Chapter 2.

Energy-Interaction Model

Representations

Algebraic Representations

Diagrammatic Representation

Generic Example involving two physical systems, three energy systems, and with Heat Input

Physical system:

Identification of beginning and end of interval:

Definitions of the Constructs of this Model

Energy

The scientific meaning of energy is rather tricky to convey in a sentence or two. There is a good reason for this: energy is an abstract concept that took scientists a long time to figure out and make sense of. Although the concept of energy is truly universal in the sense that energy changes are associated with nearly all interactions and processes, energy is not related to a single property of matter. For example, we all have an intuitive sense of “hotness” and we associate the concept of temperature with this property of matter. We associate the concept of force with the intuitive notion of push and pull. Energy, on the other hand, is associated with many properties or conditions of matter including temperature, force, motion, atomic level, mass, charge, and on and on. It is the fact that energy is so universal that makes it so difficult to define it precisely.

Another reason energy is difficult to pin down, is that the value of energy itself is seldom of importance; rather, it is the changes in the values of energy that seem to matter. In fact, we will see that change in energy is directly related to “how much” interaction occurred.

The following are useful statements about energy. Taken together they constitute a working definition of the construct (or concept of) energy. This concept will be developed further throughout the course.

Statements that help define energy

Energy is an abstract concept that characterizes the interactions of matter.

The change in energy of a physical system is a quantifiable measure of the degree of its interactions with other physical systems.

The importance of always being conscious of the word “change” as in the phrase “change in energy” and of the way this idea is represented mathematically cannot be overstressed. We signify “change in energy” mathematically by writing “∆E.” The uppercase Greek symbol “∆” often is used in science to indicate a change in some quantity. Does “∆E” look like “E” to you? No, of course not. Yet thousands of students have seriously messed up on exam questions because they don’t seem to “see” the difference between the following two expressions:

would make absolutely no sense. The point is, whether we are talking about a change in energy or the value of the energy matters a lot. You must be always conscious of which you are thinking about and why it is the one you want to think about!

Physical Systems

The word system is used in multiple ways in both scientific and non-scientific contexts. For example, we will talk about physical systems, particle systems, as well as energy systems, and doubtless a few other kinds of systems, and will, like other authors, not always include the modifier, but simply refer to “the system.” Make sure you understand how the word “system” is being used in particular discussions. In general, when we say physical system it means exactly what you would think it would mean: some stuff (objects, substances, apparatus, or just a beaker of water) that we wish to focus on. That is, we decide precisely specify what we have decided to include in the particular physical system we are talking about.

Energy Systems

Historically, many “forms” or “kinds” of energy have been identified. Sometimes, the form of energy labels the interaction that resulted in a change of energy, e.g., chemical, mechanical, or nuclear. It is often misleading to think of there being different forms, kinds, or types of energy, even though we have these “names of energy” in our vocabulary. Energy is energy, regardless of how it manifests itself. Thus, instead of speaking of forms of energy or kinds of energy, we speak of energy-systems. There is usually a one-to-one correspondence between an historically identified type or form of energy and what we will identify as an energy-system.

Our language encourages us to call the “thing” we refer to as an “energy system” more than just “an energy.” But it can be misleading to call it by any of the historical names that arose when scientists did not know that they were talking about the same thing. So in this course,we will usually refer to the total energy of a particular physical system as being the sum of the energies in the various energy systems belonging to that physical system. Likewise the change in energy of a particular physical system is the sum of the separate changes that occur in the various energy systems.

Energy is not a real thing that resides in physical systems, even though we often talk about it as if it were a “real thing.” Be careful here. Remember there is nothing at all physical about this “thing” we call energy, even when our language sometimes suggests that there is.

Interactions are modeled by treating energy as something that resides in (can be identified with) particular energy-systems. When interactions occur, energy-systems change. Frequently, it is useful to envision energy as being transferred into and out of particular energy-systems and from one energy-system to another as interactions occur. However, it is the changes in energy-systems (and not the particular transfers among energy-systems) that are the basis of the model.

When an interaction occurs, there is a change in one or more energy systems. For each energy system that changes, there is an observable (in the sense of being detectable) and quantifiable change in some part of the physical system. The magnitude of the change of each energy system is characterized by the change in its associated observable parameter. We will refer to this observable parameter as the indicator associated with a particular energy system. The questions of how to “divide up” the energy of some physical system and which energy-systems to include in the model get to the heart of the modeling process. The particular phenomenon under consideration will exhibit parameters that change as a result of the interaction or process. These parameters frequently turn out to be the indicators of the energy systems that need to be included in the particular model.

