Abstract

In this paper, it analyzes both monoprotic and diprotic titration curves of different acids with the same base, 0.1M NaOH. The main parts in this report are the titration curves of both acids along with their respective fraction bound curves. The titration curve measures the pH of the acid vs. its volume. The fraction bound curve represents its pH values vs. fraction bound. The monoprotic curve shows one equivalence point, where the graph appears vertical. The diprotic curve represents two equivalence points, one for each steep part in the graph. This represents both Ka1 and Ka2 of that reaction.

Introduction

This paper is built upon the foundations of titration reactions between an acid and a base. These reactions involve a weak acid with a known strong base, 0.1M NaOH. The monoprotic titration involved acetic acid in an unknown vinegar solution. This was done by manual titration and potentiometric titration. The volume at equivalence point was used to find the moles of NaOH, which was used to find the concentration of acetic acid. These calculations were performed because it was the manual titration. The general process and purpose of titration is to add a known concentration of the titrant of one solution to another solution of unknown concentration until it is neutralized and reaches the equivalence point. This is indicated by a color change and the endpoint has been reached. This volume is important and used to calculate the concentration of unknown solution. Because this titration experiment is monoprotic, that just means a monoprotic is being used. A monoprotic acid only dissociates once, and will thus only have one equivalence point. It only has one H+ and can only be donated once, as opposed to polyprotic acids, which will have multiple endpoints. The manual titration is done by adding in this case, the known base NaOH to a beaker of acetic acid from an unknown vinegar solution until a color change is observed and remains unchanged. That is when the equivalence point has been hit and neutralization occurs. This volume of the base added is the observed and measured volume. In the potentiator titration, it is more automatic, because a pH meter is used. This directly measures the pH of the solution as the base, NaOH is being added in a set amount of increments. The endpoint is observed by the rapid changes in pH values in smaller increments, which usually occur around pH 8. In this case, it starts around 2-3 and goes up slowly, but around this point, it jumps from like 6-8, then 8-10, and settles and slows down again from 11-12, where the titration ends. Both processes, although the method is different, both observe and measure the point of neutralization between the acid and base, one by the color change, and one by the pH change. This is also shown in a titration curve between NaOH and pH value, where the steep slope in the middle of the curve represents the equivalence point, where the pH values change more drastically.

Monoprotic Acid

\(F = \frac {H}{H+KD}\) This equation represents the fraction bound for both titration reactions. This was used to generate the pH vs. Fb curves for both reactions. This also utilizes the KD value: \(KD = \frac {P*L}{PL}\) Ka value is also known as the kd value with the same equation using the conjugate base and acid. The Ka values are expected to be small because weak acids are used in these experiments. Also, at half equivalence point, pH equals Pka, so the Pka can be used to find the Ka by using the equation: \(Ka = \exp {10}{-Pka}\) This has an inverse relationship, where 10^-Pka is going to be the opposite value of the Pka value. The binding curves are shown below using the Fb equation and the various volume and pH values. A traditional titration analysis uses the increasing volume of the reagent, which is the acid in this case with a known concentration of the base. The pH changes as more of the reagent is added. This is useful and better than a binding curve, because it provides information about the reaction rate and equivalence point, which is used to determine the concentration of the unknown acid. The binding curve helps define the linear range and gives a linear response to the range of concentrations specifically, not volume, like a calibration curve like in the potentiometric method and autotitrator. The values are not as concrete, though and dont generate exact quantitative concentrations.

