We begin our study of the energetics of chemical reactions with our understanding of mass relationships, determined by the stoichiometry of balanced reactions and the relative atomic masses of the elements. We will assume a conceptual understanding of energy based on the physics of mechanics, and in particular, we will assume the law of conservation of energy. In developing a molecular understanding of the reaction energetics, we will further assume our understanding of chemical bonding via valence shell electron pair sharing and molecular orbital theory.

The heat released or consumed in a chemical reaction is typically amongst the most easily observed and most readily appreciated consequences of the reaction. Many chemical reactions are performed routinely specifically for the purpose of utilizing the heat released by the reaction.

We are interested here in an understanding of the energetics of chemical reactions. Specifically, we wish to know what factors determine whether heat is absorbed or released during a chemical reaction. With that knowledge, we seek to quantify and predict the amount of heat anticipated in a chemical reaction. We expect to find that the quantity of heat absorbed or released during a reaction is related to the bonding of the molecules involved in the reaction.

Prior to answering these questions, we must first answer a few questions regarding the nature of heat. Despite our common familiarity with heat (particularly in Houston), the concept of heat is somewhat elusive to define. We recognize heat as "whatever it is that makes things hot," but this definition is too imprecise to permit measurement or any other conceptual progress. Exactly how do we define and measure heat?

We can define in a variety of ways a temperature scale which permits quantitative measurement of "how hot" an object is. Such scales are typically based on the expansion and contraction of materials, particularly of liquid mercury, or on variation of resistance in wires or thermocouples. Using such scales, we can easily show that heating an object causes its temperature to rise.

It is important, however, to distinguish between heat and temperature. These two concepts are not one and the same. To illustrate the difference, we begin by measuring the temperature rise produced by a given amount of heat, focusing on the temperature rise in 1000g of water produced by burning 1.0g of methane gas. We discover by performing this experiment repeatedly that the temperature of this quantity of water always rises by exactly 13.3°C. Therefore, the same quantity of heat must always be produced by reaction of this quantity of methane.

If we burn 1.0g of methane to heat 500g of water instead, we observe a temperature rise of 26.6°C. If we burn 1.0g of methane to heat 1000g of iron, we observe a temperature rise of 123°C. Therefore, the temperature rise observed is a function of the quantity of material heated as well as the nature of the material heated. Consequently, 13.3°C is not an appropriate measure of this quantity of heat, since we cannot say that the burning of 1.0g of methane "produces 13.3°C of heat." Such a statement is clearly revealed to be nonsense, so the concepts of temperature and heat must be kept distinct.

Our observations do reveal that we can relate the temperature rise produced in a substance to a fixed quantity of heat, provided that we specify the type and amount of the substance. Therefore, we define a property for each substance, called the heat capacity, which relates the temperature rise to the quantity of heat absorbed. We define qq to be the quantity of heat, and Δ⁢TΔT to be the temperature rise produced by this heat. The heat capacity CC is defined by

This equation, however, is only a definition and does not help us calculate either qq or CC, since we know neither one.

Next, however, we observe that we can also elevate the temperature of a substance mechanically, that is, by doing work on it. As simple examples, we can warm water by stirring it, or warm metal by rubbing or scraping it. (As an historical note, these observations were crucial in establishing that heat is equivalent to work in its effect on matter, demonstrating that heat is therefore a form of energy.) Although it is difficult to do, we can measure the amount of work required to elevate the temperature of 1g of water by 1°C. We find that the amount of work required is invariably equal to 4.184J. Consequently, adding 4.184J of energy to 1g of water must elevate the energy of the water molecules by an amount measured by 1°C. By conservation of energy, the energy of the water molecules does not depend on how that energy was acquired. Therefore, the increase in energy measured by a 1°C temperature increase is the same regardless of whether the water was heated or stirred. As such, 4.184J must also be the amount of energy added to the water molecules when they are heated by 1°C rather than stirred. We have therefore effectively measured the heat qq required to elevate the temperature of 1g of water by 1°C. Referring back to Equation 1, we now can calculate that the heat capacity of 1g of water must be 4.184⁢J°C4.184J°C. The heat capacity per gram of a substance is referred to as the specific heat of the substance, usually indicated by the symbol cscs. The specific heat of water is 4.184⁢J°C4.184J°C.

