Abstract

This experiment examined the titration of a diprotic acid with a known strong base in order to determine the acid’s dissociation constants, Ka1 and Ka2, and pKa values. The autotitrator gradually added a known base and monitored pH changes automatically. This allowed for both a titration curve and a binding curve to be generated, which were experimental and functional results, respectively. This analysis revealed two distinct buffering regions that correlated with the sequential deprotonation of the diprotic acid’s two acidic protons. For the experimental data, a first derivative of the titration curve was created to determine the precision of the endpoint determinations, which identified points with full neutralization of each proton site. The fraction bound was calculated for each protonation state, which revealed the equilibrium dynamics between the protonated and deprotonated species through hydronium ion consumption and base addition. A non-linear least squares method (nls2) was used to establish a curve of best fit to the fraction bound data along with the acid dissociation equilibrium equation to predict Ka1 and Ka2 values. This process helped determine accurate dissociation constants, highlighting the unique features of diprotic acid during a titration and its behavior in aqueous solutions.

Introduction

Through auto-potentiometric titration using an NaOH titrant, we aimed to identify an unknown diprotic acid. NaOH was gradually added and recorded using precise pH measurements through the use of an autotitrator. This produced a titration curve which shows two distinct equivalence points (EQP1 and EQP2). Two equivalence points are unique to diprotic acids since it represents the complete neutralization of each acidic proton. The first equivalence point (EQP1) represents the neutralization of the acid’s first proton (H2A → HA-). The second equivalence point (EQP2) represents the neutralization of the second proton (HA→ A2-). This information allowed us to calculate key properties of the acid like molar mass, acid dissociation constants, and pKa values, which allowed us to use literature to determine the identity of the unknown acid.

The acid dissociation constants Ka1 and Ka2 reveal the deprotonation behavior of the acid at each equivalence point. For a diprotic acid (H2A), the equilibrium expressions for dissociation are:

First Dissociation: \(H_2A \Leftrightarrow H^+ + HA^-, K_a1 =\frac{[H^+][HA^-]}{[H_2A]}\)

Second Dissociation: \(HA^- \Leftrightarrow H^+ + A^2-, K_a2 =\frac{[H^+][A^2-]}{[HA^-]}\)

Each dissociation constant Ka is linked to a pKa value. pKa1 correlates with the pH at which the concentrations of the acid’s first conjugate base (HA-) and the protonated form (H2A) are equal. pKa2 corresponds with the pH at which the concentration of the fully deprotonated form (A2-) equals that of HA-. These values were determined by examining the midpoint pH between each equivalence point. The first midpoint was determined by finding the average pH halfway between the initial and first equivalence points for pKa1. For the second midpoint (pKa2), the pH between the first and second equivalence points was determined.

Bromocresol Green was used as a pH indicator for the Fast Titration to show physically when the second equivalence point was approaching. At a pH near 4.66, there was a color change from yellow to green to blue. These pKa values and the molar mass of the unknown acid were calculated. The autotitrator was useful in reducing human error and was also practical in terms of laboratory use.

For the functional data analysis, the fraction bound at each protonation state was calculated. This is helpful in understanding the equilibrium dynamics between protonated and deprotonated species. Hydronium ion consumption and volume of base added were important factors in determining fraction bound. Fraction bound examines the degree to which each protonation site is occupied. This works to show how a diprotic acid responds to changes in pH throughout the titration process. A non-linear least squares method (nls2) was used to analyze the data. This function is used to create a curve of best fit to the fraction bound data points. It involves the acid dissociation equilibrium equation and develops the relationship between the concentration of each species and the acid dissociation constants. By predicting the best-fit Ka values, we were able to accurately calculate Ka1 and Ka2. These values show how a diprotic acid is unique in its two-step dissociation and its buffering behavior in aqueous solutions. This analysis was helpful in estimating Ka and showing sequential deprotonation during titration.

Diprotic Acid Titration Curve

To experimentally determine Ka1 and Ka2, we analyzed the titration curve generated by gradually adding NaOH titrant to the unknown diprotic acid solution and recording precise pH measurements at each step using the autotitrator. The curve displayed two equivalence points (EQP1 and EQP2), characteristic of a diprotic acid, where each acidic proton is neutralized by the base. The titration curve was then differentiated to plot the first derivative against volume, allowing us to accurately locate the equivalence points as peaks in the derivative plot.

