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3.4 Particle Model of Bond Energy

New Construct Definitions

 

Internal energy and Mechanical energy

        The internal energy, U, is the energy associated with all the kinetic and potential energies of the particles constituting a substance. This will include the energies associated with the formation of the various phases as well as the energies internal to the particles themselves, such as the molecular, atomic and nuclear energies. Mechanical energy refers to the potential and kinetic energies associated with the motion of objects as a whole. Thus, it is often the case that the mechanical energy of an object can be small or zero, yet the internal energy can be quite high. For example even a baseball thrown at 90 miles/hour has much more thermal energy at room temperature than kinetic energy due to its being thrown.

 

Thermal energy

        Thermal energy is the sum of the potential and kinetic energies that are associated with the disordered motions of the particles that make up an object. We will significantly expand our understanding of this construct in Section 3-5: Particle Model of Thermal Energy.

 

Bond energy and Binding energy

        In solid and liquid phases there is a bond energy associated with the attractive part of all the pair-wise potential energies acting between pairs of particles. By convention, the binding energy is the positive energy that must be added to separate the particles sufficiently far apart so that the bond energy has the value zero. Since the maximum value of the bond energy occurs when the particles are widely separated, and because of the way the pair-wise potential is defined, the bond energy of liquids and solids must be less than zero; that is, the bond energy is negative. Binding energy is simply the magnitude of the bond energy and is always a positive number, even though the bond energy is negative.

 

Nearest neighbor (n-n) pair

        In a solid or liquid, each atom or molecule will “be almost touching” about 12 other atoms or molecules. It is exactly 12 for many substances, if the atoms or molecules are spherically shaped. You will get a chance in the discussion/lab activities to check on this. It is these 12 or so atoms that form nearest neighbor pair-wise bonds with a particular atom or molecule. These are all at nearly the “right” distance apart so that the PE of each pair is very close to its minimum value.

 

Non-nearest neighbor pairs

        Do we need to worry about interactions between atoms or molecules that are not nearest neighbors? A little bit, depending on how accurate we want our numerical predictions to be. Look back at the pair-wise potential energy curve. Has the slope gone totally horizontal when the particles are located two diameters from each other. No, not totally. There are a lot of nearby neighbors that are within two diameters of each other, so these non-nearest neighbors will still be attracting each other a little bit and will make a contribution to the binding energy, but typically significantly less than the nearest neighbors.

 

Empirically determined values

        The concepts of bond energy and thermal energy are very useful in models that help us make sense of the particulate nature of matter. However, the quantities that are actually measured, although closely related to these ideas, are not quite the same. That is, the ∆H’s we encountered in Chapter 1 and the bond-energy systems we used there based on these ∆H’s are not precisely the same as the bond energy defined in terms of the pair-wise interactions. However, it is rather tricky to understand precisely how they are related. When we proceed through Chapter 4 on thermodynamics, it will be possible to sort much of this out. Until then, we will accept that when making comparisons of the concepts in our models to empirical data, we are making some approximations, which will always be pointed out. These approximations typically allow us to still make numerical comparisons to within 10 to 20 percent of the best we can do with extremely complicated models. From a modeling perspective, this is initially a price well worth paying in order to have a model sufficiently simple and broadly applicable to enable us to develop a meaningful understanding of a great deal of the “how and why” matter behaves the way it does from a particulate perspective. The models we develop in this chapter apply, in the sense that they allow us to make sense of phenomena and get pretty close when making numerical predictions, to a very wide range of phenomena without getting bogged down in so many details that we never get anywhere in our understanding. Thermodynamics is the “science” of understanding the subtleties and the details of precisely determined empirical data. In Chapter 4 we will get a brief introduction and a taste of the power it provides, but at a cost of the loss of the simplicity of the models in Chapter 3.

 

Meaning of the Model Relationships

 

1)   From a macroscopic perspective, the total internal energy of a substance, excluding nuclear and atomic energies, is comprised of thermal energy and bond energy. (Einternal = Ethermal + Ebond) Excluding atomic and nuclear energies, the bond energy is often referred to as the “chemical energy,” because it is the changes in this part of the internal energy that result from changes in chemical bonds.

        This relationship emphasizes that we are often only interested in changes in energies when using the Energy-Interaction Model, so we don’t usually care what the absolute values of the internal energy actually are.

2)   If all of the particles of a substance were sitting at rest at their equilibrium positions, the magnitude of the bond energy would be the amount of energy that would have to be added to completely separate all of the particles, still at rest. This positive quantity is customarily referred to as “binding energy.”

