CHAPTER 1. ACIDS, BASES, SALTS, BUFFERS

The following is in large part a review of material presented in a beginning college inorganic chemistry course. An understanding of this material is extremely important for students of biochemistry, biology, and chemistry, and therefore, it is presented here in summary form. Most of this material will not be covered in lecture but a thorough understanding of it will be assumed for the course. The problems in this section and the other sections are assigned at the time that section of reading is assigned. We assume you are able to solve these and similar problems. A (461) behind a problem number means the problem was given in Biochemistry 461.

If you wish to review the subjects in this section in greater depth, study the appropriate chapters in your inorganic chemistry text before coming to lecture. For those students that have used General Chemistry by P.W. Adkins the chapters to review are: 11 (first part) and 13, but particularly 14 and 15. For those students that have used General Chemistry, 2ed ed. by D.D. Ebbing review chapters 10, 16, 17 and 18. If you have used other texts, look at the chapters on: (1) Reactions in aqueous solutions, (2) Chemical equilibrium, (3) Acid-base concepts and (4) Acid-base equilibria. If you are uncertain about what chapters to review, bring in your book and ask a faculty instructor.

SOME BASIC DEFINITIONS

1. substance - matter that cannot be separated into other kinds of matter by a physical process.

The materials that surround us in the world are substances or mixtures of substance. Substances have definite physical and chemical properties. Substances are either elements or compounds. Sodium, hydrogen, carbon dioxide and sodium chloride are substances. Sea water, air, milk, and wood are mixtures of substances.

2. compound - a substance that is composed of molecules or ions held together by chemical bonds and is electrically neutral. Chemical bonds consist of ionic bonds and covalent bonds, including delocalized bonds. A compound always contains a definite (constant) proportion of its elements by mass and consists of either molecules or ions. Examples of compounds are water, carbon dioxide, sucrose and sodium chloride (and other salts) and inorganic acids and bases such as HCl and NaOH.

3. molecule - a substance that is composed of two or more atoms that are bonded chemically by strong forces (for example ionic and covalent bonds) and is not electrically charged through addition or removal of one or more electrons.

Most compounds and some elements consist of molecules. Salts do not consist of molecules. Water, carbon dioxide, nitrogen (N2) and oxygen (O2) are examples of molecules. Sodium chloride is not an example of a molecule. Sodium chloride and other salts are compounds consisting of ions rather than molecules.

4. salt - an ionic compound containing an anion other than OH- and O2-.

5. ion - an atom or a group of chemically combined atoms that is electrically charged through addition or removal of one or more electrons. Examples are sodium ion (Na+), chloride ion (Cl-), ferric ion (Fe3+), sulfate ion (SO42-), and hydride ion (H-).

6. Flow diagram of terms

7. molecular weight and formula weight Many compounds consist of molecules and are called molecular compounds. Many other compounds consist of ions (not molecules) and are called ionic compounds. The term molecular weight (M.W.) is used for molecules and molecular compounds. It is incorrect to use the term molecular weight for an ionic compound. The formula of a molecule gives the type and number of each atom present in the molecule. Although ionic compounds do not consist of molecules, we still speak of the smallest unit of an ionic compound which is called the formula unit rather than the formula. The formula unit of an ionic compound states the type of different ions present, written with no charge symbol, and the number of each ion, expressed as the ratio of the smallest whole number integers possible. For example, the formula unit for lithium bromide is LiBr, not Li+Br-. The formula unit for disodium hydrogen phosphate is Na2HPO4, not Na4H2(PO4)2. The formula weight (F.W.) is the sum of the atomic weights of the atoms of the formula unit of compound whether molecular or not.

8. atomic mass unit (amu)(also called dalton particularly by biochemists and biologists) - the mass of 1/12 the mass of a carbon-12 atom.

The mass of one (1) atom of carbon-12 is 1.99268 x 10-23g.

9. atomic weight (properly the term should be atomic mass but for historical reasons atomic weight is still used) - the average mass of an atom in an element (contains a mixture of isotopes) expressed in amu.

Hydrogen (mixture of isotopes) has an atomic weight (mass) of 1.00794 amu. When speaking of a specific isotope the term atomic mass is now being used. Hydrogen-2 (deuterium) has an atomic mass of 2 amu.

10. molecular weight - Two definitions for molecular weight are used:

- - a. molecular weight is the sum of the atomic weights (masses) of the atoms of a molecule.
- - b. molecular weight is the ratio of the molecular mass of the substance to 1/12 the mass of carbon-12. The symbol used for this definition is Mr and being a ratio it does not have units (this definition is used particularly by biochemists and biologists).

11. molecular mass - the sum of the atomic weights (masses) of the atoms of a molecule expressed in amu (daltons). This term has been used commonly only recently, particularly by biochemists and biologists.

THE CONCEPT OF MOLE

A mole is an amount of a substance, also it is a specific number of molecules, formula units, ions, etc. of a substance. A practical definition for a mole is that one mole of a substance is the molecular mass (weight) or formula weight of the substance expressed in grams. The formal definition for mole is that it is the amount of a substance that contains exactly the number of molecules or formula units as the number of atoms in 12 g of carbon-12 (12C). This number is 6.02 x 1023 and is called Avogadro's number. One mole of a molecular compound therefore contains 6.02 x 1023 molecules or Avogadro's number of molecules. The number of moles in a particular mass of a substance is:

____________________________mass of a substance_____________________________________  =  moles
molecular mass (weight) or formula weight of a substance expressed in grams (g/mole)

In practice, the term mole is also commonly used with other species such as ions, elements, electrons, and so forth. For example, one gram-atom of sodium is often called a mole of sodium and contains 6.02 x 1023 atoms. The species under discussion must be specifically stated. For example, one gram-atom of nitrogen contains one mole of nitrogen atoms (N) but 1/2 mole of N2 molecules. Therefore it must be stated whether the nitrogen atom or the N2 molecule is the species under discussion.

