Introduction

Previously, manual titration and potentiometric titration laboratory techniques were utilized to determine the concentration of a monoprotic acid solution. In this experiment, a more systematic approach to titration was performed through an Autotitrator, in which the base of a known concentration is added in timely increments to neutralize the acid with an unknown concentration. At the same time, a computer-assisted system and pH probe incredibly measures precise volume additions and associated pH changes, subsequently creating a titration curve. The titrant used was sodium hydroxide (NaOH) with a concentration of .09917M and the initial volume of the unknown diprotic acid was 25mL with its first endpoint recorded at 4.96mL.

Titration Curve

Data collection shows the pH of the solution with each volume addition of NaOH. A graphical representation of the results is shown below, plotting pH vs. Volume NaOH added (mL).

library(readr)
t1data <- read.csv ("Autotitration Lab.csv")

#Volume Values
VolVect <- t1data$Volume
VolVect
##  [1]  0.000  0.005  0.010  0.023  0.054  0.132  0.328  0.528  0.728  0.928
## [11]  1.128  1.328  1.528  1.728  1.928  2.128  2.328  2.528  2.728  2.928
## [21]  3.129  3.329  3.529  3.729  3.929  4.129  4.329  4.529  4.729  4.929
## [31]  5.129  5.329  5.529  5.729  5.929  6.129  6.329  6.529  6.633  6.723
## [41]  6.793  6.971  7.012  7.115  7.253  7.430  7.630  7.830  8.030  8.230
## [51]  8.430  8.630  8.830  9.030  9.230  9.431  9.631  9.831 10.031 10.231
## [61] 10.431 10.631 10.831 11.031 11.231 11.431 11.631 11.831 12.031 12.231
## [71] 12.431 12.631 12.831 13.031 13.231 13.431 13.631 13.831 14.031 14.231
## [81] 14.431 14.631
length(VolVect)
## [1] 82
#pH Values 
pHVect <- t1data$pH
pHVect
##  [1] 2.09 2.08 2.08 2.08 2.07 2.07 2.07 2.08 2.08 2.09 2.09 2.11 2.12 2.13 2.14
## [16] 2.15 2.18 2.20 2.22 2.24 2.26 2.29 2.31 2.34 2.37 2.40 2.44 2.49 2.54 2.59
## [31] 2.64 2.71 2.78 2.86 2.96 3.10 3.29 3.54 3.71 3.89 3.97 4.38 4.45 4.61 4.79
## [46] 4.96 5.12 5.25 5.30 5.44 5.52 5.59 5.66 5.72 5.78 5.83 5.88 5.94 5.98 6.03
## [61] 6.06 6.12 6.15 6.21 6.25 6.29 6.34 6.37 6.43 6.46 6.52 6.56 6.62 6.68 6.73
## [76] 6.78 6.86 6.94 7.03 7.12 7.24 7.39
length(pHVect)
## [1] 82
# Graphing the Titration Curve 
plot(VolVect,pHVect,xlab = 'Volume NaOH added (mL)',ylab = 'pH',main = 'Titration Curve')

Transforming Diprotic Data: Fraction Bound Curve

Titration data can be transformed into a binding curve, ultimately representing the fraction or proportion of a binding site that is occupied by the the ligand or titrant at each volume addition. A formula is used to calculate the fractional binding at each point during the titration and is as shown:

\[ F = 2 - \frac{{(Volume \times Bc) + (Hc \times (VI+ Volume))}}{{VE\times Bc}} \]

Where:
\(F\) is the Fraction bound
\(Bc\) is the concentration of the base NaOH
\(Hc\) is the concentration of protons or H+ ions in the solution as calculated by \(10^{-pH}\)
\(VI\) is the initial volume of the unknown acid
\(VE\) is the volume added at the 1st endpoint
\(Volume\) is the volume of titrant (NaOH) added

The values for each component of the equation are as follows:

#[H+]

Hc <- 10^-(pHVect)
Hc
##  [1] 8.128305e-03 8.317638e-03 8.317638e-03 8.317638e-03 8.511380e-03
##  [6] 8.511380e-03 8.511380e-03 8.317638e-03 8.317638e-03 8.128305e-03
## [11] 8.128305e-03 7.762471e-03 7.585776e-03 7.413102e-03 7.244360e-03
## [16] 7.079458e-03 6.606934e-03 6.309573e-03 6.025596e-03 5.754399e-03
## [21] 5.495409e-03 5.128614e-03 4.897788e-03 4.570882e-03 4.265795e-03
## [26] 3.981072e-03 3.630781e-03 3.235937e-03 2.884032e-03 2.570396e-03
## [31] 2.290868e-03 1.949845e-03 1.659587e-03 1.380384e-03 1.096478e-03
## [36] 7.943282e-04 5.128614e-04 2.884032e-04 1.949845e-04 1.288250e-04
## [41] 1.071519e-04 4.168694e-05 3.548134e-05 2.454709e-05 1.621810e-05
## [46] 1.096478e-05 7.585776e-06 5.623413e-06 5.011872e-06 3.630781e-06
## [51] 3.019952e-06 2.570396e-06 2.187762e-06 1.905461e-06 1.659587e-06
## [56] 1.479108e-06 1.318257e-06 1.148154e-06 1.047129e-06 9.332543e-07
## [61] 8.709636e-07 7.585776e-07 7.079458e-07 6.165950e-07 5.623413e-07
## [66] 5.128614e-07 4.570882e-07 4.265795e-07 3.715352e-07 3.467369e-07
## [71] 3.019952e-07 2.754229e-07 2.398833e-07 2.089296e-07 1.862087e-07
## [76] 1.659587e-07 1.380384e-07 1.148154e-07 9.332543e-08 7.585776e-08
## [81] 5.754399e-08 4.073803e-08
#Volume added at 1st endpoint
VE <- 7.43

