# 6.3: Buffers

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##### Learning Objectives
• Describe the composition and function of acid–base buffers
• Calculate the pH of a buffer before and after the addition of added acid or base

A mixture of a weak acid and its conjugate base (or a mixture of a weak base and its conjugate acid) is called a buffer solution, or a buffer. Buffer solutions resist a change in pH when small amounts of a strong acid or a strong base are added (Figure $$\PageIndex{1}$$). A solution of acetic acid ($$\ce{CH3COOH}$$ and sodium acetate $$\ce{CH3COONa}$$) is an example of a buffer that consists of a weak acid and its salt. An example of a buffer that consists of a weak base and its salt is a solution of ammonia ($$\ce{NH3(aq)}$$) and ammonium chloride ($$\ce{NH4Cl(aq)}$$).

## How Buffers Work

A mixture of acetic acid and sodium acetate is acidic because the Ka of acetic acid is greater than the Kb of its conjugate base acetate. It is a buffer because it contains both the weak acid and its salt. Hence, it acts to keep the hydronium ion concentration (and the pH) almost constant by the addition of either a small amount of a strong acid or a strong base. If we add a base such as sodium hydroxide, the hydroxide ions react with the few hydronium ions present. Then more of the acetic acid reacts with water, restoring the hydronium ion concentration almost to its original value:

$\ce{CH3CO2H}(aq)+\ce{H2O}(l)⟶\ce{H3O+}(aq)+\ce{CH3CO2-}(aq)$

The pH changes very little. If we add an acid such as hydrochloric acid, most of the hydronium ions from the hydrochloric acid combine with acetate ions, forming acetic acid molecules:

$\ce{H3O+}(aq)+\ce{CH3CO2-}(aq)⟶\ce{CH3CO2H}(aq)+\ce{H2O}(l)$

Thus, there is very little increase in the concentration of the hydronium ion, and the pH remains practically unchanged (Figure $$\PageIndex{2}$$).

A mixture of ammonia and ammonium chloride is basic because the Kb for ammonia is greater than the Ka for the ammonium ion. It is a buffer because it also contains the salt of the weak base. If we add a base (hydroxide ions), ammonium ions in the buffer react with the hydroxide ions to form ammonia and water and reduce the hydroxide ion concentration almost to its original value:

$\ce{NH4+}(aq)+\ce{OH-}(aq)⟶\ce{NH3}(aq)+\ce{H2O}(l)$

If we add an acid (hydronium ions), ammonia molecules in the buffer mixture react with the hydronium ions to form ammonium ions and reduce the hydronium ion concentration almost to its original value:

$\ce{H3O+}(aq)+\ce{NH3}(aq)⟶\ce{NH4+}(aq)+\ce{H2O}(l)$

The three parts of the following example illustrate the change in pH that accompanies the addition of base to a buffered solution of a weak acid and to an unbuffered solution of a strong acid.

##### Example $$\PageIndex{1}$$: pH Changes in Buffered and Unbuffered Solutions

Acetate buffers are used in biochemical studies of enzymes and other chemical components of cells to prevent pH changes that might change the biochemical activity of these compounds.

1. Calculate the pH of an acetate buffer that is a mixture with 0.10 M acetic acid and 0.10 M sodium acetate.
2. Calculate the pH after 1.0 mL of 0.10 M NaOH is added to 100 mL of this buffer, giving a solution with a volume of 101 mL.

Solution

1. Calculate the pH of an acetate buffer that is a mixture with 0.10 M acetic acid and 0.10 M sodium acetate.

To determine the pH of the buffer solution we use a typical equilibrium calculation (as illustrated in earlier Examples):

1. Determine the direction of change. The equilibrium in a mixture of H3O+, $$\ce{CH3CO2-}$$, and CH3CO2H is:

$\ce{CH3CO2H}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{CH3CO2-}(aq)$

The equilibrium constant for CH3CO2H is not given, so we look it up in Table E1: Ka = 1.8 × 10−5. With [CH3CO2H] = $$\ce{[CH3CO2- ]}$$ = 0.10 M and [H3O+] = ~0 M, the reaction shifts to the right to form H3O+.

