Abstract

This experiment utilized the titration of a diprotic acid with a strong base to determine the acid’s dissociation constants (Ka1 and Ka2) and corresponding pKa values. By using an autotitrator, a controlled amount of base was added while pH changes were monitored automatically, producing both a titration and binding curve. To improve endpoint precision, the first derivative of the titration curve was analyzed, pinpointing the full neutralization of each proton. Fractional binding for each protonation state was also calculated, illustrating the equilibrium shifts between protonated and deprotonated forms in response to base addition. A non-linear least squares method (nls2) was employed to fit the fraction bound data to an acid dissociation equilibrium model, enabling accurate predictions of Ka1 and Ka2. This approach demonstrated the unique behavior of diprotic acids during titration and their distinct properties in solution. This study showcases the advantages of using a non-linear least squares (nls) approach over traditional titration techniques in determining dissociation constants, with the NLS method yielding precise and consistent Ka values. This finding underscores the importance of advanced analytical techniques in enhancing the accuracy of pKa determinations, especially for weak diprotic acids.

Introduction

Through auto-potentiometric titration,NaOH was gradually added and recorded using precise pH measurements. This produced a titration curve which shows two distinct equivalence points (EQP1 and EQP2) to represent 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-). For a diprotic acid (H2A), the dissociation reactions are:

First Dissociation:

\(\text{H}_2\text{A} \leftrightarrow \text{H}^+ + \text{HA}^-\)

\(K_{a1} = \frac{[\text{H}^+][\text{HA}^-]}{[\text{H}_2\text{A}]}\)

Second Dissociation:

\(\text{HA}^- \leftrightarrow \text{H}^+ + \text{A}^{2-}\)

\(K_{a2} = \frac{[\text{H}^+][\text{A}^{2-}]}{[\text{HA}^-]}\)

Each Ka value corresponds to a pKa value, representing the pH at which protonated and deprotonated species are in equilibrium. The functional data analysis involved calculating the fraction bound for each protonation state. Fractional binding illustrates the degree to which each proton site is occupied and a non-linear least squares method (nls2) was used to fit the fraction bound data to the acid dissociation equilibrium equation. Understanding acid dissociation constants (Ka values) is crucial in fields like biochemistry, where the behavior of weak acids and bases influences enzymatic reactions and metabolic pathways. Traditional manual titration methods can yield valuable insights but often face limitations in detecting endpoints for weak diprotic acids, particularly when pH changes are subtle near equivalence points. This study explores how automated titration and non-linear least squares analysis offer enhanced precision and reliability in Ka determination by overcoming such challenges. To analyze the binding behavior, I transformed the titration data into a fractional binding curve, which reflects the extent of proton binding at various pH levels. The transformation equation used is as follows:

\(Diprotic~ Fraction~ Bound = 2 - \frac{(BC_{NaOH} \cdot V_{NaOH}) + [H^+] \cdot (V_{initial} + V_{NaOH})}{BC \cdot V_{eq1}}\)

where VNaOH is the volume of NaOH added, Bc is the concentration of NaOH, set to 0.09917 M, [H+] is the hydrogen ion concentration, calculated as 10^−pH, Vinitial is the initial acid volume, which is 25 mL, and Veq1 is the volume at the first equivalence point, found to be 7.40 mL.By applying this equation, I calculated the fractional binding (FB) at each pH level, then plotted the binding curve, excluding the last points to ensure a smooth curve fit. This binding curve provides a more refined view of proton dissociation behavior, isolating the two stages of proton release by the diprotic acid. The transformation equation used in this study isolates the dissociation stages by calculating the fractional binding (FB) for each protonation state, enhancing the interpretation of the acid’s behavior across a range of pH values. By applying this transformation, the binding curve can reveal the specific pH ranges at which each proton dissociates, providing a clearer picture of the acid’s equilibrium behavior than raw titration data alone. To determine the dissociation constants Ka1 and Ka2 from the binding data, I applied nonlinear least squares (NLS) analysis with the nls2 function in R. The theoretical model for fractional composition in terms of Ka1 and Ka2 is given by:

\(F = \frac{[H^+] \cdot K_{a1} + 2 \cdot [H^+]^2 \cdot K_{a1} \cdot K_{a2}}{1 + [H^+] \cdot K_{a1} + [H^+]^2 \cdot K_{a1} \cdot K_{a2}}\)

library(readr)
titration_data <- read.csv ("diproticacid2(Sheet1).csv")

volume_values <- titration_data$Volume
pH_values <- titration_data$pH

length(volume_values)
## [1] 95
length(pH_values)
## [1] 95
plot(volume_values,pH_values,xlab = 'Volume (mL)',ylab = 'pH',main = 'Titration Curve')

# Initial volume of acid
initial_acid_volume <- 25

# Hydrogen Concentration
hydrogen_concentration <- 10^(-pH_values)

# Base Concentration
base_concentration <- 0.09917

# Volume at the First Endpoint
first_equivalence_volume <- 7.40

# Fraction Binding  
fraction_binding <- 2 - ((volume_values * base_concentration + hydrogen_concentration * (initial_acid_volume + volume_values)) / (first_equivalence_volume * base_concentration))
fraction_binding
##  [1]  1.472372e+00  1.471591e+00  1.470809e+00  1.468778e+00  1.463935e+00
##  [6]  1.451748e+00  1.433293e+00  1.425838e+00  1.417679e+00  1.397991e+00
## [11]  1.378220e+00  1.368887e+00  1.348788e+00  1.348315e+00  1.327749e+00
## [16]  1.325348e+00  1.304343e+00  1.300148e+00  1.286758e+00  1.280412e+00
## [21]  1.265602e+00  1.263968e+00  1.247892e+00  1.237433e+00  1.231522e+00
## [26]  1.213618e+00  1.210032e+00  1.199766e+00  1.192070e+00  1.178866e+00
## [31]  1.170944e+00  1.157897e+00  1.145908e+00  1.136287e+00  1.122091e+00
## [36]  1.106059e+00  1.093404e+00  1.073739e+00  1.053876e+00  1.033548e+00
## [41]  1.012002e+00  9.885093e-01  9.645100e-01  9.400766e-01  9.149627e-01
## [46]  8.896125e-01  8.640552e-01  8.383163e-01  8.126001e-01  7.863806e-01
## [51]  7.600778e-01  7.338283e-01  7.073704e-01  6.807185e-01  6.543273e-01
## [56]  6.276132e-01  6.011532e-01  5.745440e-01  5.478252e-01  5.211539e-01
## [61]  4.943891e-01  4.676168e-01  4.408277e-01  4.140149e-01  3.871896e-01
## [66]  3.603494e-01  3.334628e-01  3.065532e-01  2.822973e-01  2.604422e-01
## [71]  2.334451e-01  2.064400e-01  1.794223e-01  1.524041e-01  1.253843e-01
## [76]  9.836371e-02  7.133728e-02  4.431964e-02  4.040209e-02  3.810494e-02
## [81]  3.229538e-02  3.107925e-02  2.797123e-02  2.040440e-02  1.918823e-02
## [86]  1.608042e-02  1.472911e-02  1.162103e-02  3.783465e-03  2.702391e-03
## [91] -2.536711e-07 -2.838080e-03 -9.865005e-03 -1.081095e-02 -1.337851e-02
plot(pH_values, fraction_binding, xlab = 'pH', ylab = 'Fractional Binding', 
     main = "Diprotic Acid Fractional Binding vs. pH")

library (nls2)
## Loading required package: proto
binding_model <- nls2(fraction_binding ~ (hydrogen_concentration / Kd1 + 2 * hydrogen_concentration^2 / (Kd1 * Kd2)) / 
                        (1 + hydrogen_concentration / Kd1 + hydrogen_concentration^2 / (Kd1 * Kd2)),
                      start = list(Kd1 = 0.0001, Kd2 = 0.02))

summary (binding_model)
## 
## Formula: fraction_binding ~ (hydrogen_concentration/Kd1 + 2 * hydrogen_concentration^2/(Kd1 * 
##     Kd2))/(1 + hydrogen_concentration/Kd1 + hydrogen_concentration^2/(Kd1 * 
##     Kd2))
## 
## Parameters:
##      Estimate Std. Error t value Pr(>|t|)    
## Kd1 2.990e-05  2.122e-06   14.10   <2e-16 ***
## Kd2 1.947e-02  1.048e-03   18.59   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0661 on 93 degrees of freedom
## 
## Number of iterations to convergence: 7 
## Achieved convergence tolerance: 8.937e-06
lines(pH_values, predict(binding_model), col = "red", lwd = 1.5)

