pH...

pH is one of the most commonly made measurements in water testing, but one of the least understood. Here is an attempt at an explanation: In order to understand the pH scale, we have to discuss the ideas of moles and of logarithms.

Moles : When chemists talk about the amount of a substance, they often like to use the unit of moles, rather than grams. A mole of a substance is, simply, the number of grams of that substance equal to its molecular weight. A mole of water, weighs about 18 grams, because water has a molecular weight of about 18. A mole of calcium carbonate, CaCO₃, weighs about 100 grams; a mole of methyl alcohol, CH₃OH, 32 grams, etc. The advantage for chemists of using moles is that an equal number of moles of any substance contains the same number of molecules, so it is easier to calculate the amounts of substances which react with one another.

In a liter of pure water at room temperature the number of moles of hydrogen ions is about 0.0000001. (For hydrogen, with an atomic weight of 1, this is also about equal to the number of grams of hydrogen ions.) In scientific notation, this is written as 1 x 10^-7, where the superscript,-7, is known as a power, an exponent, or a logarithm. (All three terms mean the same thing. The seven indicates the number of places to the right of the decimal point that the "one" is located.) It turns out, when measuring hydrogen ion concentration electrochemically, that the electrical potential (voltage) generated at the measuring electrode is directly related not to the H⁺ concentration, but to the logarithm of the H⁺ concentration. This makes it more convenient to refer to the H⁺ concentration in terms of its logarithm. And since H⁺ concentrations in water solutions are almost always less than one mole per liter, the exponent is almost always going to be negative, because that is the way scientific notation expresses numbers less than one. So, the negative of the logarithm of the hydrogen ion concentration in a solution is given a special name. It is called the pH, which stands for the potential of the hydrogen ion.

Of course, in the pure water, the concentration of hydroxide ions is also 1 x 10^-7 moles per liter, since each water molecule that dissociates produces one ion of each type. The water is said to be neutral. It has a pH of 7 and also a pOH of 7, where the term pOH refers to the negative logarithm of the hydroxide ion concentration. There are substances which, when dissolved in water, will upset that balance, and produce an excess of either H⁺ or OH^-. They may contain those ions and release them (dissociate) when they dissolve, or they may react with the water (hydrolyze) and produce them that way. Those substances which increase the concentration of H+ are called acids; those which decrease it (and increase the OH- ) are bases or alkalis. For instance, if a strong acid solution increases the H⁺ concentration to 0.1 moles per liter (1 x 10^-1), which has a million times as many H⁺ ions as a neutral solution, then the pH is equal to 1. Similarly, if a strong base solution contains 0.1 moles per liter of OH^- ions, it has a pOH of 1. According to the laws of chemical equilibrium, the pH and the pOH always add up to 14 (at about room temperature), so the solution with the pOH of 1 has a pH of 13. Most solutions have a pH between 0 and 14, and 7 is the neutral point. pH's below 7 are increasingly acidic as the number decreases; pH's above 7 are increasingly alkaline. And since the scale is logarithmic, each unit change in pH represents 10 times as many ions in solution.

A strong acid or base is one which dissociates completely when it dissolves in water. The amount of it in solution can be estimated from the pH. Most acids and bases, however, are weak; they dissociate or hydrolyze only partially. Many solutions also contain mixtures of several acidic or basic substances. In these cases, it is difficult to estimate the total amount of acid or base by measuring the pH, so this must be done by titration. As every high school chemistry student knows, acids react with bases to form water and salts. Therefore, an acid is titrated using a standard base, and visa versa. In water and wastewater analysis, the amount of acid needed to titrate a solution to a particular pH is a measure of the acid neutralizing capacity of that solution, and is referred to as the solution's alkalinity. In natural waters, the pH is most often controlled by the concentrations of carbonate, bicarbonate, and carbon dioxide, since these are products of respiration and fermentation. Because of this, alkalinity is usually measured in terms of the amount of acid needed to reach the pH of a pure solution of one or another of these substances. Similarly, acidity is defined as base neutralizing capacity, and is measured by titration against a standard base.

While a chemist might prefer to measure these quantities in moles per liter, engineers seem more comfortable with standard weight units. So acidity and alkalinity are usually expressed in units of milligrams per liter of calcium carbonate. Calcium carbonate, or limestone, is a weakly alkaline material, 50 grams of which react with one mole of hydrogen ions.

Titrations and Buffer Solutions...

