Chemical Energetics: an introduction to chemical thermodynamics and the First Law

# Molecules as energy carriers and converters

On this page:
Chemical energy: what is it?
How molecules take up thermal energy
Energetics of chemical reactions
Heat changes at constant pressure: the enthalpy
Changes in enthalpy and internal en ergy
How the enthalpy depends on the temperature
Concept Map

All molecules at temperatures above absolue zero possess thermal energy— the randomized kinetic energy associated with the various motions the molecules as a whole, and also the atoms within them, can undergo. Polyatomic molecules also possess potential energy in the form of chemical bonds. Molecules are thus both vehicles for storing and transporting energy, and the means of converting it from one form to another when the formation, breaking, or rearrangement of the chemical bonds within them is accompanied by the uptake or release of heat.

Chemical Energy

When you buy a liter of gasoline for your car, a cubic metre of natural gas to heat your home, or a small battery for your flashlight, you are purchasing energy in a chemical form. In each case, some kind of a chemical change will have to occur before this energy can be released and utilized: the fuel must be burned in the presence of oxygen, or the two poles of the battery must be connected through an external circuit (thereby initiating a chemical reaction inside the battery.) And eventually, when each of these reactions is complete, our source of energy will be exhausted; the fuel will be used up, or the battery will be “dead”.

Chemical substances are made of atoms, or more generally, of positively charged nuclei surrounded by negatively charged electrons. A molecule such as dihydrogen, H2, is held together by electrostatic attractions mediated by the electrons shared between the two nuclei. The total potential energy of the molecule is the sum of the repulsions between like charges and the attractions between electrons and nuclei:

PEtotal = PEelectron-electron + PEnucleus-nucleus + PEnucleus-electron

In other words, the potential energy of a molecule depends on the time-averaged relative locations of its constituent nuclei and electrons. This dependence is expressed by the familiar potential energy curve which serves as an important description of the chemical bond between two atoms.

> Translation refers to movement of an object as a complete unit. Translational motions of molecules in solids or liquids are restricted to very short distances comparable to the dimensions of the molecules themselves, whereas in gases the molecules typically travel hundreds of molecular diameters between collisions.

In gaseous hydrogen, for example, the molecules will be moving freely from one location to another; this is called translational motion, and the molecules therefore possess translational kinetic energy KEtrans = mv2/2, in which v stands for the average velocity of the molecules; you may recall from your study of gases that v, and therefore KEtrans, depends on the temperature.

In addition to translation, molecules can possess other kinds of motion. Because a chemical bond acts as a kind of spring, the two atoms in H2 will have a natural vibrational frequency. In more complicated molecules, many different modes of vibration become possible, and these all contribute a vibrational term KEvib to the total kinetic energy. Finally, a molecule can undergo rotational motions which give rise to a third term KErot. Thus the total kinetic energy of a molecule is the sum

KEtotal = KEtrans + KEvib + KErot

The total energy of the molecule (its internal energy U) is just the sum

U = KEtotal + PEtotal (1)

Although this formula is simple and straightforward, it cannot take us very far in understanding and predicting the behavior of even one molecule, let alone a large number of them. The reason, of course, is the chaotic and unpredictable nature of molecular motion. Fortunately, the behavior of a large collection of molecules, like that of a large population of people, can be described by statistical methods.

How molecules take up thermal energy

As noted above, the heat capacity of a substance is a measure of how sensitively its temperature is affected by a change in heat content; the greater the heat capacity, the less effect a given flow of heat q will have on the temperature.

We also pointed out that temperature is a measure of the average kinetic energy due to translational motions of molecules. If vibrational or rotational motions are also active, these will also accept thermal energy and reduce the amount that goes into translational motions. Because the temperature depends only on the latter, the effect of the other kinds of motions will be to reduce the dependence of the internal energy on the temperature, thus raising the heat capacity of a substance.

