Buffer Capacity: Definition and Formula Derivation

A buffer solution can maintain an almost constant pH when a small amount of acid or base is added. The quantitative measure of this resistance to pH changes is called the buffer capacity.

1. Definitions of Buffer Capacity

Buffer capacity can be defined in multiple ways. One approach is: "the maximum amount of either strong acid or strong base that can be added before a significant change in the pH will occur." However, this begs the question of what constitutes a "significant" change. In some cases, even 1 pH unit might not matter, while in biological systems a 0.1 unit change can be critical.

Another definition states: "The quantity of strong acid or base that must be added to change the pH of one liter of solution by one pH unit." This has practical uses but can give different values for acid vs. base additions $unlessthebufferisequimolarwithpH=pKa$.

Ideally, buffer capacity should be the same whether acid or base is added. One suitable definition is:

$$beta;=;fracdnd(textpH),tag19.1$$

where $n$ is the number of equivalents of strong base added per 1 L of solution. Addition of $dn$ moles of acid changes the pH in the opposite direction by the same magnitude.

2. Derivation of the Buffer Capacity Formula

The derivation presented here is based on Adam Hulanicki's book Reakcje kwasów i zasad w chemii analitycznej $2nded.,PWN,Warszawa1980$ and its English edition (Reactions of acids and bases in analytical chemistry, Chichester, West Sussex, England: E. Horwood; New York: Halsted Press, 1987).

Assume:

• The strong base added is monoprotic.
• The total volume is 1 L, so "concentration" and "number of moles" are numerically the same.

2.1 Charge Balance

Charge balance of the solution $Equation19.2$ is:

$$[mathrm{B}^{+}];+;[mathrm{H}^{+}];=;[mathrm{A}^{-}];+;[mathrm{OH}^{-}],tag19.2$$

Here, $[mathrm{B}^{+}]$ $theconcentrationofstrongbase$ is effectively $n$ in Equation $19.1$.

2.2 Total Buffer Concentration

The total concentration of the buffer, $C_{textbuf}$, is:

$$C_{textbuf};=;[mathrmHA];+;[mathrm{A}^{-}].tag19.3$$

2.3 Acid Dissociation Relationship

From the acid dissociation constant:

$$K_a;=;frac[mathrm{H}^{+}][mathrm{A}^{-}][mathrmHA],tag19.4$$

we can rearrange to get:

$$[mathrm{A}^{-}];=;frac{K}_a,[mathrmHA][mathrm{H}^{+}].tag19.5$$

2.4 Combining Equations

Combining charge balance $19.2$, the above relationship $19.5$, and water ionization $\$K_w=[mathrm{H}^{+}][mathrm{OH}^{-}]\$$ leads to:

$$n;=;f(mathrmpH,mathrmp{K}_a,C_{textbuf},ldots),tag19.7$$

where $n$ is the amount of strong base in moles per liter.

2.5 Derivative and Buffer Capacity

By differentiating $n$ with respect to pH, we get:

$$beta;=;fracdnd(mathrmpH),tag19.8$$

and through further manipulations:

$$beta;=;2.303bigg([mathrm{H}^{+}]+frac{K}_w[mathrm{H}^{+}]+text(termsinvolving{C}_{textbuf})bigg).tag19.10$$

The exact final form $Equation19.10$ shows that buffer capacity depends on the buffer species, but also on the hydronium ion concentration $[mathrm{H}^{+}]$ and hydroxide contribution via $K_w$.

3. Multiple Buffers

If several buffering systems are present, their individual capacities add up:

$$bet{a}_{texttotal};=;bet{a}_1;+;bet{a}_2;+;dots;+;bet{a}_n.tag19.11$$

4. Practical Takeaways

• Buffer capacity is highest around $mathrmpH=mathrmp{K}_a$ for a simple acid/base buffer pair.
• Extremely acidic or basic solutions can still have significant "capacity" to resist small pH changes, due to the large presence of either $[mathrm{H}^{+}]$ or $[mathrm{OH}^{-}]$ $thewaterautoionizationterms$.
• In real-world scenarios $especiallybiological$, different buffers work together, and the total buffer capacity is the sum of each buffer’s contribution.

Thus, while "classical" buffers $likeaceticacid/acetate$ show maximum capacity near $mathrmp{K}_a$, the overall picture can be more complex if you consider all acid-base species in the solution.