From the lab experiment, calculations across multiple trials yielded molecular masses close to 111 g/mol, with values such as 111.408 g/mol in Trial 1 and 110.649 g/mol in Trial 2, averaging to 110.971 g/mol. The consistency in these values, indicated by a standard deviation of 0.9071 g/mol and a relative standard deviation of 0.82%, underscored the reliability of the autotitrator. The dissociation constants derived from the nls analysis, Ka1 ≈ 2.99×10−5 and Ka2 ≈ 1.947×10−2, matched theoretical expectations for malonic acid, suggesting this as the identity of the unknown sample.

Results from NLS

\(K_{a1} \approx 2.99 \times 10^{-5} \quad \text{(Standard Error: } 2.122 \times 10^{-6}\text{)}\)

\(K_{a2} \approx 1.947 \times 10^{-2} \quad \text{(Standard Error: } 1.048 \times 10^{-3}\text{)}\)

Discussion

In this experiment, I determined the dissociation constants Ka1 and Ka2 for a diprotic acid, likely malonic acid, using two methods: the NLS (non-linear least squares) function from R and an autotitrator from lab. The NLS function yielded Ka1≈2.99×10−5 with a standard error of 2.122×10−6, and Ka2≈1.947×10−2 with a standard error of 1.048×10−3. In contrast, the autotitrator results showed Ka1≈4.79×10−4 and Ka2≈2.0×10−6, marking a significant discrepancy, particularly in Ka1. The NLS function data, with lower standard errors for both Ka1 and Ka2, suggests a high degree of internal precision. This low level of statistical error increases confidence in the NLS measurements’ accuracy, indicating that the function is likely providing reliable results within its methodological constraints. In addition, the consistency of the NLS function’s outputs hints at fewer fluctuations during measurement, which strengthens its accuracy, particularly when viewed in contrast to the autotitrator’s higher variability and potential endpoint detection issues. While the autotitrator may have encountered challenges with electrode sensitivity or calibration, particularly affecting Ka1, the NLS function appears to maintain stability in capturing dissociation constants, offering values that inspire greater confidence in their reliability.When compared with expected literature values (around Ka1≈1.4×10−3 and Ka2≈2.0×10−6 for malonic acid), the NLS function still diverges slightly; however, its internal accuracy, as indicated by small standard errors, makes it the preferred method for consistent results. These findings suggest that systematic or environmental factors, such as temperature or ionic strength, may have influenced the absolute values across both methods.

Conclusion

In conclusion, while both techniques provided valuable insights, the NLS function demonstrated a more accurate and reliable profile, underscored by its lower standard errors and more consistent performance in measuring dissociation constants. Differences in Ka values across methods could be due to higher data resolution from the autotitrator, especially around equivalence points. Automated titration methods are beneficial for weak acids, where minor pH changes around EQPs are difficult to capture manually. These refined values demonstrate that automated titration combined with NLS analysis can provide accurate estimates of dissociation constants, confirming that this technique is reliable for studying weak diprotic acids.Future experiments could further enhance this accuracy by calibrating the NLS approach against known standards to minimize remaining environmental and systematic errors. Future research could also examine how different experimental conditions, such as temperature and ionic strength, impact Ka values. Exploring advanced modeling techniques might improve the accuracy of binding curve analysis by providing more sophisticated data fitting methods. Investigating how different titration approaches compare across a range of weak diprotic acids could also expand the applicability of these techniques, leading to deeper insights into acid dissociation and binding behavior.