Undergraduate Journal of Mathematical

Modeling: One + Two

Volume 10 | 2019 Fall 2019 Article 8

2019

Acid-Base Titration
David Pierre
University of South Florida

Advisors:
Arcadii Grinshpan, Mathematics and Statistics
Ruidong Ni, Chemistry

Problem Suggested By: Ruidong Ni

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Recommended Citation
Pierre, David (2019) "Acid-Base Titration," Undergraduate Journal of Mathematical Modeling:
One + Two: Vol. 10: Iss. 1, Article 8.
DOI: https://doi.org/10.5038/2326-3652.10.1.4913
Available at: https://scholarcommons.usf.edu/ujmm/vol10/iss1/8
Acid-Base Titration

Abstract
Titration is an analytical method used in biomedical sciences and analytical chemistry laboratories to
determine the quantity or the concentration of a known or unknown substance. There are many kinds
of titrations, but this investigation is fundamentally based on acid-base titration, in which the
development of a mathematical titration model is explained and determined. In a titration experiment,
the concentration and volume of the base added is proportionally equal to those of the acid. The
concentration of the substance is determined by following this principle. Moreover, these calculations
are usually approximate. Thus, the construction of a titration model is done to minimize the percent
errors, simultaneously describing the effect of titration, and also allowing the determination of the
characteristics of the unknown substance used. Furthermore, the values generated by the titration
model are compared to the theoretical ones. It allows the verification of the mathematical model
constructed. Subsequently, the efficiency of titration model is further tested by using a theoretical
problem to exclude the errors from the experiment. This verification displays the accuracy of the
accepted titration model. The concentration of the unknown acid is found by using this mathematical
model.

Keywords
titration model, titration experiment, acid, base

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 Pierre: Acid-Base
Titration
1

PROBLEM STATEMENT

A mathematical titration model is constructed to display the effects of the titration which is also

used to accurately determine the concentration of the unknown acid. The accurate determination

of the concentration of the unknown substance will prevent confusion between two mislabeled

substances that have different concentrations.

MOTIVATION

Titration is an analytical method used to quantify a substance from a solution. There are many

kinds of titration, the most common ones are acid-base, reduction-oxidation, and precipitation as

well as complexometric titrations. This inquiry is based on acid-base titration which is used in

the determination of the unknown concentration of the considered substance. In acid-base

titration, the data recorded are named as paired values (volume of base added and the pH of

solution). Our intention is to construct a graph (volume of base added vs the pH of the solution)

from the paired values (titration curve) which is used to calculate the unknown concentration of

the substance. The usable part of this curve is restricted to the equivalence point; which is found

by locating the region where the most rapid increase occurs. This determination is essentially

done by observation, which means that the calculation is based on estimation (Salem). Thus, if

two substances having different concentrations are mislabeled then their dissociation constants

are unknown; the variation between the concentrations allows the experimenter to identify them.

So, the objective is to use more data to minimize the percent errors. The additional data allow the

determination of the more accurate concentration. Having an accurate concentration prevents

confusion between two substances even if they have close concentration values.

Produced by The Berkeley Electronic Press,

2019
Furthermore, to facilitate the integration of the new data, a mathematical titration model is

constructed to demonstrate the correlation between the volume of base added and the pH of the

solution.

MATHEMATICAL DESCRIPTION AND SOLUTION APPROACH

Our objective is to create a mathematical model that represents the interaction between acid and

base, and between the volume of the titrant added and pH of the solution. In this paper, a weak

acid is used as solvent with a strong base.

