Abstract:

The technique of titration of an acid with a base of known concentration is a common laboratory technique used to determine the acid’s concentration by identifying the endpoint, representing complete neutralization, and the midpoint, which indicates the acid’s pKa values. Diprotic acid titration by the use of an auto-titrator was utilized in this experiment. This method of titration provides a very accurate reaction by adding very precise increments of the titrant into the analyte. While simultaneously, having a computer software automatically tabulate the data such as the volume of base added and the corresponding pH. While also creating a plot of the titration curve and the first derivative. The aim of this experiment was to study diprotic acid equilibrium through the use of an auto-titrator and using the results from the experiment to determine the unknown diprotic acid. From the experimental data, the first derivative of the titration curve was plotted so that the endpoints could be determined. The endpoints revelaed the full neutralization of both protons at each proton site. The fraction bound was also calculated for each protonation state and it revealed information about the equilibrium dynamics between the protonated and deprotonated species. Using R-code a non-linear least squares (nls2) function was used to create a curve of the best fit to the fraction bound data along with the acid dissociation equilibrium equation t to predict the Ka1 and Ka2 values. The binding curve of the diprotic acid titration was also graphed with the nls2 function. This method of analysis helped provide accurate values of the acid dissociation constants and also highlighted the unique features of a diprotic acid titration. This analysis gave functionally predicted Ka values, which were then compared to the experimentally determined Ka values that was done for the class lab report and then compared to the known literature values.

Introduction:

Titration is a widely utilized nalytical chemistry technique used to determine the concentration of a known volume of acid solution by neutralizing; in other words deprotonating; it with a base solution of known volume and concentration. In this experiment, we aimed to identify an unknown diprotic acid through the use of the auto-titrator machine using NaOH as the titrant. NaOH, which is a strong base was gradually added into the beaker of the diprotic acid and the data was recorded using very precise pH measurements by the auto-titrator machine software. In previous lab experiments like for the manual titration of a protic acid, a color change was observed with phenolphthalein. However, when conducting titration experiments and the pH is required for data analysis the point of completion is determined when the pH shifts to around 12. Diprotic acids, like H2SO4, H2CO3, and H2C2O4 can donate two protons and dissociate in two steps, each with its own equilibrium constants denoted as Ka1 and Ka2. Through the examination of these two dissociation steps, the acid dissociation constants denoted a Ka can be calculated.

Additionally, for diprotic acids there are two distinct equivalence points that are observed throughout the titration process, each equivalence point corresponds to the complete reaction of one of the two dissociable protons. At the first equivalence point (EQP1), the first proton had been neutralized in the reaction which allows for the determination of the Ka1 value. During further titration the second proton eventually dissociated at the second equivalence point (EQP2) which also allowed for the determination of the Ka2 value. Specifically for this experiment, a Mettler Toledo Easy pH Titrator was used, which acted as the auto titration device that added the strong base sodium hydroxide (NaOH) to the unknown diprotic acid in very precise increments while also simultaneously capturing the pH data to generate a titration curve. The samples were prepared by added a known weight of the unknown diprotic acid and also a known volume of the solvent. There consisted of four total trials, but for the purposes of this data analysis, the represented data is only for trial #1.

Technically, the first trial conducted through auto-titration was labeled the “fast-titration” which was the only trial in which a color change indicator specifically Bromocresol Green was used. The indicator allowed us to observe a clear color change at the second equivalence point. The color change went from a yellowish color to a green and then finally to a blue.

As mentioned earlier, the acid dissociation constants Ka1 and Ka2 convey information about the deprotonation behavior of the acid at each of the two equivalence points. For a diprotic acid which has two protons the equilibrium expressions and the formula to determine the Ka values are shown below:

Ka1 (First Dissociation of H) : \(H_2A \Leftrightarrow H^+ + HA^-, Ka1 =\frac{[H^+][HA^-]}{[H_2A]}\)

Ka2 (Second Dissociation of H) : \(HA^- \Leftrightarrow H^+ + A^2-, Ka2 =\frac{[H^+][A^2-]}{[HA^-]}\)

To add on, the pKa values can also be determined from the data, by plotting the titration curve with pH on the y-axis and volume of NaOH added on the x-axis. Then locating the buffering regions on the curve, this region shows the diprotic acid’s resistance to pH changes as it reacts with increasing volumes of titrant being added. The midpoint of each buffering region is important because it corresponds to the point where half of the acid is deprotonated, meaning that the concentration of the protonated form is equal to the concentration of the deprotonated form. The Henderson-Hasselbalch equation states that at the point where the deprotonated form equals the protonated form is where the pH is also equal to the pKa.

