A living cell is a dynamic structure.
It grows, it moves, it synthesizes complex macromolecules, and it selectively shuttles substances in and out and between compartments. All of this activity requires energy, so every cell and every organ-ism must obtain energy from its surroundings and expend it as efficiently as pos-sible. Plants gather radiant energy from sunlight; animals use the chemical energy stored in plants or other animals that they consume. The processing of this energy to do the things necessary for a cell or organism to maintain the living state is what much of biochemistry is about. Much of the elegant molecular machinery that exists in every cell is dedicated to this task.Because of the central role of energy in life, it is appropriate that we begin a study of biochemistry with an introduction to bioenergetics—the quantitative analysis of how organisms capture, transform, store, and utilize energy. Bioenergetics may be regarded as a special part of the general science of energy transformations, which is called thermodynamics. In this chapter we shall review just a bit of that field, focus-ing attention on fundamental concepts, such as enthalpy, entropy, and free energy, that are important to the biochemist or biologist.
We will introduce in this chapter the basic approaches for determining changes in free energy in biochemical systems. In subsequent chapters these basic approaches will be discussed further in the context of processes such as protein folding, transport of ions across membranes, and extraction of chemical energy from nutrients to make ATP.
Energy, Heat, and Work
A word we shall often use in our discussion is system. In this context, a system is any part of the universe that we choose for study. It can be a single bacterial cell, a Petri dish containing nutrient and millions of cells, the whole laboratory in which this dish rests, the earth, or the entire universe. A system must have defined boundaries, but otherwise there are few restrictions. Anything not defined as part of the system
The Energetics of Life
is considered to be the surroundings. The system may be isolated, and thus unable to exchange energy and matter with its surroundings; it may be closed, able to exchange energy but not matter; or it may be open, so that both energy and matter can pass between the system and surroundings. For example, our planet displays the essential features of a closed system: Earth can exchange energy (e.g., in the form of electromagnetic radiation) with its surroundings, but except for a few bits of metal (spacecraft and satellites) and some geological samples (meteorites and lunar rocks), material is not exchanged between the planet and its surroundings.
Internal Energy and the State of a System
Any system contains a certain amount of internal energy, which we denote by U.
It is important for our understanding to be specific as to what this internal energy includes. The system’s atoms and molecules have energy of vibration and rotation, and kinetic energy of motion. In addition, we include all of the energy stored in the chemical bonds between atoms and the energy of noncovalent interactions between molecules. We also include any kind of energy that might be changed by chemical or nonnuclear physical processes. We need not include energy stored in the atomic nucleus, for this is unchanged in any chemical or biochemical reaction.
The internal energy is a function of the state of a system. The thermodynamic state is defined by describing the amounts of all substances present and any two of the following three variables: the temperature (T), the pressure on the system (P), or the volume of the system (V). It is essentially a recipe for producing the system in a defined way. For example, a system composed of 1 mole of gas in 1 liter at 273 K has a defined state and therefore a definite internal energy value. This value is independent of any past history of the system.
Unless a system is isolated, it can exchange energy with its surroundings and thereby change its internal energy; we define this change as For a closed system, this exchange can happen in only two ways. First, heat may be transferred to or from the system. Second, the system may do work on its surroundings or have work done on it. Work can take many forms. It may include expansion of the system against an external pressure such as expansion of the lungs, electrical work such as that done by a battery or required for the pumping of ions across a membrane, expansion of a surface against surface tension, flexing of a flagellum to propel a pro-tozoan, or lifting of a weight by contraction of a muscle. In all of these examples, a force is exerted against a resistance to produce a displacement, so work is done.
Note that heat and work are not properties of the system. They may be thought of as “energy in transit” between the system and its surroundings. Certain conven-tions have been adopted to describe these ways of exchanging energy:
1. We denote heat by the symbol q. A positive value of q indicates that heat is absorbed by the system from its surroundings. A negative value means that heat flows from the system to its surroundings.
2. We denote work by the symbol w. A positive value of w indicates that work is done by the system on its surroundings. A negative value means that the surroundings do work on the system.