2. Determine x and equilibrium concentrations. A table of changes and concentrations follows:
3. Solve for x and the equilibrium concentrations. We find:

$x=1.8×10^{−5}\:M$

and

$\ce{[H3O+]}=0+x=1.8×10^{−5}\:M$

Thus:
$\mathrm{pH=−log[H_3O^+]=−log(1.8×10^{−5})}$

$=4.74$

4. Check the work. If we calculate all calculated equilibrium concentrations, we find that the equilibrium value of the reaction coefficient, Q = Ka.

(b) Calculate the pH after 1.0 mL of 0.10 M NaOH is added to 100 mL of this buffer, giving a solution with a volume of 101 mL.

First, we calculate the concentrations of an intermediate mixture resulting from the complete reaction between the acid in the buffer and the added base. Then we determine the concentrations of the mixture at the new equilibrium:

1. Determine the moles of NaOH. One milliliter (0.0010 L) of 0.10 M NaOH contains:

$\mathrm{0.0010\cancel{L}×\left(\dfrac{0.10\:mol\: NaOH}{1\cancel{L}}\right)=1.0×10^{−4}\:mol\: NaOH}$

2. Determine the moles of CH2CO2H. Before reaction, 0.100 L of the buffer solution contains:

$\mathrm{0.100\cancel{L}×\left(\dfrac{0.100\:mol\:CH_3CO_2H}{1\cancel{L}}\right)=1.00×10^{−2}\:mol\:CH_3CO_2H}$

3. Solve for the amount of NaCH3CO2 produced. The 1.0 × 10−4 mol of NaOH neutralizes 1.0 × 10−4 mol of CH3CO2H, leaving:

$\mathrm{(1.0×10^{−2})−(0.01×10^{−2})=0.99×10^{−2}\:mol\:CH_3CO_2H}$

and producing 1.0 × 10−4 mol of NaCH3CO2. This makes a total of:

[\mathrm{(1.0×10^{−2})+(0.01×10^{−2})=1.01×10^{−2}\:mol\:NaCH_3CO_2} \]

4. Find the molarity of the products. After reaction, CH3CO2H and NaCH3CO2 are contained in 101 mL of the intermediate solution, so:

$\ce{[CH3CO2H]}=\mathrm{\dfrac{9.9×10^{−3}\:mol}{0.101\:L}}=0.098\:M$

$\ce{[NaCH3CO2]}=\mathrm{\dfrac{1.01×10^{−2}\:mol}{0.101\:L}}=0.100\:M$

Now we calculate the pH after the intermediate solution, which is 0.098 M in CH3CO2H and 0.100 M in NaCH3CO2, comes to equilibrium. The calculation is very similar to that in part (a) of this example:

This series of calculations gives a pH = 4.75. Thus the addition of the base barely changes the pH of the solution.

(c) For comparison, calculate the pH after 1.0 mL of 0.10 M NaOH is added to 100 mL of a solution of an unbuffered solution with a pH of 4.74 (a 1.8 × 10−5-M solution of HCl). The volume of the final solution is 101 mL.

Solution

This 1.8 × 10−5-M solution of HCl has the same hydronium ion concentration as the 0.10-M solution of acetic acid-sodium acetate buffer described in part (a) of this example. The solution contains:

$$\mathrm{0.100\:L×\left(\dfrac{1.8×10^{−5}\:mol\: HCl}{1\:L}\right)=1.8×10^{−6}\:mol\: HCl}$$

As shown in part (b), 1 mL of 0.10 M NaOH contains 1.0 × 10−4 mol of NaOH. When the NaOH and HCl solutions are mixed, the HCl is the limiting reagent in the reaction. All of the HCl reacts, and the amount of NaOH that remains is:

$$(1.0×10^{−4})−(1.8×10^{−6})=9.8×10^{−5}\:M$$

The concentration of NaOH is:

$$\dfrac{9.8×10^{−5}\:M\:\ce{NaOH}}{0.101\:\ce{L}}=9.7×10^{−4}\:M$$

The pOH of this solution is:

$$\mathrm{pOH=−log[OH^- ]=−log(9.7×10^{−4})=3.01}$$

The pH is:

$$\mathrm{pH=14.00−pOH=10.99}$$

The pH changes from 4.74 to 10.99 in this unbuffered solution. This compares to the change of 4.74 to 4.75 that occurred when the same amount of NaOH was added to the buffered solution described in part (b).

