Rapid determination of the acidity, alkalinity and carboxyl content of aqueous samples by 1 H NMR with minimal sample quantity

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S1 Supporting Information for:

Rapid determination of the acidity, alkalinity and carboxyl content of aqueous samples by 1_H NMR with minimal sample quantity

Matthew Wallace,* Kevin Lam, Agne Kuraite and Yaroslav Z. Khimyak

School of Pharmacy, University of East Anglia, Norwich Research Park, Norwich, NR4 7TJ, UK

Apple squash double strength concentrate was purchased from Sainsbury's Supermarkets Ltd., UK. The list of ingredients of the apple squash are as follows: water, apple juice from concentrate (20%), citric acid, malic acid, sodium citrate, fruit and vegetable extracts: apple, hibiscus, molasses; potassium sorbate, sodium metabisulphite, sweetener sucralose, ascorbic acid. Ingredients list obtained from: https://www.sainsburys.co.uk/gol-ui/product/sainsburys-double-strength-apple-squash--no-added-sugar-15l (accessed 24th April 2020).

S1.2. Preparation of carboxylic acid functionalized cellulose nanocrystals

Cellulose nanocrystals (CNC) functionalized with citric or malic acid were prepared following a procedure based on that of Yu et al.1 Microcrystalline cellulose (20 m particle size, 1.0 g) was combined with HCl (5 mL, 6 M) and either citric acid (45 mL, 3 M) or DL-malic acid (45 mL, 5 M) in a 100 mL conical flask equipped with a magnetic stirrer bar. The mixture was heated to 80 °C and stirred at 600 rpm for four hours. The mixture was then cooled in an ice bath to room temperature and the mixture transferred to four 15 mL centrifuge tubes. The tubes were centrifuged for 1 minute at 2000 rpm on a Hettich 1011 hand centrifuge. The supernatant was discarded and the white solid resuspended in deionised water and centrifuged for 1 minute at 2000 rpm. After five washings, no low molecular weight acid could be detected in the supernatant by 1_{H NMR. The solid was resuspended in 2 mL H}

2O per centrifuge tube, combined into 4 mL aliquots and the pH adjusted to 7 using 1 M NaOH, the pH being monitored using a Hanna Instruments HI-1110 electrode. The 4 mL aliquots were then sonicated at 20 kHz for 3 minutes at 5% amplitude in pulse mode (1 s on, 1 s off) using a Fisherbrand Q700 sonicator equipped with a 3 mm diameter microtip probe. 900 J of energy was typically applied to the samples during sonication. The tip was immersed in the liquid to a depth of 15 mm. The centrifuge tubes served as sample vessels for sonication and were held in an ice bath to prevent overheating and hydrolysis of the cellulose esters. Following sonication, the suspensions were again adjusted to pH 7 with a small addition of NaOH and centrifuged at 2000 rpm for 1 minute. The supernatant was transferred to 2 mL microcentrifuge tubes and centrifuged at 12,000 rpm for 5 minutes on a Boeco M24 centrifuge. The supernatant was collected, adjusted to pH 7.4, and stored at 4 °C. The wt% of functionalized cellulose in the supernatant was determined by drying an aliquot overnight in air in a fume cupboard. The stock dispersions contained between 3 and 4 wt% functionalized cellulose and were diluted for analysis. Dried yields with respect to the starting microcrystalline cellulose were approximately 10-20%.

S1.3. Use of DMSO as an alternative

1

_{H chemical shift reference}

The use of 0.005 vol% DMSO as an alternative to 0.1 mM DSS was explored with the CNC samples. DMSO has been shown to be a suitable reference for the measurement of pH and ionic concentrations by 1_{H NMR.}2-3_{DMSO can also provide a reference linewidth to check for} any interaction of the indicator with the sample (Figure S-11). The 1_{H chemical shift of 0.005} vol% DMSO in 2 mM MPA (pH 9.4) was measured as 2.7162±0.00005 ppm (average of two samples) against 0.1 mM DSS at 0 ppm. The chemical shift of 0.005 vol% DMSO in 20 mM HCl, 0.2 mM MPA (pH 1.75) was measured as 2.7167±0.0001 ppm (average of two samples) against 0.1 mM DSS at 0 ppm. Taking the reference chemical shift of DMSO relative to DSS as 2.7164 ppm, the limiting chemical shifts of the indicators relative to DMSO (2.7200 ppm) may be obtained by adding 0.0036 ppm to the values in Table 1. Adjusted limiting chemical shifts of MPA are provided in Table S-1:

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S3 Indicator pKa (ionic strength/mM) H/ppm L/ppm MPA 7.75 (41) 2.32 (20) 1.2855 1.5142 1.0747 1.2855

Table S-1. Limiting 1_{H chemical shifts of MPA quoted relative to 0.005 vol% DMSO (2.7200}

ppm)

Values of pH, TA and COO(H) in Table 5 were recalculated using DMSO as a reference and are provided in Table S-2:

Sample TA/mM pHCNC (2 mM MPA) pHref (2 mM MPA) COO(H)/mM pHCNC (20 mM HCl) pHref (20 mM HCl) Citrate CNC 0.15±0.02 0.15±0.02 8.78 8.79 9.40 9.42 3.52±0.30 3.47±0.30 1.83 1.83 1.75 1.75 Malate CNC 0.09±0.02 0.09±0.02 9.03 9.02 9.42 9.43 2.65±0.31 2.68±0.31 1.81 1.81 1.75 1.74

Table S-2. TA, pH and COO(H) of 1 wt% CNC dispersions determined using DMSO (black,

2.7200 ppm) and DSS (blue, 0 ppm) as chemical shift references. The pH was determined from the 1_{H chemical shifts of MPA in 1 wt% CNC dispersions and in reference samples as} indicated. Uncertainties are calculated assuming an uncertainty of 0.0005 ppm and 0.002 ppm in the measured (Figure S-11) and limiting chemical shifts respectively (Section S4 and S7). No significant differences are observed between the pH, TA and COO(H) values calculated using DMSO or DSS as a reference, confirming that neither of these reference compounds interacts significantly with the CNC dispersions under study. DMSO can therefore be used as an alternative chemical shift reference to DSS.

