3.5: Chemical Potential Energy

Figure 3-13. The combustion of propane in a camping stove releases CO2 and water and heat - lots and lots of heat. But if energy can't be created, what form was the heat in prior to burning the fuel? ("Circle of Flame" by carefulweb is licensed under CC BY-SA 2.0.)

 

 

Let’s shift our attention to a very different sort of energetic system: combustion. We mentioned in Chapter 1 that alkanes, at least the gaseous and liquid members of that family, are highly flammable compounds. They are outstanding fuels, releasing large amounts of heat per unit mass burned. Using propane as an example, the fuel often used by backyard barbecue enthusiasts, we can describe its combustion with the following balanced equation:

\[ \ce{C3H8 (g) + 5 O2 (g)  -> 4 H2O (g) + 3 CO2 (g)} \]

This is typical of hydrocarbon combustion reactions in that it generates water and carbon dioxide as products. But these compounds are really just nuisance byproducts (with the CO₂ being a particularly bad nuisance) — it is the heat released that is really desired from this reaction: people burn propane for the sole reason of using the resulting heat to keep their homes warm in the winter, or to boil water on a cold camping trip to take the chill off a cold morning. The First Law tells us that energy is not created during chemical reactions, so the obvious question becomes: in what form is that energy, that which ends up as precious heat, before the combustion takes place? An unilluminating answer, at least presently, is “chemical potential energy”.

What is chemical potential energy? This is not a simple question. It can’t be recognized as easily as the potential energy in water behind a dam or a boulder teetering ominously on a hillside. We can glean a hint of the essential concept if we step back and consider why atoms bond together to form molecules in the first place. Let’s take a structured approach to the problem. Consider the following: hydrogen exists as diatomic molecules H₂ while helium exists as discrete atoms; no He₂ molecules are present in a sample of pure helium gas (such as that found in birthday party balloons), nor do isolated hydrogen atoms exist (to any measurable extent) in pure elemental hydrogen. Why? Using the above concepts about the directionality of physical changes, we can infer that hydrogen atoms react with each other to form H₂ molecules because this leads to a decrease in potential energy, while an analogous reaction among helium atoms results in no such decrease. This is absolutely correct. But why? 

We’ll examine the hydrogen case first. As we already described, an individual hydrogen atom consists of a nucleus containing only one proton (or a proton and a neutron) that is surrounded by the vaguely-described “electron cloud”, composed of only one electron that moves very quickly and somewhat randomly in the surrounding space. Now, imagine two hydrogen atoms that are initially far apart from each other but move closer together (Figure 3-14). As the two atoms approach each other (that is, you are approaching the origin from the right in the figure), the negatively charged electron clouds of each begin to distort under the influence of the positively-charged nucleus of the other. The clouds become elongated, “reaching out” to the attractive positive charge on the other atom. This is an instance of oppositely charged particles getting closer together and thereby decreasing their potential energy. As they do this, they "fall" into the potential energy well, and as they get closer together they go deeper into the well. But there comes a point at which to move any closer would be destabilizing and the potential energy would start rising. This is because when the nuclei get too close to each other they repel each other because of their like charges. Decreasing the distance between them at that point would require work, the atoms would literally have to be forced together. Thus the “energy well” in this case is shaped more like an actual physical well than that seen in Figure 3-11, with a clearly discernible minimum that you can imagine an object rattling around in.

 

Figure 3-14: Potential energy of two neutral hydrogen atoms as a function of the distance between them; the minimum energy and stable bond distance is at the point where the attractive forces between the electron clouds and nuclei balances the repulsive forces between the two nuclei. This graph illustrates the bond strength and bond length for the H₂ molecule, but bonds between atoms of other elements will have different lengths and strengths. 