CONCENTRATIONS BASED ON THE VOLUME OF SOLVENT

1. Molarity (M) is the number of moles of solute dissolved in one liter of solution. Molarity is also called molar concentration. The molarity of a solution can be calculated when the number of grams of the substance in the solution, the volume of the solution, and the molecular weight or formula weight of the substance are known. Conversely, the number of grams of the substance needed to prepare a solution of a particular molarity can be calculated if the final volume of the solution is defined and the molecular weight or formula weight of the substance is known. Dilute solutions are named by using the appropriate prefix. For example:

10-3M = 1mM (millimolarity) = 1 mmole L-1 = 1µmole mL-1

10-6M = 1µM = 1 µmole L-1 = 1 nmole mL-1

The equation defining molarity is:

molarity (M) = moles of solute / liters of solution

or rearranged

molarity (M) x liters of solution = moles of solute

The following are true for a solution that is diluted. Before dilution

molarity (M1) x liters (V1) = moles (m1)
M1,V1 = molarity, volume before dilution

After dilution (number of moles does not change)

M2 x V2 = m1
M2,V2 = molarity, volume after dilution

therefore

M1 x V1 = M2 x V2

Any units can be used for volume and molarity but V1 and V2 must be in the same units as must M1 and M2.

2. Normality (N) is the number of equivalents of a solute dissolved in one liter of solution. For an acid or base, an equivalent is the molecular weight or formula weight of the acid or base expressed in grams divided by the number of moles of hydronium (H3O+) or hydroxyl ions produced by this amount of acid or base. For a compound that undergoes oxidation or reduction, an equivalent is the molecular weight or formula weight of the reductant or oxidant expressed in grams divided by the number of faradays of electrons released or accepted by this amount of reductant or oxidant. A faraday is a mole of electrons, i.e. 6.02 x 1023 electrons. Equivalents can be calculated from moles.

equivalents = moles / n

n = number of hydronium or hydroxyl ions produced from one molecule (ion) of an acid or base
or
n = number of electrons released or accepted from one molecule (ion) of an reductant or oxidant.

The term equivalent weight is used and related to molecular weight and formula weight by

equivalent weight = molecular weight / n = formula weight / n

An equivalent is also defined as the equivalent weight expressed in grams. The number of equivalents in a certain mass of a substance is calculated from:

equivalents = mass (grams) of a substance / equivalent weight (g/eqiv.)

Normality and molarity are related by the formula, N = nM, where n has the meanings stated above.

3. Percent based on volume.

percent, weight/volume (%, w/v) -- the weight in grams of a solute dissolved in 100 mL of solution

percent, volume/volume (%, v/v) -- the volume in mL of a solute dissolved in 100 mL of solution

4. Osmolarity is the number of moles of particles in one liter of solution. A 1 M NaCl solution is 2 Osmolar. A 1 M D-glucose solution is 1 Osmolar. A 1 M Na3PO4 solution containing NaCl at 1 M concentration is 6 Osmolar.

5. Ionic strength is a measure of the intensity of the charge (electric field) in solution due to the ions present.

Mi - the molarity of the ith ion

zi - the net electrical charge on the ith ion

The equation was established empirically by Lewis and Randell (1921). Ionic strength is related to and sometimes equal to the molar concentration of charge in solution. The equation deemphasizes the lower charged species and emphasizes the higher charged ones which takes into account what was observed experimentally during the development of the formula. The theoretical explanation or justification for ionic strength is given by the Debye-Hückel theory.

Problem 1: Calculate the ionic strength and the total molar concentration of charge of:

(a) 0.05 M NaCl, (b) 0.05 M Mg3(PO4)2.

EXPRESSIONS OF CONCENTRATION BASED ON THE WEIGHT OF SOLVENT

1. Percent based on weight

percent, weight/weight (%, w/w) -- the weight in grams of a solute dissolved in 100 g of solution.

The concentration of many solutions is expressed as percent (w/w). Commercial strong acids and bases are expressed in this way. The volume amount of these solutions needed to prepare a solution of known molarity can be calculated if the specific gravity is known. For commercial strong acids and bases, the specific gravity is printed on the label. Specific gravity is the ratio of the weight of a substance to the weight of an equal volume of water. Since it is a ratio, it does not have units. Another way of stating this definition is that specific gravity is the ratio of the density of a substance to the density of water. The density of water is 1 when the units, gm/cm3 (mL), are used. Therefore, the numerical value for specific gravity and density are the same when these units are used. Density (d, also ) is mass per unit volume.

Problem 2: How many milliliters of HCl [37.0% (w/w); specific gravity, 1.18] are needed to prepare 500 mL of 0.16 M HCl?

percent, volume/weight (%, v/w) is the volume in mL of solute dissolved in 100 g of solution.

2. molality (m) -- the number of moles of a solute dissolved in 1000 g of solvent

3. mole fraction -- the number of moles of a solute in solution divided by the total number of moles of all the substances comprising the solution.

ACTIVITY OF ELECTROLYTES

Many substances, for example those that yield ions in solution, act as though they are at a different concentration in solution than their actual concentration. This phenomenon has led to the term activity. Activity is important not only for electrolytes in solutions but also for other substances such as gases. The activity of an electrolyte is the effective or apparent concentration of an electrolyte in aqueous solution.

The relationship between activity and concentration is

a = c or = a / c

a = activity
= activity coefficient
c = concentration

The activity coefficient, a ratio, is the fraction of the total amount of the electrolyte in solution that is active (effective). Often only a fraction of the total ions of an electrolyte in solution are active. For example, consider a 0.1 M HCl solution. The pH meter measures the activity, not the concentration, of hydronium ion in solution. A 0.1 M HCl solution should have a pH of 1.0 but when measured with a pH meter the pH is 1.07. This is because the activity of the H3O+ is less than the concentration. The activity coefficient in this case is 0.86. Activity is the result of the interaction of the ions in solution with each other, or with the solvent, or both. The interactions render a fraction of the total ions ineffective. Activity is detected when the ions in solution are measured by certain methods (e.g. with electrodes). Conditions are called ideal when activity is equal to concentration.