Mydata <- read.csv("monoprotic1.csv")
Mydata
##     Vol     pH
## 1   0.0  3.093
## 2   1.0  3.553
## 3   2.0  3.729
## 4   3.0  3.863
## 5   4.0  3.971
## 6   5.0  4.087
## 7   6.0  4.217
## 8   7.0  4.322
## 9   8.0  4.423
## 10  9.0  4.543
## 11 10.0  4.614
## 12 11.0  4.674
## 13 12.0  4.858
## 14 13.0  4.953
## 15 14.0  5.086
## 16 15.0  5.353
## 17 16.0  5.597
## 18 17.0  6.046
## 19 18.0  6.362
## 20 18.2  6.770
## 21 18.4  6.984
## 22 18.6  7.577
## 23 18.8  8.337
## 24 19.0  9.000
## 25 19.2  9.468
## 26 19.4 10.000
## 27 19.6 10.362
## 28 19.8 10.543
## 29 20.0 10.679
## 30 20.2 10.777
## 31 20.4 10.860
## 32 20.6 10.937
## 33 20.8 11.003
## 34 21.0 11.074
## 35 21.2 11.131
## 36 21.4 11.179
## 37 21.6 11.202
## 38 21.8 11.216
## 39 22.0 11.357
## 40 23.0 11.419
## 41 24.0 11.530
## 42 25.0 11.544
## 43 26.0 11.619
## 44 27.0 11.701
## 45 28.0 11.735
## 46 29.0 11.772
## 47 30.0 11.811
## 48 31.0 11.838
## 49 32.0 11.863
## 50 33.0 11.902
## 51 34.0 11.915
volume <- Mydata$Vol
pH <- Mydata$pH

plot(volume,pH,main="Volume of NaOH vs. pH of Solution",xlab="Volume (ml)",
     ylab="pH",xlim=c(0,34),ylim=c(0,12))

H <- 10^-(pH)
H
##  [1] 8.072350e-04 2.798981e-04 1.866380e-04 1.370882e-04 1.069055e-04
##  [6] 8.184648e-05 6.067363e-05 4.764310e-05 3.775722e-05 2.864178e-05
## [11] 2.432204e-05 2.118361e-05 1.386756e-05 1.114295e-05 8.203515e-06
## [16] 4.436086e-06 2.529298e-06 8.994976e-07 4.345102e-07 1.698244e-07
## [21] 1.037528e-07 2.648500e-08 4.602566e-09 1.000000e-09 3.404082e-10
## [26] 1.000000e-10 4.345102e-11 2.864178e-11 2.094112e-11 1.671091e-11
## [31] 1.380384e-11 1.156112e-11 9.931160e-12 8.433348e-12 7.396053e-12
## [36] 6.622165e-12 6.280584e-12 6.081350e-12 4.395416e-12 3.810658e-12
## [41] 2.951209e-12 2.857591e-12 2.404363e-12 1.990673e-12 1.840772e-12
## [46] 1.690441e-12 1.545254e-12 1.452112e-12 1.370882e-12 1.253141e-12
## [51] 1.216186e-12
Vini <- 25  #initial volume
Vini
## [1] 25
Vend <- 18.4/2 #endpoint volume
Vend
## [1] 9.2
tbase <- 0.10 #concentration of base 
tbase
## [1] 0.1
Fb <- (2-(((volume*tbase)+((H)*(Vini+volume)))/(Vend*tbase)))
Fb
##  [1]  1.978064e+00  1.883394e+00  1.777131e+00  1.669741e+00  1.561848e+00
##  [6]  1.453853e+00  1.345782e+00  1.237473e+00  1.129080e+00  1.020681e+00
## [11]  9.121182e-01  8.035189e-01  6.950945e-01  5.864963e-01  4.779131e-01
## [16]  3.693723e-01  2.607568e-01  1.521328e-01  4.345795e-02  2.173116e-02
## [21] -4.894428e-06 -2.174039e-02 -4.347848e-02 -6.521744e-02 -8.695654e-02
## [26] -1.086957e-01 -1.304348e-01 -1.521739e-01 -1.739130e-01 -1.956522e-01
## [31] -2.173913e-01 -2.391304e-01 -2.608696e-01 -2.826087e-01 -3.043478e-01
## [36] -3.260870e-01 -3.478261e-01 -3.695652e-01 -3.913043e-01 -5.000000e-01
## [41] -6.086957e-01 -7.173913e-01 -8.260870e-01 -9.347826e-01 -1.043478e+00
## [46] -1.152174e+00 -1.260870e+00 -1.369565e+00 -1.478261e+00 -1.586957e+00
## [51] -1.695652e+00
plot(pH,Fb) #can exclude some data points for Fb, volume up to 21ml, ph=11

