Introduction

The Quantitative Analysis Lab Experiment 7, “Study of Diprotic Acid Equilibrium by an Autotitrator”, introduced the class to a new procedure of potentiometric titration other than manual titration. In the study conducted by Martínez et al.(2000), automated poteniometric titration was utilized to determine acid-base equilibrium constants of beta-blockers atenolol, oxprenolol, timolol, and labetalol. The study found that automatic titration had the advantage of producing large data sets, thereby increasing the definition of titration curves and subsequently increasing accuracy in pKa values (Martinez et al., 2000). In addition, the implementation of auto-titration eliminates human error, leaving only random error and systematic error as sources of bias.

Mydata <- read.csv("Diprotic Titration - Fast Titration - Sheet2.csv",header = TRUE, sep = ",")

volume <- Mydata$Consumption..mL.

pH <- Mydata$Potential..pH.

Xdata <- data.frame(volume,pH)

knitr::kable(Xdata[,], col.names = c('Volume (mL)[NaOH]', 'pH'), caption = "Diprotic Titration Curve")
Diprotic Titration Curve
Volume (mL)[NaOH] pH
0.000 2.61
0.005 2.61
0.010 2.61
0.023 2.61
0.054 2.61
0.132 2.62
0.328 2.64
0.528 2.67
0.728 2.70
0.928 2.73
1.128 2.76
1.328 2.79
1.528 2.82
1.728 2.85
1.928 2.88
2.128 2.92
2.328 2.95
2.528 2.98
2.728 3.02
2.928 3.06
3.129 3.09
3.329 3.12
3.529 3.16
3.729 3.19
3.929 3.23
4.129 3.27
4.329 3.31
4.529 3.35
4.729 3.39
4.929 3.43
5.129 3.48
5.329 3.53
5.529 3.58
5.729 3.63
5.929 3.69
6.129 3.76
6.329 3.83
6.529 3.90
6.729 3.98
6.929 4.08
7.129 4.18
7.329 4.30
7.529 4.43
7.729 4.57
7.929 4.69
8.129 4.80
8.329 4.91
8.529 5.00
8.729 5.07
8.929 5.16
9.129 5.23
9.329 5.29
9.529 5.35
9.730 5.40
9.930 5.46
10.130 5.51
10.330 5.55
10.530 5.60
10.730 5.64
10.930 5.69
11.130 5.74
11.330 5.78
11.530 5.82
11.730 5.86
11.930 5.89
12.130 5.94
12.330 5.99
12.530 6.03
12.730 6.07
12.930 6.12
13.130 6.16
13.330 6.21
13.530 6.26
13.730 6.31
13.930 6.36
14.130 6.42
14.330 6.48
14.530 6.54
14.730 6.60
14.930 6.68
15.130 6.76
15.330 6.85
15.530 6.96
15.730 7.07
15.930 7.23
16.130 7.42
16.296 7.61
16.416 7.74
16.583 7.96
16.691 8.28
16.717 8.39
16.747 8.49
16.816 8.74
16.857 8.86
16.935 9.05
17.017 9.18
17.188 9.43
17.319 9.58
17.519 9.76
17.719 9.89
17.919 10.03 
plot(volume,pH,main = "Titration Curve", xlab = "Volume of NaOH (ml)", ylab = "pH")

Diprotic Binding Analysis

The fraction bound equation for a diprotic system represents the full initial binding of a Bronsted-Lowry acid molecule and its ligand. The formula is expressed as:

FB = 2-(((\(V_{add}\) * \(C_{base}\))+(([\(H^+\)]) * (\(V_{int}\)+\(V_{add}\))))/(\(V_{EP1}\)*\(C_{base}\)))

The established parameters are: \(V_{add}\)=Volume of Base added, \(C_{base}\)=Concentration of Base, [\(H^+\)]=Concentration of \(H^+\), \(V_{int}\)=Initial Volume of Acid, and \(V_{EP1}\)=First Equivalence Point Volume. The fraction bound of a diprotic system is equivalent to 2, which is denotes the molecule’s full inital binding. The ligand in this case is \(H^{+}\) ions. The subsequent unbinding of the ligand from the molecule is driven foward by the addition of base into solution. As the volume of base added increases, the moles of \(H^{+}\) ions in solution increases, signifying the unbinding of the ligand from the rest of the molecule. The dissociation of \(H_{2}A\) in solution into \(HA^{-}\) and \(A^{2-}\) is an additional factor in reducing the binding effect of the molecule.

