Monoprotic and Diprotic Analysis in R
Noureen Ismail
April 21, 2024
Monoprotic Acid Titration Report
Abstract
The experiment aimed to determine the concentration of a commercial sodium hydroxide (NaOH) solution and the concentration of acetic acid in a vinegar sample using titrimetric methods. The average concentration of acetic acid in the vinegar sample was determined from titration data. Titration curves plotted pH against the volume of NaOH added, with equivalence volumes of 20.00 mL and the corresponding half-equivalence volumes were 10.00 mL. At the half-equivalence point, the calculated \(K_1\) value was \(K_1 = 2.627 \times 10^{-5}\).
Introduction
Background
The experiment focuses on the determination of the concentration of acetic acid in vinegar through acid-base titrations, specifically utilizing the technique of potentiometric titration. Acid-base titrations are fundamental analytical techniques used to determine the concentration of an unknown acid or base solution by reacting it with a solution of known concentration.
Purpose
This paper discusses a monoprotic acid titration method, focusing on the titration of acetic acid in vinegar. The aim is to compare the traditional titration method with the method of transforming data into a binding curve. This will be done by analyzing the titration curve and transforming it into a binding curve, and then employing non-linear least squares analysis to determine \(K_1\), the dissociation constant for acetic acid.
Results
Manual Titration of NaOH
Mydata <- read.csv("Titration.csv")
Mydata## Volume..ml...measured. pH..measured.
## 1 0.0 3.130
## 2 1.0 3.579
## 3 2.0 3.757
## 4 3.0 3.909
## 5 4.0 4.032
## 6 5.0 4.032
## 7 6.0 4.150
## 8 7.0 4.257
## 9 8.0 4.355
## 10 9.0 4.445
## 11 10.0 4.505
## 12 11.0 4.628
## 13 12.0 4.710
## 14 13.0 4.813
## 15 14.0 4.914
## 16 15.0 5.033
## 17 16.0 5.165
## 18 16.5 5.293
## 19 17.0 5.354
## 20 17.5 5.436
## 21 18.0 5.556
## 22 18.5 5.715
## 23 19.0 5.928
## 24 19.5 6.255
## 25 20.0 7.019
## 26 20.5 9.452Titration Curve pH vs. Volume of NaOH Added
Volume <- Mydata$Vol #volume Vector
Volume## [1] 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0
## [16] 15.0 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5pH <- Mydata$pH #pH Vector
pH## [1] 3.130 3.579 3.757 3.909 4.032 4.032 4.150 4.257 4.355 4.445 4.505 4.628
## [13] 4.710 4.813 4.914 5.033 5.165 5.293 5.354 5.436 5.556 5.715 5.928 6.255
## [25] 7.019 9.452plot(Volume,pH,main = "Volume of NaoH vs. pH of Solution",xlab = "volume (mL)", ylab = "pH")
Binding Curve Analysis
The next step in the analysis is to transform the titration data into a binding curve. This is done by plotting the transformed data and fitting it to a binding curve equation to determine the dissociation constant \(K_1\) for acetic acid.
Data Transformation
H <- 10^-(pH) # H+ from pH
H## [1] 7.413102e-04 2.636331e-04 1.749847e-04 1.233105e-04 9.289664e-05
## [6] 9.289664e-05 7.079458e-05 5.533501e-05 4.415704e-05 3.589219e-05
## [11] 3.126079e-05 2.355049e-05 1.949845e-05 1.538155e-05 1.218990e-05
## [16] 9.268298e-06 6.839116e-06 5.093309e-06 4.425884e-06 3.664376e-06
## [21] 2.779713e-06 1.927525e-06 1.180321e-06 5.559043e-07 9.571941e-08
## [26] 3.531832e-10#volume added at the endpoint:
VE <- 20.5
#initial volume of unknown acid:
VI <- 20
#concentration of base NaOH:
CB <- 0.0954This data reflects the measured and calculated data obtained during the titration. The mass of potassium hydrogen phthalate (KHP) used is recorded, allowing for the calculation of the moles of KHP, which is equivalent to the moles of sodium hydroxide (NaOH) due to the 1:1 stoichiometry of the reaction. The initial and ending burette readings provide the volume of NaOH solution used during titration, from which the concentration of NaOH can be calculated.
