**World Journal of Chemical Education**

## An Algebraic Derivation of Buffer Capacity

**Matthew S. Fox**^{1,}, **Joshua W. Greenberg**^{1}, **Jake M. Trookman**^{1}

^{1}Department of Chemistry, Cheyenne Mountain High School, Colorado Springs, Colorado 80906-1622 United States

Abstract | |

1. | Introduction |

2. | Methods |

3. | Limitations and Assumptions |

4. | Implications and Conclusions |

Notes | |

Acknowledgements | |

Footnotes | |

References |

### Abstract

Acid-base chemistry is becoming evermore infamous amongst new chemistry students—especially specialized topics such as buffers and their capacities. Currently, introductory chemistry students struggle to fully understand the formal definition of buffer capacity. In this manuscript, we algebraically derive a new and simple mathematical representation for buffer capacity from the Henderson-Hasselbalch equation. We find that three buffer capacity formulas evolve—each with a specific application depending on the concentration-ratio of weak acid or base to its corresponding conjugate.

**Keywords:** analytical chemistry, first-year undergraduate/general, high school/introductory chemistry, problem solving/decision making, acids-bases

**Copyright**© 2015 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Matthew S. Fox, Joshua W. Greenberg, Jake M. Trookman. An Algebraic Derivation of Buffer Capacity.
*World Journal of Chemical Education*. Vol. 3, No. 5, 2015, pp 124-126. https://pubs.sciepub.com/wjce/3/5/4

- Fox, Matthew S., Joshua W. Greenberg, and Jake M. Trookman. "An Algebraic Derivation of Buffer Capacity."
*World Journal of Chemical Education*3.5 (2015): 124-126.

- Fox, M. S. , Greenberg, J. W. , & Trookman, J. M. (2015). An Algebraic Derivation of Buffer Capacity.
*World Journal of Chemical Education*,*3*(5), 124-126.

- Fox, Matthew S., Joshua W. Greenberg, and Jake M. Trookman. "An Algebraic Derivation of Buffer Capacity."
*World Journal of Chemical Education*3, no. 5 (2015): 124-126.

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### 1. Introduction

A buffer is defined as a solution that resists changes in *p*H upon the addition of either a strong acid or base. ^{[1]} Buffer capacity is then defined as the amount of strong acid or base needed to change the *p*H of a one liter buffer solution by one unit. ^{[2]} Calculating buffer capacity, however, deems to be a daunting—and seemingly esoteric—task to new chemistry students. As of now, the current, formal definition of buffer capacity is as follows:

where *n* is the number of equivalents added of either strong acid or base and C_{A} denotes the analytical concentration of acid. ^{[3]} Without context, the above formula is difficult to fully understand—and introductory chemistry students are likely to agree.

It is therefore the purpose of this work to derive a more comprehensible formula for the capacity of a buffer. The proposed methods are directed towards undergraduate courses in chemistry, but may deem applicable to Advanced Placement curricula. The major contributions in this article include our derivation of buffer capacity and the formulas that result. We base our method off of the Henderson-Hasselbalch representation for acid-base reactions.

### 2. Methods

All assertions in this section will be made regarding the following reactions:

(1) |

and

(2) |

where denotes the conjugate base for some weak acid HA.

We begin by considering the relationship between *p*H and *p*OH. For the purposes of this article, we assume the general definition of

(3) |

Thus, if the *p*H of a solution decreases by some amount Δ, the *p*OH of that same solution will increase by that same amount Δ. On account of eq 3, we can generalize the buffer capacity definition to the following:

**Definition 1** The *buffer capacity* of a solution is the amount (in mol) of strong acid or strong base that must be added to one liter of solution to change either the pH or pOH by one unit.* *

To begin the derivation, consider the Henderson-Hasselbalch representation for eqs 1 and 2:

(4) |

where represents the acidic strength of HA. ^{[4]} Because is a constant value—one unique to HA—the only term in eq 4 that could cause a change in *p*H (or *p*OH) by ±1 unit must be . Let

(5) |

then

Before proceeding, note the definitions below.

