## CHEMISTRY THE CENTRAL SCIENCE

**16 ACID–BASE EQUILIBRIA**

**16.8 RELATIONSHIP BETWEEN ***K*_{a}** AND ***K*_{b}

*K*

_{a}*K*

_{b}We have seen in a qualitative way that the stronger acids have the weaker conjugate bases. To see if we can find a corresponding *quantitative* relationship, let's consider the NH_{4}^{+} and NH_{3} conjugate acid–base pair. Each species reacts with water. For the acid, NH_{4}^{+} the reaction is:

or written in its simpler form:

and for the base NH_{3}:

and each equilibrium is expressed by a dissociation constant:

When we add Equations 16.38 and 16.39, the NH_{4}^{+} and NH_{3} species cancel and we are left with the autoionization of water:

Recall that when two equations are added to give a third, the equilibrium constant associated with the third equation equals the product of the equilibrium constants of the first two equations. (Section 15.3)

Applying this rule to our present example, when we multiply *K** _{a}* and

*K*

*, we obtain*

_{b}Thus, the product of *K** _{a}* and

*K*

*is the ion-product constant for water,*

_{b}*K*

*(Equation 16.16). We expect this result because adding Equations 16.38 and 16.39 gave us the autoionization equilibrium for water, for which the equilibrium constant is*

_{w}*K*

*.*

_{w}This relationship is so important that it should receive special attention: *The product of the acid-dissociation constant for an acid and the base-dissociation constant for its conjugate base equals the ion-product constant for water:*

As the strength of an acid increases (*K** _{a}* gets larger), the strength of its conjugate base must decrease (

*K*

*gets smaller) so that the product*

_{b}*K*

*×*

_{a}*K*

*remains 1.0 × 10*

_{b}^{–14}at 25 °C.

**TABLE 16.5**demonstrates this relationship. Remember, this important relationship applies

*only*to conjugate acid–base pairs.

By using Equation 16.40, we can calculate *K** _{b}* for any weak base if we know

*K*

*for its conjugate acid. Similarly, we can calculate*

_{a}*K*

*for a weak acid if we know*

_{a}*K*

*for its conjugate base. As a practical consequence, ionization constants are often listed for only one member of a conjugate acid–base pair. For example, Appendix D does not contain*

_{b}*K*

*values for the anions of weak acids because they can be readily calculated from the tabulated*

_{b}*K*

*values for their conjugate acids.*

_{a}**TABLE 16.5 • Some Conjugate Acid–Base Pairs**

If you look up the values for acid-or base-dissociation constants in a chemistry handbook, you may find them expressed as p*K** _{a}* or p

*K*

*(that is, –log*

_{b}*K*

*or –log*

_{a}*K*

*) (Section 16.4). Equation 16.40 can be written in terms of p*

_{b}*K*

*and p*

_{a}*K*

*by taking the negative logarithm of both sides:*

_{b}**GIVE IT SOME THOUGHT**

What is the p*K** _{a}* value for HF? What is the p

*K*

*value for F*

_{b}^{–}?

**CHEMISTRY PUT TO WORKAmines and Amine Hydrochlorides**

Many low-molecular-weight amines have a fishy odor. Amines and NH_{3} are produced by the anaerobic (absence of O_{2}) decomposition of dead animal or plant matter. Two such amines with very disagreeable odors are H_{2}N(CH_{2})_{4}NH_{2}, *putrescine*, and H_{2}N(CH_{2})_{5}NH_{2}, *cadaverine*.

Many drugs, including quinine, codeine, caffeine, and amphetamine, are amines. Like other amines, these substances are weak bases; the amine nitrogen is readily protonated upon treatment with an acid. The resulting products are called *acid salts*. If we use A as the abbreviation for an amine, the acid salt formed by reaction with hydrochloric acid can be written AH^{+}Cl^{–}. It can also be written as A·HCl and referred to as a hydrochloride. Amphetamine hydrochloride, for example, is the acid salt formed by treating amphetamine with HCl:

Acid salts are much less volatile, more stable, and generally more water soluble than the corresponding amines. For this reason, many drugs that are amines are sold and administered as acid salts. Some examples of over-the-counter medications that contain amine hydrochlorides as active ingredients are shown in **FIGURE 16.14**.

*RELATED EXERCISES:* 16.75, 16.76, 16.104, 16.113, and 16.122

**FIGURE 16.14 Some over-the-counter medications in which an amine hydrochloride is a major active ingredient.**

**SAMPLE EXERCISE 16.17 Calculating K**

_{a}**or**

*K*

_{b}**for a Conjugate Acid–Base Pair**

Calculate **(a)** *K** _{b}* for the fluoride ion,

**(b)**

*K*

*for the ammonium ion.*

_{a}**SOLUTION**

**Analyze** We are asked to determine dissociation constants for F^{–}, the conjugate base of HF, and NH_{4}^{+}, the conjugate acid of NH_{3}.

**Plan** We can use the tabulated *K* values for HF and NH_{3} and the relationship between *K** _{a}* and

*K*

*to calculate the ionization constants for their conjugates, F*

_{b}^{–}and NH

_{4}

^{+}.

**Solve**

**(a)** For the weak acid HF, Table 16.2 and Appendix D give *K** _{a}* = 6.8 × 10

^{–4}. We can use Equation 16.40 to calculate

*K*

*for the conjugate base, F*

_{b}^{–}:

**(b)** For NH_{3}, Table 16.4 and in Appendix D give *K** _{b}* = 1.8 × 10

^{–5}, and this value in Equation 16.40 gives us

*K*

*for the conjugate acid, NH*

_{a}_{4}

^{+}:

**Check** The respective *K* values for F^{–}and NH_{4}^{+} are listed in Table 16.5, where we see that the values calculated here agree with those in Table 16.5.

**PRACTICE EXERCISE**

**(a)** Which of these anions has the largest base-dissociation constant: NO_{2}^{–}, PO_{4}^{3–}, or N_{3}^{–}?

**(b)** The base quinoline has the structure

Its conjugate acid is listed in handbooks as having a p*K** _{a}* of 4.90. What is the base-dissociation constant for quinoline?

*Answers:*** (a)** PO_{4}^{3–}(*K** _{b}* = 2.4 × 10

^{–2}),

**(b)**

*K*

*= 7.9 × 10*

_{b}^{–10}