We have seen how a chemical insight into the hydrogen-bonding capabilities of the bases of DNA led to a deep understanding of a fundamental biological process. To lay the groundwork for the rest of the book, we begin our study of biochemistry by examining selected concepts from chemistry and showing how these concepts apply to biological systems. The concepts include the types of chemical bonds; the structure of water, the solvent in which most biochemical processes take place; the First and Second Laws of Thermodynamics; and the principles of acid–base chemistry.

We will use these concepts to examine an archetypical biochemical process—namely, the formation of a DNA double helix from its two component strands. The process is but one of many examples that could have been chosen to illustrate these topics. Keep in mind that, although the specific discussion is about DNA and double-helix formation, the concepts considered are quite general and will apply to many other classes of molecules and processes that will be discussed in the remainder of the book. In the course of these discussions, we will touch on the properties of water and the concepts of pK_a and buffers that are of great importance to many aspects of biochemistry.

The discovery that DNA from natural sources exists in a double-helical form with Watson–Crick base pairs suggested, but did not prove, that such double helices would form spontaneously outside biological systems. Suppose that two short strands of DNA were chemically synthesized to have complementary sequences so that they could, in principle, form a double helix with Watson–Crick base pairs. Two such sequences are CGATTAAT and ATTAATCG. The structures of these molecules in solution can be examined by a variety of techniques. In isolation, each sequence exists almost exclusively as a single-stranded molecule. However, when the two sequences are mixed, a double helix with Watson–Crick base pairs does form (Figure 1.8). This reaction proceeds nearly to completion.

What forces cause the two strands of DNA to bind to each other? To analyze this binding reaction, we must consider several factors: the types of interactions and bonds in biochemical systems and the energetic favorability of the reaction. We must also consider the influence of the solution conditions—in particular, the consequences of acid–base reactions.

Atoms interact with one another through chemical bonds. These bonds include the covalent bonds that define the structure of molecules as well as a variety of noncovalent bonds that are of great importance to biochemistry.

The strongest bonds are covalent bonds, such as the bonds that hold the atoms together within the individual bases shown in Section 1.2. A covalent bond is formed by the sharing of a pair of electrons between adjacent atoms. A typical carbon–carbon (C C) covalent bond has a bond length of 1.54 Å and bond energy of 355 kJ mol⁻¹ (85 kcal mol⁻¹). Because covalent bonds are so strong, considerable energy must be expended to break them. More than one electron pair can be shared between two atoms to form a multiple covalent bond. For example, three of the bases in Figure 1.6 include carbon–oxygen (C O) double bonds. These bonds are even stronger than C C single bonds, with energies near 730 kJ mol⁻¹ (175 kcal mol⁻¹) and are somewhat shorter.

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For some molecules, more than one pattern of covalent bonding can be written. For example, adenine can be written in two nearly equivalent ways called resonance structures.

These adenine structures depict alternative arrangements of single and double bonds that are possible within the same structural framework. Resonance structures are shown connected by a double-headed arrow. Adenine’s true structure is a composite of its two resonance structures. The composite structure is manifested in the bond lengths such as that for the bond joining carbon atoms C-4 and C-5. The observed bond length of 1.40 Å is between that expected for a C C single bond (1.54 Å) and a C C double bond (1.34 Å). A molecule that can be written as several resonance structures of approximately equal energies has greater stability than does a molecule without multiple resonance structures.

Noncovalent bonds are weaker than covalent bonds but are crucial for biochemical processes such as the formation of a double helix. Four fundamental noncovalent bond types are ionic interactions, hydrogen bonds, van der Waals interactions, and hydrophobic interactions. They differ in geometry, strength, and specificity. Furthermore, these bonds are affected in vastly different ways by the presence of water. Let us consider the characteristics of each type:

1. Ionic Interactions. A charged group on one molecule can attract an oppositely charged group on the same or another molecule. The energy of an ionic interaction (sometimes called an electrostatic interaction) is given by the Coulomb energy:

E = kq₁q₂/Dr

where E is the energy, q₁ and q₂ are the charges on the two atoms (in units of the electronic charge), r is the distance between the two atoms (in angstroms), D is the dielectric constant (which decreases the strength of the Coulomb depending on the intervening solvent or medium), and k is a proportionality constant (k = 1389, for energies in units of kilojoules per mole, or 332 for energies in kilocalories per mole).