# Binding Curve Parameters

Mydata <- read.csv("Diprotic Titration - Fast Titration - Sheet1.csv",header = TRUE, sep = ",")

volume <- Mydata$Consumption..mL.

pH <- Mydata$Potential..pH.

H <- 10^(-pH)

VEP1 <- 7.629

VI <- 25.0 

CB <- 0.100

VA <- volume

fb <- (2-(((VA*CB)+((H)*(VI+VA)))/(VEP1*CB)))

plot(pH,fb,main = "Diprotic Binding Curve",xlab = "pH",ylab = "Fraction Bound")

Non-Linear Least Sqaures Analysis - \(K_{a1}\) and \(K_{a2}\)

The theoretical equation for nls2 in the analysis of a diprotic system is as expressed:

(L/KD1 + 2L^2/(KD1KD2))/(1+L/KD1 + L^2/(KD1*KD2))

The “L” parameter in this molarity of [\(H^+\)] ions.

L <- 10^(-pH)

The “KD1” and “KD2” parameters are the dissociation constants for the coupled equilibrium reactions of a diprotic system as shown below:

\(H_{2}A\)+\(H_{2}O\)⇌\(H_{3}O^+\)+\(HA^{-}\)

\(HA^{-}\)+\(H_{2}O\)⇌\(H_{3}O^+\)+\(A^{2-}\)

“KD1” is analogous to the first reaction and “KD2” the second reaction. The dissociation of the first and second protons in a diprotic acid differs in how readily the molecule unbinds the proton. Thus, the ability for the acid to unbind its first and second protons denotes two distinct KD values.

library(nls2)
## Loading required package: proto
fit <- nls2(fb ~ (L/KD1 + 2*L^2/(KD1*KD2))/(1+L/KD1 + L^2/(KD1*KD2)),
               start = c(KD1 = 0.001, KD2 = 0.01))

fit
## Nonlinear regression model
##   model: fb ~ (L/KD1 + 2 * L^2/(KD1 * KD2))/(1 + L/KD1 + L^2/(KD1 * KD2))
##    data: parent.frame()
##       KD1       KD2 
## 1.689e-06 5.789e-04 
##  residual sum-of-squares: 0.1458
## 
## Number of iterations to convergence: 9 
## Achieved convergence tolerance: 6.701e-06
plot(pH,fb,main = "Diprotic Binding Curve",xlab = "pH",ylab = "Fraction Bound")

summary(fit)
## 
## Formula: fb ~ (L/KD1 + 2 * L^2/(KD1 * KD2))/(1 + L/KD1 + L^2/(KD1 * KD2))
## 
## Parameters:
##      Estimate Std. Error t value Pr(>|t|)    
## KD1 1.689e-06  6.120e-08   27.61   <2e-16 ***
## KD2 5.789e-04  1.910e-05   30.31   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.04296 on 79 degrees of freedom
## 
## Number of iterations to convergence: 9 
## Achieved convergence tolerance: 6.701e-06
lines(pH,predict(fit), col = "red")

The \(K_{a1}\) and \(K_{a2}\) that were determined based on graphical interpretation of the auto-titration data revealed values of 6.18\(10^{-4}\) and 1.13\(10^{-6}\) for each of the equilibrium constants, respectively. These values were obtained through the process of deriving the first derivative curve of the diprotic curve, determining the volume of base added at the first and second half-equivalence points, and extracting their respective pH’s. Finally, the Henderson-Hasselbalch equation was utilized to obtain the respective \(K_{a1}\) and \(K_{a2}\) values. Consequently, the magnitudes of \(K_{a1}\)/\(K_{a2}\) and “KD2”/“KD1” values are quite similar, almost in reverse order. However, the NLS2 fit reveals a KD1 value of 1.689\(10^{-6}\) and a KD2 value of 5.789\(10^{-4}\), indicating that the dissociation of the first proton is less effective and less readily available for unbinding than the dissociation of the second proton, in the unknown diprotic acid. From graphical analysis, the magnitude of \(K_{a1}\) indicates that it is the larger of the two Ka dissociation constants, favoring the dissociation of the first proton. Therefore, one question that pertains to this issue is: why is the KD2 value larger than the KD1?