Thermodynamics: The Uno Law

Beyond the basic chemical equilibrium context, to understand Gibbs free energy you need to understand thermodynamics in a "ground up" fashion. The thing is, I'm not sure even I understand thermodynamics in a ground up fashion. This was part of my motivation to start blogging on it: To keep me thinking about the concepts such that they might eventually click.

The thermodynamic approach I was taught started with quantum mechanics, moved into statistical mechanics, and ended on thermodynamics. What's nice, from a chemist's perspective, about this approach is that the quantum model of the atom elucidates a lot of qualitative understanding of the atom you pick up in earlier courses, such as bond strengths, aromaticity, and IR spectra (or spectroscopy in general). Then statistical mechanics utilizes the energy levels found in quantum mechanics to make macroscopic predictions from the quantum model through statistics (ergo: statistical mechanics).

However, when approaching thermodynamics, then, outside of the basic chemical approach linked to equilibrium, I found the study to be very odd. I have been acquainted with explaining macroscopic observations through microscopic models, so it was hard to think "Macroscopically", even though the mathematics was simpler. So, in approaching general thermodynamics I think it's important to remember what it is thermodynamics is trying to describe, as that is where I lost a conceptual foot-hold in the race (and resorted to math to get me through, as opposed to understanding the concept behind the math)


Literally, thermodynamics is describing the movement of heat. But more is involved than heat: there is also work. So the name isn't exactly the best. What helped me was in emphasizing the macroscopic nature of thermodynamics. It models a large system of particles within some kind of surrounding environment. We are free to define the system, so the system is chosen such that something interesting can be measured or for conveniences sake.



In this case, the system is a beaker with a piston. The little green dots are supposed to be particles of gas floating around inside the beaker. The system stops where the beaker begins, and the surroundings begin just after the system stops. This allows measurements of the gaseous behavior alone to be recorded.

Thermodynamically speaking, there are three quantities that define this system: Pressure, Temperature, and Volume. Of these three, you only need know two to know the third as they are related through the ideal gas law. (Note: There are more "equations of state", as they are called, than the ideal gas law. But it's the simplest and gets the point across)

However, the ideal gas law isn't enough. That just defines the "State" of the system. It doesn't tell us very much about how much energy is transferred from or to the system in going from one state to another. And that, I think, is the best way to think about thermodynamics: the amount of energy transferred in moving from one state to another in a macroscopic system. Macroscopic states can be defined by the three variables of P, V, and T, and the movement between these states requires energy to enter or leave the system. How much energy enters or leaves depends upon the way in which one moves from one state to another.

That leads to the first law of thermodynamics. There are many ways of stating this law, but when trying to understanding how much energy passes into or out of a system due to a process the following is used:

ΔU = W + q

Where "U" is the "Internal Energy", W is work, and q is heat. Internal energy, to me, is a weird concept. It's this energy that's.... there?... inside the system 'n stuff? Yes. That's exactly it. Personally, I'm still wrapping my head around the concept -- the best I can do is to say that it represents every shred of energy that is within the system, from the vibration of bonds, the momentum of molecules, the mass of the atoms, the potentials of fields, EVERY source of energy that happens to be within the system. That.... I think is it. And, frankly, we don't even care about the total internal energy, but rather the changes in internal energy, because those are much easier to measure than absolute internal energies. (ergo: Δ)

So, changes occur in internal energy, and those changes are equal to work and heat. These are the processes by which energy is removed from or added to a system. They both transfer energy, but they do so in different ways. For now it is enough to understand that the changes in internal energy occur through the two processes (or mechanism, or "How-to", if that makes more sense?) of work and heat. What those processes encompass I'll blog about later.

Free Energy

A point I found difficult in studying thermodynamics as a chemist is that the concept of Free Energy is much more important to chemical study -- but it is a more deeply derived concept based upon thermodynamics and energy transfer in general. I often try and think of ways to build a conceptual framework within chemistry without referencing the physics that it is based upon, because in the end, the physics isn't totally necessary for gaining an understanding of the chemical picture. This is why I began blogging this month from the "end-point" of thermodynamics with respect to chemistry: Equilibrium. If you can understand equilibrium in general, then one should be able to understand chemical equilibrium. And, hopefully from this, one should be able to understand Free Energy.

In the previous two posts I attempted to explicate equilibrium as a general model and as a model for chemical systems. This works if you think about atoms as "billiard balls" connected to one another through "Bonds". When a chemical reaction occurs, the bonds in the reactants are broken and the bonds in the products are formed through some kind of process. To deduce an equilibrium expression the step-by-step process does not necessarily need to be known: All that need be known are the ending concentrations of the products and the reactants. The reason that these concentrations are constant is not that the chemicals stop moving due to some mystical equilibrium constant that brings out the golden tablet of concentration stating "Thou shalt not react!" Instead, the chemical species continue to react both in the forward direction (Towards products) and the backward direction (from products to reactants), it is just that at equilibrium these processes occur at the same rate. What those rates are is another story -- but what those concentrations are when the rates are equal is driven by Free Energy.