All of this may seem excessively abstract, yet it bears the most direct relationship to the everyday functioning of our bodies. When we ingest a nutrient like glucose, we metabolize it, ultimately oxidizing it to and water. A defined energy change ( ) is associated with oxidizing a gram of glucose, and some of the released energy is available for our use. We expend a significant portion of this energy as heat (a by-product of mitochondrial metabolism that allows insulated birds and mam-mals to maintain their normal body temperatures) and the remainder performing various kinds of work. These latter kinds of work include not only the obvious ones, like walking and breathing, but other more subtle kinds—sending impulses along nerves, pumping ions across membranes, and so forth.
⌬U CO2
⌬U.
O2
ENERGY, HEAT, AND WORK
59
The internal energy of a system includes all forms of energy that can be exchanged via simple (nonnuclear) physical processes or chemical reactions.
60
CHAPTER 3 THE ENERGETICS OF LIFEFIGURE 3.1
Exchange of heat and work in constant-volume and constant-pressure reactions. A single reaction, oxidation of 1 mole of a fatty acid, is carried out under two sets of conditions. (a) The reaction occurs in a sealed vessel, or “bomb.” Heat (q) is transferred to the surrounding water bath and is measured by the small increase in temperature of the water. No work is done because the system is at constant volume. (b) The reaction vessel is fitted with a piston held at 1 atm pressure. During the reaction, heating of the gas in the vessel causes the piston to be pushed up. However, the reaction results in a decrease in the number of moles of gas, so after the vessel and gas have cooled to the water temperature, the volume of the gas is smaller than the initial volume. Thus, net work is done on the system, and the total amount of heat delivered to the bath is slightly more than in (a).
Thermometer
Palmitic acid Sealed
bomb H2O
H2O H2O H2O H2O
H2O H2O
H2O
Palmitic acid 1 atm Reaction vessel with piston
1 atm
1 atm
1 atm O2
O2
CO2
CO2 CO2
Igniter
Initial state
Initial state
Reaction
Reaction Heat lost, work done Final state
Final state q
q w
q
H2O H2O
(a) Reaction at constant volume
(b) Reaction at constant pressure
The First Law of Thermodynamics
Because the internal energy of a closed system can change only by heat or work exchanges with the surroundings, the change in internal energy must be given by
(3.1)
This equation, which holds true for all processes, expresses the first law of thermodynamics. This first law is simply a bookkeeping rule, a statement of the conservation of energy. When a physical process or a chemical reaction has occurred, we can total up the incomes and expenditures of energy, and the books must balance. Energy can be gained and released in different ways, but at least in chemical processes, it can neither be created nor destroyed. Consider, for example, some process in which a certain amount of heat is absorbed by a system, while the system does an exactly equivalent amount of work on its surroundings. In this case, both q and w are positive, and q= w, so This agrees with common sense:
If some amount of energy went in as heat and an equal quantity came out as work, then the energy within the system must be unchanged.
Changes in internal energy, as for any function of state, depend only on the initial and final states of a system and are independent of the path; however, the amounts of heat and work exchanged in any process depend very much on the pathway taken between the initial and final states. To make this idea concrete, let us consider a specific chemical reaction—the complete oxidation of 1 mole of a fatty acid, palmitic acid:
The oxidation of palmitic acid is an important biochemical reaction that takes place, in a much more indirect way, in our bodies when we metabolize fats. We shall consider running this reaction in two different ways, as shown in Figure 3.1.
CH31CH2214COOH (solid)+ 23O2 (gas)S 16CO2 (gas) + 16H2O (liquid)
⌬U = 0.
⌬U = q - w Energy is conserved. According to the first law
of thermodynamics, in a closed system, the internal energy (U) can change only by the exchange of heat or work with the surround-ings; however, energy can be converted from one form to another.
In Figure 3.1a the reaction is carried out by igniting the mixture in a sealed vessel (a “bomb” calorimeter) immersed in a water bath. The reaction, under these con-ditions, is being carried out at constant volume. We can measure the heat passed from the reaction vessel (the system) to the water bath (the surroundings) by the temperature change in the bath, knowing the mass of water and the heat capacity (per gram) of water. Because the reaction vessel has a fixed volume, no work has been done against the surroundings or by the surroundings; therefore, w= 0, and from equation (3.1),
(3.2)
The total heat that is transferred from the reaction vessel to the surroundings equals the change in internal energy, and that energy change results mainly from the changes in chemical bonding that occurred during the reaction. The presently accepted unit for heat, work, and energy is the joule (J).* For the above reaction, the value observed for is . The negative sign indicates that the reaction releases energy stored in chemical bonds. The energy within the system decreased as this bond energy was transferred as heat to the surroundings.