library(nls2) 
## Loading required package: proto

Diprotic Acid

The package n1s2 was used to find the KD1 and KD2 values. This was done using the Fb code shown in the code for both monoprotic and diprotic bounding curves. It also utilized the H values for each pH value. This was inputted into the fit code to generate the linear line of best fit. The KD values generated were KD1 = 9.652e-08, KD2 = 7.665e-03. These values were very close to the Ka values obtained from the original lab report. Those Ka values were Ka1 = 7.59e-03, Ka2 = 1.91e-07. The low Ka values support the fact that it is a weak acid, and its second Ka at the second equivalence point occurs at higher volumes of acid added with more of it dissociating. This increase in dissociation, with support from the Henderson-HasselBalch equation: \(pH = Pka + log \frac {[A-]}{HA}\) represents the increase in pH and thus, Pka. Ka was also determined from its pka values in the original lab report.

Mydata <- read.csv("diprotic1.csv")
Mydata
##      pH     Vol
## 1  1.84  0.0000
## 2  1.84  0.0050
## 3  1.85  0.0100
## 4  1.91  0.9280
## 5  1.93  1.1280
## 6  1.94  1.3280
## 7  2.04  2.1280
## 8  2.18  3.1290
## 9  2.30  3.9290
## 10 2.33  4.1290
## 11 2.60  5.1290
## 12 3.04  6.1297
## 13 3.23  6.3290
## 14 3.48  6.5290
## 15 3.65  6.6390
## 16 3.84  6.7330
## 17 4.55  7.0780
## 18 4.67  7.1630
## 19 4.86  7.3360
## 20 5.05  7.5360
## 21 5.18  7.7360
## 22 5.28  7.9360
## 23 5.39  8.1360
## 24 5.48  8.3370
## 25 5.57  8.5370
## 26 5.63  8.7370
## 27 5.69  8.9370
## 28 5.75  9.1370
## 29 5.81  9.3370
## 30 5.82  9.5370
## 31 5.92  9.7370
## 32 5.96  9.9370
## 33 5.97 10.1370
## 34 6.07 10.3370
## 35 6.10 10.5370
## 36 6.15 10.7370
## 37 6.19 10.9370
## 38 6.23 11.1370
## 39 6.28 11.3370
## 40 6.33 11.5370
## 41 6.38 11.7370
## 42 6.43 11.9370
## 43 6.44 12.1370
## 44 6.53 12.3370
## 45 6.58 12.5370
## 46 6.63 12.7370
## 47 6.71 12.9370
## 48 6.78 13.1370
## 49 6.78 13.3370
## 50 6.90 13.5370
## 51 7.00 13.7370
## 52 7.01 13.9370
## 53 7.18 14.1370
## 54 7.20 14.3370
## 55 7.49 14.5370
## 56 7.60 14.6150
## 57 7.61 14.7750
## 58 8.22 14.9750
## 59 8.41 14.9930
## 60 8.44 14.9980
## 61 8.46 15.0120
## 62 9.12 15.2130
## 63 9.27 15.3030
## 64 9.49 15.4670
## 65 9.71 15.6390
## 66 9.83 15.7990
volume <- Mydata$Vol

pH <- Mydata$pH
pH
##  [1] 1.84 1.84 1.85 1.91 1.93 1.94 2.04 2.18 2.30 2.33 2.60 3.04 3.23 3.48 3.65
## [16] 3.84 4.55 4.67 4.86 5.05 5.18 5.28 5.39 5.48 5.57 5.63 5.69 5.75 5.81 5.82
## [31] 5.92 5.96 5.97 6.07 6.10 6.15 6.19 6.23 6.28 6.33 6.38 6.43 6.44 6.53 6.58
## [46] 6.63 6.71 6.78 6.78 6.90 7.00 7.01 7.18 7.20 7.49 7.60 7.61 8.22 8.41 8.44
## [61] 8.46 9.12 9.27 9.49 9.71 9.83
class(pH)
## [1] "numeric"
plot(volume,pH,main="Volume of NaOH vs. pH of Solution",xlab="Volume (ml)",
     ylab="pH",xlim=c(0,16),ylim=c(0,10))