From the dissociation equations and formulas shown above, it can also be mentioned that each of the dissociation constants is connected to a pKA value. The pKa1 value correlates with the pH at which the concentrations of the acid’s first conjugate base and the protonated form are equal. The pKa2 value correlates with the pH at which the concentration of the fully deprotonated form, meaning no hydrogens available. Specifically for diprotic acids, the titration curve will exhibit two distinct buffering regions because diprotic acids have two acidic protons. The first region corresponds to the first pKa1 value where the first proton is dissociated and the second buffering region corresponds to pKa2 where the second proton is dissociated.

The main purpose of this lab experiment consisted of using the auto-titration instrument and software to accurately run a diprotic acid titration and to use the recorded data to analyze and predict what the unknown diprotic acid is from its acid dissociation constants. The data was analyzed in R-script through the calculating the fraction bound. This calculation is useful in understanding the equilibrium dynamics between the protonated and deprotonated species. The fraction bound also shows how a diprotic acid responded to changes in pH throughout the trial of the titration experiment. A knwon R-function was also utilized which is called “non-linear least squarws” (nls2) to further analyze the data and plot the binding curve. The binding curve was useful in telling us valuable information about the acid’s buffering regions, pKa values, and equivalence points. After plotting of the titration curve and the binding curve, a line of best fit was also created with the fraction bound data points. Through the summary of the fit, we were able to accurately determine the Ka1 and Ka2 values. After the experiment was successfully conducted the results and data analysis helpdc deepen the understanding of titration techniques specifically with diprotic acid behavior in analytical chemistry and quantitative analysis. This titration technique with a auto titrator allowed for a very precise and high accuracy in determining the equivalence points and calculating the dissociation constants of the unknown acid.

Lastly, previous titration work was also conducted in previous lab experiments such as the manual titration of a monoprotic acid and the potentiometric titration of a monoprotic acid. To quickly summarize, the manual titration of a monoprotic titration consisted of determining the concentration of a 0.1 N sodium hydroxide (NaOH) solution by manual titration while using potassium hydrogen phthalate (KHP) as the primary standard. KHP is a well-known and popular standard due to its high purity, stability, and precise molar mass, which are vital properties that ensure accurate standardization of NaOH. The titration was conducted for a total of three trials with the endpoint identified through a light pink color change while using phenolphthalein as the indicator. For the potentiometric titration of a monoprotic acid,the concentration of an unknown solution by measuring the pH value continuously while using a calibrated pH meter was determined. While a titrant (base) was being added into the solution. As opposed to manual titration, that relied on visual indicators and a color change, in potentiometric titration the use of a pH meter helped identify the equivalence point and endpoint precisely with a lower margin of error. In this experiment, the concentration of acetic acid in an unknown vinegar sample was determined by using NaOH as the titrant in the burette. The reaction involved the neutralization of acetic acid by NaOH which forms acetate and water. A major difference to note from monoprotic titration and diprotic titration is the presence of two equivalence point values for diprotic titration while monoprotic only has one equivalence point since it can only donate one proton.

Titration Curve of the Unknown Diprotic Acid:

The procedure to obtain the titration curve of the diprotic acid consisted of adding 0.0817 grams of the unknown acid into a beaker followed by 25 mL of deionized water. This solution resulted in a 25 mL volume of diprotic acid with unknown identity and molar mass. This sample solution was then titrated through the use of an automatic titrator by adding increments of 1.0 M NaOH. The software of the auto-titrator automatically generated a titration curve after both the equivalence points were reach; EQP1 and EQP2. This meant that both acidic proton of the unknown diprotic acid had been neutralized by the strong base. In our data analysis of this trial, the difference in pH values and difference in volume of NaOH added (mL) was plotted to generate the first derivative graph which allowed us to determine the equivalence points from the plot.