Using the volumes at these equivalence points, we identified the midpoints between the initial and first equivalence points and between the first and second equivalence points. The pH values at these midpoint volumes directly provided the experimental pKa values, with pKa1 correlating to the first midpoint (where [H2A] = [HA−]) and pKa2 to the second midpoint (where [HA−] = [A2−]). These experimentally derived pKa1 and pKa2 values offered insights into the acid’s dissociation behavior and could be compared to literature values for identification. ________________________________________________________________

#Diprotic titration data analysis - Anna Vikatos

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

Volume_DT <- Diprotic_Titration_Data$VolumeDT

pH_DT <- Diprotic_Titration_Data$pHDT

plot(Volume_DT, pH_DT, main = "Diprotic Acid Titration Curve", xlab = "Volume of NaOH (mL)", ylab = "pH",pch = 3, col = "seagreen")

Diprotic Data Transformation Using Diprotic Fraction Bound Equation

Fraction Bound Equation for Diprotic Acid: \(Diprotic~ Fration~ Bound~ =~ 2~ - \frac{(BC_{DT}~ *~ Vadd_{DT})~ +~ [H_{DT}^+](Vi_{DT}~ +~ Vadd_{DT})}{(BC_{DT}~ *~ Ve_{DT})}\)

In the fraction bound equation, each variable was defined using constants or calculated values. Proton concentration, [H⁺], was calculated as \({10^{-pH}}\) for each recorded pH point. The base concentration was set at 0.1 M, the starting volume of the acid solution was 25 mL, and the volume of added base was recorded at each step. To ensure the fraction bound function didn’t give negative values, we estimated the volume needed to reach the second endpoint and divided it by two to set a consistent reference point. ________________________________________________________________

# DT = Diprotic titration

# H+ concentration

H_DT <- 10^(-pH_DT) 

# Base concentration

BC_DT <- 0.1

# Initial acid volume, relative volume of base required to reach endpoint, and volume added

Vi_DT <- 25

Ve_DT <- 16.527/2    

Vadd_DT <- Volume_DT

# Fraction bound formula

FB_DT <- 2 - ((BC_DT * Vadd_DT) + H_DT  * (Vi_DT + Vadd_DT)) / (BC_DT * Ve_DT)

FB_DT
##  [1]  1.436653e+00  1.435935e+00  1.435218e+00  1.420217e+00  1.428901e+00
##  [6]  1.417705e+00  1.402561e+00  1.399255e+00  1.395198e+00  1.378831e+00
## [11]  1.373783e+00  1.368041e+00  1.361629e+00  1.354570e+00  1.356443e+00
## [16]  1.347794e+00  1.338576e+00  1.328809e+00  1.326510e+00  1.315396e+00
## [21]  1.310898e+00  1.305356e+00  1.298706e+00  1.291013e+00  1.287851e+00
## [26]  1.277919e+00  1.271894e+00  1.264391e+00  1.259451e+00  1.252538e+00
## [31]  1.243856e+00  1.233583e+00  1.224364e+00  1.215428e+00  1.206088e+00
## [36]  1.195787e+00  1.184123e+00  1.172613e+00  1.161470e+00  1.147464e+00
## [41]  1.138486e+00  1.131348e+00  1.123796e+00  1.115879e+00  1.106009e+00
## [46]  1.096115e+00  1.077914e+00  1.063020e+00  1.039001e+00  1.014910e+00
## [51]  9.907678e-01  9.666085e-01  9.424339e-01  9.182548e-01  8.940670e-01
## [56]  8.697584e-01  8.455650e-01  8.213723e-01  7.971769e-01  7.729807e-01
## [61]  7.487838e-01  7.245853e-01  7.003873e-01  6.761880e-01  6.519884e-01
## [66]  6.277893e-01  6.035891e-01  5.793887e-01  5.551881e-01  5.309873e-01
## [71]  5.067864e-01  4.825853e-01  4.583848e-01  4.341834e-01  4.099821e-01
## [76]  3.857805e-01  3.615789e-01  3.373773e-01  3.131755e-01  2.889736e-01
## [81]  2.647718e-01  2.405699e-01  2.163679e-01  1.921659e-01  1.679639e-01
## [86]  1.437617e-01  1.195596e-01  9.535748e-02  7.115523e-02  4.816303e-02
## [91]  3.690890e-02  2.807500e-02  2.020918e-02  1.319043e-02  8.833944e-03
## [96] -5.769877e-08
plot(pH_DT,FB_DT, main = "Binding Curve",xlab = "pH",ylab = "Fraction Bound",pch=5,col="tomato")