Now suppose the same reaction is carried out at a constant pressure of 1 atmosphere, as shown in Figure 3.1b. In this case, the system is free to either expand or contract, and it finally contracts by an amount proportional to the decrease in the number of moles of gas, which went from 23 to 16 moles during the reaction. (We neglect the relatively tiny volume of solids and liquids.) The decrease in gas volume means that a certain amount of work has been done by the surroundings on the system. This can be calculated in the following way.
When volume (V) is changed against a constant pressure (P),
(3.3)
To calculate w, we may make an approximation. We assume that the initial and final temperatures of the system are essentially the same (say 25 °C, or 298 K) and that the gases are “ideal.” We may then use the ideal gas law, PV = nRT. This gives
(3.4)
where R is the gas constant, T the absolute temperature in kelvin, and the change in number of moles of gas per mole of palmitic acid oxidized. Then, inserting (3.4) in (3.3), we obtain
(3.5)
Because we would like w in joules per mole, we use in
Equation 3.5, resulting in palmitate.
The heat evolved in this constant-pressure combustion will then be
(3.6)
A slightly greater amount of heat is released to the surroundings under these constant-pressure conditions than under the constant-volume conditions of
= (-9941.4 kJ>mol) + (-17.3 kJ>mol) = -9958.7 kJ>mol q = ⌬U + w = ⌬U + P ⌬V = ⌬U + ⌬n RT
w = -17,300 J/mol (or -17.3 kJ/mol)
R = 8.315 J/K⭈mol
w= ⌬n RT
⌬n
⌬V = ⌬nRT P w= P ⌬V
-9941.4 kJ/mol
⌬U
⌬U = q
ENERGY, HEAT, AND WORK
61
*In the past, biochemists tended to express energy, heat, and work in calories or kilocalories.
However, the International System of Units (SI units) joules and kilojoules are now replacing these.
To convert these units: J. Similarly, 1 kcal (kilocalorie, or calories)
(kilojoules). A complication arises from the fact that the “calorie” (“C”) referred to in dietetics is really a kilocalorie.
= 4.184 kJ 103
1 cal = 4.184
The heat evolved in a reaction at constant volume is equal to U. ⌬
62
CHAPTER 3 THE ENERGETICS OF LIFEFigure 3.1a. Under constant-pressure conditions the surroundings can do work on the system and this work (called PV work) reappears as extra heat released from the system to the surroundings (which is required to keep the temperature of the system at a constant value of 298 K).
Although the heat and work exchanged with the surroundings depend on path, it is important to remember that does not—it depends only on the ini-tial and final states.
Enthalpy
Most chemical reactions in the laboratory and virtually all biochemical processes occur under conditions more nearly approximating constant pressure than con-stant volume. If we are interested in the heat obtainable by oxidizing palmitic acid in an animal, then the heat evolved at constant pressure is what we want to know.
As we showed in Equation 3.6, this heat is not exactly equal to U because of the PV work done; thus, to express the heat change in a constant-pressure reaction, we need another function of state. We define a new quantity, the enthalpy, which we give the symbol H:
(3.7)
Because U and PV are functions of state, H is also a function of state. The change depends only on the initial and final states of the process for which it is calcu-lated. For reactions at constant pressure, is defined as follows:
(3.8)
The value of is the same as the amount of heat (q) calculated in Equation 3.6.
In other words, when the heat of a reaction is measured at constant pressure, it is that is determined.
The energy changes you will find tabulated throughout this book and other books on biochemistry will almost always be given as values. That is most appropriate, for in vivo these reactions occur under nearly constant-pressure con-ditions. If a nutritionist wishes to know the energy available from the oxidation of palmitic acid in the body, is the appropriate quantity.