Monoprotic Binding Curve Equation:
\[ pH = pKa + \log \left( \frac{[A^-]}{[HA]} \right) \]
Graph of Binding Curve
Binding Curve \(pH\) vs. \(\frac{[A^-]}{[HA]}\)
# fraction bound for each data point
F <- (1-(((Volume*CB)+ ((H)*(VI+Volume)))/(VE*CB)))
F## [1] 9.924190e-01 9.483887e-01 9.004706e-01 8.522083e-01 8.037380e-01
## [6] 7.549100e-01 7.063759e-01 6.577726e-01 6.091239e-01 5.604434e-01
## [11] 5.117156e-01 4.630413e-01 4.143151e-01 3.655941e-01 3.168612e-01
## [16] 2.681268e-01 2.193863e-01 1.950269e-01 1.706480e-01 1.462712e-01
## [21] 1.218972e-01 9.752303e-02 7.314719e-02 4.876926e-02 2.438829e-02
## [26] -7.313963e-09plot(pH,F)
lines(pH,F,col="blue")
Non-linear Least Squares Analysis to Determine \(K_1\)
The dissociation constant (\(K_1\)) for acetic acid is determined by employing non-linear least squares analysis to fit the monoprotic binding curve equation to the experimental data.
library(nls2)## Loading required package: protofit <- nls(F ~ H/(KD+H), start=c(KD=0.0001))
fit## Nonlinear regression model
## model: F ~ H/(KD + H)
## data: parent.frame()
## KD
## 2.627e-05
## residual sum-of-squares: 0.01417
##
## Number of iterations to convergence: 5
## Achieved convergence tolerance: 6.467e-07Results
Dissociation Constant \(K_1\)
- \(K_1\) = \(2.627 \times 10^{-5}\)
Discussion
The experiment employed volumetric titration methods to determine the concentration of a commercial sodium hydroxide (NaOH) solution and the concentration of acetic acid in a vinegar sample. Overall, the method proved to be effective in providing quantitative results, as evidenced by the determined concentrations of NaOH and acetic acid.
Diprotic Acid Titration Report
Abstract
The experiment aimed to determine the dissociation constants (\(K_{a1}\) and \(K_{a2}\)) of an unknown acid sample through titration with a standardized sodium hydroxide (NaOH) solution. Using volumetric titration data and pH measurements, the equivalence points were identified, allowing for the calculation of \(K_{a1}\) and \(K_{a2}\). The unknown acid was found to have dissociation constants \(K_{a1}\) = \(5.324 \times 10^{-7}\) and \(K_{a2}\) = \(1.099 \times 10^{-2}\)
Purpose
This paper discusses a diprotic acid titration method, focusing on the titration of oxalic acid. The aim is to compare the traditional titration method with the method of transforming data into a binding curve. This will be done by analyzing the titration curve and transforming it into a binding curve, and then employing non-linear least squares analysis to determine \(K_{a1}\) and \(K_{a2}\), the dissociation constants for oxalic acid.
Results
Manual Titration of NaOH
Data <- read.csv("Diprotic Acid Data.csv")
Data## Consumption..mL. Potential..pH.
## 1 0.000 1.89
## 2 0.005 1.89
## 3 0.010 1.89
## 4 0.023 1.89
## 5 0.054 1.88
## 6 0.132 1.89
## 7 0.328 1.89
## 8 0.528 1.90
## 9 0.728 1.91
## 10 0.928 1.94
## 11 1.128 1.96
## 12 1.328 1.98
## 13 1.528 2.00
## 14 1.728 2.02
## 15 1.928 2.03
## 16 2.128 2.05
## 17 2.328 2.10
## 18 2.528 2.10
## 19 2.728 2.15
## 20 2.928 2.16
## 21 3.129 2.19
## 22 3.329 2.22
## 23 3.529 2.25
## 24 3.729 2.29
## 25 3.929 2.32
## 26 4.129 2.37
## 27 4.329 2.41
## 28 4.529 2.44
## 29 4.729 2.49
## 30 4.929 2.55
## 31 5.129 2.60
## 32 5.329 2.66
## 33 5.529 2.75
## 34 5.729 2.82
## 35 5.929 2.94
## 36 6.129 3.06
## 37 6.329 3.23
## 38 6.529 3.44
## 39 6.674 3.65
## 40 6.772 3.84
## 41 6.846 3.99