Determining the heat capacity (or specific heat) of water is an extremely important measurement for two reasons. First, from the heat capacity of water we can determine the heat capacity of any other substance very simply. Imagine taking a hot 5.0g iron weight at 100°C and placing it in 10.0g of water at 25°C. We know from experience that the iron bar will be cooled and the water will be heated until both have achieved the same temperature. This is an easy experiment to perform, and we find that the final temperature of the iron and water is 28.8°C. Clearly, the temperature of the water has been raised by 3.8°C. From Equation 1 and the specific heat of water, we can calculate that the water must have absorbed an amount of heat q=(10.0⁢g)⁢((4.184⁢Jg⁢°C))⁢(3.8⁢°C)=159⁢Jq 10.0g 4.184Jg°C 3.8°C 159J. By conservation of energy, this must be the amount of heat lost by the 1g iron weight, whose temperature was lowered by 71.2°C. Again referring to Equation 1, we can calculate the specific heat of the iron bar to be cs=-159⁢J(-71.2⁢°C)⁢(5.0⁢g)=0.45⁢Jg⁢°Ccs -159J -71.2°C 5.0g 0.45Jg°C. Following this procedure, we can easily produce extensive tables of heat capacities for many substances.

Second, and perhaps more importantly for our purposes, we can use the known specific heat of water to measure the heat released in any chemical reaction. To analyze a previous example, we observed that the combustion of 1.0g of methane gas released sufficient heat to increase the temperature of 1000g of water by 13.3°C. The heat capacity of 1000g of water must be (1000⁢g)⁢(4.184⁢Jg⁢°C)=4184⁢J°C 1000g 4.184Jg°C 4184J°C. Therefore, by Equation 1, elevating the temperature of 1000g of water by 13.3°C must require 55,650⁢J=55.65⁢kJ55,650J55.65kJ of heat. Therefore, burning 1.0g of methane gas produces exactly 55.65kJ of heat.

The method of measuring reaction energies by capturing the heat evolved in a water bath and measuring the temperature rise produced in that water bath is called calorimetry. This method is dependent on the equivalence of heat and work as transfers of energy, and on the law of conservation of energy. Following this procedure, we can straightforwardly measure the heat released or absorbed in any easily performed chemical reaction. For reactions which are difficult to initiate or which occur only under restricted conditions or which are exceedingly slow, we will require alternative methods.

Hydrogen gas, which is of potential interest nationally as a clean fuel, can be generated by the reaction of carbon (coal) and water:

Calorimetry reveals that this reaction requires the input of 90.1kJ of heat for every mole of C(s)C(s) consumed. By convention, when heat is absorbed during a reaction, we consider the quantity of heat to be a positive number: in chemical terms, q>0q0 for an endothermic reaction. When heat is evolved, the reaction is exothermic and q<0q0 by convention.

It is interesting to ask where this input energy goes when the reaction occurs. One way to answer this question is to consider the fact that the reaction converts one fuel, C(s)C(s), into another, H2(g)H2(g). To compare the energy available in each fuel, we can measure the heat evolved in the combustion of each fuel with one mole of oxygen gas. We observe that

produces 393.5kJ for one mole of carbon burned; hence q=-393.5⁢kJq-393.5kJ. The reaction

produces 483.6kJ for two moles of hydrogen gas burned, so q=-483.6⁢kJq-483.6kJ. It is evident that more energy is available from combustion of the hydrogen fuel than from combustion of the carbon fuel, so it is not surprising that conversion of the carbon fuel to hydrogen fuel requires the input of energy.

Of considerable importance is the observation that the heat input in Equation 2, 90.1kJ, is exactly equal to the difference between the heat evolved, -393.5kJ, in the combustion of carbon and the heat evolved, -483.6kJ, in the combustion of hydrogen. This is not a coincidence: if we take the combustion of carbon and add to it the reverse of the combustion of hydrogen, we get C(s)+O2(g)→CO2(g)C(s)+O2(g)→CO2(g) 2H2O(g)→2H2(g)+O2(g)2H2O(g)→2H2(g)+O2(g)

Canceling the O2(g)O2(g) from both sides, since it is net neither a reactant nor product, Equation 5 is equivalent to Equation 2. Thus, taking the combustion of carbon and "subtracting" the combustion of hydrogen (or more accurately, adding the reverse of the combustion of hydrogen) yields Equation 2. And, the heat of the combustion of carbon minus the heat of the combustion of hydrogen equals the heat of Equation 2.