Measuring changes in energy such as and in a calorimeter is of prac-tical use to biochemists and dieticians even though the oxidation of a substance like palmitic acid occurs very differently in the human body than it does in a reac-tion vessel like that shown in Figure 3.1. The values of and for the oxida-tion of palmitic acid are exactly the same in both pathways because a quantity like or depends only on the final and initial states. Thus, the calorimeter pro-vides an exact measurement of the energy available to a human from each gram of palmitic acid oxidized completely to and
The average human requires the expenditure of about 6000 kJ per day (roughly 1500 kcal or 1500 of the “calories” used in dietetics) just to sustain basal metabolic rates. With moderate exercise, this need for metabolic energy may easily double.
Although we have pointed out the distinction between and we should emphasize that for most biochemical reactions the quantitative difference between them is of little consequence. Most of these reactions occur in solution and do not involve the consumption or formation of gases. The volume changes are thus exceedingly small, and is a tiny quantity relative to or Even for the example given, the oxidation of palmitic acid, the difference between and is only 0.2%. Thus, we are justified in most cases in thinking of as a direct measure of the energy change in a process, and we commonly refer to as the energy change.
⌬H⌬H
⌬U P⌬V ⌬U ⌬H. ⌬H
⌬H,
⌬U H2O.
CO2
⌬H
⌬U
⌬H
⌬U
⌬H
⌬H ⌬U
⌬H
⌬H ⌬H
⌬H = ⌬U + P⌬V
⌬H ⌬H
H= U + PV
⌬U
The heat evolved in a reaction at constant pressure is equal to the change in enthalpy,
H.
⌬
The enthalpy change in a reaction is the energy change of most interest to bio-chemists.
Entropy and the Second Law of Thermodynamics
The Direction of Processes
However useful the first law may be for keeping track of energy changes in processes, it cannot give us one very important piece of information: What is the favored direction for a process? The first law cannot answer questions like these:
We place an ice cube in a glass of water at room temperature. It melts. Why doesn’t the rest of the water freeze instead?
We place an ice cube in a jar of supercooled water. All the water freezes. Why?
We touch a lit match to a piece of paper. The paper burns to carbon dioxide and water. Why can’t we mix carbon dioxide and water to form paper?
One characteristic of such processes is their irreversibility under the conditions described above. An ice cube in a glass of room-temperature water at 1 atm will continue to melt—there is no way to turn that process around without major changes in the conditions. But there is a reversible way to melt ice—to have it in contact with water at C and 1 atm. Under these conditions, adding a bit of heat to the glass will result in a small amount of ice melting, whereas removing a little heat will cause a small amount of the water to freeze. A reversible process like melting ice at C is always near a state of equilibrium. The defining feature of the equilibrium state is that it is the lowest energy state for a system. As discussed below, lower energy states are favored over those of higher energy; thus, systems tend to adopt states of lower energy. The irreversible processes we just described happen when systems are set up far from an equilibrium state. They then drive toward a state of equilibrium.
In the jargon of thermodynamics, an irreversible process is often called a
“spontaneous” process, but we prefer the word favorable. The word spontaneous tends to imply, perhaps falsely, that the process is rapid. Thermodynamics has nothing to say about how fast processes will be (this is described by “kinetics”—
see Chapter 11), but it does indicate which direction for a process is favored. The melting of ice, rather than freezing, is favored at C and 1 atm. Here, the result is intuitive; you would not expect the ice cube to grow, or even remain unmelted, when placed in C water.
Knowing whether a process is reversible, favorable, or unfavorable is vital to bioenergetics. This information can be expressed most succinctly by the second law of thermodynamics, which tells us which processes are thermodynamically favorable. To present the second law, we must consider a new concept—entropy.