Free Energy, as a concept, is simple enough to understand from the words alone. It's the amount of energy available to do stuff. The reason why we need this concept is a more difficult issue, and is directly related to the second law of thermodynamics. However, the concept itself can be understood in stating that there is some quantity we call energy, and of this quantity we can not use all of it because of the second law. That quantity which can be used in a process, however, is called Free Energy.

I skip around the second law because I myself found it hard to understand, there are several ways of explaining it, and with reference to chemical thermodynamics I don't know which is the best way to go about explaining it. In fact, I think it unnecessary to understand the second law when first approaching chemical phenomena so long as we conceive that there is this concept that limits the amount of energy that can be obtained from any process, and that concept is the second law of thermodynamics.

The way in which Free Energy applies to equilibrium is through a relation (or equation, expression, what-have-you) of a certain type of Free Energy, that is conveniently defined for standard laboratory conditions. This type of free energy is called "Gibbs Free Energy", because it was invented by Josiah Willard Gibbs. This is the energy available to do work when the system is under constant pressure (or nearly so), such as you find in a laboratory at a given height above the earth. In a chemical reaction taking place in a beaker, the change in free energy can be measured with a simple thermometer. Further, the change in free energy is directly related to the equilibrium constant at a given temperature through:

ΔG = -RT ln (K)

One thing you can notice from this relationship is that if the change in free energy is positive, then raising e to the power of a negative number will give you a number less than one. Similarly, raising e the power of a positive number will give you a number greater than one. This indicates that negative changes in Gibbs Free Energy are indicative of chemical reactions where products are favored more than reactions. The converse of this is also true: Positive changes in Gibbs Free Energy are indicative of chemical reactions that favor reactants.

Naturally, one needs to know how one measures Gibbs Free Energy. The above equation is not what one would call the definition of Gibbs, or give someone a good way of measuring the change in Gibbs free energy, but only its relation to the equilibrium constant. However, I think I'll save that discussion for later. Currently what is more important to grasp is that Free Energy and equilibrium and linked together, and that Free Energy is a thermodynamic concept which is why questions of equilibrium and answered through the concepts of thermodynamics. However, in first understanding chemical reactions, one need not have the grounding principles of thermodynamics down: One need only understand equilibrium as a ratio of products of reactants in a chemical reaction, and that this ratio of equilibrium is governed by the concept of available free energy in a chemical system. That, I think, is the basic beginning to understanding the thermodynamics of chemistry without basing that understanding in the thermodynamics proper.

Equilibrium as a General Model

I think I'm going to have a series of posts on the basics of thermodynamics and its application to chemistry because, well, it's so darn interesting.

In the previous post I outlined some basic concepts of chemical equilibrium. But the case that I gave was very specific and would only apply to a system that operates in a similar manner -- namely, that one molecule of X would combine with one molecule of Y to form one molecule of XY. This is not always the case. But before going into specific chemical equilibrium, I think it better to look at where the model for equilibrium comes from.

A model for physical systems can be constructed from the concept of a system and surroundings. Both are selected by the modeler, usually specifically selected for convenience of calculation, and in that sense are arbitrary. The system is simply what you are interested in. The surroundings includes everything else, but usually only the immediate surroundings are all that are taken into account -- a chemical example would be what is in a beaker for a system, and the lab that the beaker is in for the surroundings.

In designating a system/surroundings, you have some quantities that can describe both: Energy, Pressure, Temperature, Volume, and moles of gas. Any of these quantities can be exchanged between the system and the surroundings, and which quantities can be exchanged often describe the type of system that you are looking at.

A common example from physics is the mechanical equilibrium of a spring represented by the following force diagram.
(Do forgive my Paint abilities). According to Newton's Second Law

ΣF = Fs + Fg
Fs = -kx
Fg = mg
And, because this system is in equilibrium ΣF = 0
Therefore, Fs + Fg = 0, and Fs = -Fg, which means by substitution kx = mg

Which happens to usually be a highly convenient situation. In particular, note that the previous solution had no reference to time. This is something unique to equilibrium solutions: There is no reference to time, only to what each respective variable is at when no variable of interest is changing.

However, when dealing with the system/surroundings model, usually forces aren't the variables of interest. A close analogue. A common example of pressure equilibrium would be a balloon which has been tied off. The gas within the balloon would have a higher pressure than the exterior pressure, but pressure would not be exchanged between the system (the gas in the balloon) and the surroundings (the room the balloon is in).

This situation can occur between any variables of interest -- Pressure, Volume, Temperature, Energy, and chemical concentration. The final one is the one that chemists are first introduced to. This is, ultimately, just the application of thermodynamics/statistical mechanics to chemical systems. Ergo, the study of the model of equilibrium is actually the study of thermodynamics, which is one of the main branches of chemistry. The above explication of equilibrium should also clarify why it is that kinetics, despite being related to energy just as thermodynamics is related to energy, is a separate case of study: The concept of equilibrium requires things to not change with time, and the concept of equilibrium is the model within thermodynamics most often used to model chemical systems as, regardless of the time it takes, the system will tend towards concentrations which satisfy the equilibrium constant.