In an acid-base reaction, the molecule of acid would react with the base to generate an anion of

the acid and water molecule:

HA (acid) + OH- (base) A- + H2O (Salem)

Therefore, the amount of mole of the acid prior the reaction occurs is equal to the volume times

the concentration (A∙a) and the base also follows the same principle (B∙b). Moreover, the reaction

reaches its equivalence point when enough base is added so A∙a=B∙b, and the solution is

monitored by measuring the pH of every drop base added. However, a weak acid does not

completely dissociate before reaching its equilibrium (Salem),

Dissociation Reaction: HA H+ + A- (Salem),

which means that the amount of moles of the acid present in this solution is equal to the amount

of moles of hydrogen ion. Therefore, to keep track of the concentration of the substance, the

amount of moles of each compound is divided by the total volume (a+b) used in this experiment

(Salem).
The concentration of the compound prior to the dissociation is determined by formulas (1) and

(2):

𝐴∙𝑎−𝐵∙𝑏
[𝐻𝐴] = 𝑎+𝑏 (1)

𝐵∙𝑏
[A−] =
𝑎+𝑏 (2)

The titration model is created by using the dissociation constant formula:

𝐾𝑎 = [𝐻+]∙[𝐴−] (3)
[𝐻𝐴]

By equations (1), (2), and (3), the dissociation constant formula can be rewritten in terms of A, a,

B, b, and X= [𝐻+] :

𝐵∙𝑏
𝑋( )
𝑋 ∙𝐵 ∙𝑏
𝐾𝑎 = 𝐴 𝑎+𝑏
= (4)
∙𝑎−𝐵∙𝑏
𝐴∙𝑎−𝐵∙𝑏
𝑎+𝑏

By solving (4) for X we have:

𝐾𝑎(𝐴∙𝑎−𝐵∙𝑏)
X= 𝐵∙𝑏 (5)

By knowing 𝑋 = [𝐻+], the pH of the solution (a measure of the molar concentration of hydrogen

ions) is:

pH = − log [𝐻 +] (6)

and the p𝐾𝑎 of the solution is:

𝑝𝐾𝑎 = −log (𝐾𝑎) (7)

By equations (5), (6), and (7), we obtain:

𝐾𝑎[𝐴 ∙ 𝑎 − 𝐵 ∙ 𝑏]
−log 𝑋 = − log ),
[𝐵 ∙ 𝑏]
(

− log[𝐻+] = − log(𝐾𝑎[𝐴 ∙ 𝑎 − 𝐵 ∙ 𝑏]) + log (𝐵 ∙ 𝑏),

pH = log(𝐵 ∙ 𝑏) − log(𝐴 ∙ 𝑎 − 𝐵 ∙ 𝑏) + 𝑝𝐾𝑎 (8)

In titration, any data beyond the equivalence point are irrelevant and unusable; thus, the model is

limited to 0< b < {b at the equivalence point}, and the equation for P (b) represents the pH of the

solution at the different volume of base added. Hence and by (8) the titration model is

𝑃(𝑏) = log(𝐵 ∙ 𝑏) − log(𝐴 ∙ 𝑎 − 𝐵 ∙ 𝑏) + 𝑝𝐾𝑎, (9)

where 0< b < {b at the equivalence point} and 𝐴 ∙ 𝑎 > 𝐵 ∙ 𝑏.

Experimental Problem

In this inquiry, the titration of an unknown acid is done with NaOH (base) which has a

concentration of 0.09141 M and the volume of solution before adding any base is 0.10 L

(Salem).

Data :
Volume of Acid: 0.10 L
Volume total of Base added: 0 .05400 L
Concentration of Base: 0.09141 M
Concentration of Acid: Unknown
𝑝𝐾𝑎: Unknown
𝐾𝑎: Unknown
Determination of the equivalence point

By observing the titration curve, a rapid increase between pH 6 and pH 10 is noticed, and the

equivalence point is found around this area. Moreover, the estimation is done by searching the

volume that is responsible for this increase. We obtain that the volume of NaOH at the

equivalence point is 0.04L (see Appendix).

Determination of the dissociation constant

Determination of the dissociation constant 𝐾𝑎 is done by knowing that at half-equivalence

dissociation point the number of moles of base is equal to the number of moles of acid.