Since the volumes of the equivalence points were now determined, the midpoints were also calculated and the correlating pH values provided the experimental pKa values, thanks to the Henderson-Hasselbalch equation mentioned earlier. It is also important to state that pKa1 value correlates with the first midpoint and the pka2 value correlates with the second midpoint. The first midpoint (pKa1) is where [\(H_2A] = [HA-] and the second midpoint (pKa2) is where [HA-] = [\)A_2 - 1].

With the recorded data of the pH and the corresponding volumes of NaOH, a diprotic acid titration curve was coded on an R script file after uploading the dataset as a “csv” file directly into PositCloud. For the purpoes of the in-class lab report, the titration curve was plotted by manually on the downloaded excel file from the auto-titrator software. While the auto-titrator software also provided the titration curve and the first derivative plot, that was used to compare our manual results to the automated generated results. The purpse of plotting the curve on an R-script file was to be able to compare the two methods and conclude which method generated a closer set of dissociation constants that is closer to what the literature states.

# Loading the lab data
Diprotic_Titration_Data <- read.csv("Diprotic_Titration_Data.csv")

Volume <- Diprotic_Titration_Data$Volume
pH <- Diprotic_Titration_Data$pH

# Plot of the titration curve
plot(Volume, pH, main = "Titration Curve of Diprotic Titration", xlab = "Volume of NaOH (mL)", ylab = "pH", pch = 19, col = "red")

Utilizing the Fraction Bound to Transform the Diprotic Acid Data:

The fraction bound formula for a diprotic acid which is:

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

Each variable in the equation above corresponds to a certain calcuated value or even a constant. The proton concentration [H⁺] was calculated using 10 to the power of the negative pH value. Base concentration was set at a constant of 1.0 M. The starting volume of the acid solution was 25 mL, it technically also includes the mass of the unknown diprotic acid which was 0.0817 grams. The end volume was the calculated volume that was needed to reach the second endpoint divided by two. This was done so that the fraction bound equation did not result in any negative values. The volume add variable was based on the “volume” column from the diprotic acid data-set which correleated with the volume of base added.

For the binding curve that was plotted through the transformed data by the fraction bound formula, the last points after the volume reached the second endpoint was omitted so that a good curve could be obtained. Additionally, the line of fit does not seem to match-up exactly and go through all the points of the curve, multiple attemps were done to fix this issue but it still presisted.

# Proton Concentration:
H <- 10^(-pH)

# Base Concentration: 
BC <- 0.1 

# Initial Acid Volume 
Vini <- 25.0 

# Volume of Base Needed to Reach Endpoint: 
Vend <- (16.527)/(2)

# Amount of NaOH volume added: 
Vadd <- Volume 

# Calculating the Fraction Bound: 

FB <- 2 - ((BC * Vadd) + H  * (Vini + Vadd)) / (BC * Vend)

FB 
##  [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,FB, main = "Binding Curve of Diprotic Titration",xlab = "pH",ylab = "Fraction Bound",pch=20,col="purple")

# Using the nls2 (non-linear fitting) function:

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

# Summary of the fit 
summary(fit)
## 
## Formula: FB ~ (((H/Ka1) + ((2 * H^2)/(Ka1 * Ka2)))/(1 + (H/Ka1) + (H^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
# Adding the fitted line to the binding plot
lines (pH, predict(fit), col = "black")

Determination of Ka1 and Ka2 Using NLS2 Function:

After the generation of the titration and binding curves for the diprotic acid titration data that was obtained in the lab through the auto-titrator. The nls2 function was applied in the R-script to generate the approximation values of the Ka1 and Ka2 values of the acid. The purpose of using this nls2 function for a diprotic acid titration is because it provided insightful information about the diprotic acid’s behavior by fitting a model that describes the deprotonation and the equilibrium constants that characterize the acid, as well as estimating the Ka values. These two values helps quantify the strength of the acid and how it behaves as each proton is being neutralized through the addition of the strong base. Through the nls2 function the approximate Ka1 and Ka2 values were:

