There are many ways to view the physical world. In science and engineering, it is often useful to view the world in terms of a concept called energy. Energy is a funny thing because most people have at least some idea of what it means, but when pressed to apply their understanding in some useful way, it seems to vanish right in front of them. There are a number of reasons for this; one is that most people never learned (or do not remember) how scientists and engineers define energy. Beginning with the concept of force, herein I set out to provide a tenacious understanding of energy and why it is a useful perspective for understanding physical systems.

Force is one of those words that most people have some idea of what it means, but may not understand the subtleties of how scientists and engineers use it. Intuition and everyday experience tell us that when we “force” something, we exert our influence in order to cause something to happen. Similarly, when something is “forced,” some external factor causes it to happen even though it normally would not have. Colloquially, when something is “forced,” it tends to imply something actually happened. Scientists and engineers generally use the concept of force to mean only a pushing or pulling, without implying that this push or pull did anything at all.

The presence of forces does not mean that anything actually happens; it merely means that there was some factor exerting its influence. Just knowing the force does not tell you anything about the magnitude of its influence. Here consider if you push as hard as you can against a sturdy brick wall: You exert a force and nothing happens (i.e. the wall does not move or does ever so slightly), so your force has little or no influence. Here it is important to note that by simply considering the force you exerted, we do not get the whole story. We also had to consider the distance the wall moved. We'll see later on that in the case of you versus the wall, there was little or no energy transferred. A critical observer will ask: “If I did not use any energy, then why do I feel tired after I spend all day pushing on a brick wall?” Well, in reality your muscles contracted, so something did happen—there was motion between different parts of your muscle, and it is the energy associated with this motion (transferred to the heat of the atmosphere) that you lost. Read on to find out more about how motion, force, and energy are related.

Energy is the way that scientists and engineers talk about what actually happened. Roughly,

In other words, energy is a measure of the influence of a force on a system. When a force is exerted and its influence causes something to move, we say that energy is transferred (from the thing exerting a force to the thing experiencing the force—the thing that has moved). The transfer of energy is always associated with a movement, and movement is always associated with a distance over which a thing moves. Scientists and engineers have worked hard to come up with a number of clever ways to determine how much a collection of forces will exert its influence—how much energy it will transfer. All of the methods used to calculate energy are based on experiments and theories that relate the magnitude and direction of a force and the distance over which that force makes some object (or group of objects) move. In general, energy is calculated by multiplying a force times the distance over which this force causes something to move. For most cases this is not trivial, and if you are not fluent in the language of mathematics, looking at the relationship between force, distance, and energy written in the Math language, can obfuscate the simple underlying concept: The greater the distance over which something moves and the strength of the force causing it to move, the greater the energy that can be transferred. That is to say, two ways of increasing the transfer of energy are to:

These words are easy to say and (hopefully) fairly straightforward to understand without too much contemplation; what is harder is understanding how to apply them to different systems. How do we use this concept of energy to learn something about the world? Why do we even need the concept of energy? Why can't we simply talk about forces and distances?

These questions are difficult to answer completely without at least some training in science or engineering. The basic idea is that since neither the exertion of forces nor the movement of objects alone imply that anything new has happened, in many situations we need to talk about the energy that can be transfered, which relates a specific force to the distance it can make something move. Saying the same thing differently, in order to glean useful information about the world, scientists and engineers often talk about the influence of a force or the energy it can transfer, before they talk about the specific forces and distances themselves.

One of the reasons scientists and engineers like to talk about the various energies associated with a system is because it makes understanding the influence of various parts of the system more manageable. As a note, here and throughout this document I have used the words: energy, energies, energy transferred, et cetera to all mean the same thing: the energy that can be transferred—or force-induced motion—from some object to some other object (or group of objects). It is common (and ubiquitous) practice to use the shorthand “energy” to refer to the energy that can be transferred. Be confident that you understand what energy really means, and when someone mentions the energy of a system, understand that part of the system must be (or at least have the potential to be) in motion; try to think of the part that is exerting a force and the part that is (or has to potential to be) moving. For example, by considering the thermal energy, the electrostatic energy, and the hydrophobic energy, we can begin to understand how various factors affect a solution of proteins (e.g. milk). Let's look at each of these energies individually and consider if it is easier to think about the forces exherted and distances travelled or the associated energy.