A solution of an pure acid will have a pH determined by its concentration and by how strong an acid it is - that is, how easily it releases a proton (hydrogen ion) when dissolved in water. As we titrate a solution of an acid with strong base, hydrogen ions are consumed by reacting with the added hydroxide ions to form water, which leads to an increase in the pH of the solution. For an acid with one proton which can dissociate ("monoprotic" acid), the titration will be complete when the number of moles of hydroxide added equals the number of moles of acid originally present. If we call the fraction of acid which has been neutralized f, then the titration is complete when f = 1. (For an acid with two replaceable hydrogens ("diprotic" acid), the titration is complete at f = 2, and so forth.) If more base is added after the acid is all neutralized, then the pH of the solution will be determined by the concentration of hydroxide - essentially as if it were being added to plain water. The initial, final, and intermediate pH's will be a function of the acid's concentration, strength, and the value of f. The calculations are different for strong and weak acids, so let's consider them separately.

For a strong acid, essentially all of the acid dissociates, so that the concentration of hydrogen ions (H⁺) is equal to the concentration of the acid. Therefore, the initial pH of the acid solution is equal to the negative logarithm of the concentration of the acid in moles per liter (by the definition of pH). When 90 % of the acid has been neutralized (f = 0.9), the concentration of H⁺ is only one-tenth of its what it was originally - so the pH will be one unit higher, since -log(0.1) = 1. When 99 % has been neutralized (f = 0.99), the pH is 2 units higher, and so on. When f = 1, the pH should equal 7 - and any further addition will raise the pH to a value equal to [14 minus pOH], just as though it were being added to pure water. (Note that we have made the simplifying assumption here of ignoring the increase in the volume of the solution due to adding the base - but this could easily be accounted for. We also assumed that the original acid concentration was a lot higher than 10^-7 molar, so that we could ignore the H⁺ contributed by the dissociation of water.)

For a weak acid, an approximate formula can be derived for the pH of a solution of the pure acid which states that ;

pH = 1/2 ( pK_a + pC )

The pK_a is the negative logarithm of the "acid dissociation constant", and pC is the negative logarithm of the concentration of the acid in moles per liter. (The pK_a is a property of each particular acid, and is a number which can be looked up in reference books.) So, for example, a 0.1 molar solution (pC = 1) of an acid which has a pK_a of 5, would have a pH of about 1/2(5+1), or 3.

For the hypothetical acid with the formula, HA, the reaction which occurs as the titration with strong base proceeds can be written as ;

HA + OH^- ---> A^- + H₂O

The major chemical species in the solution are the remaining acid, HA, which has not been neutralized, and the anion (negative ion), A^-, which is called the "conjugate base." It is the ratio of these which determines the pH, according to the formula ;

pH = pK_a + log [ ( A^-) / ( HA ) ]

where ; (A^-) means the molar concentration of A^- and (HA) is the molar concentration of the remaining HA. Note that this formula can also be expressed as ;

pH = pK_a + log [ f / ( 1 - f ) ]

When the concentration of the two species is equal, the ratio [ ( A^- ) / ( HA ) ] equals 1 - and since the logarithm of 1 equals zero, the pH is equal to the pK_a. At an earlier point in the titration, when, say, one-tenth of of the acid had been neutralized, the pH would be equal to pK_a + log ( 0.1 / 0.9 ). This works out to about 0.95 pH units below the pK_a. When 90% of the titration is complete, the pH should be about equal to pK_a + log ( 0.9 / 0.1 ), or about 0.95 units above the pK_a. So the pH change during the middle 80% of the titration will vary less than one unit below or above the value of the pK_a. Likewise, you can easily show that between the 1% and 99% points of the titration, the pH will vary between 2 units below and two units above the pK_a. (Note that the same assumptions are made as for the strong acid case discussed above.)

For a monoprotic acid (also called a "monobasic" acid - how's that for a confusing term) at the end of the titration (f = 1), there is another approximate formula for the pH ;

pH = 7 + 1/2 ( pK_a - pC )

For the previous example of a 0.1 molar solution (pC = 1) of an acid which has a pK_a of 5, the endpoint pH would be about 7 + 1/2 ( 5 - 1 ), or 9.