Entropy
Why do chemical and physical processes have thermodynamically favored directions? A first guess at an explanation might be that systems simply go toward a lowest-energy state. Water runs downhill, losing energy as it sponta-neously falls in the earth’s gravitational field; the oxidation of palmitic acid, like the burning of paper, releases energy as heat. Certainly, energy minimization is the key to the favored direction for some processes. But such an explanation cannot account for the melting of ice at C; in fact, energy is absorbed in that process. Another, very different factor must be at work, and a simple experi-ment gives a clear indication of what this factor may be. If we carefully layer pure water on top of a sucrose solution, we will observe as time passes that the solution becomes more and more uniform (Figure 3.2). Eventually the sucrose molecules will be evenly distributed throughout the solution. Though there is practically no energy change, in terms of heat and work, the process is clearly a
25 ⬚ 25 ⬚
25⬚ 0 ⬚
0⬚
Reversible processes occur always near a state of equilibrium; irreversible processes drive toward equilibrium.
ENTROPY AND THE SECOND LAW OF THERMODYNAMICS
63
64
CHAPTER 3 THE ENERGETICS OF LIFEfavorable one. We know from experience that the opposite process (self-segregation of the sucrose molecules into a portion of the solution volume) never occurs. What is clearly important here is that systems of molecules have a natural tendency to randomization.
The degree of randomness or disorder of a system is measured by a function of state called the entropy (S). There are several ways of defining entropy, but the most useful for our applications depends on the fact that a given thermodynamic state may have many substates of equal energy. Those substates correspond, for example, to different ways in which molecules can be arranged or distributed within the system (see Figure 3.2). If the thermodynamic state has a number (W) of substates of equal energy, the entropy (S) is defined as
(3.9)
where kBis the Boltzmann constant, the gas constant R divided by Avogadro’s number. A consequence of this definition is that entropy is seen to be a measure of disorder. There will always be many more ways of putting a large number of mol-ecules into a disorderly arrangement than into an orderly one; therefore, the entropy of an ordered state is lower than that of a disordered state of the same sys-tem. In fact, the minimal value of entropy (zero) is predicted only for a perfect crystal at the absolute zero of temperature (0 K or 273.15 C). The process of diffusion evens out the concentrations in our sucrose solution simply because there are more ways to distribute molecules over a large volume than over a small one. In Figure 3.2 the state shown in panel (c) has more substates of equal energy (i.e., different random arrangements of the sucrose molecules in the cells) than does the state shown in panel (b). We could also say that state (c) has more degrees of freedom than state (b); thus, the value of W is greater for state (c).
Rotations around bonds increase the degrees of freedom available to a molecule.
We will consider the entropy associated with bond rotations when we examine the folding of proteins (Chapter 6). To make the concept of entropy a bit more famil-iar, consider the examples given in Table 3.1.
The Second Law of Thermodynamics
The preceding example shows that the driving force toward equilibrium for the sucrose solution in Figure 3.2 is just the increase in entropy. This observation can be generalized as the second law of thermodynamics. The entropy of an isolated
⬚
-S = kB ln W Pure
water
Final state Water layer added
Sucrose solution:
Initial state
(a) High entropy (b) Low entropy (c) High entropy
NA molecules in NF cells NA molecules
in NI cells
NF cells Time
FIGURE 3.2
Diffusion as an entropy-driven process. The gradual mixing of a dilute sucrose solution and pure water is the result of random move-ment of their molecules. We can visualize the increase in entropy if we imagine the volume of the two liquids to be made up of cells, each big enough to hold one sucrose molecule. (a) Initially, the sucrose solution is at equilibrium because its NAmolecules are distributed ran-domly into its NIcells. (b) When a layer of pure water is added without mixing, the system is no longer at equilibrium. It has become more ordered, with all the occupied cells located in one-half of the solution. (c) As sucrose and water molecules continue to move randomly, their arrangement becomes less ordered because every cell has an equal chance of being occupied. Eventually, the solution reaches a new equilibrium, with sucrose molecules randomly distributed throughout. The drive toward equilibrium is a consequence of the tendency for entropy to increase. A system would never go spontaneously from state (c) to state (b).
Entropy is a measure of the randomness or disorder in a system.
TABLE3.1 Examples of low-entropy and high-entropy states
Low Entropy High Entropy Ice, at 0 °C Water, at 0 °C A diamond,
at 0 K
Carbon vapor, at 1,000,000 K A protein
molecule in its regular, native structure
The same protein molecule in an unfolded, random coil state A Shakespearean
sonnet
A random string of letters A bank manager’s
desk A professor’s desk