The word “energy” is sometimes used in everyday or non-scientific speech to mean something other than what we talking about here. The ideas mentioned in the previous paragraph can help us distinguish between the concept of “scientific” energy that is used in biology, chemistry and physics, from other uses of this same word. Specifically, if we can’t identify a quantifiable and consistent way to measure changes in what we are calling energy system when a process or interaction occurs, then the “thing” or concept that the word energy is applied to is not energy in the scientific sense.

Closed and open physical systems

There are two useful ways to express the principle of conservation of energy, one corresponding to closed physical systems and the other to open physical systems.

Closed physical system

In a closed physical system, there is no transfer of energy into (or out of) that physical system from some other physical system. (This condition would generally preclude mass transfers as well, since mass transfers would also result in energy transfers.) Another way to state the condition of being closed is that all interactions occur within the identified physical system. There are no interactions with other physical systems.

Open physical system

In an open physical system, there can be a net transfer of energy into (or out of) that physical system from some other physical system. Another way to state the condition of being open is that interactions can occur between matter in the identified physical system and matter in other physical systems (which might simply be the environment.

Energy transfers: Heat, Q & Work, W

Even though energy is not a physical thing that can be transferred, it is conventional and sometimes useful to model the process as if transfers of energy actually take place among various physical systems. When energy is transferred into a physical system from another physical system, it is customary to name the energy transferred as either heat, or work. The name heat, Q, is given to energy transfers that occur as result of a difference in temperatures. Energy “flows” from the physical system at the higher temperature to the physical system at the lower temperature. We will use the word “heat” only as a label for this kind of energy transfer . Note, however, that historically, the word “heat” was also used to mean what is now commonly called thermal energy. The notation ∆Q will never be used, since this implies a change in a state quantity, which Q is not. However, the notation dQ will be used to mean an infinitesimally small amount of energy transferred as heat.

When you are reading science journal articles or other textbooks, especially older ones, be sure you understand how the authors are using the word “heat.” This is important, because the concept of energy transfer as heat is very different from the concept of thermal energy. We (and most modern authors) now restrict the use of the word “heat” to its meaning as a transfer of energy between physical systems as a result of temperature differences.

The term work, W, is used to describe the energy transferred between physical systems (objects), which exert forces on one another and move relative to one another. Specifically, work is a transfer of energy into or out of one physical system by a force exerted by another physical system. The change in energy results from an interaction in which an object moves through a distance parallel to the force exerted on it. The physical system containing the object doing the pushing loses energy if the motion is in the direction of the push, while the physical system containing the object that is pushed, increases in energy by the same amount. Like heat, work is a scalar quantity; i.e., it has no direction associated with it.

Where the subscript “parallel” means to multiply only the part of the force and ∆x that are parallel to each other. Heat and work, which are energy transfers, are measured in the same units as is energy. In the SI system of units, joule, J, is the unit of energy, which according to the above relation for W must be equivalent to the product of the SI unit of force and distance. That is, a joule is equivalent to the product of the SI unit of force, newton, N, and the unit of distance, meter, m. That is, a joule, J is equivalent to a newton meter, N m.

One of the great advantages of the energy interaction model is that we don’t have to be concerned with the details of how energy transfers occur. We don’t need to have a microscopic explanation, for example, of how friction causes increases in thermal energy-systems and decreases in other non-thermal energy-systems. On a microscopic scale, all kinds of energy transfers are taking place between individual atoms and molecules. The energy transfers we are talking about that occur between different physical systems are always the net transfers that occur as a result of an interaction between those physical systems.

Processes and Interactions

One of the frequent ways beginning students go astray is to not be perfectly clear in their own mind how the beginning and end of a process is defined or determined. Sometimes, we think of an interaction instead of a process. The same problem arises. What determines the beginning of the interaction and what determines the end of the interaction?

Often there is a point in time that can be identified with the beginning and the end of the interaction. This is not usually an actual clock time, such as 10 minutes after one, but rather, something happens at a particular time. The something that happens can be easily pictured in our mind and remembered. It is when the “ball was let go” or “just before the ball hit the ground.” Or perhaps it was when the coffee pot was turned on and when the coffee pot was turned off. Something physically happens that we make a conscious decision to identify with the beginning of the time interval and something else happens that we use to identify with the end of the time interval that corresponds to the interaction or process we are interested in. It is crucial to always clearly identify these starting and ending “events,” which precisely determine the interval over which the process or interaction occurs. We often use phrases like, “Determine the interval.” to indicate precisely defining the beginning and end of a process.