EQUILIBRIUM CONSTANT

K (sometimes Keq) is the symbol for the equilibrium constant under ideal conditions. This equilibrium constant is often referred to as the thermodynamic or true equilibrium constant. K is the equilibrium constant obtained when activities are used to calculate the equilibrium constant (even when non-ideal conditions exist).

- - (1) Ideal conditions are where there are no interactions between the species (ions, molecules) in solution and other species in solution and between the species and the solvent (usually water in biological systems). Often only at infinite dilution are there no interactions of this type. Under ideal conditions, activity equals concentration.

- - (2) Non-ideality is usually the case in experimental research conditions. When this is the case, then K is obtained by extrapolation from the non-ideal conditions.

- - (3) K is reported and used because, by formal agreement, it serves as the standard for reporting equilibrium constants and, consequently, results from different labs can be directely compared.

- - (4) The equilibrium constant reported in reference books and tables and used in other books is K unless it is stated otherwise.

K' is the symbol for the equilibrium constant under non-ideal conditions. This equilibrium constant is sometimes referred to as the apparent equilibrium constant. K' is the equilibrium constant obtained when concentrations are used to calculate the equilibrium constant and non-ideal conditions exist.

- - (1) Non-ideal conditions are often the conditions under which experimental work is performed. Non-ideality occurs because of the interactions stated above.

- - (2) Since an infinite number of experimental conditions are possible for a reaction and, therefore, an infinite number of K' values are possible, it is not surprising that equilibrium constants are reported at infinite dilution.

Other Symbols

- - (1) Ki - equilibrium constant for an ionization reaction

- - (2) Kc - equilibrium constant (c means equilibrium concentrations are used)

- - (3) Kp - equilibrium constant (p means equilibrium pressures are used)

- - (4) Ka - Ka (a = acid) and Kb (b = base)

- - (5) Kb are not true equilibrium constants, but are related to the corresponding equilibrium constant by the value for the concentration of water in an aqueous system. For example, K[H2O] = Ka. Ka and Kb are often called dissociation constants and are treated as equilibrium constants.

(6) Kw - Kw (w = water) is related to K by the [H2O]2. Kw has the name, ion product of water or simply ion product, and is treated as an equilibrium constant.

ACIDS AND BASES, GENERAL

Arrhenius defined an acid as a substance that increases the concentration of hydronium ion, H3O+ (usually the designation hydrogen ion or proton, H+, is used) when dissolved in water while a base is a substance that increases the concentration of hydroxyl ion when dissolved in water. The hydrogen ion, H+ does not exist in aqueous solution but is bonded to water to form the hydronium ion. In 1923 Bronsted and Lowry developed a more general definition for acids and bases. They were more general because they applied to non-aqueous as well as aqueous systems. Bronsted-Lowry defined an acid as a substance that donates a proton in a reaction while a base is a substance that accepts a proton in a reaction. The definition of Bronsted-Lowry will be used throughout. Acids and bases are classified as strong and weak.

When a Bronsted-Lowry acid donates a proton, a Bronsted-Lowry base is formed from this acid. For example, when the Bronsted-Lowry acid, NH4+ donates a proton to water the Bronsted-Lowry base, NH3 is formed.

NH4+ + H2O <==> NH3 + H3O+

Similarly when a Bronsted-Lowry base accepts a proton, a Bronsted-Lowry acid is formed. The Bronsted-Lowry acid/base pair are joined together (conjugated) in the sense that each is formed from the other through the addition or removal of a proton. Such an acid/base pair is called a conjugate acid/base pair and consists of a conjugate acid and conjugate base. In the example cited, one conjugate acid/base pair is NH4+/NH3. The reaction of a Bronsted-Lowry acid or base in water always involves at least two conjugate acid/base pairs. In the reaction above, the two conjugate acid/base pairs are NH4+/NH3 and H3O+/H2O.

IONIZATION OF WATER

Understanding how solutes such as weak acid and bases behave in water first requires an understanding of the ions present in pure water and their interrelationships. A very small portion of the water molecules in pure liquid water react with each other as follows:

H2O + H2O <==> H3O+ + OH-

The reaction results in the formation of both an acid, H3O+, whose conjugate base is H2O and a base, OH-, whose conjugate acid is H2O. Water acts as both an acid and base as defined by Bronsted-Lowry. The equilibrium expression for this reaction is

The activity of H3O+ and OH- in pure water have been determined experimentally to be 1.0 x 10-7M. Now one liter of pure water basically weighs 1000g at 20°C and therefore the concentration of water is

(1000g/18.02(M.W.) / 1 liter = 55.5 M

In pure water the concentration of water is the same as the activity of water therefore

Ki = (1 x 10-7)2 / (55.5)2 = 3.25 x 10-18

However by convention the activity of pure water is set equal to one and a new constant is defined. The expression is

Kw = (aH3O+)(aOH-)

where Kw is called the ion product of water. Virtually all aqueous solution of biological and chemical interest contain some electrolytes which lower the activity of water. However as long as their concentration is not too great, aH2O is not significantly different from unity. For example, for a 0.1 N solution of a strong electrolyte, aH2O is about 0.996; for a 1.0 N solution, aH2O is approximately 0.97. In dilute solution activities are approximately equal to concentrations (except for aH2O which, as already stated, is set equal to one) therefore

Kw = (aH3O+)(aOH-) = [H3O+] [OH-]

At 25°C the value of Kw for pure water is 1 x 10-14. Kw is considered an equilibrium constant and consequently varies with temperature. At 37°C it is 2.5 x 10-14.

pH AND pOH

The terms pH and pOH were devised as a means of expressing H3O+ and OH- activity without using exponents. pH is the negative logarithm of the H3O+ activity. The original meaning of "p" was "potential", in this case potential of H3O+.

Similarly, pOH is the negative logarithm of the OH- activity.

In dilute solution the activities of H3O+ and OH- are considered to be the same as their concentrations. Therefore in dilute solutions:

pH = -log[H3O+] and pOH = -log[OH-]

The ion product expression for the ionization of water may be rearranged as follows to yield a useful equation.