Vibrational and rotational motions are not possible for monatomic species such as the noble gas elements, so these substances have the lowest heat capacities. Moreover, as you can see in the leftmost column of Table 3, their heat capacities are all the same. This reflects the fact that translational motions are the same for all particles; all such motions can be resolved into three directions in space, each contributing one degree of freedom to the molecule and 1/2 R to its heat capacity. (R is the gas constant, 8.314 J K–1).

Molar heat capacities of some gaseous substances at constant pressure. Values are |
monatomic | diatomic | triatomic
He | 20.5 | CO | 29.3 | H2O | 33.5
Ne | 20.5 | N2 | 29.5 | D2O | 34.3
Ar | 20.5 | F2 | 31.4 | CO2 | 37.2
Kr | 20.5 | Cl2 | 33.9 | CS2 | 45.6

Whereas monatomic molecules can only possess translational thermal energy, two additional kinds of motions become possible in polyatomic molecules. A linear molecule has an axis that defines two perpendicular directions in which rotations can occur; each represents an additional degree of freedom, so the two together contribute a total of 1/2 R to the heat capacity. For a non-linear molecule, rotations are possible along all three directions of space, so these molecules have a rotational heat capacity of 3/2 R. Finally, the individual atoms within a molecule can move relative to each other, producing a vibrational motion. A molecule consisting of N atoms can vibrate in 3N –6 different ways or modes. Each vibrational mode contributes R (rather than ½ R) to the total heat capacity. (These results come from advanced mechanics and will not be proven here.)

monatomic | 3/2R | 0 | 0
diatomic | 3/2 R | R | R
polyatomic | 3/2 R | 3/2 R | 3N – 6
separation between adjacent levels, in kJ mol–1. | 6.0 × 10–17 J (O2) | 373 J (HCl) | 373 J (HCl)

Table 4: Contribution of molecular motions to heat capacity

Now we are in a position to understand why more complicated molecules have higher heat capacities. The total kinetic energy of a molecule is the sum of those due to the various kinds of motions:

KEtotal = KEtrans + KErot + KEvib

When a monatomic gas absorbs heat, all of the energy ends up in translational motion, and thus goes to increase its temperature. In a polyatomic gas, by contrast, the absorbed energy is partitioned among the other kinds of motions; since only the translational motions contribute to the temperature, the temperature rise is smaller, and thus the heat capacity is larger.

There is one very significant complication, however: classical mechanics predicts that the energy is always partitioned equally between all degrees of freedom. Experiments, however, show that this is observed only at quite high temperatures. The reason is that these motions are all quantized. This means that only certain increments of energy are possible for each mode of motion, and unless a certain minimum amount of energy is available, a given mode will not be active at all and will contribute nothing to the heat capacity.

| Distribution of thermal energy in a typical diatomic molecule at 300K The shading indicates the average thermal energy available at this temperature; only those levels within this rage will have significant occupancy as indicated by the thickness of the lines in the two rightmost columns. At 300K, only the lowest vibrational state and the first few rotational states will be active. Most of the thermal energy will be confined to the translational levels whose minute spacing (10–17 J) causes them to appear as a continuum.

Heat capacity of dihydrogen as a function of temperature This plot is typical of those for other substances, and shows the practical consequences of the spacings of the various forms of hermal energy. Thus translational motions are available at virtually all temperatures, but contributions to heat acapacity by rotational or vibrational motions can only develop at temperatures sufficiently large to excite these motion. |

It turns out that translational energy levels are spaced so closely that they these motions are active almost down to absolute zero, so all gases possess a heat capacity of at least 3/2 R at all temperatures. Rotational motions do not get started until intermediate temperatures, typically 300-500K, so within this range heat capacities begin to increase with temperature. Finally, at very high temperatures, vibrations begin to make a significant contribution to the heat capacity

The strong intermolecular forces of liquids and many solids allow heat to be channeled into vibrational motions involving more than a single molecule, further increasing heat capacities. One of the well known “anomalous” properties of liquid water is its high heat capacity (75 J mol–1 K–1) due to intermolecular hydrogen bonding, which is directly responsible for the moderating influence of large bodies of water on coastal climates.