State of a Physical System

Related to the discussion regarding the beginning and ending of a process or interaction is the idea also mentioned previously that we are not concerned about the details of the interaction in an energy conservation approach. In fact, all we care about is how the state of the system changed from the beginning to the end of the process or interaction. By state of the physical system we mean the values of certain parameters that changed. For the Energy-Interaction Model we care only about the indicators that tell us how much the energy changed in each energy system. This notion of state of the physical system will become more obvious as we work through more and varied phenomena using the Energy Interaction Model.

Energy Systems Related to Thermal and Chemical Processes

When dealing with thermal and chemical processes from a macroscopic perspective, it is convenient and useful to define energy systems that correspond to the empirically determined heats, i.e., the change in enghalpies, ∆H’s, that correspond to the process or interaction. These include heats associated with physical phase changes and with the formation of various molecular species. The indicator for an energy system associated with these processes would be the amount of substance that changed phase or for a chemical reaction, the amount of substance that was formed or that “disappeared” during the interaction.

The general form of the expression for the amount of energy change in these processes will be

The indicator for the energy change is the amount of substance that changed, ∆m. If ∆H, which is typically given as an intensive quantity, has units of J/kg, then ∆m will have units of kg. If ∆H is given as a molar quantity, with units of J/mol, then ∆m will have units of moles.

When there are no phase changes or chemical reactions occurring and heat is added to a substance, its temperature changes with the amount of change dependent on its heat capacity. The energy change associated with this process is exactly equal to the heat, Q, added, since we have stipulated that there are no other energy systems that are changing. From our discussion of heat capacity in the Three-Phase Model of Matter we had

Equating ∆E with Q and rearranging, we have an expression for the change in the energy system associated with this process:

Temperature is the indicator for this energy system.

These two energy systems we have just defined are very useful for two reasons. (1) They correspond to changes in indicators that we can directly observe and (2) depend on parameters, (∆H and C) that are tabulated for most substances. Additionally, as we have defined them, during a physical or chemical reaction, only one of them will be changing at a given time.

Both phase changes and chemical reactions involve the making and breaking of atomic and/or molecular bonds. That is, they are closely related to “bond energy.” When heat is added to a substance and its temperature changes, its “thermal energy” changes. Consequently, these two macroscopic energy systems we have defined above are closely related to bond energy and thermal energy that we will develop from a particle model later. Rather than use a different name for the energy systems involving the empirically determined parameters, ∆H and C, and macroscopic indicators, ∆m and ∆T, we will simply refer to them as the “bond energy system” and “thermal energy system,” remembering that after particle models for the bond and thermal energies are developed, our understanding of these macroscopic energy systems will be deepened and we will see how they are specifically related to the microscopically defined bond and thermal energies using particle models.

So, to summarize, our understanding now of the constructs bond energy system and thermal energy system is:

At a physical phase change, only the bond-energy system changes and the change is given by

Between phase changes, only the thermal-energy system changes and the change is given by

In chemical reactions, there will typically be several bond energy changes, corresponding to each of the molecular species present in either the reactants or products. The energy change for each reactant or product is given by

The temperature is assumed to have the same value after the reaction as it did before the reaction.

These energy systems, defined above, are the appropriate energy systems to use in the Energy-Interaction Model whenever we are using empirically determined values for the various ∆H’s and C’s. The salient features of the phenomenon that we want to model are precisely the ∆H’s and C’s characterizing the particular substance(s) and the indicators ∆m and ∆T. These salient features appear in both the phenomenon and in the particular model we construct. It is this particular model that we then use to create explanations and to make numerical predictions.

The algebraic signs of thermal and bond energies

It is very important to make sense of the algebraic sign of the change in the energy system based on what physically is happening. This is actually very simple to do, once you get the hang of it. However, it is easy to make a simple algebraic slip-up when actually calculating numerical values. You should always check to see if the final algebraic sign makes sense.

Any thermal energy-system for which the temperature increases during the process will always have a positive change in energy. Likewise, any thermal energy system for which the temperature decreases will always have a negative change in energy. This is consistent with the simple notion that thermal energy increases with increases in temperature, because at higher temperatures there is “more vigorous” motion of the particles.