Kw = [H3O+] [OH-]

Substitute 1 x 10-14 for Kw at 25°C and take logarithms.

log[H3O+] + log[OH-] = -14

multiply by -1

-log[H3O+] + (- log[OH-]) = 14

therefore

pH + pOH = 14

Problem 3 (461): Calculate the H3O+ ion concentration and the OH- concentration in solutions having pH values of (a) 2.73, (b) 5.29, (c) 8.65, (d) 11.41, and (e) 0.0.

Problem 4 (461): Lake Michigan water has a pH of 6.2. What is the molar concentration of H3O+ in the lake?

Problem 5 (461): Lake Lansing water has a pH of 5.8. What is the molar concentration of H3O+ in the lake. What is the molar concentration of OH- in the lake? How many times more acidic is Lake Lansing than Lake Michigan?

Problem 6 (461): Lake Lansing for all practical purposes is 0.75 miles in diameter and, on average, 8 feet deep. How many pounds of NaOH would have to be added to the lake to adjust its pH to pH 6.2, the pH of Lake Michigan?

Problem 7 (461): Vinegar is approximately 4.5% (v/v) aqueous acetic acid [sp. gr. 1.05, purity = 99.6% w/w]. How much NaOH would have to be added to 100 mL of vinegar to completely neutralize the acid?

STRONG ACIDS AND BASES

Strong acids are acids that ionize completely in water. Examples are HCl, HBr, HI, H2SO4, HNO3 and HClO4.

HCl + H2O ==> H3O+ + Cl-

Strong bases are bases that ionize completely in water. Examples are NaOH, KOH, Ba(OH)2, Sr(OH)2, Ca(OH)2, and Mg(OH)2.

KOH ==> K+ + OH-

The single arrows (==>) indicate the reverse reactions are not significant for our purposes.

Problem 8 (461): How many mL of 0.12 M H2SO4 are required to neutralize exactly half of the OH- ions present in 540 mL of 0.18 N NaOH?

WEAK ACIDS AND BASES

Weak acids are acids that ionize only partly in water. An example is acetic acid.

CH3COOH + H2O <===> CH3COO- + H3O+

Weak bases are bases that ionize only partly in water. An example is ammonia.

NH3 + H2O <===> NH4+ + OH-

There is no clear demarcation line between strong and weak acids and between strong and weak bases. Rather there is a continuum in the strengths of each. For our purposes, the following definitions will be used:

- - weak acid - any acid that ionizes less than 10% in water at 0.5 M (for many weak acids, the percent ionization is 1 or 2%, or less). Based on this definition, weak acids have pKa's from 2.27 - 14.

- - weak base - any base that ionizes less than 10% in water at 0.5 M (for many, the percent ionization is 1 or 2%, or less). Based on this definition, weak bases have pKb's from 2.27-14.

These definitions for weak acid and base are somewhat arbitrary. There is general agreement that weak acids are those with pKa's between about 2 and 14, and weak bases are those with pKb's between about 2 and 14.

An equilibrium expression can be written for any weak acid, and one can be written for any weak base. These expressions are useful for obtaining information about a system involving the weak acid or weak base in aqueous solutions. For example, the degree of ionization of a weak acid or base and the ratio of conjugate base/conjugate acid can be obtained. These expressions are derived.

Acids when dissolved in water produce hydronium ions. Let HA represent any weak acid. In water, the following occurs:

HA + H2O <===> A- + H3O+

The equilibrium expression for the ionization of the weak acid is (assume concentration equals activity)

Since the [H2O] remains essentially constant in the solution when ionization occurred, by convention, it is combined with Ki to give a new constant called Ka, where "a" refers to acid.

Ka is called a dissociation constant since HA appears to dissociate into A- and H+. However, dissociation of HA to A- and H+ does not actually occur in solution but rather HA reacts with H2O as shown in the above reaction. Ka is considered as an equilibrium constant.

Bases produce hydroxyl ions when dissolved in water. In water, some bases release OH- without reaction with water while others react with water to form OH-. For example

NaOH in water releases Na+ + OH- into the solution

On the other hand, methylamine ionizes in water as follows:

CH3NH2 + H2O <===> CH3NH3+ + OH-

The following equilibrium expression can be written for bases that contain an amino group.

Combining Ki and [H2O] for the same reason given above under weak acids gives a new constant, Kb, called the dissociation constant of the base, where "b" refers to base.

Although this is the expression for a base containing an amino group, the analogous expression for any base producing an OH- is the same. ROH is any base and the expression is

There is an important relationship between the Ka and Kb of any conjugate acid/base pair. Let HA represent any acid. In water it ionizes as follows and a Ka can be determined.

HA + H2O <===> A- + H3O+

The conjugate base of HA, namely A-, when placed in water reacts as follows and a Kb value can be determined.

A- + H2O <===> HA + OH-

Recall that [H3O+][OH-] = 1 x 10-14. Therefore, for any particular acid, HA, and its corresponding conjugate base, A-, the following expression is valid:

Cancelling gives

Ka x Kb = 1 x 10-14

Take logarithms and multiply by -1 to give

-log Ka + (-log Kb) = 14

The definitions for pKa and pKb are: pKa = -log Ka; pKb = -log Kb. Substituting gives

pKa + pKb = 14

This expression holds true for any conjugate acid/base pair. For example, the pKa of NH4+ is 9.26 while the pKb of NH3 is 4.74. The sum of the two is 14 as expected since NH4+/NH3 are a conjugate acid/base pair. This expression is particularly useful for calculating the pK of one member of a conjugate acid/base when the pK of the other member is known. Therefore, once you have determined the pKa or pKb of one member of a conjugate acid/base pair, you automatically have determined the pK value of the other member. Ka, Kb, pKa, and pKb are all constants and lists of these constants can be found in reference books.

EXPERIMENTAL DETERMINATION OF Ka and Kb. Degree of Ionization of Weak Acids and Bases.