By convention, an attractive interaction has a negative energy. The ionic interaction between two ions bearing single opposite charges separated by 3 Å in water (which has a dielectric constant of 80) has an energy of −5.8 kJ mol⁻¹ (−1.4 kcal mol⁻¹). Note how important the dielectric constant of the medium is. For the same ions separated by 3 Å in a nonpolar solvent such as hexane (which has a dielectric constant of 2), the energy of this interaction is −232 kJ mol⁻¹ (−55 kcal mol⁻¹).

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2. Hydrogen Bonds. These interactions are largely ionic interactions, with partial charges on nearby atoms attracting one another. Hydrogen bonds are responsible for specific base-pair formation in the DNA double helix. The hydrogen atom in a hydrogen bond is partially shared by two electronegative atoms such as nitrogen or oxygen. The hydrogen-bond donor is the group that includes both the atom to which the hydrogen atom is more tightly linked and the hydrogen atom itself, whereas the hydrogen-bond acceptor is the atom less tightly linked to the hydrogen atom (Figure 1.9). The electronegative atom to which the hydrogen atom is covalently bonded pulls electron density away from the hydrogen atom, which thus develops a partial positive charge (δ⁺). Thus, the hydrogen atom with a partial positive charge can interact with an atom having a partial negative charge (δ⁻) through an ionic interaction.

Hydrogen bonds are much weaker than covalent bonds. They have energies ranging from 4 to 20 kJ mol⁻¹ (from 1 to 5 kcal mol⁻¹). Hydrogen bonds are also somewhat longer than covalent bonds; their bond lengths (measured from the hydrogen atom) range from 1.5 Å to 2.6 Å; hence, a distance ranging from 2.4 Å to 3.5 Å separates the two nonhydrogen atoms in a hydrogen bond.

The strongest hydrogen bonds have a tendency to be approximately straight, such that the hydrogen-bond donor, the hydrogen atom, and the hydrogen-bond acceptor lie along a straight line. This tendency toward linearity can be important for orienting interacting molecules with respect to one another. Hydrogen-bonding interactions are responsible for many of the properties of water that make it such a special solvent, as will be described shortly.

3. van der Waals Interactions. The basis of a van der Waals interaction is that the distribution of electronic charge around an atom fluctuates with time. At any instant, the charge distribution is not perfectly symmetric. This transient asymmetry in the electronic charge about an atom acts through ionic interactions to induce a complementary asymmetry in the electron distribution within its neighboring atoms. The atom and its neighbors then attract one another. This attraction increases as two atoms come closer to each other, until they are separated by the van der Waals contact distance (Figure 1.10). At distances shorter than the van der Waals contact distance, very strong repulsive forces become dominant because the outer electron clouds of the two atoms overlap.

Energies associated with van der Waals interactions are quite small; typical interactions contribute from 2 to 4 kJ mol⁻¹ (from 0.5 to 1 kcal mol⁻¹) per atom pair. When the surfaces of two large molecules come together, however, a large number of atoms are in van der Waals contact, and the net effect, summed over many atom pairs, can be substantial.

We will cover the fourth noncovalent interaction, the hydrophobic interaction, after we examine the characteristics of water; these characteristics are essential to understanding the hydrophobic interaction.