Therefore, 𝐾𝑎 is calculated at half-equivalence point, [𝐴−] = [HA], V=V (equiv.)/2 (Reijenga

Jetse).

pH at half-equivalence point:

V= 0.04L/2=0.02 L

Hence the estimation for the pH at half-equivalence point is 5.40, where the volume is 0.02L or

close to it (see Appendix).

Determination of dissociation constant of the unknown acid by assuming that the pH at

half-equivalence point is 5.40

𝑝𝐾𝑎 = −log (𝐾𝑎)

𝐾𝑎 = 10−𝑝𝐾𝑎

𝐾𝑎 = 10−5.40 ≈ 3.981 × 10−6𝑀

Determination of the theoretical concentration of the unknown acid

𝐶1 × 𝑉1(𝑀1) = 𝐶2(𝑀2) × 𝑉2

0.09141𝑀 × 0.054𝐿 = 𝐶2 × 0.1𝐿

0.054𝐿
𝐶2(𝑀2) = 0.09141𝑀 × = 0.04936𝑀
0.1𝐿

(C2: Theoretical concentration of the unknown acid)

Determination of the concentration of the unknown acid at pH 5.50 using the titration

model ( see (9) and Appendix)

𝑃(𝑏) = log(𝐵 ∙ 𝑏) − log(𝐴 ∙ 𝑎 − 𝐵 ∙ 𝑏) + 𝑝𝐾𝑎

5.50 = log(0.09141𝑀 × 0.02711𝐿) − log (A×0.1L−0.09141M×0.02711L) +5.40

𝑆𝑜𝑙𝑣𝑒𝑑 𝑓𝑜𝑟 𝐴 (Concentration of acid)

−5.50 + log(0.09141𝑀 × 0.02711𝐿) + 5.40 = log ((A×0.10L)−(0.09141M ×0.02711L))

−2.705877 𝑚𝑜𝑙𝑒𝑠 = log (0.1𝐿 𝐴 − 2.4781 × 10−3𝑚𝑜𝑙𝑒𝑠)

10−2.705877𝑚𝑜𝑙𝑒𝑠 = 0.1𝐿 𝐴 − 2.4781 × 10−3𝑚𝑜𝑙𝑒𝑠

0.1𝐿𝐴 = 10−2.705877𝑚𝑜𝑙𝑒𝑠 + 2.4781 × 10−3 𝑚𝑜𝑙𝑒𝑠

4.4465437×10 𝑚𝑜𝑙𝑒𝑠
A= −3 = 0.044465437 M (moles/L)
0.1 𝐿
 VERIFICATION

Verification (Step 1) of the titration model (Experimental Problem)

Determination of percent error (Titration model deviation+ Experimental errors):

𝑇ℎ𝑒𝑜𝑟𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒−𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙
𝑉𝑎𝑙𝑢𝑒 0.04936𝑀−0.044465𝑀 | × 100% ≈ 9.92%
| | × 100% = | 0.04936𝑀
𝑇ℎ𝑒𝑜𝑟𝑖𝑐𝑎𝑙 𝑉𝑎𝑙𝑢𝑒

Therefore, the percent of efficiency of titration model with the experimental errors is

approximately 90.08%.

Verification (Step 2) of the titration model (Theorical Problem)

For the titration of 25.00 mL of CH3CO2H with 12.50 mL of 0.100 M NaOH:

𝐾𝑎 = 1.8 × 10−5 for CH3CO2H, pH =4.74. What is the concentration of the weak acid? (B.C)

Determination of the concentration of the acid (traditional method)

pH = 𝑝𝐾𝑎 + log [𝐵𝑎𝑠𝑒] (Bletry),

[𝐴𝑐𝑖𝑑]

𝑝𝐾𝑎 = − log(1.8 × 10−5) = 4.7447,

0.100𝑀
4.74 = 4.7447 + log ,
[CH3CO2H]