Ka and Kb can be determined experimentally by several methods. A common method is to determine the pH of a solution of the weak acid or base and calculate the Ka or Kb. Other methods use electrical conductivity or spectroscopic measurements or the measurement of a colligative property of the weak acid or base. Only the first method will be discussed.

When a solution of a weak acid or base of known concentration is prepared with pure water and the pH measured, the activity of the H3O+ in the solution can be measured with a pH meter. From this information, the K'a or K'b can be calculated. When pH measurements are made with a sufficient number of solutions of different concentration of the weak acid or base, the equilibrium constant at infinite dilution (Ka or Kb, no prime) is obtained through extrapolation. For precise determinations of Ka, activities must be used or experimental values of K'a must be extrapolated to infinite dilution. Ka and Kb values are used to calculate the degree of ionization (percent ionization) of a weak acid or base. The degree of ionization of a weak acid or base is the fraction of the total molecules of the acid or base that react with water to form hydronium or hydroxyl ions. Percent ionization is the degree of ionization expressed as a percent.

Problems involving weak acids and bases follow.

Problem 9: A weak acid has a Ka of 6.2 x 10-4. (a) What are the pH and percent ionization of the acid in a 3 x 10-3 M solution? (b) What is the pH and percent ionization of the acid when the solution in (a) is diluted 1000-fold?

Solution for (a): The equilibrium expression for the ionization is

at equilibrium: [H+] = [A-] = x
[HA] = 0.003 M - x

In problems of this type, the denominator may be simplified (from 0.003 M - x to 0.003 M) if the simplification does not significantly change the answers. The [H+] is normally significant to 3 digits. The only way to determine which way to solve the problem is to calculate [H+] both with and without the simplification until you have sufficient experience to judge when simplification is justified. Generally, if the acid or base is less than 10% ionized, simplification is justified. When simplification is not justified, the quadratic equation must be used.

With simplification: Substituting yields:

6.2 x 10-4 = x2 / 0.003 M
x = 0.00136 M = [H+] and pH = 2.87

Percent ionization : (0.00136 M / 0.003 M) x 100 = 45.3%

Without simplification: Substituting yields:

6.2 x 10-4 = x2 / 0.003 M - x

Rearranging: x2 + 6.2 x 10-4x + (-1.86 x 10-6) = 0 (a quadratic equation)
ax2 + bx + c = 0 (general quadratic equation)

Solve for x using the quadratic formula.

The answers clearly show that simplification is not justified and the quadratic formula must be used.

Problem 10 (461): A 0.05 M solution of the weak acid HA has a pH of 3.9 at 25°C. (a) What is the ionization constant of the acid at 25°C, the percent ionization of HA, and the ratio of A-/HA in the solution? (b) What is the pH of a 0.05 M HCl solution? (c) How much stronger in acidity is the HCl solution than the HA solution? (d) What is the Kb and pKb of the conjugate base, A-?

Problem 11 (461): A 0.02 M solution of an amine has a pH of 10.2 at 25°C. (a) Calculate the ionization constant of the amine and its conjugate acid, (b) the percent ionization of the amine and the ratio of conjugate base/conjugate acid in the solution, and (c) the amount of 0.045 M HCl required to neutralize 320 mL of the weak base solution.

Problem 12 (461): What is the pH of a 2.6 x 10-2 M amine (RNH2) solution? The protonated amine of the amine hydrochloride (RNH3Cl) has a pKa of 8.7.

Problem 13: What is the ionic strength of a 0.06 M formic acid (pKa = 3.75) solution?

SALTS

A salt is an ionic compound containing an anion other than OH- or O2-. Almost all salts dissolve completely in water at concentrations commonly used in biological research. A salt consists of ions both when in the solid form and when dissolved in water.

For many salts dissolved in water, one more of the ions react with water to form significant amounts of either H3O+ or OH-, that is, the ions act as weak acids or bases or both. The reaction is termed a hydrolysis. For example, when NH4Cl is dissolved in water the ions, NH4+ and Cl- react with water. A small fraction of the NH4+ reacts as follows:

NH4+ + H2O <===> NH3 + H3O+

Although only a small fraction reacts, it is sufficient to make the solution acidic. Cl- does not react to any significant extent.

Cl- + H2O ==> no significant reaction

Although many salts can act as weak acids or bases or both others do not, for example NaCl. Some salts contain more than one ion that acts as a weak acid or base in water, for example, (NH4)2CO3. NH4+ acts as a weak acid while CO32- acts as a weak base. Other salts contain an ion that acts as both an acid and a base, for example, (NH4)2HPO4. HPO42- acts as both a weak acid and a weak base. Whether the solution is acidic, basic, or neutral depends on the relative strength of the ions that act as acids or bases. This can be determined from the Ka and Kb values of the acids and bases.

Problem 14 (461): a. State whether a 0.05 M solution of each of the following will be acid, neutral or basic, explain why (use reactions) and write the structural formulas of the ions present in each solution: (a) H3PO4, (b) NaH2PO4, (c) Na2HPO4, (d) Na3PO4, (e) NH4Cl, (f) monosodium citrate, (g) phenol, (h) NaHCO3, (i) disodium succinate, (j) Tris(hydroxymet hyl)aminomethane.hydrochloride, (k) methylamine.

b. Calculate the pH of each solution.

Problem 15 (461): What are the final hydrogen ion concentration and pH of a solution obtained by mixing 100 mL of 0.2 M KOH with 150 mL of 0.1 M acetic acid (HAc)? (pKa of HAc = 4.77)

EFFECT OF OTHER SUBSTANCES ON THE EQUILIBRIUM CONCENTRATION OF SPECIES PRESENT WHEN WEAK ACIDS AND BASES IONIZE

The equilibrium concentrations of the ions present in a solution of a weak acid or base will be changed when an appropriate solute is added to the solution. One way the equilibrium is affected is through the common-ion effect. The common-ion effect is the shift in equilibrium of an ionic reaction caused by the addition of an ion that participates in the reaction. Consider the ionization of formic acid.