 

This “unit” of two so-joined hydrogen atoms is the hydrogen molecule, H₂, and the interactions between the electrons and the two nuclei that hold the atoms in close proximity to one another is a covalent chemical bond [13]. In the H₂ molecule, the two electrons are, on average, equidistant from both nuclei; they have lower potential energy than they did in the well-separated atoms because they are in close proximity to two nuclei instead of just one, as they were initially. Figure 3-14 gives us two important quantities that are useful in the characterization of chemical bonds: bond length and bond strength, defined below:

• Bond length is defined as the average distance between two nuclei joined by a covalent bond in a molecule. It corresponds to the distance at which the potential energy is minimized. Just like a ping-pong ball may rattle around in a hole if it is agitated a bit, the atoms joined by a covalent bond will often oscillate, as if connected by a spring, constantly expanding and compressing around an average value. For H₂ that average is 74 pm (pm is the picometer, or 1 × 10^-12 m), the value of that aligns with the bottom of the energy well in the figure. In comparison, the O-H bond of water is 97 pm. Why is this significant? Bond lengths are often used to describe the structure of molecules and can give clues about exactly how the atoms are interacting, or, in other words, how they are bonded together. When bond lengths give such insights we will present them and explain the underlying logic.
• Bond strength is defined as the energy required to separate the two atoms involved; this corresponds to the energy gap between the bottom of the energy well and the potential energy at infinite separation - think of it as the energy necessary to climb out of the energy well up to the plateau on the right side of in Figure 3-14 . So, how much energy does it take to climb out of the well? We'll answer that question, but first we need to define some common units of energy and do so in the box below.

 

Quantifying Energy

We started this chapter with an off-hand comment that we would need to discuss energy in quantitative terms but, since then, we have assiduously avoided doing so. We are at a point now, however, where putting units on energy is necessary. As is true with length and mass, there are many different units of energy that are used in various countries and contexts. Perhaps the most familiar in the United States is the calorie, which is the amount of energy (in the form of heat) needed to increase the temperature of 1 gram of water 1 degree Celsius. In our day-to-day lives, this is a relatively small amount of energy. An average adult human consumes roughly 2 million such calories daily. [14] In this text we will use the joule (J) as the basic unit of energy as it is preferred by the Internal System of Units [15].

What is a joule? It is about one quarter of a calorie, thus it is a small amount of energy for activities on a human scale. Roughly speaking, it is the amount of energy needed to raise an object with a mass of 100 grams one meter [16]. Alternatively, in terms of heat, it is the amount of heat necessary to increase the temperature of 4 mL of water about 1 degree Celsius. To compare this to something that might be more familiar or tangible: the amount of heat released by the combustion of one gallon of gasoline is roughly one hundred million joules (100,000,000 J).  On the molecular level, however, one joule is a tremendous amount of energy. For example, the energy released by burning that one gallon of gasoline is the result of a tremendous number of involving individual molecules of octane and oxygen. One gallon of gasoline consists of about 23 moles of octane (C₈H₁₈), and recall that a mole is 6.02 × 10²³ molecules. On a per molecule basis, therefore, the reaction of each octane molecule with oxygen releases about 9 x 10^-18 J, or 9 aJ (where aJ is the abbreviation for one attojoule, equal to 1 x 10^-18 joules).

Getting back to Figure 3-14, we can now describe the depth of the energy well in quantitative terms. It is 0.72 aJ. It is given as a positive value because it expresses the energy required to break the bond, not the energy released upon its formation. Think of it as being similar to the energy required to pull two magnets apart. In those terms, the logic where the value of zero on the vertical axis was in set n Figure 3-14 becomes clearer. Specifically, we define the potential energy of the isolated atoms as zero, so the depth of the well corresponds exactly with the bond energy. Because this value — 0.72 aJ —  is so small as to lose much intuitive value, we describe the energy changes associated with reactions on a molar basis rather than for individual molecules. Bond energies are therefore usually reported with units of kJ/mol, or thousand of joules per mole of bonds. In this case of H₂, the bond strength is usually expressed as 436 kJ/mol. This means that it requires 436 kJ to convert one mole of H₂ molecules to two moles of isolated hydrogen atoms according to the equation below.

\[ \ce{H2 (g)   -> 2 H (g) } \]