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S4

S2. Determination of limiting chemical shifts of 2MI by chemical shift imaging

A solution was prepared in H2O containing 0.2 mM DSS and 2 mM each of 2-methylimidazole (2MI), sodium glycinate, disodium methylphosphonate (MPA), methylamine hydrochloride and NaOH. Approximately 1-2 mg of KH2PO4 was weighed into a 5 mm diameter Wilmad 528-PP NMR tube and four 2 mm glass beads (Karl Hecht, Germany) placed on top. The solution was carefully layered on top of the glass beads to a height of 40 mm from the base of the NMR tube and the sample transferred immediately to a Bruker Avance I Spectrometer operating at 499.69 MHz (1_{H). The temperature of the instrument was maintained at 298 K, calibrated using} a methanol standard.4_{The temperature can be assumed accurate to 0.5 K, the variation with} time being less than 0.1 K. A 1_{H chemical shift image was recorded on the sample 4 hours 50} minutes after preparation. The chemical shift image was acquired using a gradient phase encoding sequence based on that of Trigo-Mouriño et al.5_{incorporating the perfect-echo} WATERGATE sequence of Adams et al.6_{based on the double echo W5 sequence of Liu et} al.7_{The phase encoding gradient pulses were in the form of smoothed squares and were of} 238 μs duration. Their amplitude was varied from −27 to 27 G/cm in 128 steps giving a theoretical spatial resolution of 0.2 mm. 8 scans were acquired at each gradient step with an acquisition time of 1 s (16K data points) and a relaxation delay of 0.1 s. The delay between successive pulses in the selective pulse train was set at 333 s. 16 dummy scans were acquired prior to signal acquisition.

The chemical shift image was processed with 32K points in phase-sensitive mode with phase correction, baseline correction and peak-picking performed automatically on each spectrum using macro scripts written in house. An exponential line broadening factor of 3 Hz was applied to extract the chemical shifts of MPA and 2MI (aromatic), 1 Hz to extract the chemical shifts of DSS, glycine, methylamine, and 2MI (methyl) above pH 7. Below pH 7, the methyl resonance of 2MI is very close to the resonance of methylamine and a line broadening factor of 0.3 Hz was used to separate the two resonances. The pH at each position along the sample could thus be determined from the chemical shifts of DSS, methylamine, glycine and MPA using Equation 1, following the procedure described in our previous work.8_{The pK}

a and limiting chemical shifts of 2MI were extracted by fitting the extracted chemical shifts of 2MI to Equation S1 in Microsoft Excel using the procedure of Brown:9

δobs=

δL+ δH10pKa−pH

1 + 10pKa−pH S1

L and H were treated as free variables in the fit as clear limiting values are approached on Figure 1.8_{The pK}

a value was obtained as 8.02 and 8.03 for the methyl and aromatic resonances respectively (Fits shown on Figure 1, a). The ionic strength, I, at the midpoint of the titration was obtained as 25 mM using the CurTiPot package,10_{based upon the theoretical} quantity of KH2PO4 required to attain a pH of 8.02. The thermodynamic pKa value, pKa,0, can be obtained from the fitted pKa value using Equation S2:8, 11

pKa,0 = pKa(I) −0.51√I_1+√I + 0.1I S2

A pKa,0 of 7.96 is thus obtained for 2MI, in good agreement with the value of 7.86 reported by Kirby and Neuberger.12_{Our previous work has demonstrated that the chemical shift imaging} approach is a robust and accurate method to determine the pKa values and limiting chemical shifts of analyte molecules.8_Example1_{H spectra extracted from the chemical shift image are} provided on Figure S-1.

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S5

Figure S-1. Example 1_{H spectra extracted from chemical shift image of a pH gradient at pH}

values indicated. Spectra have been processed with an exponential line broadening factor of 3 Hz (a) or 0.3 Hz (b). The resonances of 2MI, methylamine, glycine and MPA are broadened in places along the pH gradient due to the relatively large pH change over a single slice (0.2 mm) and the high pH sensitivity of these resonances over the respective pH ranges, as described in Section S10 of our previous work.8_{The measurement of titratable acidity is} performed on homogeneous samples in the absence of a pH gradient where the resonances of 2MI and MPA are sharp (Figure 1b, S-8 and S-11). The resonances of methylamine and 2MI (methyl) overlap at pH ≈ 6.6 but can be resolved at lower pH with a line broadening factor of 0.3 Hz, allowing H to be obtained (Figure 1, a).

S3. Propagation of uncertainty analysis of Equation 1

The pKa values quoted in Table 1 are defined in terms of the activity of H+ and the concentrations of protonated and deprotonated indicator.8_{The pK}

a of the indicators (Table 1) at ionic strengths, I, below 0.1 M may be extrapolated from the thermodynamic pKa values, pKa,0 (I = 0), using Equation S3:8, 11

pKa(I) = pKa,0+ (ZH2 − ZL2) (0.51√I_1+√I − 0.1I) S3

where ZH and ZL are the charges of the protonated and deprotonated forms of the indicator respectively. For acetic acid, 2MI, and the lower pKa of MPA, the pKa value will not deviate from the values quoted in Table 1 by more than 0.1 units when the ionic strength is below 0.1 M (Equation S3). The upper pKa of MPA (ZH = -1, ZL = -2) will not deviate by more than 0.1 units from 7.75, provided the ionic strength lies between 0.01 M and 0.1 M. The uncertainty in the pKa values of the indicators, pKa, is thus taken as 0.1 for the calculation of pH using Equation 1. A propagation of uncertainty analysis of Equation 1 yields Equation S4:

∆pH= √( ∂pH ∂pKa ) 2 ∆pKa 2_{+ (}∂pH ∂δH ) 2 ∆δH 2_{+ (}∂pH ∂δL ) 2 ∆δL 2_{+ (}∂pH ∂δobs ) 2 ∆δobs 2

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S6 ∆pH= √∆pKa 2_{+ (} ∆δlim 2.303(δobs− δH) ) 2 + ( ∆δlim 2.303(δL− δobs) ) 2 + ( ∆δobs(δL− δH) 2.303(δobs− δH)(δL− δobs) ) 2 S4 where  is the uncertainty in the subscripted variable and lim are the limiting (H and L) chemical shifts. obs is taken as 0.0005 ppm at 500 MHz in this work (Figure S-11), lim is taken as 0.002 ppm.3, 13 

pHis plotted on Figure S-2 as a function of pH for the indicators listed in Table 1. MPAH refers to the lower pKa of methylphosphonate, corresponding to the equilibrium between the fully protonated and monoprotonated forms. obs was calculated for the indicators at each pH value using Equation S1 and the limiting chemical shifts and pKa values listed in Table 1. The ratio [HInd]/[Ind]total was calculated using Equation S5:

[HInd]/ [Ind]_total= 10

pK𝑎−pH

1 + 10pK_𝑎−pH S5

From Figure 2, pKa is the dominant source of uncertainty in the pH determined using Equation 1 when the pH is ±1 units either side of the pKa value of the indicator.

Figure S-2. Plot of uncertainty in pH determined using Equation 1 from (a) 2MI aromatic

(black) and methyl (red) chemical shifts, (b) acetic acid, (c) MPAH and (d) MPA. The ratio [HInd]/[Ind]total is also plotted (blue line). Uncertainties are plotted for obs of 0.0005 ppm (solid), 0.001 ppm (dotted), 0.002 ppm (long dash) and 0.005 ppm (short dash). In all plots; lim = 0.002 ppm, pKa = 0.1.

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S7

S4. Propagation of uncertainty analysis of Equation 3

The acidity of the reference sample (Table 2) is calculated using Equation S6: TA = [Ind]total( δref− δL δH− δL ) − (δH− δref δref− δL ) 10pKa−pKw _S6

Equation 3 can be re-written as:

TA = [HInd]sample−[HInd]ref+ [OH−]ref S7

where [OH-_]

ref is the free concentration of hydroxide ions in the reference sample, given by the right-hand term of Equation 3. The pKa of the indicator and pKw are defined in terms of the activity of H+_{, γ}

H+[H+], and the concentration of OH- and indicator:

Kw= γH+[H+][OH−] = 10 −pKw,0+(0.51√I 1+√I−0.1I) S8 Ka= γH+[H+][Ind] [HInd] S9

pKw,0 is expressed in terms of the activities of both H+ and OH- and has a value of 14.00 at 25 °C. The use of fixed pKa values (Table 1) and a fixed pKw of 13.95 in Equation 3 will cause the term (pKa - pKw) to vary (Section S3) by no more than 0.1 units when the ionic strength is between 0 and 100 mM (2MI) or 10 and 100 mM (MPA, upper pKa). A propagation of uncertainty analysis of the right-hand term, r, of Equation 3 yields Equation S10:

∆r= √( 10pKa−pKw δref− δL) 2 [(2.303∆pKa−pKw(δH− δref)) 2 + (∆δlim δH− δref δref− δL) 2 + ∆δlim 2 + (∆δs δL− δH δref− δL) 2 ] S10

where  is the uncertainty in the subscripted variable and s and lim are the user-measured (obs and ref) and limiting (H and L) chemical shifts respectively. Based on our previous work, s can be taken as 0.0005 ppm at 500 MHz (Figure S-11), lim is taken as 0.002 ppm.3, 8 

pKa-pKw is taken as 0.1. ref, calculated in the absence of CO2, is provided in Table S-3:

Indicator

(concentration/mM)

pHref [HInd]/mM ref/ppm

2MI (20 mM) 10.13 0.137 2.3468 (methyl)

6.9607 (aromatic)

2MI (2 mM) 9.62 0.043 2.3506 (methyl)

6.9652 (aromatic)

MPA (2 mM) 9.57 0.040 1.0753

Table S-3. ref, pH and [HInd] of reference samples calculated in absence of CO2 using CurTiPot package10_{. pK}

a,0 2MI = 7.96, pKa,0 MPA = 7.99.8 δref= δH [HInd]

[Ind]_total+ δL(1 − [HInd] [Ind]_total)

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S8 ∆TA= √(∆[Ind]total( δobs− δref δH− δL )) 2 + 2 (∆δs[Ind]total δH− δL ) 2

+ 2 (∆δlim[Ind]total(δobs− δref)

(δH− δL)2 ) 2

+ ∆r2 S11

[Ind]total is taken as 2%, arising from the preparation of the stock solution and sample.

For the calculation of TA as a function of TA/[Ind]total (Figure S-3), obs was calculated using Equation S12:

δobs= δL+

TA + 10pH−pKw

[Ind]total

(δH− δL) S12

where the pH was calculated using Equation S13:

pH = pKa+ log10[

([Ind]total−TA)(1−(10pKa−pKw_TA ))

TA+([Ind]total−TA

TA )10pKa−pKw

] S13

The pKa values and limiting chemical shifts of 2MI and MPA listed in Table 1 were used in the calculation. The pH calculated using Equation S13 is in good agreement with more accurate calculations using the CurTiPot package (Figure S-3).10_{The pH calculated using Equation S13} is therefore sufficiently accurate for the calculation of obs using Equation S12. Uncertainties calculated using Equations S10 and S11 are plotted on Figure S-3 as a function of the ratio TA/[Ind]total for s of 0.0005, 0.001, 0.002 and 0.005 ppm. TA is expressed as a percentage of TA. Uncertainties quoted in Tables 2 and 5 are calculated with s = 0.0005 ppm, [Ind]total = 2%, lim = 0.002 ppm, pKa-pKw = 0.1, pKw = 13.95. In the calculations, the acid in the sample is assumed to be strong and to be quantitatively deprotonated by the indicator (Figure S-5). Finally, we note that we note that it is likely that the titratable acidity could be determined using commercial low field benchtop NMR instruments where 1_{H chemical shifts can now be} measured with uncertainties of < 0.005 ppm at indicator concentrations of 10 mM or less.14-15 Calculations performed at s = 0.005 ppm suggest that the titratable acidity could be determined to within 20% uncertainty when the ratio TA/[Ind]total is between 0.2 and 0.9 (Figure S-3).

