Abstract

Titration of an acid with a base of known concentration is a common method used to determine the concentration of the acid by determination of the endpoint, which describes the full neutralization of the acid, and the midpoint, which reveals the pKa values of the acids. In this paper, both monoprotic acid and diprotic acid titrations conducted in the laboratory and analyzed experimentally using the first derivative of the titration curve and midpoint method were used to generate binding curves in order to functionaly determine the Ka values of the acids. The binding curves were graphed by establishing a fraction bound function that generated fraction bound values per volume of base added based on the consumption of hydronium ions and addition of base. A non-linear least squares curve of best fit was used to fit the fraction bound data points by establishing a function that uses the acid dissociation equilbirum equation and a prediction of the Ka values that best fit the data. This analysis gave functionally predicted Ka values, which were then compared to the experimentally determined Ka values as determined by the midpoint method, and then compared to the literature values describing the acids.

Introduction

Titration is an Analytical Chemistry technique used to determine the concentration of a known volume of acid solution by neutralizing (deprotonating) it with a base solution of known volume and concentration. In the laboratory, we conducted two separate titrations: one for a Monoprotic acid and one for a Diprotic acid.

A Monoprotic Acid is a compound that contains a single acidic Hydrogen atom, meaning that it can be removed from the compound with the addition of a base in the solution. The Hydrogen atom is referred to as a proton because it is removed as a cation containing only 1 proton and no electron. When the acid is first dissolved in an aqueous solution, the assumption is made that all of the acid molecules are still protonated, meaning that they still contain their acidic proton. However, this assumption is erroneous because the very reason the pH of the acid solution is lower than the neutral value of 7 is because some of the acid molecules partially dissociate to form their conjugate base and hydronium according to the following equation:

\(HA + H_2O~ \rightleftharpoons ~A^- + H_3O^+\)

As base is added to the solution, more of the hydronium ions become consumed which causes the acid dissociation equilibrium described above to be perturbed and cause more of the acid molecules to become deprotonated, causing the hydronium ion concentration to decrease. The following equation describes this reaction where hydroxide from the dissociation of sodium hydroxide acts as the base:

\(OH^- +H_3O^+~ \rightleftharpoons~ 2~H_2O\)

The two equations above can be combined to form the following net equation:

\(HA + OH^-~ \rightleftharpoons~ A^- +H_2O\)

The pH value of the solution is inversely proportional to the concentration of hydronium ions in solution, and can be measured using a pH meter which measures this difference in hydronium concentration. The pH value at which half of the acid molecules are deprotonated is known as the acid’s pKa value. Particular acids have a particular identifiable pKa value. The point at which all of the acid molecules have been deprotonated is known as the endpoint, and is easily identifiable because at this point is where the sharpest increase in pH occurs with each incremental addition of base. Thus, by identifying the volume of base of known concentration needed to reach the endpoint, this identifies the number of moles of base needed for neutralization, which is equal to the number of moles of acid in solution. For this reason, the endpoint is also called the equivalence point. Because we know the initial volume of acid, we can determine the concentration of the acid solution. By dividing the volume of base needed for equivalence by 2 and identifying the pH at this value, we can determine the acid’s pKa value.

A Diprotic Acid is a compound that contains 2 acidic protons with distinct pKa values. It will have 2 equivalence points, where the first equivalence point represents the point at which all the acid molecules have had their more acidic proton removed, and the second equivalence point represents the point at which all acid molecules have had the second acidic proton removed. Once again, by conducting titration and identifying the equivalence points, the pKa1 and pKa2 values can be identified. In the experiment conducted in the laboratory, the identify of the diprotic acid was unknown, but by using a known volume of the solvent and known weight of the acid compound, the determination of the moles of acid helped identify the molecular weight of the compound which in tandem with the two pKa values was used to identify the diprotic acid.

Titration curves were generated for both the Monoprotic Titration and Diprotic Titration, which included the volume of base (NaOH) added in milliliters on the x-axis and the corresponding pH of the solution as detected by the pH meter on the y-axis.

The purpose of this report is to convert the titration curves of the monoprotic and diprotic acids into binding curves using Posit Cloud functions. We can use the binding curve to functionally determine the pKa values and compare them to the experimental method done in the laboratory.