By studying many chemical reactions in this way, we discover that this result, known as Hess' Law, is general.

(Although we have not considered the restriction, applicability of this law requires that all reactions considered proceed under similar conditions: we will consider all reactions to occur at constant pressure.)

A pictorial view of Hess' Law as applied to the heat of Equation 2 is illustrative. In Figure 1, the reactants C(s)+2H2O(g)C(s)+2H2O(g) are placed together in a box, representing the state of the materials involved in the reaction prior to the reaction. The products CO2(g)+2H2(g)CO2(g)+2H2(g) are placed together in a second box representing the state of the materials involved after the reaction. The reaction arrow connecting these boxes is labeled with the heat of this reaction. Now we take these same materials and place them in a third box containing C(s)C(s), O2(g)O2(g), and 2H2(g)2H2(g). This box is connected to the reactant and product boxes with reaction arrows, labeled by the heats of reaction in Equation 3 and Equation 4.

Figure 1

A Pictorial View of Hess' Law

This picture of Hess' Law reveals that the heat of reaction along the "path" directly connecting the reactant state to the product state is exactly equal to the total heat of reaction along the alternative "path" connecting reactants to products via the intermediate state containing C(s)C(s), O2(g)O2(g), and 2H2(g)2H2(g). A consequence of our observation of Hess' Law is therefore that the net heat evolved or absorbed during a reaction is independent of the path connecting the reactant to product. (This statement is again subject to our restriction that all reactions in the alternative path must occur under constant pressure conditions.)

A slightly different view of Figure 1 results from beginning at the reactant box and following a complete circuit through the other boxes leading back to the reactant box, summing the net heats of reaction as we go. We discover that the net heat transferred (again provided that all reactions occur under constant pressure) is exactly zero. This is a statement of the conservation of energy: the energy in the reactant state does not depend upon the processes which produced that state. Therefore, we cannot extract any energy from the reactants by a process which simply recreates the reactants. Were this not the case, we could endlessly produce unlimited quantities of energy by following the circuitous path which continually reproduces the initial reactants.

By this reasoning, we can define an energy function whose value for the reactants is independent of how the reactant state was prepared. Likewise, the value of this energy function in the product state is independent of how the products are prepared. We choose this function, HH, so that the change in the function, Δ⁢H=Hproducts−HreactantsΔHHproductsHreactants, is equal to the heat of reaction qq under constant pressure conditions. HH, which we call the enthalpy, is a state function, since its value depends only on the state of the materials under consideration, that is, the temperature, pressure and composition of these materials.

The concept of a state function is somewhat analogous to the idea of elevation. Consider the difference in elevation between the first floor and the third floor of a building. This difference is independent of the path we choose to get from the first floor to the third floor. We can simply climb up two flights of stairs, or we can climb one flight of stairs, walk the length of the building, then walk a second flight of stairs. Or we can ride the elevator. We could even walk outside and have a crane lift us to the roof of the building, from which we climb down to the third floor. Each path produces exactly the same elevation gain, even though the distance traveled is significantly different from one path to the next. This is simply because the elevation is a "state function." Our elevation, standing on the third floor, is independent of how we got to the third floor, and the same is true of the first floor. Since the elevation thus a state function, the elevation gain is independent of the path.

Now, the existence of an energy state function HH is of considerable importance in calculating heats of reaction. Consider the prototypical reaction in Figure 2(a), with reactants RR being converted to products PP. We wish to calculate the heat absorbed or released in this reaction, which is Δ⁢HΔH. Since HH is a state function, we can follow any path from RR to PP and calculate Δ⁢HΔH along that path. In Figure 2(b), we consider one such possible path, consisting of two reactions passing through an intermediate state containing all the atoms involved in the reaction, each in elemental form. This is a useful intermediate state since it can be used for any possible chemical reaction. For example, in Figure 1, the atoms involved in the reaction are C, H, and O, each of which are represented in the intermediate state in elemental form. We can see in Figure 2(b) that the Δ⁢HΔH for the overall reaction is now the difference between the Δ⁢HΔH in the formation of the products PP from the elements and the Δ⁢HΔH in the formation of the reactants RR from the elements.