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S9

Figure S-3. Plot of uncertainty in TA determined using Equation 3 versus TA/[Ind]total with s of 0.0005 ppm (solid), 0.001 ppm (dotted), 0.002 ppm (long dash) and 0.005 ppm (short dash) using methyl (red) and aromatic (black) resonances of 2MI at (a) [2MI]total = 20 mM, (b) [2MI]total = 2 mM. (c) uncertainty with MPA as an indicator, [MPA]total = 2 mM (Table 5). The pH calculated using Equation S13 (blue line) is plotted along with more accurate calculations using the CurTiPot package10 (blue cross). In all plots; 

[Ind]total = 2%, lim = 0.002 ppm,  pKa-pKw = 0.1, pKw = 13.95.

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S10

S5. Effect of CO2 and indicator concentration on TA measurements

Figure S-4. (a) Plot of simulated error in TA determined using Equation 3 versus percentage

CO2 contamination of 2MI stock solution at 1 mM HCl (white triangle), 2 mM (green cross), 5 mM (black cross), 10 mM (black diamond), 15 mM (red triangle) and 18 mM HCl (blue square). The concentration of protonated 2MI, [H2MI], at each concentration of HCl and CO2 was calculated using the CurTiPot package10_{with 0.2 mM DSS, [2MI]}

total = 20 mM, and 1 mol% CO2 equivalent to 0.2 mM H2CO3. The apparent TA reported by the NMR method was calculated as TA = [H2MI]– [H2MI]ref+ 10pHref−pKw (Equation 3) where [H2MI]ref was calculated in the absence of HCl at the background concentration of CO2 indicated. pHref was calculated as pHref= 8.02 + log10(

[2MI]total−[H2MI]ref

[H2MI]ref ). pKw = 13.95. Percentage error was

calculated as 100([H2MI]– [H2MI]ref−[HCl]+10pHref−pKw)

[HCl] . The error in the calculation of TA using

Equation 2 is plotted at 10 mM HCl (grey vertical cross) as 100([H2MI]−[HCl] )

[HCl] . (b) Plot of pH in

samples, symbols as for (a), (white diamond, 0 mM HCl), calculated as pH = 8.02 + log10(

[2MI]total−[H2MI]

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S11

Figure S-5. Plots of simulated error in TA determined using Equation 3 versus concentration

of HCl (diamond) or citric acid (triangle) with 2MI (black) or MPA (red) as indicator. The corresponding sample pH calculated using the CurTiPot package10_{is plotted as open symbols.} Calculations at [2MI]total = 20 mM, [H2CO3] = 0.2 mM (a) and [Ind]total = 2 mM, [H2CO3] = 0.02 mM (b). The concentration of protonated indicator, [HInd], at each concentration of HCl and citric acid was calculated using the CurTiPot package10_{with 0.2 mM DSS, pK}

a,0 2MI = 7.96, pKa,0 MPA = 7.99. The apparent acidity reported by the NMR method was calculated as TA = [HInd]– [HInd]ref+ 10pHref−pKw where [HInd]ref was calculated in the absence of HCl or citric acid. Percentage error is calculated as 100([HInd]– [HInd]ref−TA+10pHref−pKw)

TA where TA = [HCl] or

3[Citric acid], pKw = 13.95. pHref was calculated as pHref= pKa+ log10[

[Ind]total−[HInd]ref [HInd]ref ]

(Equation 1) using the pKa values of the indicators listed in Table 1. The significant error below pH 7.5 with citric acid is due to incomplete deprotonation of the acid (Section S10). The error at TA < 0.2 mM, [Ind]total = 2 mM, is due to the presence of OH- ions in the sample arising from the deprotonation of water by the indicator, and partially to the error in the calculation of pHref.

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S12

S6. Propagation of uncertainty analysis of Equation 4

The effective pKa of acetic acid can be computed at any ionic strength (I) from the thermodynamic pKa, pKa,0, using Equation S14:8, 11

pKa(I) = pKa,0− (0.51√I_1+√I − 0.1I) S14

This pKa is defined in terms of the activity of H+ and the concentrations of acetate and acetic acid in order that Equation 1 yields the activity-based pH.8_{I is taken as 10 mM in this work} yielding a pKa value of 4.72, based upon pKa,0 = 4.76.16 The pKa will not deviate from this value by more than 0.1 units when the ionic strength is less than 0.1 M. However, in Equation 4, the pKa is, strictly, expressed wholly in terms of concentration in order that the right-hand term yields the free concentration of H+_{. The concentration-based pK}

a, pKa,conc, for a carboxylic acid indicator is given by Equation S15:

pKa,conc (I) = pKa,0− 2 (0.51√I_1+√I − 0.1I) S15

Ionic strengths up to 0.1 M or small concentrations of Ca2+_{or Mg}2+_{will not cause pK}

a,conc to deviate by more than 0.2 units from 4.72.2, 8_{A pK}

a of 4.72 can thus be used in Equation 4 without significant loss of accuracy. The uncertainty in the pKa value used in Equation 4, pKa, can be taken as 0.2. To test the sensitivity of acetate to Ca2+_{or Mg}2+_{, the}1_{H chemical shift of} acetate was measured in 20 mM solutions of either calcium or magnesium acetate in H2O. 0.01 vol% DMSO was included as a chemical shift reference (2.7164 ppm, Section S1.3). The acetate chemical shift was measured as 1.9076 ppm and 1.9073 ppm in calcium and magnesium acetate respectively. The presence of 20 mM Ca2+_{or Mg}2+_{therefore affects the} chemical shift of acetate by less than 0.002 ppm (Table 1) which has negligible effect on the alkalinity values determined (Figure S-6). Ca2+_{or Mg}2+_{concentrations in natural fresh waters} are typically < 3 mM (Section S12).17

