Abstract

diprotic data

\(F = \frac {H}{H+KD}\)

library(readr)

mydata <- read.csv("Diprotic.csv")
mydata
##      pH Volume
## 1  2.46  0.000
## 2  2.46  0.005
## 3  2.42  0.010
## 4  2.41  0.015
## 5  2.41  0.028
## 6  2.39  0.059
## 7  2.41  0.138
## 8  2.38  0.335
## 9  2.39  0.535
## 10 2.38  0.735
## 11 2.38  0.935
## 12 2.40  1.135
## 13 2.42  1.335
## 14 2.44  1.536
## 15 2.49  1.736
## 16 2.49  1.936
## 17 2.51  2.136
## 18 2.57  2.336
## 19 2.61  2.536
## 20 2.63  2.736
## 21 2.62  2.936
## 22 2.70  3.136
## 23 2.75  3.336
## 24 2.77  3.536
## 25 2.81  3.736
## 26 2.84  3.936
## 27 2.89  4.136
## 28 2.93  4.336
## 29 2.97  4.536
## 30 3.02  4.736
## 31 3.06  4.936
## 32 3.10  5.136
## 33 3.13  5.336
## 34 3.23  5.536
## 35 3.25  5.736
## 36 3.31  5.936
## 37 3.38  6.136
## 38 3.47  6.336
## 39 3.56  6.536
## 40 3.67  6.736
## 41 3.78  6.936
## 42 3.94  7.136
## 43 4.05  7.336
## 44 4.24  7.536
## 45 4.35  7.710
## 46 4.41  7.910
## 47 4.51  8.110
## 48 4.62  8.310
## 49 4.70  8.510
## 50 4.77  8.710
## 51 4.84  8.910
## 52 4.92  9.110
## 53 4.98  9.311
## 54 5.05  9.511
## 55 5.06  9.711
## 56 5.14  9.911
## 57 5.19 10.111
## 58 5.24 10.311
## 59 5.28 10.511
## 60 5.33 10.711
## 61 5.38 10.911
## 62 5.42 11.111
## 63 5.45 11.311
## 64 5.51 11.511
## 65 5.55 11.711
## 66 5.59 11.911
## 67 5.63 12.111
## 68 5.69 12.311
## 69 5.73 12.511
## 70 5.77 12.711
## 71 5.83 12.911
## 72 5.88 13.111
## 73 5.92 13.311
## 74 5.99 13.511
## 75 6.03 13.711
## 76 6.10 13.911
## 77 6.17 14.111
## 78 6.25 14.311
## 79 6.30 14.511
## 80 6.39 14.711
## 81 6.53 14.911
## 82 6.66 15.111
## 83 6.74 15.311
## 84 6.90 15.511
## 85 7.30 15.711
## 86 7.03 15.754
## 87 7.02 15.762
## 88 7.04 15.785
## 89 7.11 15.840
## 90 7.27 15.937
## 91 7.41 16.025
## 92 7.68 16.133
## 93 7.71 16.180
## 94 7.91 16.298
## 95 7.95 16.356
## 96 8.41 16.501
## 97 8.40 16.517
## 98 8.50 16.557
Volume <- mydata$Volume

Volume
##  [1]  0.000  0.005  0.010  0.015  0.028  0.059  0.138  0.335  0.535  0.735
## [11]  0.935  1.135  1.335  1.536  1.736  1.936  2.136  2.336  2.536  2.736
## [21]  2.936  3.136  3.336  3.536  3.736  3.936  4.136  4.336  4.536  4.736
## [31]  4.936  5.136  5.336  5.536  5.736  5.936  6.136  6.336  6.536  6.736
## [41]  6.936  7.136  7.336  7.536  7.710  7.910  8.110  8.310  8.510  8.710
## [51]  8.910  9.110  9.311  9.511  9.711  9.911 10.111 10.311 10.511 10.711
## [61] 10.911 11.111 11.311 11.511 11.711 11.911 12.111 12.311 12.511 12.711
## [71] 12.911 13.111 13.311 13.511 13.711 13.911 14.111 14.311 14.511 14.711
## [81] 14.911 15.111 15.311 15.511 15.711 15.754 15.762 15.785 15.840 15.937
## [91] 16.025 16.133 16.180 16.298 16.356 16.501 16.517 16.557
pH <- mydata$pH