## 42 6.929 4.18
## 43 7.002 4.32
## 44 7.108 4.51
## 45 7.223 4.67
## 46 7.396 4.88
## 47 7.572 5.04
## 48 7.772 5.16
## 49 7.972 5.18
## 50 8.172 5.40
## 51 8.315 5.45
## 52 8.515 5.50
## 53 8.715 5.59
## 54 8.915 5.66
## 55 9.115 5.70
## 56 9.315 5.78
## 57 9.515 5.85
## 58 9.715 5.86
## 59 9.915 5.93
## 60 10.115 5.98
## 61 10.315 5.99
## 62 10.515 6.07
## 63 10.715 6.09
## 64 10.916 6.18
## 65 11.116 6.19
## 66 11.316 6.26
## 67 11.516 6.28
## 68 11.716 6.30
## 69 11.916 6.39
## 70 12.116 6.40
## 71 12.316 6.48
## 72 12.516 6.55
## 73 12.716 6.56
## 74 12.916 6.63
## 75 13.116 6.65
## 76 13.316 6.77
## 77 13.516 6.79
## 78 13.716 6.91
## 79 13.916 6.99
## 80 14.116 7.01
## 81 14.316 7.03
## 82 14.516 7.32
## 83 14.597 7.33
## 84 14.797 7.60
## 85 14.893 7.62
## 86 15.094 8.07
## 87 15.128 8.26
## 88 15.141 8.29
## 89 15.173 8.31
## 90 15.253 8.70
## 91 15.260 8.73
## 92 15.277 8.83
## 93 15.291 8.86
## 94 15.328 8.91
## 95 15.420 9.08
## 96 15.481 9.29
## 97 15.511 9.32
## 98 15.587 9.47Experimental Details
The titration experiment involved titrating a solution of a diprotic acid with a known initial concentration. The titrant used was sodium hydroxide (NaOH) with a concentration of 0.10 M. pH readings were recorded at each step of titration as the volume of NaOH was incrementally added to the solution. The initial volume of the unknown acid was 25 mL, and the volume added at the endpoint was determined to be 15.181 mL.
Graphical Representation
With the data collected the plotting of pH against the volume of NaOH added was graphed. This will provide insights into the acid-base behavior of the diprotic acid and help identify any inflection points corresponding to the pK values.
Volume <- Data$Consumption #volume Vector
Volume## [1] 0.000 0.005 0.010 0.023 0.054 0.132 0.328 0.528 0.728 0.928
## [11] 1.128 1.328 1.528 1.728 1.928 2.128 2.328 2.528 2.728 2.928
## [21] 3.129 3.329 3.529 3.729 3.929 4.129 4.329 4.529 4.729 4.929
## [31] 5.129 5.329 5.529 5.729 5.929 6.129 6.329 6.529 6.674 6.772
## [41] 6.846 6.929 7.002 7.108 7.223 7.396 7.572 7.772 7.972 8.172
## [51] 8.315 8.515 8.715 8.915 9.115 9.315 9.515 9.715 9.915 10.115
## [61] 10.315 10.515 10.715 10.916 11.116 11.316 11.516 11.716 11.916 12.116
## [71] 12.316 12.516 12.716 12.916 13.116 13.316 13.516 13.716 13.916 14.116
## [81] 14.316 14.516 14.597 14.797 14.893 15.094 15.128 15.141 15.173 15.253
## [91] 15.260 15.277 15.291 15.328 15.420 15.481 15.511 15.587pH <- Data$Potential #pH Vector
pH## [1] 1.89 1.89 1.89 1.89 1.88 1.89 1.89 1.90 1.91 1.94 1.96 1.98 2.00 2.02 2.03
## [16] 2.05 2.10 2.10 2.15 2.16 2.19 2.22 2.25 2.29 2.32 2.37 2.41 2.44 2.49 2.55
## [31] 2.60 2.66 2.75 2.82 2.94 3.06 3.23 3.44 3.65 3.84 3.99 4.18 4.32 4.51 4.67
## [46] 4.88 5.04 5.16 5.18 5.40 5.45 5.50 5.59 5.66 5.70 5.78 5.85 5.86 5.93 5.98
## [61] 5.99 6.07 6.09 6.18 6.19 6.26 6.28 6.30 6.39 6.40 6.48 6.55 6.56 6.63 6.65
## [76] 6.77 6.79 6.91 6.99 7.01 7.03 7.32 7.33 7.60 7.62 8.07 8.26 8.29 8.31 8.70
## [91] 8.73 8.83 8.86 8.91 9.08 9.29 9.32 9.47# R code for plotting titration curve
plot(Volume,pH,main = "Volume of NaoH vs. pH of Solution in Diprotic Titration",xlab = "volume (mL)", ylab = "pH")
Titration Curve of Diprotic Acid
Converting to Binding Curve
In this section, the titration data will be converte into a binding curve. Points should be excluded beyond the second endpoint to ensure accuracy in the analysis.