H <- 10^(-pH)
H
##  [1] 1.445440e-02 1.445440e-02 1.412538e-02 1.230269e-02 1.174898e-02
##  [6] 1.148154e-02 9.120108e-03 6.606934e-03 5.011872e-03 4.677351e-03
## [11] 2.511886e-03 9.120108e-04 5.888437e-04 3.311311e-04 2.238721e-04
## [16] 1.445440e-04 2.818383e-05 2.137962e-05 1.380384e-05 8.912509e-06
## [21] 6.606934e-06 5.248075e-06 4.073803e-06 3.311311e-06 2.691535e-06
## [26] 2.344229e-06 2.041738e-06 1.778279e-06 1.548817e-06 1.513561e-06
## [31] 1.202264e-06 1.096478e-06 1.071519e-06 8.511380e-07 7.943282e-07
## [36] 7.079458e-07 6.456542e-07 5.888437e-07 5.248075e-07 4.677351e-07
## [41] 4.168694e-07 3.715352e-07 3.630781e-07 2.951209e-07 2.630268e-07
## [46] 2.344229e-07 1.949845e-07 1.659587e-07 1.659587e-07 1.258925e-07
## [51] 1.000000e-07 9.772372e-08 6.606934e-08 6.309573e-08 3.235937e-08
## [56] 2.511886e-08 2.454709e-08 6.025596e-09 3.890451e-09 3.630781e-09
## [61] 3.467369e-09 7.585776e-10 5.370318e-10 3.235937e-10 1.949845e-10
## [66] 1.479108e-10
Vini <- 25  #initial volume
Vini
## [1] 25
Vend <- 15.799
Vend
## [1] 15.799
tbase <- 0.10 #concentration of base 
tbase
## [1] 0.1
Fb <- (2-(((volume*tbase)+((H)*(Vini+volume)))/(Vend*tbase)))
Fb
##  [1] 1.771277 1.770914 1.775761 1.739361 1.734301 1.724612 1.708709 1.684318
##  [9] 1.659543 1.652417 1.627457 1.594050 1.587728 1.580138 1.575300 1.570931
## [17] 1.551425 1.546182 1.535384 1.522824 1.510212 1.497580 1.484945 1.472239
## [25] 1.459592 1.446940 1.434287 1.421634 1.408980 1.396321 1.383669 1.371012
## [33] 1.358353 1.345699 1.333041 1.320384 1.307726 1.295068 1.282411 1.269753
## [41] 1.257095 1.244437 1.231778 1.219121 1.206463 1.193804 1.181146 1.168488
## [49] 1.155829 1.143171 1.130512 1.117853 1.105195 1.092536 1.079878 1.074941
## [57] 1.064814 1.052155 1.051016 1.050699 1.049813 1.037091 1.031394 1.021014
## [65] 1.010127 1.000000
plot(pH,Fb,main="Fraction Bound vs. pH of diprotic acid") #can exclude some data points for Fb, volume up to 21ml, ph=11

library(nls2) 

fit <- nls2(Fb ~ (H/KD1 + (2*H^2)/(KD1*KD2))/(1+H/KD1+H^2/(KD1*KD2)),
               start=c(KD1=0.000001,KD2=0.0001))

summary(fit)
## 
## Formula: Fb ~ (H/KD1 + (2 * H^2)/(KD1 * KD2))/(1 + H/KD1 + H^2/(KD1 * 
##     KD2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## KD1 -3.188e-12  2.062e-11  -0.155    0.878    
## KD2  2.733e-06  5.462e-07   5.003 4.68e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1993 on 64 degrees of freedom
## 
## Number of iterations to convergence: 17 
## Achieved convergence tolerance: 6.083e-06
lines(pH,predict(fit),col="blue")

This program is helpful in the analysis of data obtained from labs and experiments. This was demonstrated through chemistry and how certain curves are used. In future studies, like in biochemistry and organic chemistry experiments, certain data can be analyzed and graphs can be generated using this program. An example of this binding curve, actually, that can be applied in biochemistry, is the myoglobin and hemoglobin curves. They represent its presense and binding affinity in different oxygenative states, like hypoxic and normoxic conditions. It switches states when oxygen levels are high and low, where myoglobin has usually high O2 binding affinity even at low partial O2 levels, and hemoglobin has lower binding affinities to O2 so it can release the oxygen to our cells. Graphs like these binding curves can also be analyzed and generated using this program and can further analyze more quantitatively the results from this concept and many other concepts and experiments.