Fundamentally, it comes from our sharp separation between the system and surroundings. This separation forces us to introduce parameters to ‘mediate’ (I can’t think of a better word at the moment) between the variables of the system with those of the surroundings. In biology, by trying to define what makes an organism distinct (even though it interacts with its environment), we use genotype to mediate the function (phenotypic behavior) of an organism with its environment. For practical reasons, or maybe because we’re enamored by the idea, or because it’s easier to teach students, we employ reductionism and turn genotype into genome with a focus on DNA. The blueprint of life, we call it. But we now know that that the genome is not enough. We need to account for the proteome, the metabolome, the metallome, the interactome, and likely more. (I introduced the metallome in my elective class this semester!)

 

As a chemist, I think of the state of a system primarily in terms of its structure – what chemical bonds are present, how strong they are, and what degrees of freedom the structures possess – which then determines its function, its phenotypic behavior given some environmental conditions or ‘forces’. The genotype has something to do with the idiosyncracies of each atom, but also accounts for its context dependence. The parameters I might invoke include polarizability, electronegativity, acidity, and more. But unlike the one-way structure-determines-function of the chemist, the biologist needs to employ a two-way relationship between structure and function. I don’t think this is simply practical language when discussing biology, but that it may be paradigmatic to biology that structure and function loop around each other. States and parameters in some sense cannot be cleanly separated, and we use such terms because we tried to separate system from surrounding as a starting model. I don’t see this as bad or wrong. But I do think as professors we should be more cognizant of these issues, and perhaps even expose our college-level students to these underpinnings in our fields of science.