pH
##  [1] 2.46 2.46 2.42 2.41 2.41 2.39 2.41 2.38 2.39 2.38 2.38 2.40 2.42 2.44 2.49
## [16] 2.49 2.51 2.57 2.61 2.63 2.62 2.70 2.75 2.77 2.81 2.84 2.89 2.93 2.97 3.02
## [31] 3.06 3.10 3.13 3.23 3.25 3.31 3.38 3.47 3.56 3.67 3.78 3.94 4.05 4.24 4.35
## [46] 4.41 4.51 4.62 4.70 4.77 4.84 4.92 4.98 5.05 5.06 5.14 5.19 5.24 5.28 5.33
## [61] 5.38 5.42 5.45 5.51 5.55 5.59 5.63 5.69 5.73 5.77 5.83 5.88 5.92 5.99 6.03
## [76] 6.10 6.17 6.25 6.30 6.39 6.53 6.66 6.74 6.90 7.30 7.03 7.02 7.04 7.11 7.27
## [91] 7.41 7.68 7.71 7.91 7.95 8.41 8.40 8.50
length(Volume)
## [1] 98
length(pH)
## [1] 98
H <- 10^(-pH)
H
##  [1] 3.467369e-03 3.467369e-03 3.801894e-03 3.890451e-03 3.890451e-03
##  [6] 4.073803e-03 3.890451e-03 4.168694e-03 4.073803e-03 4.168694e-03
## [11] 4.168694e-03 3.981072e-03 3.801894e-03 3.630781e-03 3.235937e-03
## [16] 3.235937e-03 3.090295e-03 2.691535e-03 2.454709e-03 2.344229e-03
## [21] 2.398833e-03 1.995262e-03 1.778279e-03 1.698244e-03 1.548817e-03
## [26] 1.445440e-03 1.288250e-03 1.174898e-03 1.071519e-03 9.549926e-04
## [31] 8.709636e-04 7.943282e-04 7.413102e-04 5.888437e-04 5.623413e-04
## [36] 4.897788e-04 4.168694e-04 3.388442e-04 2.754229e-04 2.137962e-04
## [41] 1.659587e-04 1.148154e-04 8.912509e-05 5.754399e-05 4.466836e-05
## [46] 3.890451e-05 3.090295e-05 2.398833e-05 1.995262e-05 1.698244e-05
## [51] 1.445440e-05 1.202264e-05 1.047129e-05 8.912509e-06 8.709636e-06
## [56] 7.244360e-06 6.456542e-06 5.754399e-06 5.248075e-06 4.677351e-06
## [61] 4.168694e-06 3.801894e-06 3.548134e-06 3.090295e-06 2.818383e-06
## [66] 2.570396e-06 2.344229e-06 2.041738e-06 1.862087e-06 1.698244e-06
## [71] 1.479108e-06 1.318257e-06 1.202264e-06 1.023293e-06 9.332543e-07
## [76] 7.943282e-07 6.760830e-07 5.623413e-07 5.011872e-07 4.073803e-07
## [81] 2.951209e-07 2.187762e-07 1.819701e-07 1.258925e-07 5.011872e-08
## [86] 9.332543e-08 9.549926e-08 9.120108e-08 7.762471e-08 5.370318e-08
## [91] 3.890451e-08 2.089296e-08 1.949845e-08 1.230269e-08 1.122018e-08
## [96] 3.890451e-09 3.981072e-09 3.162278e-09
plot(Volume, pH, main = "Diprotic", xlab = "Volume" , ylab ="pH")

Vinital <- 25.00

Vadd <- Volume

Vend <- 14.278/2

CB <- 0.100

FB <-  2 -(CB * Vadd + H * (Vinital + Vadd)) / (CB *  Vend) 

FB
##  [1]  1.878576534  1.877851871  1.865407805  1.861577752  1.859685924
##  [6]  1.848738726  1.843678150  1.805135371  1.779346472  1.746769385
## [11]  1.717586392  1.695272014  1.672751257  1.649885989  1.635640846
## [16]  1.606719166  1.583333440  1.569721536  1.550087036  1.525676523
## [21]  1.494867914  1.482086146  1.462124492  1.436810364  1.414334228
## [26]  1.390075297  1.368070544  1.344352438  1.320285202  1.296823561
## [31]  1.272064482  1.247040376  1.221055628  1.199354350  1.172315278
## [36]  1.147287018  1.122314547  1.097607479  1.072299012  1.046946300
## [41]  1.021011267  0.995251847  0.968368190  0.941767402  0.917970161
## [46]  0.890208226  0.862553303  0.834852148  0.807019733  0.779139266
## [51]  0.751239461  0.723336472  0.695252444  0.667309736  0.639301975
## [56]  0.611356061  0.583377650  0.555395443  0.527403888  0.499415838
## [61]  0.471424987  0.443427244  0.415423958  0.387431251  0.359429240
## [66]  0.331426144  0.303422053  0.275422077  0.247415816  0.219408821
## [71]  0.191404854  0.163397899  0.135388626  0.107382815  0.079372283
## [76]  0.051364465  0.023355593 -0.004653462 -0.032665363 -0.060675413
## [81] -0.088684380 -0.116695301 -0.144708412 -0.172720409 -0.200731251
## [86] -0.206756974 -0.207877704 -0.211099201 -0.218802592 -0.232388568
## [91] -0.244714380 -0.259841518 -0.266424994 -0.282953506 -0.291077832
## [96] -0.311388376 -0.313629591 -0.319232570
plot(pH,FB,main ="Fraction Bound vs. pH of Diprotic Acid ",xlim = c(2,10),ylim = c(0.0,2.5) )

library(nls2)
## Loading required package: proto
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 5.528e-06  5.985e-07   9.236 6.51e-15 ***
## KD2 1.722e-03  1.563e-04  11.020  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1248 on 96 degrees of freedom
## 
## Number of iterations to convergence: 7 
## Achieved convergence tolerance: 2.098e-06
lines(pH,predict(fit),col="green")