Any bond energy-system for which bonds are broken during the process or interaction will always have a positive change in energy. Likewise, any bond energy-system for which bonds are formed during the process or interaction will always have a negative change in energy. This is consistent with the common experience of having to add energy (and thus increase the bond energy) to vaporize liquid water that is at 100 °C. The bonds that had existed in the liquid phase disappear (are broken) in the process of the liquid changing to a vapor. You will probably need to struggle mentally with this last point: broken bonds have more energy than intact bonds. Work on this until it “seems obvious” to you.

Commonly Used Energy Systems

Useful Groupings of Energy Systems

Mechanical Energy

Sum of kinetic and potential energies associated with the physical “objects” as a whole, not with the internal energies of the objects.

Internal Energy, U

Sum of kinetic and potential energies associated with the individual molecules/atoms comprising a substance, as well as the energies associated with their atomic and nuclear energies.

Meaning of the Model Relationships

Energy System Diagrams

Energy-system diagrams illuminate the types of energy transformations that occur when two or more physical systems interact, or when a constraint is removed and changes occur in two or more energy systems within a single physical system. The diagram helps make clear the physical systems involved, the particular energy-systems involved, and the changes in those energy-systems resulting from the interaction. The initial and final states of the systems should be clearly indicated on the diagram. These diagrams are “before-to-after” diagrams. That is, they indicate the state of the systems before the interaction occurs and the state of the systems after the interaction has occurred. (Note that there is one diagram that includes both the initial and final states. The focus is the change that occurs because of the interaction.) We use energy-system diagrams because they are useful. They help us to systematically apply the energy conservation principle to a particular physical situation using the Energy-Interaction Model.

Drawing Energy-System Diagrams When Modeling the Interaction as a CLOSED System

Listed here are the conventions we adopt in this course.

Drawing Energy-System Diagrams When Modeling the Interaction as an OPEN System

Listed here are the conventions we adopt in this text.

The important difference between open and closed systems is that energy from outside the open physical system can be transferred into or out of the physical system as heat, Q, or work, W. (By definition, these transfers do not occur for a closed system.) The only difference, then, in the energy-system diagrams for the two types, is that the diagram for an open system needs to explicitly show the transfer of Q and/or W. This is done by drawing a dashed oval around all of the energy-systems (to indicate the open physical system boundary) and drawing arrows terminating on the boundary to show a Q or a W transfer.

Generic Example of an Energy System Diagram involving two physical systems, three energy systems, and with Heat Input

Physical system:

Identification of beginning and end of interval:

Comments:

General Process of Constructing an Energy-System Diagram

Listed here are some general questions you need to ask yourself as you use the Energy-Interaction Model. The order suggested is logical, but it is often necessary to cycle back to previous questions. The energy-system diagram is a tool to help you use the Energy-Interaction Model. The energy-system diagram helps you keep track of the many important details you need as you construct the particular model corresponding to the particular physical situation you are interested in.

Units for Energy

The historical development of the energy concept separately as heat and mechanical energy, as well as the widespread use of several different systems of units, has created a multitude of energy units. But, energy is energy and all forms, types, whatever, can and should be expressed in the same basic energy unit. Fortunately, this is now becoming common. The SI unit of energy is the joule, J, (rhymes with cool). All other energy units are related to the joule through an appropriate conversion factor.

A concept closely related to energy is power–the time rate of energy change or energy transfer. The SI unit of power is a joule per second and is given the name watt, W.

Fortunately, essentially everyone within the scientific and technical community has now embraced the International System (SI) system of units. We will generally use SI units in this course. However, in those instances where non-SI units are commonly used, we will use both, and expect you to be able to convert back and forth. An advantage of working exclusively in SI is that you don’t have to be concerned about unit conversions (and keeping track of them).

SI base units

The SI base units for mass, length, and time are kilogram (kg), meter (m), and second (s). Other SI units can be expressed in base units when desired. For example, a joule is a kg m²/s².

SI unit of temperature

The kelvin (K), the SI unit for temperature, is another independent base unit. The zero of the kelvin scale is at thermodynamic zero, the so-called “absolute zero” of temperature. (Note that “kelvin” is used without the word “degree” attached.) Although the zero of the Celsius, or centigrade, scale is not at absolute zero, the kelvin is the same size as the Celsius degree. Thus, when dealing with temperature differences, it is sometimes convenient to use Celsius degrees for ∆T. For example, the SI unit for specific heat, J/kg K is equal to (and will sometimes be written as) J/kg C°.