library(nls2)
## Loading required package: proto
fit_DT <- nls2(FB_DT ~ ( ( (H_DT/Ka1) + ((2*H_DT^2)/(Ka1*Ka2)) ) / ( 1 + (H_DT/Ka1) + (H_DT^2/(Ka1*Ka2)) ) ), 
              start=c(Ka1=0.0001,Ka2=0.0001))

summary(fit_DT)
## 
## Formula: FB_DT ~ (((H_DT/Ka1) + ((2 * H_DT^2)/(Ka1 * Ka2)))/(1 + (H_DT/Ka1) + 
##     (H_DT^2/(Ka1 * Ka2))))
## 
## Parameters:
##      Estimate Std. Error t value Pr(>|t|)    
## Ka1 4.076e-07  2.034e-08   20.04   <2e-16 ***
## Ka2 2.118e-02  1.009e-03   21.00   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.05912 on 94 degrees of freedom
## 
## Number of iterations to convergence: 12 
## Achieved convergence tolerance: 7.942e-07
lines(pH_DT,predict(fit_DT),col="darkblue")

Functional Determination of Ka1 and Ka2 (NLS2)

The NLS2 function was applied to estimate the Ka1 and Ka2 values of the diprotic acid, producing the following results:

Experimental Determination of Ka1 and Ka2

To determine Ka1 and Ka2 experimentally, we generated a titration curve by adding NaOH titrant to the unknown diprotic acid solution and recording pH measurements with an autotitrator. The curve’s two equivalence points (EQP1 and EQP2) were located by plotting the first derivative, and the pH values at the midpoints between the initial and first equivalence points, and between the first and second equivalence points, provided the experimental pKa values.

Ka Determination Comparison

The comparison between experimental and NLS2-derived Ka values shows that NLS2 provides more precise and reliable results. For Ka1, the experimental value of 0.00772 is much higher than the NLS2-derived \(4.076×10^{-7}\) , which has a low error and high confidence. This difference likely comes from difficulties in pinpointing the first equivalence point in the experiment, where pH changes are small. For Ka2, the experimental value \(4.57×10^{−6}\) is much lower than the NLS2 result of \(2.118×10^{−2}\), which is statistically stronger. These findings suggest that NLS2’s binding curve analysis is more accurate, especially for weak acids where subtle pH changes make the visual method challenging.

Discussion | Conclusion

In this experiment, we identified the unknown diprotic acid as malonic acid through auto-potentiometric titration, and supported this with binding curve analysis. The autotitrator’s precise data points allowed us to generate a pH versus volume titration curve and a corresponding binding curve, which enhanced our understanding of the acid’s dissociation behavior. By applying the non-linear least squares (NLS2) function to the binding curve, we achieved a good fit that accurately predicted the Ka values and the pKa values of the acid. This approach was important because the first acidic proton of malonic acid displayed a narrow titration curve which is difficult to produce correctly. So, the binding curve analysis allowed for a more sensitive measurement of these slight pH changes. This meant that our acid dissociation constant values were relatively close to literature references. The binding curve shows less tightly packed lines because the autotitrator consistently ended the titration too early, causing errors. To address this, we combined multiple sets of data to create a more accurate curve. By calculating the first derivative of the data (dpH/dV), we could more reliably identify EQP1 and EQP2, resulting in a clearer and more accurate representation of the binding curve than the autotitrator alone provided. This binding curve approach minimized manual errors and provided a functional way to examine the acid’s binding dynamics over a large range of pH values. This in turn made the endpoint determination results more reliable. This method speaks to broader implications in field that require precise binding data analysis, like enzyme-substrate affinity studies, where the fraction bound is a key indicator. For future studies, refinement of data points around the equivalence volumes and advanced computational analysis could improve accuracy in determining binding constants and exploring acid-base properties in more complex biological contexts.