Figure 2

Calculation of ΔH
(a) (b)

The Δ⁢HΔH values for formation of each material from the elements are thus of general utility in calculating Δ⁢HΔH for any reaction of interest. We therefore define the standard formation reaction for reactant RR, aselements in standard state→Relements in standard state→R

and the heat involved in this reaction is the standard enthalpy of formation, designated by Δ⁢Hf°ΔHf°. The subscript ff, standing for "formation," indicates that the Δ⁢HΔH is for the reaction creating the material from the elements in standard state. The superscript ° indicates that the reactions occur under constant standard pressure conditions of 1 atm. From Figure 2(b), we see that the heat of any reaction can be calculated from

Extensive tables of Δ⁢Hf°ΔHf° have been compiled and published. This allows us to calculate with complete confidence the heat of reaction for any reaction of interest, even including hypothetical reactions which may be difficult to perform or impossibly slow to react.

The bond energy for a molecule is the energy required to separate the two bonded atoms to great distance. We recall that the total energy of the bonding electrons is lower when the two atoms are separated by the bond distance than when they are separated by a great distance. As such, the energy input required to separate the atoms elevates the energy of the electrons when the bond is broken.

We can use diatomic bond energies to calculate the heat of reaction Δ⁢HΔH for any reaction involving only diatomic molecules. We consider two simple examples. First, the reaction

is observed to be endothermic with heat of reaction 70⁢kJmol70kJmol. Note that this reaction can be viewed as consisting entirely of the breaking of the H2H2 bond followed by the formation of the HBrHBr bond. Consequently, we must input energy equal to the bond energy of H2H2 (436⁢kJmol436kJmol), but in forming the HBrHBr bond we recover output energy equal to the bond energy of HBrHBr (366⁢kJmol366kJmol). Therefore the heat of Equation 7 at constant pressure must be equal to difference in these bond energies, 70⁢kJmol70kJmol.

Now we can answer the question, at least for this reaction, of where the energy "goes" during the reaction. The reason this reaction absorbs energy is that the bond which must be broken, H2H2, is stronger than the bond which is formed, HBrHBr. Note that energy is released when the HBrHBr bond is formed, but the amount of energy released is less than the amount of energy required to break the H2H2 bond in the first place.

The second example is similar:

This reaction is exothermic with Δ⁢H°=-103⁢kJmolΔH°-103kJmol. In this case, we must break an H2H2 bond, with energy 436⁢kJmol436kJmol, and a Br2Br2 bond, with energy 193⁢kJmol193kJmol. Since two HBrHBr molecules are formed, we must form two HBrHBr bonds, each with bond energy 366⁢kJmol366kJmol. In total, then, breaking the bonds in the reactants requires 629⁢kJmol629kJmol, and forming the new bonds releases 732⁢kJmol732kJmol, for a net release of 103⁢kJmol103kJmol. This calculation reveals that the reaction is exothermic because, although we must break one very strong bond and one weaker bond, we form two strong bonds.

There are two items worth reflection in these examples. First, energy is released in a chemical reaction due to the formation of strong bonds. Breaking a bond, on the other hand, always requires the input of energy. Second, Equation 8 does not actually proceed by the two-step process of breaking both reactant bonds, thus forming four free atoms, followed by making two new bonds. The actual process of the reaction is significantly more complicated. The details of this process are irrelevant to the energetics of the reaction, however, since, as we have shown, the heat of reaction Δ⁢HΔH does not depend on the path of the reaction. This is another example of the utility of Hess' law.

We now proceed to apply this bond energy analysis to the energetics of reactions involving polyatomic molecules. A simple example is the combustion of hydrogen gas discussed previously here. This is an explosive reaction, producing 483.6kJ per mole of oxygen. Calculating the heat of reaction from bond energies requires us to know the bond energies in H2OH2O. In this case, we must break not one but two bonds:

The energy required to perform this reaction is measured to be 926.9⁢kJmol926.9kJmol. Equation 4 can proceed by a path in which we first break two H2H2 bonds and one O2O2 bond, then we follow the reverse of Equation 9 twice:

4H(g)+2O(g)→2H2O(g)4H(g)+2O(g)→2H2O(g) 2H2(g)+O2(g)→2H2O(g)2H2(g)+O2(g)→2H2O(g) Therefore, the energy of Equation 4 must be the energy required to break two H2H2 bonds and one O2O2 bond minus twice the energy of Equation 9. We calculate that Δ⁢H°=2×(436⁢kJmol)+498.3⁢kJmol−2×(926.9⁢kJmol)=-483.5⁢kJmol ΔH° 2436kJmol 498.3kJmol 2926.9kJmol -483.5kJmol . It is clear from this calculation that Equation 4 is strongly exothermic because of the very large amount of energy released when two hydrogen atoms and one oxygen atom form a water molecule.