HCOOH + H2O <===> HCOO- + H3O+

Addition of HCl to this solution causes the concentration of formate ion (HCOO-) to decrease because H3O+ reacts with HCOO- to yield HCOOH and water. This reaction will proceed until equilibrium is again established. Some H3O+ also reacts with OH- to form water. When equilibrium is again established, the [H3O+] is higher than before addition of HCl. This occurs because a higher concentration of H3O+ must be present to maintain the more repressed ionization of formic acid. When HCl was added some of the H3O+ reacted with the formate ions and some reacted with OH-, in both cases to form H2O, but the majority remained in solution as H3O+ to maintain the new equilibria.

When the salt, sodium formate, is added to a solution of formic acid the equilibria are also disturbed. The concentration of H3O+ decreases because the formate ion reacts with H3O+ to yield formic acid and water. In this case, the pH of the solution increases, that is, becomes more basic. When the new equilibrium of this reaction is established, the concentration of formate is higher and that of H3O+ is lower than before the sodium formate was added. The solution became more basic because when the [H3O+] decreased, some H3O+ was produced from the ionization of water, 2H2O <==> H3O+ + OH-, but not enough to fully replace the H3O+ that reacted with formate ion and this resulted in a net increase in OH-.

A second way equilibrium concentrations will be changed is by removing a product. If NaOH is added to a solution of formic acid, the reaction involving HCOOH will proceed in the direction of formate ion formation because H3O+ are removed by reaction with OH-. The reactions are:

HCOOH + H2O <===> HCOO- + H3O+

H3O+ + OH- + Na+ <===> 2H2O + Na+

When the new equilibria are established, the concentration of H3O+ has decreased and, therefore, the concentration of OH- must have increased since the ion product of water (1 x 10-14) is a constant. In addition, the concentration of formate ion has increased while that of formic acid has decreased.

A third way equilibrium concentrations will change is by dilution, that is, by adding water. A weak acid, HAc, ionizes in water as follows

HA + H2O <===> A- + H3O+

When water is added to a solution of the weak acid, the equilibrium shifts to the right. One reason is because the concentration of water has increased. This is analogous to the common-ion effect, although water is not an ion. However this is a minor effect. The addition of water has a second more profound effect. Ka is dependent on the rate of the forward and reverse reactions and at equilibrium these two rates are equal. When water is added, the concentrations of all species in solution are decreased. In the forward direction, the rate of reaction is proportional to the product of [HA][H2O]. In the reverse direction, it is proportional to the product of [A-][H3O+]. The concentration of water itself, before and after dilution, does not change nearly as much as the concentration of the other species. Consequently in the forward reaction only the [HA] is changed substantially on dilution, while in the reverse direction both the [A-] and [H3O+] are changed. Therefore dilution decreases the rate of the reverse reaction much more than the rate of the forward reaction and the system is no longer at equilibrium. This is because the product of two smaller numbers will be considerably less than the product of one smaller and a number that does not change substantially, the [H2O]. The forward reaction now proceeds more rapidly than the reverse reaction until equilibrium is again achieved. This results in a net conversion of HA to A- plus H3O+ and therefore a greater ionization of HA. On further dilution, HA ionizes further until eventually ionization is 100%.

Dilution results in a decrease in the concentration of all ions including the [H3O+]. Although the degree of ionization of a weak acid increases on dilution, the pH does not decrease because the dilution effect on the [H3O+] is more significant than formation of H3O+ from increased ionization of the weak acid. On continued dilution, the pH of a solution of a weak acid approaches the pH of water even as the percent ionization approaches 100%. The above also holds true for weak bases but on dilution the pH decreases from an alkaline pH toward the pH of water as the percent ionization approaches 100%.

Besides changing the concentration of species, dilution also changes the activity of ionic species and this in turn affects the pH. The effect of changes in activity are detectable when the extent of dilution is relatively small, in general less than 10-fold. When dilution is greater than 10-fold, the dilution of species outweighs changes in activity. The effect of dilution on the activity of ions will be discussed further in class. Also recall that temperature changes equilibrium concentrations, and this results in a change in Keq and therefore a change in pKa and pKb.

The above discussion emphasizes that while Ka and Kb (and pKa and pKb) are constants at a particular temperature, the degree of ionization (percent ionization) of a weak acid or base is not a constant but varies depending on the conditions.

Problem 16: A solution of a weak acid HA (pKa = 5.8) was prepared by diluting 25 mL of a 0.03 M solution of HA to 50 mL (solution A). A second solution of HA was prepared by mixing 25 mL of 0.03 M HA with 20 mL of 0.02 M NaOH and diluting to 50 mL (solution B). A third solution of HA was prepared by mixing 25 mL of 0.03 M HA with 20 mL of 0.02 M HCl and diluting to 50 mL (solution C). What is the: (a) percent ionization of HA and (b) the ratio of A-/HA in solutions A, B, and C?

Problem 17: A solution of a weak acid HA (pKa = 4.1) was prepared by diluting 50 mL of a 0.1 M solution of HA to 100 mL (solution A). A second solution was prepared from 50 mL of 0.1 M HA, 5 g of the sodium salt of HA, namely NaA (F.W. = 63), and enough water to give a final volume of 100 mL (solution B). Calculate: (a) the concentration of HA and (b) the pH of solutions A and B.

Problem 18 (461): See problem 10. Dilute the 0.05 M HA solution, 1 to 1,000, 1 to 10,000, and 1 to 1 x 105 and calculate the percent ionization of HA and pH of each solution.

HENDERSON-HASSELBALCH EQUATION

Many biological compounds are weak acids or weak bases or both. Researchers often want to know what proportion of such biological compounds are in the conjugate acid and conjugate base forms at a particular pH. The Henderson-Hasselbalch equation relates these values. The derivation of this equation is described.

