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Buffer SolutionsA buffer is a solution that resists changes in pH

upon addition of small quantities of acids or bases. In other words, a buffer is a solution that keeps its pH almost constant. A buffer is a solution containing a weak acid and its conjugate base or a weak base and its conjugate acid. Below is an explanation of how buffers work:

Let us look at a buffer formed from a weak acid (like acetic acid) and its conjugate base (e.g. sodium acetate); we have the following equilibria:

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HOAc H+ + OAc-

OAc- + H2O HOAc + OH-

In the first equilibrium, acetate is produced from the dissociation of acetic acid. However, there is a lot of acetate added to solution from the sodium acetate. Therefore, the first equilibrium does not occur to any significant degree. We can fairly assume that HOAc will practically not dissociate. The same applies for the second equilibrium where acetic acid is produced. Since a great excess of acetic acid is present in solution from the first equilibrium, one can say that acetate will not practically associate in water.

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Therefore, neither acetic acid nor acetate equilibria will proceed to any significant extent. The equilibrium constant for dissociation of acetic acid can be written as:

Ka = [OAc-][H+]/[HOAc]

Where the [OAc-] = COAc- and [HOAc] = CHOAc

When an acid is added to the buffer above, H+ from the acid will combine with OAc- to form HOAc which is reflected by an increase in HOAc

and a decrease in OAc-.

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In case the amount of H+ added is not very large, the ratio [OAc-]/[HOAc] will only change slightly. A similar argument can be presented when a small amount of base is added to the buffer solution. However, we will treat the buffer problem as we described for the common ion effect

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Buffer Capacity

The degree of buffer resistance to changes in pH is referred to as buffer capacity. The buffer capacity is defined as the concentration of acid or base ( in moles ) needed to cause a pH change equal to dpH where:

= dCbase/dpH = - dCacid/dpH

The minus sign is because addition of an acid causes a decrease in pH. A more practical relation to use for calculation of buffer capacity is:

= 2.303 Cacid Cconjugate base / (Cacid + Cconjugate base)

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Example

Calculate the buffer capacity of a solution containing 0.10 M acetic acid and 0.10 M acetate. Find the pH change

when you add NaOH so that the solution becomes 0.005 M in NaOH

Solution

= 2.303 Cacid Cconjugate base / (Cacid + Cconjugate base)

= 2.303 * 0.10 * 0.10/( 0.10 + 0.10) = 0.115 M

= dCbase/dpH

0.115 = 0.005/dpH

dpH = 0.005/0.115 = 0.043

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Hasselbalch-Henderson Equation

In the acetic acid/acetate buffer described above, we have:

Ka = [OAc-][H+]/[HOAc]

Pka = pH – log [OAc-]/[HOAc]

pH = pka + log [OAc-]/[HOAc]

The above equation is referred to as Hasselbalch-Henderson equation.

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this equation can be useful to describe buffer limits where the maximum pH limit for a buffer is when

the salt to acid ratio is 10 and the minimum pH limit of the buffer is when the acid to salt ratio is 10. Inserting these values, one in a time, in the

Hasselbalch equation gives

pH = pka + 1

Therefore, the buffer acts well within two pH units and the midpoint pH value is equal to

pka. One should first look at the pka or pkb to

choose the correct buffer system which can be used within a specific pH range.

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Example

Calculate the pH of the buffer containing 0.50 M formic acid (HA, ka = 1.8x10-4) and 0.25 M sodium formate (NaA).

Solution

First, write the acid equilibrium equation

HA H+ + A-

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Ka = x (0.25 + x)/(0.50 – x)

Assume that 0.25 >> x

1.8x10-4 = 0.25 x /0.50

x = 3.6x10-4

Relative error = (3.6x10-4/0.25) x 100 = 0.14%

The assumption is therefore valid and we have

[H+] = 3.6x10-4 M

pH = 3.44

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Example

Calculate the pH of the buffer solution prepared by mixing 10 mL 0.10 M HOAc (ka = 1.75x10-5) with

20 mL of 0.20 M sodium acetate.

Solution

Let us first calculate the concentrations after mixing (final concentrations of the acid and its

conjugate base)

mmol HOAc = 0.10 x 10 = 1.0 mmol

[HOAc] = 1.0/30

mmol OAc- = 0.20 x 20 = 4.0 mmol

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[OAc- = ]4.0/30

The equilibrium equation is

HOAc = H+ + OAc-

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Ka = x (4.0/30 + x)/ (1.0/30 – x)

Assume 1.0/30 >> x

1.75x10-5 = x (4.0/30)/1.0/30

x = 1.75x10-5 /4.0

x = 4.38x10-6

Relative error = {4.38x10-6/(1.0/30)} x 100 = 0.013%

The assumption is valid therefore:

[H+] = 4.38x10-6 M

pH = 5.36

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Example

Calculate the pH of the solution resulting from adding 25 mL of 0.10 M NaOH to 30 mL of 0.20

M acetic acid.

Solution

Let us find what happens when we mix the two solutions. Definitely the hydroxide will react with

the acid to form acetate which also results in a decrease in the acid concentration.

mmol OH- = 0.10 x 25 = 2.5 mmol

mmol HOAc = 0.20 x 30 = 6.0 mmol

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Now the mmol base added will react in a 1:1 mole ratio with the acid . Therefore we have

mmol OAc- formed = 2.5 mmol

[OAc-] = 2.5/55 M

mmol HOAc left = 6.0 – 2.5 = 3.5 mmol

[HOAc] = 3.5/55 M

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Ka = x (2.5/55 + x)/ (3.5/55 – x)

Assume 2.5/55 >> x

1.75x10-5 = x (2.5/55)/3.5/55

x = 1.75x10-5 x 2.5/3.5

x = 2.45x10-5

Relative error = {2.45x10-5/(2.5/55)} x 100 = 0.054%

The assumption is valid

pH = 4.61

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Example

A buffer solution is 0.20 M in HOAc and in NaOAc. Find the change in pH after addition of 0.10 mmol of HCl to 10 mL of the buffer without

change in volume.

Solution

We should find the initial pH (before addition of HCl) and then calculate the pH after addition.

HOAc H+ + OAc-

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Initial pH

Ka = x (0.20 + x)/ (0.20 – x)Assume 0.20 >> x

1.75x10-5 = 0.20 x/ 0.20x = 1.75x10-5

Relative error = (1.75x10-5/0.20) x 100 = 8.8x10-3%[H+] = 1.75x10-5 M

pH = 4.76

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After addition of HCl, the acetate concentration will decrease while the acetic acid concentration will

increase.

mmol HOAc = 0.20 x 10 + 0.10 = 2.1 mmol

[HOAc] = 2.1/10 = 0.21 M

mmol OAc- left = 0.20 x 10 – 0.10 = 1.9 mmol

[OAc-] = 1.9/10 = 0.19 M

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Ka = x (0.19 + x)/ (0.21 – x)

Assume 0.19 >> x

1.75x10-5 = 0.19 x/ 0.21

x = 1.93x10-5

Relative error = (1.93x10-5/0.19) x 100 = 0.01%

[H+] = 1.93x10-5 M

pH = 4.71

pH = 4.71 – 4.76 = - 0.05

Try the same problem replacing NaOH for the HCl and get the change in pH using the same procedure. You

should realize that in this case the acetic acid will decrease while acetate will increase.