It is tempting to use the heat of Equation 9 to calculate the energy of an O-H bond. Since breaking the two O-H bonds in water requires 926.9⁢kJmol926.9kJmol, then we might infer that breaking a single O-H bond requires 926.9⁢kJmol2=463.5⁢kJmol926.9kJmol2463.5kJmol. However, the reaction

has Δ⁢H°=492⁢kJmolΔH°492kJmol. Therefore, the energy required to break an O-H bond in H2OH2O is not the same as the energy required to break the O-H bond in the OHOH diatomic molecule. Stated differently, it requires more energy to break the first O-H bond in water than is required to break the second O-H bond.

In general, we find that the energy required to break a bond between any two particular atoms depends upon the molecule those two atoms are in. Considering yet again oxygen and hydrogen, we find that the energy required to break the O-H bond in methanol (CH3OHCH3OH) is 437⁢kJmol437kJmol, which differs substantially from the energy of Equation 11. Similarly, the energy required to break a single C-H bond in methane (CH4CH4) is 435⁢kJmol435kJmol, but the energy required to break all four C-H bonds in methane is 1663⁢kJmol1663kJmol, which is not equal to four times the energy of one bond. As another such comparison, the energy required to break a C-H bond is 400⁢kJmol400kJmol in trichloromethane (HCCl3HCCl3), 414⁢kJmol414kJmol in dichloromethane (H2CCl2H2CCl2), and 422⁢kJmol422kJmol in chloromethane (H3CClH3CCl).

These observations are somewhat discouraging, since they reveal that, to use bond energies to calculate the heat of a reaction, we must first measure the bond energies for all bonds for all molecules involved in that reaction. This is almost certainly more difficult than it is desirable. On the other hand, we can note that the bond energies for similar bonds in similar molecules are close to one another. The C-H bond energies in the three chloromethanes above illustrate this quite well. We can estimate the C-H bond energy in any one of these chloromethanes by the average C-H bond energy in the three chloromethanes molecule, which is 412⁢kJmol412kJmol. Likewise, the average of the C-H bond energies in methane is 1663⁢kJmol4=416⁢kJmol1663kJmol4416kJmol and is thus a reasonable approximation to the energy required to break a single C-H bond in methane.

By analyzing many bond energies in many molecules, we find that, in general, we can approximate the bond energy in any particular molecule by the average of the energies of similar bonds. These average bond energies can then be used to estimate the heat of a reaction without measuring all of the required bond energies.

Consider for example the combustion of methane to form water and carbon dioxide:

We can estimate the heat of this reaction by using average bond energies. We must break four C-H bonds at an energy cost of approximately 4×412⁢kJmol×4412kJmol and two O2O2 bonds at an energy cost of approximately 2×496⁢kJmol×2496kJmol. Forming the bonds in the products releases approximately 2×743⁢kJmol×2743kJmol for the two C=O double bonds and 4×463⁢kJmol×4463kJmol for the O-H bonds. Net, the heat of reaction is thus approximately Δ⁢H°=1648+992−1486−1852=-698⁢kJmolΔH°164899214861852-698kJmol. This is a rather rough approximation to the actual heat of combustion of methane, -890⁢kJmol-890kJmol. Therefore, we cannot use average bond energies to predict accurately the heat of a reaction. We can get an estimate, which may be sufficiently useful. Moreover, we can use these calculations to gain insight into the energetics of the reaction. For example, Equation 12 is strongly exothermic, which is why methane gas (the primary component in natural gas) is an excellent fuel. From our calculation, we can see that the reaction involved breaking six bonds and forming six new bonds. The bonds formed are substantially stronger than those broken, thus accounting for the net release of energy during the reaction.