I think this is done mostly to simplify predictions: thermodynamics, I must admit, is still a bit of an impenetrable thick fog as it is. Not having to worry about time-dependence makes things a little easier.

Equilibrium, Basic Chemical Approach

In a previous post I mentioned that there are two things one must consider in analyzing a chemical reaction: thermodynamics and kinetics. The model of equilibrium covers the first of these.


Equilibrium, in its most basic sense, is the ratio between what is created in a chemical reaction and what was used in a chemical reaction. Suppose the following general chemical reaction:

This reads as "Chemical X and Chemical Y react to yield Chemical XY in equilibrium" Many chemical reactions will include a simpler notation for their "react to yield" symbol such as "→" because most reactions that one is introduced to in a general chemistry course fully react to products. The vast majority of chemical reactions do not have this feature, however, and the double arrow symbol above is used to indicate that both products and reactants are being formed when the system has reached chemical equilibrium.

However, the relative amounts of product and reactant can be predicted through the use of an equilibrium constant. Each chemical system has its own equilibrium constant, but the equilibrium constant remains the same for systems prepared with the same chemicals and under the same surrounding conditions. This constant can be found from the following:

This states that the concentration of chemical XY divided by the product of the concentrations of chemical X and Y equals a constant -- specifically, the equilibrium constant. This means that given a certain reaction and its equilibrium constant, one should be able to predict the concentrations each species will have when equilibrium is reached -- and you can do so. This is a very tidy result because chemists have to model ~10^23 particles all interacting at once. This allows one to predict effects in the physical world while ignoring things such as electric fields, momentum, and position. For example, if one has the equilibrium constant one can be a qualitative prediction about the relative concentrations of products to reactants. If the equilibrium constant is greater than one, then products are highly favored. This is the case with many introductory chemical systems one studies, which is when the symbol → is used since products are so highly favored. However, if the equilibrium constant is smaller than one, then reactants are more favored than products in this particular chemical reaction.

This equilibrium relationship governs the reverse reaction as well: if one where to look at the reaction , then the equilibrium constant for this reaction would be the reciprocal of the previous reaction. This is easily proven if you simply place the concentration of the products of this reaction multiplied together over the concentration of the reactant.


Of course, it also has its own set of limitations. The equilibrium constant is only constant for a given temperature. It can also be somewhat difficult to actually obtain the equilibrium constant, though there are several methods of doing so. Further, as I've previously mentioned, the equilibrium constant says nothing about the kinetics of the reaction: Only the relative energy of the products and reactants, or how favorable the reaction is thermodynamically. This can be important in synthesizing chemicals as two different products could be likely to form, but because of one product is slow to form, the other product is the major chemical created.

Choosing a Method

Every field has a methodology. Adhering to a methodology may seem the best way of retaining objectivity, but last night I found experienced methodologies, arriving at valid conclusions within the reference frame of their method, that conflicted. This was the result of a philosophical discussion after watching a debate on whether or not the resurrection of Jesus is true. Now, the conflict is not irreconcilable on either side -- both sides could explain the other in their own terms -- so there wasn't a want of consistency on either party. But there was a lack of a definitive answer, as is often the case in philosophical questions.

Though this is not, by necessity, a philosophical question. The answer can require a philosophical method, but it does not necessitate it. The Pro side used a historical method. I can't meaningfully comment on the historical method, but I can say that it seeemed the Pro side should be treated as an expert. The argument, to my philosophical knowledge, was valid. The case that his argument failed, and I knew that it failed, was in my knowledge of the scientific method. Resurrection, to the best of our knowledge, is highly unlikely to be possible. The best inference, at this point, is to conclude that Resurrection can not happen. Naturally this doesn't mean that it can't happen. But I would also point out that nothing has brought us closer to knowledge than the method of science. However, in the case of historical details, that remains problematic -- thereby the need for a historical method.

I would place the scientific method above the historical method in its ability to ascertain the truth due to my experience with the method. But, supposing that your world-view allowed resurrection, and you apply the historical method, everything seemed to fall well into place. So, in some sense, the debate becomes an argument of metaphysics -- hardly the appropriate place for truth-determination.

So, in what way does one choose a method? The whole point of creating method is to become as objective as possible. Yet there is a multiplicity of methods. While, normally, the choice of method is obvious (Are you investigating the world? (science) Are you investigating the past? (history) Are you investigating wisdom? (philosophy).), there can come a point where two individuals can come to an impasse on some questions, such as the resurrection example, and both think their method is the most objective way to truth, yet come to different conclusions. I believe that both debaters were making good-faith efforts for their position, as opposed to playing polemics to "win", so I don't think an ego-driven hypothesis resolves the question. Surely anyone interested in proper conclusions can admit that they don't want the ranking of methods to be dictated by how comforting we find their conclusions, or how well they justify everything we already believe -- that would entirely defeat the point of method! 

How do you rank methods, and why?