Water is the solvent in which most biochemical reactions take place, and its properties are essential to the formation of macromolecular structures and the progress of chemical reactions. Two properties of water are especially relevant:

1. Water is a polar molecule. The water molecule is bent, not linear, and so the distribution of charge is asymmetric. The oxygen nucleus draws electrons away from the two hydrogen nuclei, which leaves the region around each hydrogen atom with a net positive charge. The water molecule is thus an electrically polar structure.

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2. Water is highly cohesive. Water molecules interact strongly with one another through hydrogen bonds. These interactions are apparent in the structure of ice (Figure 1.11). Networks of hydrogen bonds hold the structure together; similar interactions link molecules in liquid water and account for many of the properties of water. In the liquid state, approximately one in four of the hydrogen bonds present in ice are broken. The polar nature of water is responsible for its high dielectric constant of 80. Molecules in aqueous solution interact with water molecules through the formation of hydrogen bonds and through ionic interactions. These interactions make water a versatile solvent, able to readily dissolve many species, especially polar and charged compounds that can participate in these interactions.

A final fundamental interaction called the hydrophobic effect is a manifestation of the properties of water. Some molecules (termed nonpolar molecules) cannot participate in hydrogen bonding or ionic interactions. The interactions of nonpolar molecules with water molecules are not as favorable as are interactions between the water molecules themselves. The water molecules in contact with these nonpolar molecules form “cages” around them, becoming more well ordered than water molecules free in solution. However, when two such nonpolar molecules come together, some of the water molecules are released, allowing them to interact freely with bulk water (Figure 1.12). The release of water from such cages is favorable for reasons to be considered shortly. The result is that nonpolar molecules show an increased tendency to associate with one another in water compared with other, less polar and less self-associating, solvents. This tendency is called the hydrophobic effect and the associated interactions are called hydrophobic interactions.

Let us now see how these four noncovalent interactions work together in driving the association of two strands of DNA to form a double helix. First, each phosphate group in a DNA strand carries a negative charge. These negatively charged groups interact unfavorably with one another over distances. Thus, unfavorable ionic interactions take place when two strands of DNA come together. These phosphate groups are far apart in the double helix with distances greater than 10 Å, but many such interactions take place (Figure 1.13). Thus, ionic interactions oppose the formation of the double helix. The strength of these repulsive ionic interactions is diminished by the high dielectric constant of water and the presence of ionic species such as Na⁺ or Mg²⁺ ions in solution. These positively charged species interact with the phosphate groups and partly neutralize their negative charges.

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Second, as already noted, hydrogen bonds are important in determining the formation of specific base pairs in the double helix. However, in single-stranded DNA, the hydrogen-bond donors and acceptors are exposed to solution and can form hydrogen bonds with water molecules.

When two single strands come together, these hydrogen bonds with water are broken and new hydrogen bonds between the bases are formed. Because the number of hydrogen bonds broken is the same as the number formed, these hydrogen bonds do not contribute substantially to driving the overall process of double-helix formation. However, they contribute greatly to the specificity of binding. Suppose two bases that cannot form Watson–Crick base pairs are brought together. Hydrogen bonds with water must be broken as the bases come into contact. Because the bases are not complementary in structure, not all of these bonds can be simultaneously replaced by hydrogen bonds between the bases. Thus, the formation of a double helix between noncomplementary sequences is disfavored.

Third, within a double helix, the base pairs are parallel and stacked nearly on top of one another. The typical separation between the planes of adjacent base pairs is 3.4 Å, and the distances between the most closely approaching atoms are approximately 3.6 Å. This separation distance corresponds nicely to the van der Waals contact distance (Figure 1.14). Bases tend to stack even in single-stranded DNA molecules. However, the base stacking and associated van der Waals interactions are nearly optimal in a double-helical structure.

Fourth, the hydrophobic effect also contributes to the favorability of base stacking. More-complete base stacking moves the nonpolar surfaces of the bases out of water into contact with each other.