Monoprotic Acid Titration Curve

In the laboratory, 25 mL of an unknown concentration of Acetic Acid was titrated against 0.1 M of NaOH. A table was generated including the total volume of NaOH added to the solution and the corresponding pH value at that volume. This data was entered as separate vector values of Volume and pH. A plot of pH of solution vs. Volume of NaOH in mililiters was graphed. ________________________________________________________________

Monoprotic_Titration_Data <- read.csv("Monoprotic_Titration_Data.csv")

Volume <- Monoprotic_Titration_Data$Volume

pH <- Monoprotic_Titration_Data$pH

plot(Volume, pH, main = "Titration Curve", xlab = "Volume of NaOH (mL)", ylab = 
       "pH",pch = 20, col="violet")

Monoprotic Acid Binding Curve Transformation

The fraction bound equation for a monoprotic acid was given as:

\(Fraction~Bound~ = 1- \frac{(BC~ * ~Vadd)~ +~ [ [H^+] (Vini~ + ~Vadd)]}{(BC~ * ~Vend)}\)

Where:

As the numerator’s first term increases at a fast rate, the numerator’s second term decreases at a slower rate, causing their overall sum to increase and approach the denominator term and the full fraction to approach 1. This causes the fraction bound value to approach zero. Thus, this equation helps us view the decrease in binding affinity between the acidic protons at their original acid molecules as base is added to solution.

All variables of the fraction bound equation were defined either by constant values or as functions of the vectors: Proton concentration was defined as 10 to the power of the negative pH value, the base concentration was defined as 0.1, the initial volume of the acid was defined as 25, and the added volume was defined as the volume of base added. An estimate of the volume needed to reach the endpoint was established to ensure that the fraction bound function did not produce values lower than 0. This estimate was used based on the volume at which the sharpest increase in pH occurred in the titration curve.

The fraction bound equation was used to generate an output of fraction bound values based on the pH and volume input values. A graph of the fraction bound vs. the pH was generated to produce the Binding Curve.

Posit Cloud’s nls2 function was downloaded and used to generated a non-linear least squares curve of best fit to fit the data using the following fraction bound equation derived from the equilibrium formula of the dissociation of the acid:

\(Fraction~ Bound~ \approx~ \frac{[H^+]}{[H^+]~+~K_a}\)

The nls2 function works by beginning at a given start value of Ka and trying various values until a proper value is found that creates a proper curve of fit. By plotting this prediction generated by the nls2 function on the Binding Curve graph, the curve can be viewed superimposed on top of the points corresponding to the fraction bound values at given pH. This function gives a prediction of the Ka value of the acid. ________________________________________________________________

#Proton Concentration (M): 

H <- 10^(-pH) 

#Base Concentration (M):

BC <- 0.1

#Initial Acid Volume (mL):

Vini <- 25

#Estimated Volume of Base (mL) needed to reach Endpoint:

Vend <- 19.5

#Volume of Base Added (mL):

Vadd <- Volume

#Fraction Bound Function and Values:

FB <- 1 - ((BC * Vadd) + H  * (Vini + Vadd)) / (BC * Vend)

FB
##  [1]  9.983971e-01  8.969558e-01  7.946210e-01  6.921218e-01  5.896307e-01
##  [6]  4.870930e-01  3.845550e-01  2.820086e-01  1.794631e-01  1.281880e-01
## [11]  1.025513e-01  7.691371e-02  5.127610e-02  4.102065e-02  3.076452e-02
## [16]  2.051012e-02  1.025606e-02  5.127951e-03 -8.097023e-08
#Plotting the Fraction Bound Graph with a Non-Linear Least Squares Curve of Best Fit:

plot(pH,FB, main = "Binding Curve",xlab = "pH",ylab = "Fraction Bound",pch=20,col="green")

library(nls2)
## Loading required package: proto
fit <- nls2(FB ~ H/(Ka+H), start=c(Ka=0.0001))

summary(fit)
## 
## Formula: FB ~ H/(Ka + H)
## 
## Parameters:
##     Estimate Std. Error t value Pr(>|t|)    
## Ka 4.970e-06  1.131e-07   43.95   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01382 on 18 degrees of freedom
## 
## Number of iterations to convergence: 6 
## Achieved convergence tolerance: 1.03e-06
lines(pH,predict(fit),col="blue")

Determination of Ka

As shown in the summary of the “fit” value, the name given to the output of the nls2 function, the function predicted a Ka value of \(4.970 *10^{-6}\) M.