        This concept can be applied to phase changes of substances without causing chemical changes as well as to the energy required to separate a particular molecular species into separate atoms. It also is applied in exactly the same way to changes in nuclear processes.

        There is one tricky aspect associated with directly relating bond energies to heats involved in phase changes. It is that there can be, as we shall see in our Particle Model of Thermal Energy, changes in the thermal energy at a phase change as well as in the bond energy. The empirically determined ∆H’s that we used in the bond energy system in Chapter 1, however, do incorporate any changes of energy in thermal energy at a phase change. Thus, the ∆H’s are not precisely a measure of the particle model bond energy change. For the most part we will ignore this until we have sufficient background to make sense of it. There are also several other rather subtle effects that we will ignore until we are ready to make sense of them in Chapter 4.

        It is important to understand that “this is how science works.” And it is certainly the way we begin to learn science! We create models that are sufficiently simple to make a start at making sense of the phenomena, and then the discrepancies with empirically determined data allow us to refine the models (as well as making them a lot more complicated) to whatever degree we need to answer the questions we are interested in answering. In the beginning phases of making sense of phenomena, when doing science and when learning science, simple and more broadly applicable models are almost always the better way to begin.

3)   In terms of particle potential energies, the bond energy of a substance is the sum of all of the pair-wise potential energies of the particles comprising the substance calculated when all of the particles are at their equilibrium positions corresponding to a particular physical and chemical state. In molecular substances there will be both inter- and intra-molecular contributions to the bond energy.   

        This definition of bond energy avoids the issue of the thermal energy possibly changing, because the calculation is carried out at essentially zero kelvin (because all particles are in their equilibrium positions as they would be at absolute zero, if the phase actually existed at absolute zero) in both the bound state as well as when the particles are separated. We take this to be our technical definition of bond energy. An equivalent definition would be to say that the energy required to separate the particles is carried out so that the thermal energy is the same after the separation as before the separation.

 

4)   By convention, all pair-wise potentials are defined to be zero when the particles are separated sufficiently so that the force acting between the particles is zero. Therefore, the bond energy of any condensed substance is always negative. The maximum value of the bond energy is zero when the particles that comprised the substance are all completely separated to large distances.

        This is sometimes hard to get our minds around. For example, think of the oxidation of hydrogen to form water. When oxygen atoms are far away form the hydrogen atoms, the bond energy of two hydrogen atoms and one oxygen atom have their maximum value, which is zero. As they move close to one another “and bond,” their bond energy becomes some negative number. It seems like we might be saying they were bound when they were far apart, because that is when they had their greatest bond energy. No, they were not bonded. There were no chemical or any other kind of bonds when the atoms are greatly separated. It is just a lot more sensible to measure bond energies this way. It takes some getting used to, however.

5)  For molecular substances that don’t disassociate, the total number of n-n pairs times the well-depth of the pair-wise potential energy (e) between molecules is a rough approximation for the sum of all intermolecular pair-wise potential energies of a substance. This approximation, however, will tend to underestimate the binding energy. The underestimation comes in because we have not added in the contributions of the many neighbor pairs that are in the one to two diameter separation range.

6)   The empirically determined heats of melting and heats of vaporization are reasonable approximations to the changes in bond energy at the respective physical phase changes.

        But see comments following relationship (2).

7)   The empirically determined heats of formation of various chemical species can be used to calculate changes in bond energy when chemical reactions occur.

        Chemists use a very useful system to enable these calculations to be easily carried out. It involves carefully defining the “starting state” of the elements and compounds. It is something you must understand precisely, but when you do, it is an extremely powerful method. This works, because what we are interested in is changes in bond energy. Where the zero of energy is assigned to be for each and every distinct element or substance, doesn’t matter, as long as everyone agrees on the assignment and sticks with it.

Algebraic Representations

The three approximate relationships mentioned in the numbered relationships above can be expressed algebraically. For reference purposes, we list them here.

 

Relationship (3)     \( E_{bond} = \Sigma_{all-pairs} (PE_{pair-wise}) \)      (calculated with all particles at their equilibrium positions)

 

Relationship (5)         Intermolecular  \( E_{bond} \approx  – (\text{total number of pairs})  \times \epsilon \)

 

Relationship (6)           \( \left |{\Delta E_{bond}}  \right |\approx \left |{\Delta H \Delta m}  \right |_{\text{at a phase change}} \)