The principles of double-helix formation between two strands of DNA apply to many other biochemical processes. Many weak interactions contribute to the overall energetics of the process, some favorably and some unfavorably. Furthermore, surface complementarity is a key feature: when complementary surfaces meet, hydrogen-bond donors align with hydrogen-bond acceptors and nonpolar surfaces come together to maximize van der Waals interactions and minimize nonpolar surface area exposed to the aqueous environment. The properties of water play a major role in determining the importance of these interactions.

We can look at the formation of the double helix from a different perspective by examining the laws of thermodynamics. These laws are general principles that apply to all physical (and biological) processes. They are of great importance because they determine the conditions under which specific processes can or cannot take place. We will consider these laws from a general perspective first and then apply the principles that we have developed to the formation of the double helix.

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The laws of thermodynamics distinguish between a system and its surroundings. A system refers to the matter within a defined region of space. The matter in the rest of the universe is called the surroundings. The First Law of Thermodynamics states that the total energy of a system and its surroundings is constant. In other words, the energy content of the universe is constant; energy can be neither created nor destroyed. Energy can take different forms, however. Heat, for example, is one form of energy. Heat is a manifestation of the kinetic energy associated with the random motion of molecules. Alternatively, energy can be present as potential energy —energy that will be released on the occurrence of some process. Consider, for example, a ball held at the top of a tower. The ball has considerable potential energy because, when it is released, the ball will develop kinetic energy associated with its motion as it falls. Within chemical systems, potential energy is related to the likelihood that atoms can react with one another. For instance, a mixture of gasoline and oxygen has a large potential energy because these molecules may react to form carbon dioxide and water and release energy as heat. The First Law requires that any energy released in the formation of chemical bonds must be used to break other bonds, released as heat or light, or stored in some other form.

Another important thermodynamic concept is that of entropy, a measure of the degree of randomness or disorder in a system. The Second Law of Thermodynamics states that the total entropy of a system plus that of its surroundings always increases. For example, the release of water from nonpolar surfaces responsible for the hydrophobic effect is favorable because water molecules free in solution are more disordered than they are when they are associated with nonpolar surfaces. At first glance, the Second Law appears to contradict much common experience, particularly about biological systems. Many biological processes, such as the generation of a leaf from carbon dioxide gas and other nutrients, clearly increase the level of order and hence decrease entropy. Entropy may be decreased locally in the formation of such ordered structures only if the entropy of other parts of the universe is increased by an equal or greater amount. The local decrease in entropy is often accomplished by a release of heat, which increases the entropy of the surroundings.

We can analyze this process in quantitative terms. First, consider the system. The entropy (S) of the system may change in the course of a chemical reaction by an amount ΔS_system. If heat flows from the system to its surroundings, then the heat content, often referred to as the enthalpy (H), of the system will be reduced by an amount ΔH_system. To apply the Second Law, we must determine the change in entropy of the surroundings. If heat flows from the system to the surroundings, then the entropy of the surroundings will increase. The precise change in the entropy of the surroundings depends on the temperature; the change in entropy is greater when heat is added to relatively cold surroundings than when heat is added to surroundings at high temperatures that are already in a high degree of disorder. To be even more specific, the change in the entropy of the surroundings will be proportional to the amount of heat transferred from the system and inversely proportional to the temperature (T) of the surroundings. In biological systems, T [in kelvins (K), absolute temperature] is usually assumed to be constant. Thus, a change in the entropy of the surroundings is given by

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The total entropy change is given by the expression

Substituting equation 1 into equation 2 yields

Multiplying by −T gives

The function −TΔS has units of energy and is referred to as free energy or Gibbs free energy, after Josiah Willard Gibbs, who developed this function in 1878:

The free-energy change, ΔG, will be used throughout this book to describe the energetics of biochemical reactions. The Gibbs free energy is essentially an accounting tool that keeps track of both the entropy of the system (directly) and the entropy of the surroundings (in the form of heat released from the system).