SI Units related to energy:

Some common energy units and conversions to SI:

1 kWh = 3.6 MJ

1 erg = 10^–7 J

1 cal = 4.184 J

1 food Calorie (big “C” calorie) = 1 kcal = 4.184 kJ

1 ft•lb = 1.36 J

1 eV = 1.602 x 10^–19 J

1 BTU = 778 ft•lb = 252 cal = 1.054 kJ

Videos

Check Your Understanding of the Basics

Question Practice Question 1

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The situation refers to liquid water turning into steam. It is not usually useful to use “liquid water” as our physical system here because some of it “disappears”. On the other hand, what doesn’t disappear is are the water molecules so lets choose all of the water molecules as our physical system so that the system (some of them may make up the liquid and some make up the gas but the total number is constant). The interaction we are interested in is defined for us in the description of the situation. We start with purely liquid water at 100°C and end with purely gaseous water (steam) so those are our endpoints.

Question Practice Question 2

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In the liquid state the H2O molecules are stuck to each other (they are bound to their neighbors by hydrogen bonds). But in gas state the molecules are separated apart so bonds must have been broken in the process of boiling. You have to add energy (bond energy) to break bonds so bond energy must have been added to the water to boil it (i.e. bond energy went up).

Question Practice Question 3

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The situation explains that the initial temperature (T = 100° C) and the final temperature (T = 100° C) are the same. Since our model tells us that the indicator of thermal energy change is the temperature, the constant T tells us that the thermal energy did not change.

Question Practice Question 4

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You should understand at least two ways of thinking about this situation. i) You have probably have common knowledge that you have to add heat to something (e.g. turn on the “burner” on the range) to boil it. That would tell you that the arrow points in. ii) Your physics knowledge tells you that the bond energy went up (bonds were broken) and that the thermal energy did not change (no temperature change). From this you can say that the total internal energy (Eth + Ebond) increased and, according to our energy-interaction model, the total energy of a physical system can only increase if energy was added to it so heat must have been added. To check that we are consistent, note that the “Heat” arrow points inward and Bond E goes up.

Question Practice Question 5

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From previous questions, you have learned that the energy diagram looks something like the picture to the right. The only thing missing is the “conservation of energy” equation relating the energy changes with the energy transfers. The sum of the energy changes equals the energy transferred. ∆Eth +∆Eb=Q Note that ΔEth = 0 and ΔEbond > 0 so Q > 0 which tells us that heat was transferred in. In other words, the change (increase) in the bond energy is exactly equal to the heat added.

Question Practice Question 6

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Molecules are disappearing and new molecules are appearing but the total number and types of atoms doesn’t change so a reasonable physical system could be “all the atoms involved in the interaction”. Note that sunlight is not “matter” and so is not a part of any physical system. For energy conservation situations you will usually choose a time before the relevant interactions as the starting time (i.e. before the sunlight hits the physical system) and an ending time after all the relevant interactions are done.

Question Practice Question 7

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For the purposes of bond energy changes, we can pretend that the reaction proceeds by first breaking up the reactants into atoms and, after that, forming the products from the atoms. Let’s think about the reactants first. Breaking bonds (without forming any new bonds yet) always requires that energy be added so the bond energy of the reactant had to increase (to break them up into atoms). Hence the upward arrow. Now the products. Forming bonds always requires that energy be removed (forming bonds is the opposite of breaking them) so the bond energy of the products must decrease. Hence the downward arrow.

Question Practice Question 8

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The story of this situation doesn’t tell us about temperature. However, thinking about the real situation should convince you that a leaf is not continually heating up and heating up as photosynthesis proceeds within it. Since our model tells us that the indicator of thermal energy change is the temperature, the constant T tells us that the thermal energy did not change (this is probably an approximation but it is very likely to be a good one). Hence, in the diagram to the right no change is shown.

Question Practice Question 9

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Sunlight (energy) absorption makes this reaction possible so we conclude that energy is entering the energy system from the outside (from the sun) so we draw the arrow pointing inward.

Question Practice Question 10

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a) From previous questions, you have learned that the energy diagram looks something like the picture to the right. The only thing missing is the “conservation of energy” equation relating the energy changes with the energy transfers. The sum of the energy changes equals the energy transferred. We leave Eth out because ΔEth = 0 and get: ∆E_"b reactants" +∆E_(b products)=Q Note that Q represents the sunlight energy that was transferred in (so Q > 0). b) Sunlight energy was added so Q > 0 so (∆E_"b reactants" +∆E_"b products" )>0 and we can conclude that the increase in the bond energy of the reactants is larger than the decrease in the bond energy of the products.