### Heat capacities of metals

Metallic solids are a rather special case. In metals, the atoms oscillate about their equilibrium positions in a rather uniform way which is essentially the same for all metals, so they should all have about the same heat capacity. That this is indeed the case is embodied in the Law of Dulong and Petit. In the 19th century these workers discovered that the molar heat capacities of all the metallic elements they studied were around to 25 J mol–1 K–1, which is close to what classical physics predicts for crystalline metals. This observation played an important role in characterizing new elements, for it provided a means of estimating their molar masses by a simple heat capacity measurement.

Heat changes at constant pressure: the enthalpy

Most chemical processes are accompanied by changes in the volume of the system, and therefore involve both heat and work terms. The work term PΔV becomes especially significant when the number of moles of gaseous components changes. If the process takes place at a constant pressure, then the work is given by PΔV and the change in internal energy will be

ΔU = qP – PΔV (2)

Thus the amount of heat qP that passes between the system and the surroundings is given by

qP = ΔU +PΔV (3)

Problem Example

Hydrogen chloride gas readily dissolves in water, releasing 75.3 kJ/mol of heat in the process. If one mole of HCl at 298 K and 1 atm pressure occupies 24.5 liters, find the ΔU for the system when one mole of HCl dissolves in water under these conditions.

Solution: In this process the volume of liquid remains practically unchanged, so ΔV = –24.5 L. The work done is

w = –PΔV = –(1 atm)(–24.5 L) = 24.6 L-atm

(The work is positive because it is being done on the system as its volume decreases due to the dissolution of the gas into the much smaller volume of the solution.)

Using the conversion factor 1 L-atm = 101.33 J mol–1 and substituting in Eq. 3 we obtain

ΔU= q +PΔV = –(75300 J) + [101.33 J/L-atm) × (24.5 L-atm)] = –72.82 kJ

In other words, if the gaseous HCl simply dissolved without volume change, the heat released by the process (75.3 kJ) would cause the system’s internal energy to diminish by 75.3 kJ. But the volume decrease due to the disappearance of the gas is equivalent to compression of the system by the pressure of the atmosphere; the resulting work done on the system acts to increase its internal energy, so the net value of ΔU is –72.82 kJ instead of –75.3 kJ.

Since both ΔU and ΔV in Eq 3 are state functions, then qP, the heat that is absorbed or released when a process takes place at constant pressure, must also be a state function and is known as the enthalpy change H.

ΔH ≡ qP = ΔU +PΔV (4) must know this!

The enthalpy change associated with a chemical reaction corresponds to the quantity of heat exchanged with the surroundings when the reactants are converted to products at constant pressure.

Under the special conditions in which the pressure is 1 atm and the reactants and products are at a temperature of 298 K, ΔH becomes the standard enthalpy change ΔH °.

Since most changes that occur in the laboratory, on the surface of the earth, and in organisms are subjected to a constant pressure of one atmosphere, this is the form of the First Law that is of greatest interest to most scientists.

Energetics of chemical reactions

The rearrangement of atoms that occurs in a chemical reaction is virtually always accompanied by the liberation or absorption of heat. If the purpose of the reaction is to serve as a source of heat, such as in the combustion of a fuel, then these heat effects are of direct and obvious interest. We will soon see, however, that a study of the energetics of chemical reactions in general can lead us to a deeper understanding of chemical equilibrium and the basis of chemical change itself.

In chemical thermodynamics, we define the zero of the enthalpy and internal energy as that of the elements as they exist in their stable forms at 298K and 1 atm pressure. Thus the enthalpies H of Xe(g), O2(g) and C(diamond) are all zero, as are those of H2 and Cl2 in the reaction

H2(g) + Cl2(g) → 2 HCl(g)

The enthalpy of two moles of HCl is smaller than that of the reactants, so the difference is released as heat. Such a reaction is said to be exothermic. The reverse of this reaction would absorb the same quantity of heat from the surroundings and be endothermic.