For the analysis of uncertainty, we may split Equation 4 into two parts: The right-hand term, r, yields the free concentration of H+_{ions due to dissociation of the acetic acid. A propagation of} uncertainty analysis for r yields Equation S16:

∆r= √( 2.303∆pKa(δobs− δL)10 −pKa δH− δobs ) 2 + (∆δobs(δH− δL)10 −pKa (δH− δobs)2 ) 2 + (∆δlim10 −pKa δobs− δH ) 2 + (∆δlim(δL− δobs)10 −pKa (δH− δobs)2 ) 2 S16 The uncertainty in [B] calculated using Equation 4, [B], is given by Equation S17:

∆[B]

= √(∆[Ind]total(δH− δobs)

δH− δL

)

2

+ (∆δobs[Ind]total

δH− δL

)

2

+ (∆δlim[Ind]total(δobs− δL)

(δH− δL)2

)

2

+ (∆δlim[Ind]total(δH− δobs)

(δH− δL)2

)

2

+ ∆r2

S17 where  is the uncertainty in the subscripted variable. lim is the uncertainty in the limiting (H and L) chemical shifts. The uncertainty in the total acidic indicator concentration, [Ind]total, is taken as 2%. lim is taken as 0.002 ppm.3, 13_{For the estimation of uncertainty in [B] as a} function of [B]/[Ind]total, obs of the indicator is obtained from Equation S1 via the following procedure: It is first necessary to estimate the ionic strength of the solution at different values of [B]/[Ind]total. The approximate concentration of deprotonated indicator, [Ind-], is calculated from Equation S18 at a fixed ionic strength of 10 mM:

[Ind−_{] =}_−0.5₍₍_K

a− [B])−√(Ka− [B])2+ 4Ka[Ind]total) S18

where Ka is expressed wholly in concentration units and obtained from pKa,0, correcting for an ionic strength of 10 mM, using Equation S19:

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S13 Ka(I) = 10

−[pK_a,0−2(0.51√I

1+√I−0.1I)] S19

The ionic strength of the solution, I, is assumed equal to [Ind-_{] calculated using Equation S18} with Ka at a fixed ionic strength. A more accurate Ka is then computed at each value of [B]/[Ind]total using Equation S19. The pH of the solution is then computed from Equation S20:

pH =−log₁₀Ka+(

0.51√I

1 +√I− 0.1I)− log10

[

[Ind]_total+ 0.5((Ka− [B])−√(Ka− [B])2+ 4Ka[Ind]total)

−0.5((Ka− [B])−√(Ka− [B])2+ 4Ka[Ind]total) ]

S20

The observed chemical shift of the indicator is then computed from Equation S1, correcting the pKa for ionic strength using Equation S14. The uncertainty in [B] can then be computed from Equations S16 and S17 and is plotted on Figure S-6 at obs of 0.0005, 0.005, 0.001 and 0.002 ppm as a percentage of [B]. obs can be assumed equal to 0.0005 ppm on the 500 MHz spectrometer used in this work (Figure S-11). The uncertainty in the pKa, pKa, is taken as 0.2. The discrepancy between the value of [B] calculated using Equation 4 assuming a pKa value of 4.72, [B]Eq4, (Table 3) with the calculated value of obs, and the theoretical (inputted) value of [B] is also plotted. The error introduced by assuming a fixed pKa value (Table 1) is negligible when [B]/[Ind]total > 0.1. Analogous plots are provided for formic acid as an indicator, assuming pKa,0 = 3.75, and pKa = 3.71. The base is assumed to be strong and quantitatively protonated by the indicator. Uncertainties quoted in Table 3 were calculated using Equations S16 and S17 with obs = 0.0005 ppm, lim = 0.002 ppm, pKa = 0.2, [Ind]total = 2%.

It is likely that the titratable alkalinity could be determined using commercial low field benchtop NMR instruments where 1H chemical shifts can be measured with _s_{= 0.005 ppm at indicator} concentrations of 10 mM or less.14-15_{Calculations suggest that measurements within 15%} uncertainty would be possible if [B]/[Ind]total > 0.2 (Figure S-6).

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S14

Figure S-6. Plot of uncertainty in [B] (black) determined with acetic acid (a, Ac) and formic

acid (b, Form) as indicators versus [B]/[Ind]total with obs of 0.0005 ppm (solid), 0.001 ppm (dotted), 0.002 ppm (long dash) and 0.005 ppm (short dash). The pH calculated using Equation S20 is also plotted (blue line). The discrepancy between the inputted [B] and the value calculated from Equation 4 is plotted as a red line. In all plots: [Ind]total = 8 mM, [Ind]total = 2%, lim = 0.002 ppm, pKa = 0.2. The indicator parameters for formic acid are: H = 8.4414 ppm, L = 8.2669 ppm, pKa,0 = 3.75.8

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S15

S7. Derivation and propagation of uncertainty analysis of Equations 5 and 6

The effective dissociation constant of MPA, Keff, in Equation 5 is expressed in terms of the concentrations of fully protonated MPA (MPAH2), monoprotonated MPA (MPAH-) and free H+ ions, [H3O+]ref, in the reference sample:

Keff=

[H3O+]ref[MPAH −_] [MPAH2]