**Definition 2** We title a buffer solution as an *imperfect buffer **solution* whenever and a *perfect buffer solution* whenever * *

**Definition 3** If the buffer capacity of a solution is being tested via strong base, we title the capacity as the *base-buffer capacity*. If instead the solution is being tested via strong acid, we title the capacity as the *acid-buffer capacity*.* *

We being by considering the addition of a strong base to an imperfect buffer solution (eq 1). As shown in eqs 1 and 2, the stoichiometric ratio of is 1:1. Thus, if α mol of HA reacts completely with α mol of then α mol of will be produced (and eventually settled to equilibrium). Let Δ denote some arbitrary change in δ after α mol of HA have reacted. Then,

(6) |

Let denote the base-buffer capacity for eq 1.^{{1}} Then, and because bases attempt to increase *p*H. We then obtain the formula

Identically,

Solving for produces

Substituting in δ (eq 5) and reformulating yields

which simplifies to

Finally, multiplying both the numerator and denominator by [HA] yields the imperfect base-buffer capacity formula

(7) |

In other words, the maximum amount of base (in mol) a one liter solution of buffer can withstand before experiencing a change in *p*H or *p*OH by one unit is given by eq 7.

In continuation, consider eq 2. Here a strong base is being added to the buffer solution. Let denote the acid-buffer capacity for an imperfect buffer solution. To derive a suitable formula for , refer to eq 6. Because acids attempt to decrease *p*H, Δ= −1. Moreover, because the addition of mol of strong acid will decrease and increase [HA] by (opposite to adding strong base), Thus, utilizing eq 6, we acquire the formula

Equivalently,

We find

Substituting in δ and reformulating yields

Multiplying both the numerator and denominator by 10[HA] produces the imperfect acid-buffer capacity formula

(8) |

Comparing eqs 7 and 8, we find for all δ≠0. This occurs because the buffer is imperfect. If then we would expect because the buffer would have greater resistance to strong bases. If, however, the inverse were true, we would expect because the buffer would have greater resistance to strong acids.

We now consider a perfect buffer solution. Let denote the buffer capacity for a perfect buffer. Because a perfect buffer implies δ=0. We find, utilizing eqs 7 and 8, a perfect buffer’s capacity—in terms of [HA]—to simply be

(9) |

Identically, in terms of

(10) |

Eqs 9 and 10 yield a very trivial, yet quite surprising result. Both state that the maximum amount of strong acid or base that can be added to a perfect buffer solution, before a change of either one *p*H or *p*OH unit occurs, is simply the product of and the concentration of either HA or A^{−}.

### 3. Limitations and Assumptions

Throughout this article we have assumed a simple buffer solution—that is, a buffer with only one weak acid or base and its corresponding conjugate. If multiple weak acids or bases (and their corresponding conjugates) are present, the methods in this paper will not yield proper results. Further, we have assumed that polyprotic acids and bases do not introduce an additional level of acidity or basicity, respectively. Realistically, however, if the buffer capacity of a solution is being tested via a strong, polyprotic acid or base, it is likely the capacity will come out relatively low. This will occur because the additional level of acidity or basicity will cause a faster change in *p*H (or *p*OH) in comparison to a strong, monoprotic acid or base.

### 4. Implications and Conclusions

In this article, we introduced a new buffer capacity definition and derivation intended for entry-level chemistry courses. The derivation provides a more understandable notion on buffer capacity by utilizing one of the most fundamental equations in introductory acid-base chemistry: the Henderson-Hasselbalch equation. We believe this method will grant more intuition on buffer capacity than the current, formal definition—thus allowing students to better understand from where their computational results evolve.

### Notes

The authors declare no competing financial interests.

### Acknowledgements

This article was supported in part by Janie Mueller, Ruth Gibson, Adam Bentson, and Ally Arenson.

### Footnotes

1. For emphasis, is the amount (in mol) of OH^{−} being added to eq 1 to change either the *p*H or *p*OH by one unit.

### References

[1] | Brown, et al. Chemistry: The Central Science. 11th ed. Upper Saddle River, New Jersey: Pearson/Prentice Hall, 2008. | ||

In article | |||

[2] | Sandell, E.B.; West, T.S. Recommended Nomenclature for Titrimetric Analysis. Pure Appl. Chem. 1969, 18 (3), 430. | ||

In article | View Article | ||

[3] | Hulanicki, A. Reaction of Acids and Bases in Analytical Chemistry. PWN & Ellis Horwood, 1987. | ||

In article | |||

[4] | Po, H.N.; Senozan, N.M. The Henderson-Hasselbalch Equation: Its History and Limitations. J. Chem. Educ. 2001, 78 (11), 1499. | ||

In article | View Article | ||