Conversion Process
To convert the titration data into a binding curve, the concept of fractional saturation should be utilize, which represents the fraction of binding sites occupied by the titrant at each point during the titration. This is calculated using the formula:
\[ F = 2 - \frac{{(Volume \times CB) + (H \times (VI + Volume))}}{{VE \times CB}} \]
Where: - \(F\) is the fraction bound. - \(Volume\) is the volume of titrant added. - \(CB\) is the concentration of the base NaOH. - \(H\) is the concentration of \(H^+\) ions calculated from the pH readings. - \(VI\) is the initial volume of the unknown acid. - \(VE\) is the volume added at the endpoint.
H <- 10^-(pH) # H+ from pH
H## [1] 1.288250e-02 1.288250e-02 1.288250e-02 1.288250e-02 1.318257e-02
## [6] 1.288250e-02 1.288250e-02 1.258925e-02 1.230269e-02 1.148154e-02
## [11] 1.096478e-02 1.047129e-02 1.000000e-02 9.549926e-03 9.332543e-03
## [16] 8.912509e-03 7.943282e-03 7.943282e-03 7.079458e-03 6.918310e-03
## [21] 6.456542e-03 6.025596e-03 5.623413e-03 5.128614e-03 4.786301e-03
## [26] 4.265795e-03 3.890451e-03 3.630781e-03 3.235937e-03 2.818383e-03
## [31] 2.511886e-03 2.187762e-03 1.778279e-03 1.513561e-03 1.148154e-03
## [36] 8.709636e-04 5.888437e-04 3.630781e-04 2.238721e-04 1.445440e-04
## [41] 1.023293e-04 6.606934e-05 4.786301e-05 3.090295e-05 2.137962e-05
## [46] 1.318257e-05 9.120108e-06 6.918310e-06 6.606934e-06 3.981072e-06
## [51] 3.548134e-06 3.162278e-06 2.570396e-06 2.187762e-06 1.995262e-06
## [56] 1.659587e-06 1.412538e-06 1.380384e-06 1.174898e-06 1.047129e-06
## [61] 1.023293e-06 8.511380e-07 8.128305e-07 6.606934e-07 6.456542e-07
## [66] 5.495409e-07 5.248075e-07 5.011872e-07 4.073803e-07 3.981072e-07
## [71] 3.311311e-07 2.818383e-07 2.754229e-07 2.344229e-07 2.238721e-07
## [76] 1.698244e-07 1.621810e-07 1.230269e-07 1.023293e-07 9.772372e-08
## [81] 9.332543e-08 4.786301e-08 4.677351e-08 2.511886e-08 2.398833e-08
## [86] 8.511380e-09 5.495409e-09 5.128614e-09 4.897788e-09 1.995262e-09
## [91] 1.862087e-09 1.479108e-09 1.380384e-09 1.230269e-09 8.317638e-10
## [96] 5.128614e-10 4.786301e-10 3.388442e-10#volume added at the endpoint:
VE <- 15.181/2
#initial volume of unknown acid:
VI <- 25
#concentration of base NaOH:
CB <- 0.10
# fraction bound for each data point
F <- (2-(((Volume*CB)+ ((H)*(VI+Volume)))/(VE*CB)))
F## [1] 1.575703329 1.574959752 1.574216174 1.572282873 1.557768206
## [6] 1.556072884 1.526924647 1.507043701 1.487091036 1.485550002
## [11] 1.473963740 1.461843088 1.449206245 1.436070854 1.414917702
## [16] 1.401121725 1.407319650 1.378877971 1.381991691 1.359706800
## [21] 1.348506583 1.336540274 1.323719970 1.314617025 1.299963244
## [26] 1.292328110 1.279358342 1.262086399 1.250245493 1.239508092
## [31] 1.224582536 1.210522861 1.200067068 1.183966506 1.172108236
## [36] 1.156824681 1.141890675 1.124764524 1.111401192 1.101781897
## [41] 1.093789897 1.084369240 1.075513191 1.062259097 1.047508181
## [46] 1.025061508 1.002045899 0.975789834 0.949452811 0.923217100
## [51] 0.904396013 0.878063390 0.851740121 0.825407815 0.799067165
## [56] 0.772733089 0.746395160 0.720047533 0.693707896 0.667364772
## [61] 0.641016880 0.614675939 0.588328792 0.561855307 0.535507123
## [66] 0.509162826 0.482815145 0.456467424 0.430123129 0.403774750
## [71] 0.377429212 0.351082836 0.324734355 0.298387605 0.272039348
## [76] 0.245693292 0.219344909 0.192998138 0.166650442 0.140301927
## [81] 0.113953403 0.087607020 0.076935838 0.050588236 0.037940904
## [86] 0.011461246 0.006982122 0.005269474 0.001053690 -0.009485647
## [91] -0.010407845 -0.012647467 -0.014491872 -0.019366379 -0.031486771
## [96] -0.039523115 -0.043475422 -0.053487931Analyzing the Binding Curve
In this section, it shows the binding curve obtained from the
titration data using the nls2 package in R. A nonlinear
model was fit to the data and examined the results. If the fit appears
off, the endpoint was iteratively adjusted until a satisfactory fit is
achieved. Finally, the results obtained from the binding curve analysis
was compared with those from the titration.