The thermal energy is determined from the sum of the motion of all of the molecules in a substance—typically called the molecular vibrations. Consider a molecule that is moving around. Eventually it bounces into another molecule, and thus exerts a force on this molecule. This force causes the other molecule to move in a way that it would not have otherwise—in the language used above, something happens. In order to get the full story of what happens in terms of forces and distance, we need to consider all of the molecules along with their velocities and positions, as well as all of the instances when a molecule bounces into another molecule. Given that there are approximately 10²⁵ molecules in a liter of water, that's an awful lot of data that we need to process! Considering instead the thermal energy, we simply need to know the number of molecules, the mass of a single molecule, and an average of the velocity at which the molecules travel. Understanding the reasons for this requires some concepts from statistical mechanics and a discussion of the concept of kinetic energy, which is outside the scope of this document; there are many resources available on the Internet, in libraries, et cetera to aid your understanding. By considering the thermal energy rather than the thermal forces and movements, we have gone from an truly unmanageable set of data required to understand a system to only having to know three numbers! If fact, scientists have gone even further and done experiments to relate an average velocity of the molecules to the temperature; this allows us to determine the thermal energy of an object simply from knowing the number of molecules in the object and the temperature of the object. This is the basis of our relationship between temperature and thermal energy via the Boltzmann constant. In the Math language, we write the equation: U_Thermal=k_BT where U_Thermal is the thermal energy, k_B is the Boltzmann constant, and T is the temperature. Depending on the units used for U_Thermal, k_B, and T, the equation is sometimes written as: U_Thermal=NkBT where N stands for the number of molecules and is typically given in the units of moles. Remember that the symbol used for the units of moles is mol and that one mole of an object (or 1 mol) contains (or is equal to) approximately 6.022 × 10²³ objects. As an example, consider your homework from week one: The interaction energies were given in the units kilojoules per mole of bonds (which is typically given the symbol: kJ/mol, but should be given by the symbol: kJ/mol of bonds). In order to determine the interaction energy—which was given the symbol: U_Interaction—of a single bond, you had to multiply U_Interaction by the number of bonds you were interested in: One. So you multiplied U_Interaction by the number of bonds. “But Mike... I don't remember multiplying by the number 'one'.” That's because you didn't! (well... you technically did, but not in the way you may think; more on this soon) You multiplied by the number of bonds: “one bond!” The units you used for “number of bonds” were moles, where one bond is approximately 1.66 × 10^-24 mol of bonds. In practice, you multiplied U_Interaction by one bond divided by Avogadro's number of moles of bonds or more precisely: 1 bond/6.022 × 10²³ mol of bonds. To say this differently and be very precise, you used the conversion between “number of bonds” and “number of moles of bonds” to convert the units of U_Interaction from the units of kilojoules per mole of bonds to the units of kilojoules per bond. In symbols, you converted from kJ/mol of bonds to kJ/bond using the conversion factor:
1 bond/6.022 × 10²³ mol of bonds, which is equal to 1.66 × 10^-24 bonds/ mol of bonds (Notice that because the numerator and denominator are equal, this—and every—conversion factor is actually equal to the number one. Every time we convert units, we are simply multiplying by various factors of one. This is why when we convert units, we are not fundamentally changing the numbers in our equation; we are only looking at them with a different measuring stick. Everything else about the problem is exactly the same.) Now it should be clear that it is appropriate to more precisely write the equation from week one as: U_Interaction=Nk_BT_{Bond breaking}. Moreover, you should have a better understanding of how monumental such a simple equation really is: In five symbols we are able to communicate the macroscopically observable perception of temperature to the movement and interactions of an unfathomable number of molecules bouncing into each other at a slightly less unfathomable length scale! Since one of these symbols is a relational operator (the equals sign, =), and one of the symbols stands for a constant number (the Boltzmann constant, k_B), we only need to know two numbers to figure out the third! That is, we only need two numbers to say something intelligent about what will happen on the nanometer scale! And one of these is simply one bond, and the other is a ridiculously easy-to-observe quantity: temperature! In week one you were given the interaction energy and the number of bonds and you were able to calculate the temperature at which a certain type of bond would break. Had we told you just a little more, you could even have calculated an average of the velocity for a molecule at a certain temperature! This is the immense power of the energy perspective. It's no wonder why we clapped for our equations of the week! (it's ok if you clap now... I won't tell anyone :])