Recall that the Second Law of Thermodynamics states that, for a process to take place, the entropy of the universe must increase. Examination of equation 3 shows that the total entropy will increase if and only if

Rearranging gives TΔS_system. ΔH or, in other words, entropy will increase if and only if

Thus, the free-energy change must be negative for a process to take place spontaneously. There is negative free-energy change when and only when the overall entropy of the universe is increased. Again, the free energy represents a single term that takes into account both the entropy of the system and the entropy of the surroundings.

Let us see how the principles of thermodynamics apply to the formation of the double helix (Figure 1.15). Suppose solutions containing each of the two single strands are mixed. Before the double helix forms, each of the single strands is free to translate and rotate in solution, whereas each matched pair of strands in the double helix must move together. Furthermore, the free single strands exist in more conformations than possible when bound together in a double helix. Thus, the formation of a double helix from two single strands appears to result in an increase in order for the system, that is, a decrease in the entropy of the system.

On the basis of this analysis, we expect that the double helix cannot form without violating the Second Law of Thermodynamics unless heat is released to increase the entropy of the surroundings. Experimentally, we can measure the heat released by allowing the solutions containing the two single strands to come together within a water bath, which here corresponds to the surroundings. We then determine how much heat must be absorbed by the water bath or released from it to maintain it at a constant temperature. This experiment reveals that a substantial amount of heat is released—namely, approximately 250 kJ mol⁻¹ (60 kcal mol⁻¹). This experimental result reveals that the change in enthalpy for the process is quite large, −250 kJ mol⁻¹, consistent with our expectation that significant heat would have to be released to the surroundings for the process not to violate the Second Law. We see in quantitative terms how order within a system can be increased by releasing sufficient heat to the surroundings to ensure that the entropy of the universe increases. We will encounter this general theme again and again throughout this book.

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Throughout our consideration of the formation of the double helix, we have dealt only with the noncovalent bonds that are formed or broken in this process. Many biochemical processes entail the formation and cleavage of covalent bonds. A particularly important class of reactions prominent in biochemistry is acid–base reactions.

In acid and base reactions, hydrogen ions are added to molecules or removed from them. Throughout the book, we will encounter many processes in which the addition or removal of hydrogen atoms is crucial, such as the metabolic processes by which carbohydrates are degraded to release energy for other uses. Thus, a thorough understanding of the basic principles of these reactions is essential.

A hydrogen ion, often written as H⁺, corresponds to a proton. In fact, hydrogen ions exist in solution bound to water molecules, thus forming what are known as hydronium ions, H₃O⁺. For simplicity, we will continue to write H⁺, but we should keep in mind that H⁺ is shorthand for the actual species present.

The concentration of hydrogen ions in solution is expressed as the pH. Specifically, the pH of a solution is defined as

pH = −log[H⁺]

where [H⁺] is in units of molarity. Thus, pH 7.0 refers to a solution for which −log[H⁺] = 7.0, and so log[H⁺] = −7.0 and [H⁺] = 10^log[H+] = 10^−7.0 = 1.0 × 10⁻⁷ M.

The pH also indirectly expresses the concentration of hydroxide ions, [OH⁻], in solution. To see how, we must realize that water molecules can dissociate to form H⁺ and OH⁻ ions in an equilibrium process.

The equilibrium constant (K) for the dissociation of water is defined as

K = [H⁺][OH⁻]/[H₂O]

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and has a value of K = 1.8 × 10⁻¹⁶. Note that an equilibrium constant does not formally have units. Nonetheless, the value of the equilibrium constant given assumes that particular units are used for concentration (sometimes referred to a standard states); in this case and in many others, units of molarity (M) are assumed.