S21 [H3O+]ref is equal to the total concentration of HCl added minus the quantity of H+ absorbed by the dibasic MPA added as an indicator. From Equation 2, [H3O+]ref is obtained as:

[H3O+]ref= [HCl] − [MPA]total(1 +

δref− δL

δH− δL) S22

Combining S21 and S22 yields Equation 5. COO(H) is assumed equal to the difference in [H3O+] between reference and sample. Assuming only monovalent counterions are present in the sample, the ionic strength and therefore Keff will be equal in the sample and reference. Equation 6 is thus obtained from Equation S21. Making specific corrections for ionic strength, I, the pH may be calculated from Keff and the chemical shifts of MPA using Equation S23:

pH = log10[ δobs− δH (δL− δobs)Keff10 −(0.51√I 1+√I−0.1I) ] S23

where I = [HCl] + 2[MPA]total. The difference between the pH calculated using Equation 1 and S23 is negligible (Figure S-7). To compute the uncertainties in the measurement of COO(H), [MPA]total is assumed insignificant relative to [HCl] and is neglected in the calculation. Equations 5 and 6 can be combined to yield Equation S24:

COO(H) = [HCl] [1 −(δH− δref) (δ_ref− δL)

(δL− δobs)

(δ_obs− δH)

] S24

We define the term, f, as: f =(δH− δref)

(δ_ref− δL)

(δL− δobs)

(δ_obs− δH)

S25 The partial derivatives required for the calculation of uncertainty are thus:

∂f ∂δH= (δL− δobs) (δref− δL) (δobs− δref) (δobs− δH)2 S26 ∂f ∂δL= (δH− δref) (δobs− δH) (δref− δobs) (δref− δL)2 S27 ∂f ∂δobs =(δH− δref) (δ_ref− δL) (δH− δL) (δ_obs− δH)2 S28 ∂f ∂δref =(δL− δobs) (δobs− δH) (δ_L− δH) (δref− δL)2 S29 The uncertainty in COO(H) determined using Equations 5 and 6 may thus be estimated from Equation S30: ∆COO(H) = √(∆δlim[HCl] ∂f ∂δH ) 2 + (∆δlim[HCl] ∂f ∂δL ) 2 + (∆δs[HCl] ∂f ∂δref ) 2 + (∆δs[HCl] ∂f ∂δobs ) 2 + (∆[HCl](1 − f)) 2 S30

where  is the uncertainty in the subscripted variable and s and lim are the user-measured (obs and ref) and limiting (H and L) chemical shifts respectively. s can be taken as 0.0005

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S16 ppm by comparing the 2_J

PHcoupling of MPA in the sample and reference (Figure S-11). For the calculation of uncertainty in [H3O+] in the sample under study, [H3O+]sample, the concentration of MPA is again assumed negligible and [H3O+]sample obtained from Equation S31: [H3O+]sample= [HCl] (δH− δref) (δref− δL) (δL− δobs) (δobs− δH) S31

The uncertainty in [H3O+] can be estimated from Equation S32: ∆[H3O+]= √(∆δlim[HCl] ∂f ∂δH) 2 + (∆δlim[HCl] ∂f ∂δL) 2 + (∆δs[HCl] ∂f ∂δref) 2 + (∆δs[HCl] ∂f ∂δobs) 2 + (∆[HCl]f) 2 _S32

Two hypothetical carboxylates are considered as analytes with pKa,0 values of4.0 and 3.0. The concentration of protonated carboxylic acid, [HA], is computed from [HCl] and the total concentration of carboxylate, [COO]total, using Equation S33:

[HA] = 0.5 (([HCl]+[COO]total+ Ka) − √([HCl]+[COO]total+ Ka)2− 4[COO]total[HCl]) S33

Ka is expressed in concentration units and is obtained from pKa,0 using Equation S19 with the ionic strength assumed equal to [HCl]. The concentration of deprotonated carboxylate groups, [COO-_{], and [H}

3O+] may be computed as: [COO−_{] = [COO]}

total− [HA] S34

[H₃O+_{] = [HCl] − [HA]} _S35

The pH is calculated using Equation S36: pH =0.51√I

1+√I − 0.1I − log10[H3O

+_] _S36

where I = [HCl]. Finally, obs and ref are calculated from Equation S37:

δ

_obs

=

δL+δH10pKa(I)−pH

1+10pKa(I)−pH S37

where pKa (I) is obtained from Equation S38 using the pKa,0 value of 2.38 for MPA:8 pKa(I) = pKa,0−

0.51√I

1+√I + 0.1I S38

The spreadsheet calculations can be checked by comparing the results of Equations S23, S36, S31 and S35. Uncertainties in COO(H) and [H3O+] are plotted on Figure S-7 as a function of [COO]total. The uncertainty in COO(H) is expressed as a percentage of [COO]total. Uncertainties quoted in Tables 4 and 5 were calculated with pKa,0 = 4, s = 0.0005 ppm (Figure S-11), lim = 0.002 ppm. [HCl] = 1%, [HCl] = 20 mM and [HCl] = 3%, [HCl] = 46 mM. The higher [HCl] at 46 mM arises from the uncertainty in the concentration of the 1.0±0.05 M HCl standard used for the fumarate experiments.