plot(pH,F,main = "Binding Curve of Diprotic Titration",xlab = "pH", ylab = "Fraction Bound")
lines(pH,F,col="blue")
Non-linear Least Squares Analysis to Determine \(K_1\) and \(K_2\)
The dissociation constants (\(K_1\) and \(K_2\)) for oxalic acid are determined by employing non-linear least squares analysis to fit the diprotic binding curve equation to the experimental data.
Fitting Nonlinear Model
The nls2 package was used to fit a nonlinear model to
the binding curve data. The model represents the binding curve based on
the following equation:
\[ F = \frac{{H/KD1 + 2 \times H^2/(KD1 \times KD2)}}{{1 + H/KD1 + H^2/(KD1 \times KD2)}} \]
Where: - \(F\) is the fraction bound. - \(H\) is the concentration of \(H^+\) ions calculated from the pH readings. - \(KD1\) and \(KD2\) are the dissociation constants for the first and second acidic protons, respectively.
# R code for fitting nonlinear model
library(nls2)
fit <- nls(F ~ (H/KD1 + 2 * H^2 / (KD1 * KD2)) / (1 + H/KD1 + H^2 / (KD1 * KD2)), start=c(KD1=0.0001, KD2=0.01))
fit## Nonlinear regression model
## model: F ~ (H/KD1 + 2 * H^2/(KD1 * KD2))/(1 + H/KD1 + H^2/(KD1 * KD2))
## data: parent.frame()
## KD1 KD2
## 5.324e-07 1.099e-02
## residual sum-of-squares: 0.1428
##
## Number of iterations to convergence: 7
## Achieved convergence tolerance: 5.584e-07Dissociation Constants \(K_1\) and \(K_2\)
- \(K_{a1}\) = \(5.324 \times 10^{-7}\)
- \(K_{a2}\) = \(1.099 \times 10^{-2}\)
Discussion
The experiment conducted aimed to determine the dissociation constants (Ka1 and Ka2) of the unknown acid sample through potentiometric titration. The titration method employed involved the use of NaOH as the titrant and pH measurement to identify the equivalence points. Potentiometric titration is a frequently used and successful method for obtaining acid dissociation constants. Equivalence points can be reliably determined by monitoring pH changes during the titration process. The method allows for precise readings, particularly when employing automated titration equipment, resulting in consistent findings. Inaccuracies in measuring the quantities of solutions or samples are one possible source of error. Even little changes in volume measurements can have an impact on the accuracy of the predicted dissociation constants. Variability in the purity and concentration of reagents, such as NaOH, can cause errors in the titration process.
Comparison of Traditional Titration Analysis with Binding Curve
Pros and Cons
Traditional Titration Analysis
Pros:
- Easy and straightforward to perform.
- Provides direct measurement of equivalence point.
Cons:
- Less precise and accurate.
- Requires indicator for endpoint detection.
- Difficult to detect the midpoint.
Binding Curve Analysis
Pros:
- More accurate and precise.
- No need for an indicator.
- Midpoint can be easily determined.
Cons:
- More complex and requires additional mathematical analysis.
- Sensitive to errors in pH measurement.