The concentration of water, [H₂O], in pure water is 55.5 M, and this concentration is constant under most conditions. Thus, we can define a new constant, K_W:

Because K_W = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴, we can calculate

[OH⁻] = 10⁻¹⁴/[H⁺] and [H⁺] = 10⁻¹⁴/[OH⁻]

With these relations in hand, we can easily calculate the concentration of hydroxide ions in an aqueous solution, given the pH. For example, at pH = 7.0, we know that [H⁺] = 10⁻⁷ M and so [OH⁻] = 10⁻¹⁴/10⁻⁷ = 10⁻⁷ M. In acidic solutions, the concentration of hydrogen ions is higher than 10⁻⁷ and, hence, the pH is below 7. For example, in 0.1 M HCl, [H⁺] = 10⁻¹ M and so pH = 1.0 and [OH⁻] = 10⁻¹⁴/10⁻¹ = 10⁻¹³ M.

The reaction that we have been considering between two strands of DNA to form a double helix takes place readily at pH 7.0. Suppose that we take the solution containing the double-helical DNA and treat it with a solution of concentrated base (i.e., with a high concentration of OH⁻). As the base is added, we monitor the pH and the fraction of DNA in double-helical form (Figure 1.16). When the first additions of base are made, the pH rises, but the concentration of the double-helical DNA does not change significantly. However, as the pH approaches 9, the DNA double helix begins to dissociate into its component single strands. As the pH continues to rise from 9 to 10, this dissociation becomes essentially complete. Why do the two strands dissociate? The hydroxide ions can react with bases in DNA base pairs to remove certain protons. The most susceptible proton is the one bound to the N-1 nitrogen atom in a guanine base.

Proton dissociation for a substance HA (such as that bound to N-1 on guanine) has an equilibrium constant defined by the expression

K_a = [H⁺][A⁻]/[HA]

The susceptibility of a proton to removal by reaction with a base is often described by its pK_a value:

pK_a = −log(K_a)

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When the pH is equal to the pK_a, we have

pH = pK_a

and so

−log[H⁺] = −log([H⁺][A⁻]/[HA])

and

[H⁺] = [H⁺][A⁻]/[HA]

Dividing by [H⁺] reveals that

1 = [A⁻]/[HA]

and so

[A⁻] = [HA]

Thus, when the pH equals the pK_a, the concentration of the deprotonated form of the group or molecule is equal to the concentration of the protonated form; the deprotonation process is halfway to completion.

The pK_a for the proton on N-1 of guanine is typically 9.7. When the pH approaches this value, the proton on N-1 is lost (Figure 1.16). Because this proton participates in an important hydrogen bond, its loss substantially destabilizes the DNA double helix. The DNA double helix is also destabilized by low pH. Below pH 5, some of the hydrogen bond acceptors that participate in base-pairing become protonated. In their protonated forms, these bases can no longer form hydrogen bonds and the double helix separates. Thus, acid–base reactions that remove or donate protons at specific positions on the DNA bases can disrupt the double helix.

These observations about DNA reveal that a significant change in pH can disrupt molecular structure. The same is true for many other biological macromolecules; changes in pH can protonate or deprotonate key groups, potentially disrupting structures and initiating harmful reactions. Thus, systems have evolved to mitigate changes in pH in biological systems. Solutions that resist such changes are called buffers. Specifically, when acid is added to an unbuffered aqueous solution, the pH drops in proportion to the amount of acid added. In contrast, when acid is added to a buffered solution, the pH drops more gradually. Buffers also mitigate the pH increase caused by the addition of base and changes in pH caused by dilution.

Compare the result of adding a 1 M solution of the strong acid HCl drop by drop to pure water with adding it to a solution containing 100 mM of the buffer sodium acetate (Na⁺CH₃COO⁻; Figure 1.17). The process of gradually adding known amounts of reagent to a solution with which the reagent reacts while monitoring the results is called a titration. For pure water, the pH drops from 7 to close to 2 on the addition of the first few drops of acid. However, for the sodium acetate solution, the pH first falls rapidly from its initial value near 10, then changes more gradually until the pH reaches 3.5, and then falls more rapidly again. Why does the pH decrease so gradually in the middle of the titration? The answer is that, when hydrogen ions are added to this solution, they react with acetate ions to form acetic acid. This reaction consumes some of the added hydrogen ions so that the pH does not drop. Hydrogen ions continue reacting with acetate ions until essentially all of the acetate ion is converted into acetic acid. After this point, added protons remain free in solution and the pH begins to fall sharply again.