The example of the formation of the H-H bond from two separate hydrogen atoms exemplifies an important general principle: chemical bonds exist because their formation decreases the potential energy of the atoms involved. To look at it another way, if the formation of a bond did not decrease the potential energy of the component atoms, it would not form in the first place, which is exactly the case with helium atoms. Helium atoms do not form chemical bonds to each other because doing so would not result in a decrease of their collective potential energy. Unfortunately it is not at all obvious, based on the arguments presented above, why this should be. Specifically, if two helium atoms approached one another, wouldn’t the electron clouds of each atom be attracted to the nucleus of the atom approaching them? Simple electrostatics would indicate, yes, of course they would. So, either other effects must also be at play, or the relative magnitudes of the attractive and repulsive forces must not be the same for helium as they were for hydrogen, or both. It’s both. 

In the case of hydrogen, the destabilizing force of nuclear repulsion is outweighed by the attraction between the shared electron pair and the two protons in the nuclei, provided the nuclei do not get too close. This is never true in the case of helium: the repulsive forces always outweigh the attractive ones, so helium atoms push each other away if they get too close. One reason for this is the fact that each helium nucleus has two protons and according to the laws of electrostatics, this increases the repulsive forces between the nuclei by a factor of four, other things being equal. You could speculate that the additional electrons, each helium has two of them, should help offset this effect because there would be more attractive electron/proton interactions. But this is not as large a factor as one might surmise because with additional electrons comes more electron-electron repulsion; to put it crudely, you can’t cram four electrons where they would need to go to overcome the nuclear repulsion without destabilizing the molecule with additional repulsive forces. This is, without a doubt, an oversimplified treatment and its hand-wavy nature makes it difficult to see how it could be applied more generally; a good bonding model would be more structured and allow for predictions that could be tested. We will introduce such bonding models later. For now, however, we want to emphasize the following point: the existence of a chemical bond is an indication that the atoms so connected have a lower potential energy, that is, are more stable, than they would be if they were to be separated.

In short, molecules exist because they stabilize the atoms involved.

 

 

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Notes and References.

[13] As distinct from an ionic bond as described in Chapter 4.

[14]Who consumes 2,000,000 calories a day? Chances are, you do! In a rather unfortunate evolution of the language, dietitians decided to use the Calorie as the unit of energy contained in food (note the upper-case "C" in Calorie). By definition 1 Calorie = 1000 calories (note the lower case "c" in calorie), where a calorie has the definition provided in the text, i.e., the amount of energy required to raise the temperature of 1 gram of water 1 degree Celsius. So, a diet that provides 2,000 Calories per day can be said to provide 2,000,000 calories per day. Nothing confusing about that. 

[15] The International System of Measurement is essentially the modern form of the metric system. Established and maintained by the General Conference on Weights and Measures, most scientific work uses the units recommended by this body, referred to as SI units. These include the meter (m), second (s), kilogram (kg), mole (mol), as well as units for frequency, electric current and light intensity among others.

[16]This comes from the physical relationships between force and energy. You may know the equation, \[F = ma \] or force (F) equal mass (m) times acceleration (a). The SI unit of force is the newton (named after Sir Isaac) and has units of kg m s^-2 (kilograms meters per second squared). Work, which has the same units as energy, is defined as follows: \[w = Fd \] or work equals force times distance. Putting them together gives \[w = mad \] In terms of units, you can see that the joule is derived from the kilogram, meter and second as follows:

\[ 1\ joule = \frac {1 kg\ m^2}{s^2}  \]

On Earth, the acceleration due to gravity is roughly equal to 10 m/sec², so if we use a mass of 0.1 kg, the product mad equal 1. A typical apple has a mass of about 100 g (0.1 kg), so if you imagine Sir Isaac Newton lifting the apple that famously hit him on the head 1 meter, he will have done roughly one joule of work. (Thanks to David McAvity, professor of physics and mathematics at The Evergreen State College, for pointing out the Newton/apple/joule connection).