In comparing the internal energies and enthalpies of different substances as we have been doing here, it is important to compare equal numbers of moles, because energy is an extensive property of matter. However, heats of reaction are commonly expressed on a molar basis and treated as intensive properties.

Changes in enthalpy and internal energy

We can characterize any chemical reaction by the change in the internal energy or enthalpy:

ΔH = Hfinal – Hinitial

The significance of this can hardly be exaggerated because ΔH, being a state function, is entirely independent of how the system gets from the initial state to the final state. In other words, the value of ΔH or ΔU for a given change in state is independent of the pathway of the process.

Consider, for example, the oxidation of a lump of sugar to carbon dioxide and water:

C12H22O11 + 12 O2(g) → 12 CO2(g) + 11 H2O(l)

This process can be carried out in many ways, for example by burning the sugar in air, or by eating the sugar and letting your body carry out the oxidation. Although the mechanisms of the transformation are completely different for these two pathways, the overall change in the enthalpy of the system (the atoms of carbon, hydrogen and oxygen that were originally in the sugar) will be identical, and can be calculated simply by looking up the standard enthalpies of the reactants and products and calculating the difference

ΔH = [12 × H(CO2)] + [11 × H(H2O)] – H(C12H22O11) = –5606 kJ

The same quantity of heat is released whether the sugar is burnt in the air or oxidized in a series of enzyme-catalyzed steps in your body.

Variation of the enthalpy with temperature

The enthalpy of a system increases with the temperature by the amount ΔH = CPΔT. The defining relation

ΔH = ΔU + PΔV

tells us that this change is dominated by the internal energy, subject to a slight correction for the work associated with volume change. Heating a substance causes it to expand, making ΔV positive and causing the enthalpy to increase slightly more than the internal energy. Physically, what this means is that if the temperature is increased while holding the pressure constant, some extra energy must be expended to push back the external atmosphere while the system expands. The difference between the dependence of U and H on temperature is only really significant for gases, since the coefficients of thermal expansion of liquids and solids are very small.

### Phase changes

A plot of the enthalpy of a system as a function of its temperature is called an enthalpy diagram. The slope of the line is given by Cp. The enthalpy diagram of a pure substance such as water shows that this plot is not uniform, but is interrupted by sharp breaks at which the value of Cp is apparently infinite, meaning that the substance can absorb or lose heat without undergoing any change in temperature at all. This, of course, is exactly what happens when a substance undergoes a phase change; you already know that the temperature the water boiling in a kettle can never exceed 100 until all the liquid has evaporated, at which point the temperature of the steam will rise as more heat flows into the system.

Enthalpy of carbon tetrachloride as a function of temperature at 1 atm A plot of the enthalpy of a system as a function of its temperature provides a concise view of its thermal behavior. The slope of the line is given by the heat capacity Cp. All H-vs.-C plots show sharp breaks at which the value of Cp is apparently infinite, meaning that the substance can absorb or lose heat without undergoing any change in temperature at all. This, of course, is exactly what happens when a substance undergoes a phase change; you already know that the temperature of the water boiling in a kettle can never exceed 100°C until all the liquid has evaporated, at which point the temperature (of the steam) will rise as more heat flows into the system. The lowest-temperature discontinuity on the CCl4 diagram corresponds to a solid-solid phase transition associated with a rearrangement of molecules in the crystalline solid. |

Fusion and boiling are not the only kinds of phase changes that matter can undergo. Most solids can exist in different structural modifications at different temperatures, and the resulting solid-solid phase changes produce similar discontinuities in the heat capacity. Enthalpy diagrams are easily determined by following the temperature of a sample as heat flows into our out of the substance at a constant rate. The resulting diagrams are widely used in materials science and forensic investigations to characterize complex and unknown substances.

Concept map