Finally, we note that Equation 6 can be corrected for the different concentrations of protonated indicator in sample and reference so that higher total concentrations of indicator could be used (Equation S39). This correction is not required in the present work due to the very low concentrations of MPA used so that [Ind]total << COO(H). This modification may allow the determination of the carboxylate content using commercial low field benchtop NMR instruments, although the observation of 19_{F or}31_{P may be required to minimize the uncertainty} arising from the limited 1_{H chemical shift range of the indicators (Figure S-7).}18

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S17 COO(H) = Keff( δref− δL δH− δref −δobs− δL δH− δobs ) − [Ind]total δref− δobs δH− δL S39

Figure S-7. Plot of uncertainty in COO(H) determined by NMR, calculated using Equation S30

(solid lines), percentage of carboxylate groups remaining deprotonated (short dash) and pH of sample calculated using Equation S23 (dotted) as function of [COO]total, with pKa,0 = 4 (black) and pKa,0 = 3 (red). The uncertainties in COO(H) determined at pKa,0 = 4 and pKa,0 = 3 overlap very closely. [HCl] = 20 mM (a), [HCl] = 46 mM (b). In all plots except green lines: s = 0.0005 ppm, lim = 0.002 ppm, [HCl] = 1%. Green lines: s = 0.002 ppm, pKa,0 = 4, lim = 0.002 ppm, [HCl] = 1%. Uncertainty in [H3O+], calculated using Equation S32 is also plotted (long dash) blue: s = 0.0005 ppm, green: s = 0.002 ppm. The pH calculated using Equation 1 with the pKa value in Table 1 is plotted as grey lines, pKa,0 = 4. The discrepancy between Equation S23 and Equation 1 is not significant.

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S18

S8.

1

_{H NMR analysis of apple squash, HCl and tap water}

Figure S-8. 1_{H NMR spectra of samples analysed in Tables 2 and 3: (a) 20}L/mL apple squash with 20±0.4 mM 2MI i: scaled to show resonances of apple squash components and ii: scaled to show 2MI resonances. The resonances of citric acid are highlighted. The other resonances correspond to the expected components present in apple squash.19_{(b) 20±0.4} mM 2MI in 10.0±0.1 mM HCl, (c) reference spectrum of 20 mM 2MI in H2O, (d) 8.0±0.2 mM acetic acid in Brita®_{filtered water, (e) 8.0±0.2 mM acetic acid in tap water, (f) 8.0±0.2 mM} acetic acid in deionised water. The half-height linewidths obtained by Lorentzian deconvolution are provided along with the linewidths normalised to DSS (blue). The constant ratios of the linewidths of the indicators and DSS across all samples suggests the absence of any significant interaction between the indicators and the sample components. All spectra plotted have been processed with an exponential line broadening factor of 0.3 Hz. Linewidths were measured with this line broadening factor.

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S19

S9.

1

_{H NMR analysis of CNC dispersions}

Figure S-9. Partial 1_{H NMR spectra of citrate-functionalized CNC dispersions: (a) 3.74 wt%}

dispersion at pH 7.4 with 0.1 mM DSS. Relative integrals of the citrate resonances are indicated. (b) 1 wt % dispersion in 0.1 M NaOH and 4 mM KHP after standing overnight. The sharp resonances between 2.8 and 2.4 ppm correspond to free citrate. The spectrum has been referenced to the upfield resonance of citrate at 2.54 ppm. (c) 1 wt% CNC dispersion in 2 mM MPA (pH 8.79), (d) 1 wt% dispersion in 20 mM HCl (pH 1.83). All spectra have been processed with an exponential line broadening factor of 1 Hz. The resonance marked # belongs to DSS (0.1 mM). (e-g) photographs of samples corresponding to b, c and d respectively. (h) sample of (c) after standing for two days at 24±1 °C. The spectral acquisition parameters were identical for (c, d and h) and so their intensities can be compared directly.

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S20

Figure S-10. Partial 1_{H NMR spectra of malate-functionalized CNC dispersions. (a) 2 wt%} dispersion at pH 7.4 with 0.1 mM DSS, (b) 1 wt % dispersion in 0.1 M NaOH and 4 mM KHP after standing overnight. The sharp resonances at 4.3 ppm and between 2.8 and 2.2 ppm correspond to free malate. The spectrum has been referenced to the upfield resonance of malate at 2.34 ppm. (c) 1 wt% dispersion in 2 mM MPA (pH 9.02) and (d) 1 wt% dispersion in 20 mM HCl (pH 1.81). The spectral acquisition parameters were identical for (c) and (d) and so their intensities can be compared directly. All spectra have been processed with an exponential line broadening factor of 1 Hz. The peaks marked * correspond to free malate produced during sonication and workup of the CNC. Integration of this resonance demonstrates the dispersions contain less than 0.05 mM of free malate at 1 wt%. The resonance marked # belongs to DSS (0.1 mM). (e-g) photographs of samples corresponding to b, c and d respectively.

Citrate and malate functionalized CNCs form stable dispersions at neutral pH.20_The1_H resonances between 3 and 4.5 ppm correspond to mobile chains on the surface of the CNC particles.21_{Resonances of the citrate or malate moieties esterified to the CNC occur between} 2 and 3 ppm. The structure of the 1_{H resonances of citrate are consistent with} mono-esterification in an asymmetric manner (Scheme 1).22_{Asymmetric mono-esterification is} supported by solid-state NMR data presented by Spinella et al.20_{Alkaline hydrolysis (b, e)} causes aggregation of the dispersions while sharp resonances appear corresponding to the free citrate or malate released from the CNC. No other sharp resonances are apparent suggesting minimal degradation of the cellulose into low molecular weight products.23-24_{At pH} 9 (c, f), in the presence of 2 mM dibasic MPA, no hydrolysis is detected. Samples were analysed within one hour of preparation; however, no hydrolysis can be detected after standing for two days (Figure S-9, h). In 20 mM HCl (d, g) the dispersions are aggregated and the resonances of the surface groups severely broadened. No resonances of free citric or malic acid are apparent confirming negligible hydrolysis of the CNC dispersions during the experiments.

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S21

Figure S-11. Partial 1_{H NMR spectra (black solid line) and Lorentzian fits (red dash) of} samples used for determination of COO(H) (a - e), [HCl] = 20 mM, [MPA]total = 0.2 mM: (a) 1 wt% citrate-functionalized CNC and (b) reference spectrum. (c) 1 wt% malate-functionalized CNC and (d) reference spectrum. (e) 2.0 mM solution of disodium fumarate. Plots (f – h): spectra used for determination of TA (Table 5), [MPA]total = 2 mM: (f) 1wt% citrate-functionalized CNC and (g) reference spectrum. (h) 1 wt% malate-citrate-functionalized CNC. The fitted half-height linewidths are provided along with the linewidths normalised to DSS (blue). The 2_J

PH coupling constants of MPA are provided (grey) along with the pH calculated using Equation 1 from the chemical shift of MPA.