Comparison of Titration and Binding Analysis

In the traditional titration analysis used in the laboratory, the first derivative of the titration curve was taken and used to generate a graph of the first derivative vs the volume of base added to identify the point of largest pH increase. This point was identified as the endpoint as used to determine the midpoint, which defines the pKa value of the acid:

Initial volume: 0 mL | Final volume: 19.5 mL | Midpoint volume: 9.75 mL

pH at 9.75 mL = 5.299 | pH = pKa | Ka = 10 - pKa = 10 - 5.299 = \(5.0234 *10^{-6}\) M.

Using the binding curve method, the Ka value was determined to be \(4.970 *10^{-6}\) M. Thus, the titration analysis method and binding curve method produce similar results, confirming that the titration curve was appropriately converted to a binding curve.

The literature value for the pKa of Acetic Acid is 4.76, which means that its Ka should actually be \(1.738 *10^{-5}\) M. This means that the acid’s observed value from the lab, an average Ka of \(4.997 *10^{-6}\) M, actually had a larger pKa value of about 5.29. This could be due to a contamination of the sample or a systematic error of the pH meter.

The pros of using the titration method is that it is faster, but the cons are that it relies on an estimation of the endpoint and thus an estimation of the midpoint to determine the Ka value. The pros of using the binding curve method is that it uses two mathematical functions that define the fraction bound: one to generate the fraction bound points using Hydronium ion consumption and base addition, and one to fit the curve based on the acid dissociation equilibrium. This generates high accuracy in the prediction of the curve of best fit, which allows for a seemingly more accurate Ka value. The con of the binding curve method is that it is slightly more tedious.

Diprotic Acid Titration Curve

In the laboratory, 0.08 g of an unknown Diprotic Acid was dissolved in 25 mL of deionized water to form a 25 mL volume of diprotic acid solution of unknown identity and unknown molar concentration. This diprotic acid solution was titrated against 0.1 M of NaOH. A table was generated including the total volume of NaOH added to the solution and the corresponding pH value at that volume. This data was entered as separate vector values of Volume_D and pH_D. A plot of pH of solution vs. Volume of NaOH was graphed, just as in the Monoprotic Acid Titration. ________________________________________________________________

Diprotic_Titration_Data <- read.csv("Diprotic_Titration_Data.csv")

Volume_D <- Diprotic_Titration_Data$VolumeD

pH_D <- Diprotic_Titration_Data$pHD

plot(Volume_D, pH_D, main = "Diprotic Acid Titration Curve", xlab = 
       "Volume of NaOH (mL)", ylab = "pH",pch = 20, col = "turquoise")

Diprotic Acid Binding Curve Transformation

The fraction bound equation for a diprotic acid was given as:

\(Diprotic~ Fration~ Bound~ =~ 2~ - \frac{(BC~ *~ Vadd)~ +~ [H^+](Vini~ +~ Vadd)}{(BC~ *~ Vend)}\)

Again, all variables of the fraction bound equation were defined either by constant values or as functions of the vectors: Proton concentration was defined as 10 to the power of the negative pH value, the base concentration was defined as 0.1, the initial volume of the acid was defined as 25, and the added volume was defined as the volume of base added. An estimate of the volume needed to reach the 2nd endpoint was established and divided by 2 to ensure that the fraction bound function did not produce values lower than 0. ________________________________________________________________

#Proton Concentration:

H_D <- 10^(-pH_D) 

#Base Concentration:

BC_D <- 0.1

#Initial Acid Volume:

Vini_D <- 25

#Estimated Volume of Base Needed to reach Endpoint:

Vend_D <- 13.5/2    

#Amount of Volume Added:

Vadd_D <- Volume_D

#Fraction Bound:

FB_D <- 2 - ((BC_D * Vadd_D) + H_D  * (Vini_D + Vadd_D)) / (BC_D * Vend_D)