Using this information, the pKa values can be calulated through the equation of pKa = -log(Ka), so the pKa values for Ka1 and Ka2 are:

Laboratory Determination of Ka1 and Ka2 Experimentally:

The acid dissociation constants were experimentally calculated through the recorded data-set from the auto-titrator software. A titration curve was generated and compared to the one generated by the software, the conclusion made was that both curves were almost identical. The two equivalence points denoted as EQP1 and EQP2 were determined by the first derivative plot. The calculation of the Ka values consisted of calculating the volumes at the half-equivalence point by dividing the EQP by 2, then determining the pH at the half-equivalence points. Once the pH was determined the Ka was calculated by the formula 10^-pH. The experimentally determined Ka values were:

Once, again the pKa values were determined using the same equation as before:

Comparison of Experimental Method vs. R Computational Method:

There is a very distinct difference between the experimentally calculated acid dissociatin constants as compared to the computational method using R script and the nls2 function. Theoretically, the nls2 function is supposed to provide a more precise and accurate result. The Ka1 value from R was 4.076e-07 while the experimental value was 7.59e-03. This a very big difference between the two values and the experimental method resulted in a higher value. In the case of Ka2, the value determined from R was 2.118e-02 while the experimental value was 4.57e-06. Another big difference is observed and the R derived value is higher than the experimentally derived value. These findings illusrate opposite values for each constant when compared between the two methods. Knowing this information, we can compare these values to the known acid dissociation constants and pKa values of the known diprotic acid.

Through, data-analysis after the experiment was performed in the lab, the molecular weight of the acid was determined by taking the mass of the unknown diprotic acid (0.0817 grams), the volume of NaOH added at EQP2, and the concentration of NaOH (0.100 M). The volume of NaOh was convrted into liters and then the moles of NaOH was calculated. Following that the moles of the unknown diprotic acid was determined. Finally, using the intial mass of the acid and the moles of the acid, the molar mass was easily determined. From the list of unknown diprotic acids given in the lab manual, the unknown diprotic acid was determined to be Malonic acid (H4C3O4).

The corresponding Ka and pKa values with malonic acid through the NIH (https://pubchem.ncbi.nlm.nih.gov/compound/Malonic-acid) was determined to be:

From this, it can be seen that the experimentally calculated Ka values align more with the literature reported values. The binding curve method using the nls2 function is technically supposed to provide a more accurate and precise determination of the Ka values. This is not observed through the analysis of the recorded data from the auto-titration for several reasons.

Conclusion:

The main purpose of this experiment was to identify the unknown diprotic acid through the use of a auto-titrator which automatically recorded and tabulated the data in an excel file. Through manual data-analysis of the experimental data the titration curve, first derivative plot, determination of the unknown acid, and determination of the acid dissociation constants were conducted. The acid was concluded to be malonic acid. The acid dissociation constants were also calculated computationally, through R-script and applying the non-linear least squares function to plot the binding curve and establish a line of fit. However, the line of fit was a bit skewed from the points on the curve. Summary of the nls2 function also generated estimated Ka values based on the data-set imported into R which started at a volume of 0 mL and ended at a volume of 16.527 mL.

Comparison of the Ka values between the experimental and computational method showed that the experimental derived values were much closer to the literature reported values of malonic acid. Although, the computational method is known to be more precise and accurate for the purposes of this trial of the experiment the data derived did not favor the reported values of the Ka and pKa.

The titration and binding curves plotted computationally was accurate and showed nice curves. The titration curve plotted on R and compared to the generated curve by the auto-titrator showed almost the exact same plot.

Henceforth, binding curve analysis of a diprotic acid is an appropriate and efficent method to graphically visualize the curves based on the data, as well as calculate the fraction bound to further determine mathematical estimations through other functions such as nls2 to determine the acid dissociation constants. Further thoughts on this experiment is to figure out what went wrong in the data-analysis that showed that the estimated Ka values are not similar to the reported literature values. Future studies can be to undergo the same analysis with a different diprotic acid.