##### Exercise $$\PageIndex{1}$$

Show that adding 1.0 mL of 0.10 M HCl changes the pH of 100 mL of a 1.8 × 10−5 M HCl solution from 4.74 to 3.00.

Initial pH of 1.8 × 10−5 M HCl; pH = −log[H3O+] = −log[1.8 × 10−5] = 4.74

Moles of H3O+ added by addition of 1.0 mL of 0.10 M HCl: 0.10 moles/L × 0.0010 L = 1.0 × 10−4 moles; final pH after addition of 1.0 mL of 0.10 M HCl:

$\mathrm{pH=−log[H_3O^+]=−log\left(\dfrac{total\: moles\:H_3O^+}{total\: volume}\right)=−log\left(\dfrac{1.0×10^{−4}\:mol+1.8×10^{−6}\:mol}{101\:mL\left(\dfrac{1\:L}{1000\:mL}\right)}\right)=3.00}$

## Buffer Capacity

Buffer solutions do not have an unlimited capacity to keep the pH relatively constant (Figure $$\PageIndex{3}$$). If we add so much base to a buffer that the weak acid is exhausted, no more buffering action toward the base is possible. On the other hand, if we add an excess of acid, the weak base would be exhausted, and no more buffering action toward any additional acid would be possible. In fact, we do not even need to exhaust all of the acid or base in a buffer to overwhelm it; its buffering action will diminish rapidly as a given component nears depletion.

The buffer capacity is the amount of acid or base that can be added to a given volume of a buffer solution before the pH changes significantly, usually by one unit. Buffer capacity depends on the amounts of the weak acid and its conjugate base that are in a buffer mixture. For example, 1 L of a solution that is 1.0 M in acetic acid and 1.0 M in sodium acetate has a greater buffer capacity than 1 L of a solution that is 0.10 M in acetic acid and 0.10 M in sodium acetate even though both solutions have the same pH. The first solution has more buffer capacity because it contains more acetic acid and acetate ion.

## Selection of Suitable Buffer Mixtures

There are two useful rules of thumb for selecting buffer mixtures:

1. A good buffer mixture should have about equal concentrations of both of its components. A buffer solution has generally lost its usefulness when one component of the buffer pair is less than about 10% of the other. Figure $$\PageIndex{4}$$ shows an acetic acid-acetate ion buffer as base is added. The initial pH is 4.74. A change of 1 pH unit occurs when the acetic acid concentration is reduced to 11% of the acetate ion concentration.
1. Weak acids and their salts are better as buffers for pHs less than 7; weak bases and their salts are better as buffers for pHs greater than 7.

Blood is an important example of a buffered solution, with the principal acid and ion responsible for the buffering action being carbonic acid, H2CO3, and the bicarbonate ion, $$\ce{HCO3-}$$. When an excess of hydrogen ion enters the blood stream, it is removed primarily by the reaction:

$\ce{H3O+}(aq)+\ce{HCO3-}(aq)⟶\ce{H2CO3}(aq)+\ce{H2O}(l)$

When an excess of the hydroxide ion is present, it is removed by the reaction:

$\ce{OH-}(aq)+\ce{H2CO3}(aq)⟶\ce{HCO3-}(aq)+\ce{H2O}(l)$

The pH of human blood thus remains very near 7.35, that is, slightly basic. Variations are usually less than 0.1 of a pH unit. A change of 0.4 of a pH unit is likely to be fatal.

## Summary

A solution containing a mixture of an acid and its conjugate base, or of a base and its conjugate acid, is called a buffer solution. Unlike in the case of an acid, base, or salt solution, the hydronium ion concentration of a buffer solution does not change greatly when a small amount of acid or base is added to the buffer solution. The base (or acid) in the buffer reacts with the added acid (or base).

## Key Equations

• pKa = −log Ka
• pKb = −log Kb

## Glossary

buffer capacity
amount of an acid or base that can be added to a volume of a buffer solution before its pH changes significantly (usually by one pH unit)
buffer
mixture of a weak acid or a weak base and the salt of its conjugate; the pH of a buffer resists change when small amounts of acid or base are added
Henderson-Hasselbalch equation
equation used to calculate the pH of buffer solutions

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