[CH3CO2H] = 0.10109𝑀

Determination of the concentration using the titration model (9)

𝑃(𝑏) = log(𝐵 ∙ 𝑏) − log(𝐴 ∙ 𝑎 − 𝐵 ∙ 𝑏) + 𝑝𝐾𝑎

4.74 = log(0.1𝑀 × 0.0125𝐿) − log(𝐴 × 0.025𝐿 − 0.1𝑀 × 0.0125𝐿) + 4.7447

 2.898 = −log (𝐴 × 0.025 𝐿 − 0.1 × 0.0125 𝑚𝑜𝑙𝑒𝑠)

−2.898 = log (A × 0.025 L − 0.1 × 0.0125 moles)

10−2.898 + (0.1 × 0.0125 𝑚𝑜𝑙𝑒𝑠) = 𝐴 × 0.025𝐿

𝑚𝑜𝑙𝑒𝑠
A= 2.515×10−3 = 0.10059 M (moles/L)
0.025 𝐿

The percent of efficiency (Theoretical Problem):

Determination of the (titration model) deviation:

𝑇ℎ𝑒𝑜𝑟𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒−𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙
𝑉𝑎𝑙𝑢𝑒 0.100𝑀−0.10059𝑀 | × 100% = 0.59%
| | × 100% = | 0.100𝑀
𝑇ℎ𝑒𝑜𝑟𝑖𝑐𝑎𝑙 𝑉𝑎𝑙𝑢𝑒

Therefore, the percent of efficiency of the titration model with no experimental errors would be

99.41%.

Determination of the experimental errors :

% 𝐸𝑟𝑟𝑜𝑟 = 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛(%) + 𝐸𝑥𝑝. 𝐸𝑟𝑟𝑜𝑟(%)

𝐸𝑥𝑝. 𝐸𝑟𝑟𝑜𝑟(%) = 9.08% − 0.59% = 8.49%

LIMITATION OF THE TITRATION MODEL

The volume of the base added in the titration model is restricted between 0 and the equivalence

point. However, the amount of moles of base and acid also influences the titration model. The
titration model is constructed on weak acid /strong base reaction in which the concentration of

the weak acid is lower than the base concentration. Moreover, what happens if the number of

moles of the acid tends to be equal to those of the base in the titration model? :

𝐴 ∙ 𝑎 ≈ 𝐵 ∙ 𝑏, 𝑝𝐾𝑎 ≈ P (b),

𝑃(𝑏) = log(𝐵 ∙ 𝑏) − log(𝐴 ∙ 𝑎 − 𝐵 ∙ 𝑏) + 𝑝𝐾𝑎,

0 = log(𝐵 ∙ 𝑏) − " log(0 +)" → log(𝐵 ∙ 𝑏) + "∞"

Therefore, 𝑃(𝑏) 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 𝑤ℎ𝑒𝑛 𝐴 ∙ 𝑎 ≈ 𝐵 ∙ 𝑏.

DISCUSSION

The purpose of this experiment is to create a mathematical model that determines an accurate

concentration of an unknown substance. By using the titration model, the result is approximately

0.044465 M which is close to the theoretical value (0.04936 M) of the substance. The percent

error of the titration model is approximately 9.1 % which is acceptable. However, the percent

error of the titration model also includes the experimental errors. Thus, by applying the titration

model in a theoretical problem, the deviation of the titration model is ~0.6% (as predicted),

because the titration model is created by considering any elements that can impact the result. If

the titration model has ~0.6% deviation in the theoretical problem, the deviation of titration

should be about the same under any circumstances. Therefore, the experiment errors were

calculated by using this equation: %Error = Deviation (%) + Exp. Errors (%).

So, the acid-base titration contains ~8.5 % of experimental errors and the titration model has

~0.6% deviation. A mathematical model with no deviation would likely generate accurate
values, and there will be no confusion between two substances that have close concentration

values. However, the deviation of the titration model is independent to the experimental errors,

which means the titration model cannot influence the experimental errors, but the experimental

errors can alter the result. Therefore, an accurate concentration is guaranteed if the experimental

error is low. Furthermore, the identification of the unknown substance would be easily done.