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We can analyze the effect of the buffer in quantitative terms. The equilibrium constant for the deprotonation of an acid is

K_a = [H⁺][A⁻]/[HA]

Taking logarithms of both sides yields

log(K_a) = log([H⁺]) + log([A⁻]/[HA])

Recalling the definitions of pK_a and pH and rearranging gives

pH = pK_a + log([A⁻]/[HA])

This expression is referred to as the Henderson–Hasselbalch equation.

We can apply the equation to our titration of sodium acetate. The pK_a of acetic acid is 4.75. We can calculate the ratio of the concentration of acetate ion to the concentration of acetic acid as a function of pH by using the Henderson–Hasselbalch equation, slightly rearranged.

At pH 9, this ratio is 10^9−4.75 = 10^4.25 = 17,800; very little acetic acid has been formed. At pH 4.75 (when the pH equals the pK_a), the ratio is 10^4.75−4.75 = 10⁰ = 1. At pH 3, the ratio is 10^3−4.75 = 10^−1.25 = 0.02; almost all of the acetate ion has been converted into acetic acid. We can follow the conversion of acetate ion into acetic acid over the entire titration (Figure 1.18). The graph shows that the region of relatively constant pH corresponds precisely to the region in which acetate ion is being protonated to form acetic acid.

From this discussion, we see that a buffer functions best close to the pK_a value of its acid component. Physiological pH is typically about 7.4. An important buffer in biological systems is based on phosphoric acid (H₃PO₄). The acid can be deprotonated in three steps to form a phosphate ion.

At about pH 7.4, inorganic phosphate exists primarily as a nearly equal mixture of H₂PO₄⁻ and HPO₄²⁻. Thus, phosphate solutions function as effective buffers near pH 7.4. The concentration of inorganic phosphate in blood is typically approximately 1 mM, providing a useful buffer against processes that produce either acid or base. We can examine this utility in quantitative terms with the use of the Henderson–Hasselbalch equation. What concentration of acid must be added to change the pH of 1 mM phosphate buffer from 7.4 to 7.3? Without buffer, this change in [H⁺] corresponds to a change of 10^−7.3 −10^−7.4 M = (5.0 × 10⁻⁸ −4.0 × 10⁻⁸) M = 1.0 × 10⁻⁸ M. Let us now consider what happens to the buffer components. At pH 7.4,

[HPO₄²⁻]/[H₂PO₄⁻] = 10^7.4−7.21 = 10^0.19 = 1.55

The total concentration of phosphate, [HPO₄²⁻] + [H₂PO₄⁻], is 1 mM, Thus,

[HPO₄²⁻] = (1.55/2.55) × 1 mM = 0.608 mM

and

[H₂PO₄⁻] = (1/2.55) × 1 mM = 0.392 mM

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At pH 7.3,

[HPO₄²⁻]/[H₂PO₄⁻] = 10^7.3−7.21 = 10^0.09 = 1.23

and so

[HPO₄²⁻] = (1.23/2.23) = 0.552 mM

and

[H₂PO₄⁻] = (1/2.23) = 0.448 mM

Thus, (0.608 −0.552) = 0.056 mM HPO₄²⁻ is converted into H₂PO₄⁻, consuming 0.056 mM = 5.6 × 10⁻⁵ M [H⁺]. Thus, the buffer increases the amount of acid required to produce a drop in pH from 7.4 to 7.3 by a factor of 5.6 × 10⁻⁵/1.0 × 10⁻⁸ = 5600 compared with pure water.