The equation that describes the dissociation reaction of any weak acid at equilibrium is:

This equation describes the ionization of a weak acid in pure water at equilibrium. The concentration of H+, A-, and HA are the concentrations present at equilibrium. It was shown earlier that the equilibrium concentrations of H+, A-, and HA in this reaction can be changed by adding a common ion, adding a strong acid or base, or adding water (dilution). In each of these situations a new equilibrium was established where the initial equilibrium concentrations of H+, A-, and HA have changed. However, even in these situations the above equilibrium expression is still valid because only H+, A--, and HA are involved and the system comes to equilibrium. The above equilibrium expression therefore has this more general meaning (that is, for situations involving a common ion, a strong acid or base, or dilution by water) than just being the expression for the ionization of a weak acid in pure water and therefore has a more general applicability, that is, it can be used at any concentration of HA, A-, and H+ not just the concentrations present when a pure weak acid or weak base ionizes in pure water. Commonly the above expression is rearranged into a more useful form as follows:

Take logarithms, multiply by -1, and rearrange the log term

This is the Henderson-Hasselbalch equation. An analogous expression can be derived for weak bases that are amines.

Rearrange, take logarithms, multiply by -1, and rearrange log term.

Problem 19 (461): What is the pH of a solution prepared by adding 0.2 g of Na2CO3 and 0.2 g of NaHCO3 to water and bring to a volume of 1 liter? (pKa2 = 10.25).

TITRATION CURVES

A titration curve is a plot of data obtained from the titration of an acid or a base by a strong base or acid. Titration curves were originally made to determine the best color indicator to use to determine the equivalence (end-point) of a titration. Titration curves of weak acids and bases, however, are also extremely useful for illustrating how the ratio of conjugate acid/conjugate base changes as pH changes, that is, as the titration proceeds. The titration curve of 20 mL of a 0.05 M solution of the weak acid HA (pKa = 4.29) by NaOH is shown.

The pH of the solution before any NaOH is added is acid because some of the HA ionizes to yield A- and H3O+. In this case the pH is 2.8. When NaOH is added, it reacts with H3O+. More H3O+ is produced by conversion of HA to A- + H3O+. Since NaOH is a strong base each portion converts basically a stoichiometric amount of HA to A-. Consequently, the milliequivalents of A- in solution are basically equal to the milliequivalents of strong base added. However, a small portion of HA is always ionized; but with each addition of NaOH, less is ionized because the concentration of A- has increased and A- suppresses HA ionization. Because the concentration of A- continues to increase with added NaOH, the concentration of H3O+ ion continues to decrease because a continually lower [H3O+] is needed to satisfy the equilibrium conditions for the reaction, HA + H2O <===> A- + H3O+. This results in an increase in pH. Note that initally, the rate of increase in pH per constant amount of NaOH added continually decreases. This continues to occur until 0.5 milliequivalent of NaOH has been added. At this point, the rate of increase in pH per constant amount of NaOH added begins to increase until the rate of increase is maximal at 1 milliequivalent of strong base added. Stated in another way, there is an inflection point (a change in the direction of the curve) at the point where 0.5 milliequivalent of NaOH was added and another where 1.0 milliequivalent of base was added. There is no portion of the curve that is a straight line, that is, the curve is continually changing in one direction or another. Finally, note that the pH changes least, that is, resistance to change in pH is greatest, at the first inflection point.

In the titration, the ratio of [A-]/[HA] is continually increasing as NaOH is added. What is the concentration of the two species when different amounts of NaOH is added? When no NaOH has been added, the ratio is

It is extremely important to understand what this ratio means and how to use it to calculate the concentration of the two species in any particular solution (i.e. at any point along the titration curve). In the above example, the sum of the concentrations of the two species is 0.05 M and the ratio of their concentrations is 0.0324/1 (or 1/30.9), [A-]/[HA]. The ratio states the relative concentration of each species in that particular solution. In this example, the sum of the two relative concentrations is 1.0324 (the problem could also be solved by using the sum, 31.9). The actual concentration of each species can be calculated from knowing the relative concentration of each species, the sum of the two relative concentrations, and the sum of actual concentration of the two species. The sum of the actual concentration of the two species is 0.05 M. The actual concentrations of the A- and HA are:

[A-]: 0.05 M x 0.0324 (relative concentration of A-)/1.0324 (relative concentration of two combined) = 0.00157 M
(or 0.05 x 1/31.9 = 0.00157M)

[HA]: 0.05 M x 1.00/1.0324 = 0.0484 M (or 0.05 x 30.9/31.9 = 0.0484M)

It is important to remember that the number obtained when solving for log [A-]/[HA] is a ratio and therefore consists of that number over 1. In the above example, it was 0.0324 and therefore the ratio is 0.0324/1 (a ratio always consists of two numbers although at times, when the lower number of the ratio is 1, the 1 is not written).

An alternative method for calculating the concentrations of the individual species in the above example is to adjust the two numbers of the ratio so that the sum of the numbers is equal to 1. In the above example, the ratio obtained was 0.0324/1, [A-]/[HA], and the sum of the two was 1.0324. The ratio values can be adjusted so their sum is 1.0 (or not significantly different than 1) as follows:

relative [A-] on the basis of 1 is 1 x 0.0324/1.0324 = 0.03138

relative [HA-] on the basis of 1 is 1 x 1/1.0324 = 0.9686

sum = 0.99998 (not significantly different than 1)

The value of this adjustment is that now the concentrations can be easily expressed as a percent (by simply multiplying each number by 100). In the above example, 3.14% (3.14/100) of the total concentration of species is A- and 96.9% is HA. The actual concentrations of each are:

[A-]: 0.05 M x 3.14/100 = 0.00157 M

[HA]: 0.05 M x 96.86/100 = 0.0484 M

The ratio obtained from the Henderson-Hasselbalch equation, [A-]/[HA], expresses not only the ratio of the concentrations of the two species, but also the ratio of the amounts of the two species in solution (remember concentration alone does not tell the amount). Although any concentration term could be used, let concentrations be expressed as molarity (moles L-1). In the above example

Since A- and HA are in the same volume, 20 mL, the volume term can be cancelled.