"Diprotic" (also called "dibasic") acids can be thought of of dissociating in two steps. For a generic dibasic acid H₂Z, loss of one proton can be written as ;

H₂Z <---> H⁺ + HZ^-

for which pK_a is called pK₁ and loss of the second proton is written as ;

HZ^- <---> H⁺ + Z⁼

for which pK_a is called pK₂. Since HZ^- is negatively charged, and positive charges are attracted to negative charges, it is harder for the second proton to break away. Because of this, the value of pK₂ is usually several units higher than pK₁. Often, a solution of a dibasic acid behaves essentially like a mixture of two independent acids, one being a much weaker acid than the other. The titration curve runs from f = 0 to f = 2, and looks like one monobasic titration curve followed by another one at a higher pK. The pH's at f = 0.5 and f = 1.5 correspond to the values of pK₁ and pK₂, respectively. If the acid is concentrated enough that the contribution due to the dissociation of water can be ignored, the pH at f = 1 is about equal to the average of pK₁ and pK₂. (In cases where the pK's are fairly close the simple model does not work so well).

The pH range near the pK_a value of a particular weak acid is sometimes referred to as the buffer region. As we have seen, the pH does not change much in this region when strong acid or base is added-- even in amounts which are a significant fraction of the amount of the weak acid/base mixture itself. This property is made use of in chemical, biological and pharmaceutical work - and in nature - to keep solutions at a near-constant pH. To make a buffer solution, you do not actually need to titrate a weak acid or base. For instance, to make an acetic acid/acetate buffer you can purchase acetic acid and the salt, sodium acetate, from a chemical supplier and make a solution containing the proportions which will give the desired pH, based on the eq. given above. The buffer would be most efficient at a pH near the acid's pK_a value of 4.7.

In natural waters and in wastewater treatment plants, the water most often relies on the carbonic acid/bicarbonate system for buffering near neutral pH (pK = 6.3). The carbonic acid is formed when carbon dioxide dissolves in water. It is a product of aerobic or anaerobic respiration by microorganisms living in the water, and is also present in air; carbonates are present is some minerals, such as limestone, with which the water may come in contact.

In laboratories, phosphate buffers are often used in chemistry or bacteriology to keep pH conditions constant. Phosphoric acid is a tribasic acid, with pK's of 2.1, 7.2, and about 12.0. You can see that the middle one, corresponding to a mixture of the ions H₂PO₄^- and HPO₄⁼, would be very useful for making neutral buffers. In wastewater analysis, phosphate buffers are used in the BOD test, the DPD method for total chlorine residual and the colorimetric test for cyanide, for diluting and rinsing in coliform bacterial testing, and for calibrating pH meters.

As an example of the protective effect of buffer solutions, consider the 0.01 M (moles per liter) carbonic acid/bicarbonate/carbonate system shown in the last of the six titration curves. If we take the case of this system at a pH of 7.0, the graph shows that the value of f equals about 0.83. This means that of the total concentration of 0.01 moles per liter, 83% (0.0083 moles per liter) is in the form of bicarbonate (HCO₃^-) - so that 17% (0.0017 moles per liter) is in the form of carbonic acid (H₂CO₃). The ratio, 0.0083 / 0.0017, equals 4.88 - the logarithm of which is 0.69, or about 0.7. Add this to the pK1 of 6.3, according to the formula above, and you get a pH of 7.0. Now, let's say we add 0.001 moles of a strong acid to a liter of this solution. (Remember that adding this amount of strong acid to pure water will lower the pH from 7 down to a value of 3). The reaction which would occur ;

HCO₃^- + H⁺ ---> H₂CO₃

would convert 0.001 moles of bicarbonate ion to carbonic acid, so the new concentrations would be 0.0073 moles per liter of bicarbonate and 0.0027 moles per liter of carbonic acid. The new ratio of concentrations would be (0.0073 / 0.0027), or 2.70, the logarithm of which equals 0.43, or about 0.4. So the new pH would be 6.3 plus 0.4, or 6.7 - a decrease of only 0.3 units. This would have much less of an effect on the chemistry and microbiology of the water than the drop of four pH units that would occur in an unbuffered solution. The protective species here is bicarbonate, and its "acid neutralizing capacity" would be referred to as the bicarbonate alklinity of the solution. In terms of "mg/L as calcium carbonate", the amount of bicarbonate alkalinity in the original solution would have been equivalent to 0.0083 X 50,000, or 415 mg/L. After the addition of the strong acid, it would have been lowered to 365 mg/L. Buffering is clearly an important feature of water quality control.