Thus, the titration model has accomplished its objective because the concentration of the

unknown substance (weak acid) is determined with a low percent error and the correlation

between pH and the volume of base added is demonstrated. However, the titration model has its

limitations which are:

1. The volume used should be between 0 and the volume of the base at the equivalence

point.

2. The amount of moles of acid and base present should not be equal.

CONCLUSION AND RECOMMENDATION

The mathematical model is based on acid-base titration and its development allows the

determination of the concentration of the acid used. The model is constructed by following the

dissociation constant rules and the amount of moles present. Its purpose is to minimize the

percent error as low as possible (<1%) which is achieved. By comparing the values generated by

the titration model to theoretical one, the percent efficiency is ~99.4% which is remarkable.

However, the titration model cannot minimize the experimental errors, which means the titration

model cannot erase any errors made during the experiment. The percent error in this experiment

is ~9%, which means the calculation, or the determination of the equivalence point is slightly off.
Therefore, to minimize the experimental errors, the use of computer algebra system is needed to

lower the percent error in the dissociation constant; and the acid-base titration should be repeated

several times. Besides, the titration model has an efficiency of ~99.4%, which means that any

value generated would be accurate. Moreover, the titration model displays a correlation between

pH, 𝑝𝐾𝑎, volumes, and concentrations. This correlation allows the determination of any one of

those four elements. Therefore, this mathematical model also describes the different

characteristics (pH, 𝑝𝐾𝑎, volume, concentration) of the substances used (acid or base).

In conclusion, the accurate concentration of the acid is found by using this model, see (9).

And any other characteristics (pH, 𝑝𝐾𝑎, volume) of the substances used could be also

determined.

Hypothetically, if two acidic substances are mislabeled they have different concentration values,

and the constant of dissociation are unknown. Those two substances could be easily identified by

using the titration model constructed in this paper, and the confusion between those two

substances would be elucidated.

 NOMENCLATURE(S)

Table 1. List of symbols used in this paper:

Symbols Descriptions Units

H+ Hydrogen Ion moles

A- Base moles

HA Acid moles

NaOH Sodium hydroxide moles

A Concentration of acid used M = moles/liter

B Concentration of base used M = moles/liter

a Initial volume of acid solution L=liter

b Total volume of base added L=liter

H2O Water molecule moles

Aa # of moles of the acid moles

Bb # of moles of the base moles

a+b Total volume of the system L=liter

pH -log[H+] none
[H+] Concentration of the Hydrogen Ion M = moles/liter

𝒑𝑲𝒂 -log[𝐾𝑎] none

[Base] or Concentration of the base used M = moles/liter

[A-]

[Acid] or Concentration of the acid used M = moles/liter

[HA]

𝑲𝒂 Dissociation constant of acid M = moles/liter

V(equiv.) Volume at equivalence point L=liter

V(equiv.)/2 Volume at half-equivalence point L=liter

P(b) pH of the solution at different volume of base added None

This table describes the symbols used in our

experiment with their units (SI)

 REFERENCES

B.C Open Textbook. ‘’ Acid-Base Titrations’’, Chemistry Open Textbook, ch.14, Custom

edition B.C Campus (OpenStax),Quebec, Canada.(B.C)

https://opentextbc.ca/chemistry/chapter/14-7-acid-base-titrations/

Bletry Marc, ‘’Henderson Hasselbach relationship and weak acid titration’’ Univeristé Paris-

Est Créteil, ICMPE-CNRS, Thiais, France. Published by Academia.(Bletry)

https://www.academia.edu/36251379/Henderson_Hasselbalch_relationship_and_w

eak_acid_titration.