Now the ratio states the amount of each of the two species in the solution. The [ ] have been removed and the 20 mL terms cancelled because concentration is no longer being expressed but we must always remember these amounts are in a known volume which in this example is 20 mL. The ratio now tells the relative number of moles or molecules of A- and HA in the 20 mL. If the ratio is expressed on the basis of 1 (the sum of A- and HA is set equal to 1) then the numbers of the ratio (when multiplied by 100) state the percent of the total moles (or molecules) of A- plus HA in 20 mL that is A-, and the percent of the total that is HA. In the above example, 3.14% of the total moles (or molecules) (A- plus HA) in the 20 mL is A- and 96.9% of the total moles (or molecules) (A- plus HA) in the 20 mL is HA.

Problem 20 (461): Construct a titration curve for the titration of 10 mL of 0.1 M acetic acid with a solution of 1 M NaOH. Place pH on the ordinate. Assume the titration curve is mathematically expressed by the Henderson-Hasselbalch equation.

Problem 21 (461): (a) Calculate the ratio, A-/HA when 0.2, 0.5 and 0.8 equivalent of NaOH has been added (read the pH from the titration curve). (b) What is the pH when the ratio of A-/HA is 1/1, 1/10, 10/1, 1/100 and 100/1?

The expression,

describes the concentration of the three species at any point on the titration curve as long as the system has come to equilibrium. The equation is a mathematical description of the titration curve. Since the Henderson-Hasselbalch equation was derived from this equation, the Henderson-Hasselbalch equation is also a mathematical description of the titration curve. Although both equations can be used to calculate the ratio of A-/HA in solution once the pH and the pKa are known, the Henderson-Hasselbalch equation is easier to use. Since both pH and pKa are relatively easily obtained--the pH in the laboratory with a pH meter and the pKa from a reference book--the ratio of conjugate acid/conjugate base is also relatively easy to obtain. The titration of a weak base by a strong acid yields analogous information except the titration curve begins in the basic region of the pH scale and the expression involving pKb is used.

BUFFERS

A buffer is a solution that resists changes in pH when acid or base are added to it. A buffer consists of a solution of a weak acid and the corresponding conjugate base. One can also state it consists of a solution of a weak base and its conjugate acid. These two statements say the same thing. A buffer may be prepared from either a weak acid or a weak base. For example, a 0.05 M acetate solution at pH 5.0 is a buffer. It is prepared from acetic acid by converting some of the acetic acid to acetate ion with strong base. The conjugate weak acid/weak base pair is acetic acid/acetate ion. The word "acetate" in acetate buffer refers to both the conjugate acid (in this example, acetic acid) and the conjugate base (acetate ion). These two species together make up the acetate buffer. Each species is present in the buffer at a specific concentration and the concentration given for the acetate buffer is the sum of the concentrations of these two species, in this example 0.05 M. The ratio of the concentrations of the two species in the buffer is obtained from the Henderson-Hasselbalch equation.

How do we know what weak acid or base to select for the preparation of a buffer? First, the pH of the buffer to be prepared must be decided. This is dictated by what pH is needed in the experiment being performed. Secondly the pKa of the weak acid of the conjugate acid/base pair should be at or near the pH to be used in the experiment. Recall from the discussion of titration curves that the pH of a solution of a weak acid changes least at the pKa of the acid. When the pH is equal to the pKa of the weak acid, the ratio of A-/HA is 1/1. Finally the weak acid and conjugate base should not interfere in any way in the experiment, such as by altering the activity of an enzyme, or by reacting chemically with any of the substances present in the experimental solution. As an example, assume the activity of an enzyme is to be tested at pH 5.0. What buffer should be used? A weak acid whose pKa is about 5.0 should be used. Acetic acid has a pKa of 4.77 and hence an acetate buffer can be used.

How is a buffer prepared? All buffers should be prepared by dissolving the appropriate weak acid (or weak base) in about 3/4 of the total volume of water needed for the buffer and adjusting the solution to the desired pH by using a pH meter and either a strong base (or acid). The solution is then quantitatively transferred to a volumetric flask and brought to volume.

How does a buffer work? Consider the acetate buffer above. When an acid (H3O+) is added to the buffer, a change in pH is resisted by Ac- reacting with the acid:

In this way, H3O+ is neutralized. But all of the added H3O+ is not neutralized because as neutralization occurs, the [HAc] increases and the [Ac-] decreases and neutralization only continues until equilibrium is again restored. The [H3O+] is somewhat higher in concentration at the new equilibrium because the new equilibrium concentration of Ac- is lower. For this reason a buffer only resists changes in pH it does not completely prevent a change in pH. Buffers therefore do change in pH when an acid or base is added. They simply change much less than a solution with no buffer. When base (OH-) is added to the acetate buffer the following reaction occurs:

Again all the added OH- is not neutralized because the new equilibrium concentration of HAc is lower and therefore a higher concentration of OH- is required to maintain the new equilibrium.

Buffers have maximum buffering capacity against added acid and base when equal molar quantities of the weak acid/weak base conjugate pair are present in the buffer solution. If more of the conjugate acid is present in the buffer than conjugate base, the buffer has greater buffering capacity against added bases than acids. Conversely if more conjugate base is present than conjugate/acid, the buffer has greater buffering capacity against added acid than base.

Problem 22: (a) Describe the preparation of 500 mL of 0.08 M acetate buffer (pH 5.4) from acetic acid [F.W. = 60.05, specific gravity = 1.05, 99.6% (w/w)] and solid NaOH (F.W. = 40.0). (b) What is the concentration of Na+ in the buffer?

Problem 23: Calculate the pH of the buffer in problem 22 after 10 mL of 0.01 M HCl is added to 100 mL of the 0.08 M acetate buffer (pH 5.4).

ADDITIONAL PROBLEMS

The following problems are in Biochemical Calculations, 2nd ed. by I.H. Segel.

Problems in chapter 1: 1-3, 1-6a, 1-14, 1-22, 1-30, 1-32, 1-36, 1-37, 1-38, 1-39 (understand why pH changes without going into the calculation), 1-42, 1-44, 1-45, 1-47, 1-49.

Problems at end of chapter 1: 2a-c,h, 9, 10, 30, 31, 32, 33, 34, 37, 40-42 (one of these), 44.