All spectra on Figure S-11 were processed with an exponential line broadening factor of 1 Hz. Linewidths were extracted from these spectra by Lorentzian deconvolution. 2_J

PH coupling constants (a-e) were measured on spectra processed with an exponential line broadening factor of 3 Hz to match the processing applied to extract the chemical shifts of MPA. The linewidths at half-height of DMSO, MPA and DSS remain in constant ratio over all samples indicating the absence of a significant interaction of these species with the CNC dispersions. The resonances of DSS, MPA and DMSO in the CNC dispersions at pH 1.8 (a, c) are broad due to the aggregation of the CNC (Figure S-9, g and S-10, g) which increases the inhomogeneity of the magnetic field. Lorentzian deconvolution was performed using Bruker Topspin 3.6.2. The 2_J

PH coupling of MPA varies by ca. 0.25 Hz (0.0005 ppm) between comparable samples, indicating that the uncertainty in the determination of a chemical shift is 0.0005 ppm on our instrument. We note that the 2_J

PH coupling of MPA is pH sensitive, so care is required if the sample and reference have significantly different pH values (Figure S-12).

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S22

Figure S-12. Plot of 2_J

PH coupling constant of MPA as a function of pH. Black: samples of Table 4. Red: couplings extracted from chemical shift image used to extract limiting chemical shifts and pKa of 2MI (Section S2). Blue: couplings extracted from chemical shift image of sample containing 0.2 mM DSS, 1 mM 2,6-lutidine, 2 mM aspartic acid, sodium glycinate, MPA, 4 mM sodium formate, 10 mM NaHCO3, 12 mM sodium acetate and 20 mM disodium oxalate. This solution was placed on top of 3.4 mg oxalic acid dihydrate in a 5 mm NMR tube and the sample left to stand for 7 hours at 298 K prior to analysis in order to establish a pH gradient (Section S2). Image acquisition parameters were as described for 2MI. 2_J

PH is only plotted where the change in pH with position was sufficiently small for accurate couplings to be extracted (Figure S-1). The solid line is a fit to Equation S40.

The limiting 2_J

PH couplings of the fully protonated (J(H2L)), monoprotonated (J(HL)) and deprotonated (J(L)) forms of MPA were extracted by fitting the observed 2JPH coupling constant of MPA, J(obs), (Figure S-12) to Equation S40:

J

_(obs)

=

J(H2L)+J(HL)10pH−pKa,2+J(L)102pH−pKa,2−pKa,1

1+10pH−pKa,2+102pH−pKa,2−pKa,1 S40

where pKa,1 = 7.75, pKa,2 = 2.32 (Table 1). The pKa values were fixed at these values while the limiting J couplings were treated as free variables during the fitting. Limiting 2_J

PH couplings were obtained as: J(H2L) = 17.38 Hz, J(HL) = 16.53 Hz, J(L) = 15.57 Hz.

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S23

S10. Effect of acid pKa on TA measurements

Figure S-13. Simulated plots of apparent TA versus pKa of acid at different ratios of acid to 2MI: [Acid]total/[2MI]total = 0.9 (black), 0.5 (dashed), 0.1 (dotted). Plots were calculated using Equation S42 assuming pKa,2MI = 8.02 (Table 1).

To generate the plots on Figure S-13, it is assumed that an indicator is mixed with a monoprotic acid of the stated pKa. The concentration of protonated indicator and TA are assumed equal to the concentration of deprotonated acid, G, in solution. As the indicator and the acidic compound are necessarily at the same pH in solution we may write:

10

pKa,Acid−pKa,Ind

₌

([Ind]

total

− G)([Acid]

total

− G)

G

2 S41

where pKa,Ind is the pKa of the protonated indicator and [Acid]total is the total concentration of acidic compound. Using the quadratic formula, TA may be obtained as:

TA =q([Ind]total+ [Acid]total) − √q

2_([Ind]

total+ [Acid]total)2− 4q(q − 1)[Ind]total[Acid]total

2(q − 1) S42

where q =

10

pKa,Ind−pKa,Acid

_.

_{The effect of sample composition and ionic strength on the pK} a values of indicator and acid is ignored.

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S24

S11. Simulated titration curves of water samples and estimation of experimental

uncertainty

Figure S-14. Simulated titration of 30 mL 1.6 mM HCO3- (a, tap water) and 0.2 mM HCO3- (b, Brita®_{filtered water) with 2.0 mM HCl, calculated using the CurTiPot package.}10_The equivalence point (red) is indicated with vertical and horizontal error bars denoting the uncertainty arising from an 0.2 unit error in the determination of the pH using our conventional electrode at the low ionic strength of these natural waters.25-26_{We note that Brita}®_{filters lower} the alkalinity of water but do not lower the conductivity so the uncertainty in the determination of the pH is similar for both samples.27

The experimental uncertainty in the number of moles of HCl, HCl, added to attain a reported pH of 4.7 in our potentiometric titration is estimated as 2% and 4% for the tap and filtered water respectively, arising from the uncertainty in the concentration of HCl and the titre and sample volumes. The uncertainty in the endpoint of the titration caused by the 0.2 unit error in the pH, End, is calculated using CurTiPot10 as 2% and 5% for the tap and filtered water respectively.

We estimate the overall uncertainty in [B] obtained by potentiometric titration, [B], using Equation S43:

∆

_[B]

= √∆

_HCl2

+ ∆

_End2 S43

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S25

S12. Water quality report for the domestic supply used in this study (Table 3)

Images taken directly from Anglian Water website:

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S27

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