FB_D
##  [1] 1.371020869 1.370154333 1.354631130 1.352370516 1.346979823 1.333416142
##  [7] 1.299333046 1.264554377 1.244853145 1.210191683 1.205116234 1.185080475
## [13] 1.178788847 1.158316904 1.163689074 1.142597554 1.145423501 1.135171328
## [19] 1.113221342 1.122883094 1.110011725 1.106015037 1.109070811 1.101602846
## [25] 1.092816435 1.096538948 1.090753855 1.088631223 1.083694303 1.080635579
## [31] 1.074311335 1.068431527 1.061926738 1.053929723 1.042002308 1.028340768
## [37] 1.011479162 0.992058354 0.970590956 0.946881487 0.922095131 0.896013121
## [43] 0.868635016 0.841608422 0.813897317 0.785682854 0.757293589 0.728751071
## [49] 0.700074197 0.671279541 0.642381635 0.613393222 0.584232984 0.554876006
## [55] 0.525623801 0.496264414 0.466882749 0.437537981 0.408163742 0.378808090
## [61] 0.349418473 0.320032838 0.290612068 0.261185369 0.231725480 0.206270964
## [67] 0.187274271 0.170661405 0.150908924 0.129072600 0.099534466 0.077077974
## [73] 0.058014957 0.050032163 0.030060774 0.027104286 0.026366949 0.025034878
## [79] 0.021924711 0.020443584 0.016740198 0.015999641 0.014073826 0.009481306
## [85] 0.008592465 0.006814720 0.002222150
plot(pH_D,FB_D, main = "Binding Curve",xlab = "pH",ylab = "Fraction Bound",
     pch=20,col="darkorange")

library(nls2)

fit_D <- nls2(FB_D ~ ( ( (H_D/Ka2) + ((2*H_D^2)/(Ka1*Ka2)) ) / ( 1 + (H_D/Ka2) 
    + (H_D^2/(Ka1*Ka2)) ) ), 
    start=c(Ka1=0.0001,Ka2=0.0001))

summary(fit_D)
## 
## Formula: FB_D ~ (((H_D/Ka2) + ((2 * H_D^2)/(Ka1 * Ka2)))/(1 + (H_D/Ka2) + 
##     (H_D^2/(Ka1 * Ka2))))
## 
## Parameters:
##      Estimate Std. Error t value Pr(>|t|)    
## Ka1 4.940e-02  2.256e-03   21.90   <2e-16 ***
## Ka2 5.551e-05  2.092e-06   26.54   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.04068 on 85 degrees of freedom
## 
## Number of iterations to convergence: 9 
## Achieved convergence tolerance: 7.891e-06
lines(pH_D,predict(fit_D),col="purple")

Determination of Ka1 and Ka2 Functionally

The NLS2 function was used to predict the Ka1 and Ka2 values of the diprotic acid. It found that:

Determination of pKa1 and pKa2 Experimentally

The midpoint method was used in the laboratory to analyze the diprotic titration curve and determine pKa1 and pKa2. The first and second equivalence/endpoints were determined by graphing the first derivative of the titration curve vs the volume of base added to determine at which points are the sharpest increase in pH occur. These two endpoint volumes were used to find the midpoint volumes, which were used to identify the pH at those values and define pKa1 and pKa2.

Using the midpoint method to analyze the titration curve of the diprotic acid gave:

Comparison of Experiment and Computational Methods

In the laboratory, after the molecular weight of the acid was determined and coroborated with the experimentally determined pKa1 and pKa2 values of 1.99 and 4.20, the unknown diprotic acid was identified as Oxalic acid

The literature pKa1 and pKa2 values for Oxalic Acid are pKa1 = 1.27 and pKa2 = 4.27.

The Binding curve method that uses NLS2 to predict the Ka values (which are converted to pKa values) is much more accurate and closer to the literature values than the experimental method. This confirms that it is a reliable method for determining Ka values, especially for difficult acids. The first acidic proton in Oxalic Acid has a very small titration curve with a very small change in pH value across a rather larger amount of added base, creating a lot of room for error when determining its endpoint and midpoint. The binding curve analysis helps accurately analyze this change in pH.

Conclusion

In conclusion, binding curve analysis is an excellent way to graphically visualize the change in fraction bound percentage and to further draw mathematical estimations on the Kd value, which describes how strongly the substrate binds to the active site. As a biochemisty student interested in analyzing the binding affinities of different substrates and enzymes, for future studies I could see myself using this fraction bound technique with the NLS2 function to compare the Kd values of different substrates to an enzyme to determine which has the greatest affinity. This could be used in an instance in drug design where the goal is to synthesize a substrate molecule that exhibits competitive inhibition of an enzyme and outcompetes another substrate because of its higher affinity for the enzyme.