#Initial volume of unknown acid
VI <- 25

#[base] [NaOH]
Bc <- 0.09917

#FB Equation
F <- (2-((VolVect*Bc)+ (Hc)*(VI+VolVect))/(VE*Bc))
F
##  [1] 1.72421485 1.71706160 1.71633221 1.71443579 1.70332589 1.69192691
##  [7] 1.66328331 1.64076747 1.61159190 1.58907864 1.55995446 1.54390200
## [13] 1.52123864 1.49852530 1.47576205 1.45294899 1.44163466 1.42403280
## [19] 1.40608873 1.38781438 1.36907926 1.35477189 1.33539912 1.31989782
## [25] 1.30371733 1.28689744 1.27284199 1.26076217 1.24716425 1.23220292
## [31] 1.21601719 1.20251443 1.18709344 1.17136899 1.15599353 1.14154294
## [37] 1.12637697 1.10892441 1.09889695 1.08960846 1.08111010 1.05996780
## [43] 1.05471691 1.04132580 1.02311244 0.99951741 0.97274617 0.94591364
## [49] 0.91902163 0.89216466 0.86527348 0.83837528 0.81147425 0.78456879
## [55] 0.75766180 0.73061729 0.70370655 0.67679633 0.64988292 0.62297018
## [61] 0.59605502 0.56914232 0.54222668 0.51531305 0.48839765 0.46148205
## [67] 0.43456678 0.40765028 0.38073503 0.35381828 0.32690256 0.29998593
## [73] 0.27306978 0.24615342 0.21923664 0.19231974 0.16540326 0.13848655
## [79] 0.11156975 0.08465276 0.05773582 0.03081880

A graphical representation of the results is shown below, plotting Fraction Bound vs. pH:

#Graphing the Fraction Bound Curve
plot(pHVect,F, xlab = 'pH',ylab = 'Fraction Bound', main = "Fraction Bound Curve")

Binding Curve Analysis: Fitline for Non-Linear Model

For non-linear least squares regression, also known as fitting challenging non-linear models, the package in R markdown known as ‘nls2’ was used. The curve was best fit to the plotted data points on the Fraction Bound curve and in fitting a non-linear model to the titration data, information about the acid’s Ka1 and Ka2 can be extracted.The fraction bound fitline used to to represent the model is derived from following equation:

\[F = \frac{{Hc/KD1 + 2 \times Hc^2/(KD1 \times KD2))}}{{1 + Hc/KD1 + Hc^2/(KD1 \times KD2)}} \]

Where:
\(F\) is the Fraction bound
\(Hc\) is the concentration of protons or H+ ions in the solution as calculated by \(10^{-pH}\)
\(KD1\) and \(KD2\) are the binding dissociation constants for each of the two protons

A non-linear model will be fit to the collected data, as shown on the fraction bound curve below, again plotting Fraction Bound vs. pH.

#Graphing a fitline for a non-linear model

library (nls2)
## Loading required package: proto
fitline <- nls(F ~ (Hc/KD1+2*Hc^2/(KD1*KD2))/(1+Hc/KD1+Hc^2/(KD1*KD2)),
                start = c(KD1 = 0.0001,KD2=0.01))

summary (fitline)
## 
## Formula: F ~ (Hc/KD1 + 2 * Hc^2/(KD1 * KD2))/(1 + Hc/KD1 + Hc^2/(KD1 * 
##     KD2))
## 
## Parameters:
##      Estimate Std. Error t value Pr(>|t|)    
## KD1 5.866e-07  3.173e-08   18.48   <2e-16 ***
## KD2 6.567e-03  2.972e-04   22.10   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.06261 on 80 degrees of freedom
## 
## Number of iterations to convergence: 7 
## Achieved convergence tolerance: 1.345e-06
plot(pHVect,F, xlab = 'pH',ylab = 'Fraction Bound', 
     main = "Fraction Bound vs pH")

lines (pHVect, predict (fitline), col = "#2a5d7f")

Non-Linear Model Fitting Analysis and Conclusions

While the model is showing Kd1 and Kd2 estimate values, they truly represent the Ka values for the unknown diprotic acid. Hence, as shown through the estimate for the Ka values extracted from nls2 parameters, the Ka1 and Ka2 values were found to be \(6.567 x 10^{-3}\) and \(5.866 x 10^{-7}\) respectively. Similarly, the Ka1 and Ka2 values obtained from lab report data processing was found to be ( 4.571 x 10^{-3} ) and \(5.6234 x 10^{-7}\) respectively. Discrepancies in these values may be due to the more refined method that the program may use that analyzes and fits the whole data set with a non-linear model/curve, resulting in more precise calculations for the Ka values. Those determined for the lab report were performed graphically and were primarily approximated by viewing inflections in the first derivative graph as well as evident steep slopes in the titration curves to find the equivalence points. Since we rely on these specific characteristics of the curve and depend on various inflection points, which can be affected by even slight pH changes or volume additions, less accurate readings/human error are more likely. Overall, the two methods of obtaining Ka values are evidently different and thus result in varying values.