Burton J.O and Acree S.F, ‘’Calculation of the Concentration and Dissociation Constant of

Each Acid Group in A Mixture from the pH Titration Curve of the mixture’’.US

Department of Commerce National Bureau Standards. 1936.(Burton)

https://nvlpubs.nist.gov/nistpubs/jres/16/jresv16n6p525_a1b.pdf

Reijenga Jetse, Van Hoof Arno, Van Loon Antonie and Teunissen Bram, ‘’Development of

Methods for the Determination of pKa Values’’ , Department of Chemical

Engineering and Chemistry, Eindhoven University of Technology, Eindhoven, The

Netherlands, Published online 2013 Aug 8, NCBI.(Reijenga Jetse)

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3747999/

Salem Anita, Paula Shorter and Dale Harak.’’ Titration’’ Mathematics and Chemistry

Department of Rockhurst University, Connected Curriculum Project, 1999-2000.

Published by Duke university.(Salem)

https://services.math.duke.edu/education/ccp/index.html.
 APPENDIX

Graph 1- Titration Curve

Titration Curve
12
pH of solution

10

0 0 0.01 0.02 0.03 0.04 0.05 0.06

Volume of NaOH added (L)

The titration curve is constructed by using data from Table 2 which shows the relation between

the pH of the solution and the volume NaOH added.

Table 2: Result collected

(L of base added) pH of solution

0.00 4.54
0.00101 4.55
0.00220 4.61
0.00329 4.70
0.00419 4.72
0.00517 4.79
0.00623 4.81
0.00740 4.88
0.00858 4.90
0.00963 4.97
0.01074 5.00
0.01150 5.02
0.01290 5.04
0.01399 5.11
 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 10, Iss. 1 [2019], Art. 8
16

0.01510 5.15
0.01615 5.20
0.01730 5.22
0.01849 5.26
0.01978 5.30
0.01986 5.33
0.01990 5.40
0.02285 5.43
0.02389 5.46
0.02502 5.49
0.02711 5.50
0.02721 5.54
0.02809 5.60
0.02929 5.63
0.03051 5.69
0.03150 5.75
0.03270 5.80
0.03365 5.89
0.03483 5.95
0.03573 6.02
0.03689 6.15
0.03711 6.21
0.03734 6.22
0.03759 6.25
0.03789 6.29
0.03809 6.34
0.03835 6.39
0.03861 6.48
0.03890 6.53
0.03914 6.65
0.03935 6.71
0.03951 6.80
0.03975 6.98
0.04000 7.30
0.04021 8.60
0.04030 9.12
0.04053 9.43
0.04079 9.64
0.04091 9.71
0.04111 9.80
0.04130 9.88
0.04159 9.96
0.04180 10.00
0.04210 10.06
0.04229 10.10

https://scholarcommons.usf.edu/ujmm/vol10/iss1/8
DOI: https://doi.org/10.5038/2326-3652.10.1.4913
 Pierre: Acid-Base
Titration
17

0.04251 10.15
0.04275 10.20
0.04300 10.21
0.04401 10.29
0.04510 10.38
0.04625 10.43
0.04733 10.49
0.04839 10.50
0.04949 10.52
0.05069 10.59
0.05189 10.60
0.05289 10.61
0.05400 10.62

Determination of the dissociation constant using Table 2 (Observational Method)

The 𝐾𝑎 is found by calculating the volume at half-equivalence point:

V= 0.04L/2=0.02,

𝑝𝐾𝑎 ≈ 5.40 (observation from data),

𝐾𝑎 = 10−5.40 ≈ 3.981 × 10−6𝑀

Therefore, the dissociation constant of the acid is ≈ 3.981 × 10−6𝑀.

Data used in the determination of the concentration of the unknown acid (using the titration

model).

pH= 5.50 𝑝𝐾𝑎= 5.40

b= 0.02711 ≈0.03 L B= 0.09141≈0.09 M
a= 0. 1L A= ? M