Acid Neutralizing Capacity - GitHub Pages

Theory¶

The ANC for a typical carbonate-containing sample is defined as:

(33)¶\[ ANC = [HCO_3^-]+2[CO_3^{-2} ]+{[OH}^- ] - [H^+]\]

Equation (33) can be derived from a charge balance if ANC is considered to be the cation contributed by a strong base titrant and if other ions present do not contribute significantly.

Determination of ANC or alkalinity involves determination of an equivalence point by titration with a strong acid. The equivalence point is defined as the point in the titration where titrant volume that has been added equals the “equivalent” volume (\(V_e\)). The equivalent volume is defined as:

(34)¶\[V_{e} {\; =}\frac{V_{s} \cdot N_{s} }{N_{t} }\]

where:

Ns = normality (in this case alkalinity or ANC) of sample, equivalents/L Vs = volume of sample, liters Nt = normality of titrant, equivalents/L.

The titration procedure involves incrementally adding known volumes of standardized normality strong acid (or base) to a known volume of unknown normality base (or acid). When enough acid (or base) has been added to equal the amount of base (or acid) in the unknown solution we are at the “equivalence” point. The point at which we add exactly an equivalent or stoichiometric amount of titrant is the equivalence point. Experimentally, the point at which we estimate to be the equivalence point is called the titration endpoint.

There are several methods for determining \(V_e\) (or the equivalence point pH) from titration data (titrant volume versus pH). The shape of the titration curve (\(V_t\) versus pH) can reveal \(V_e\). It can be shown that one inflection point occurs at \(V_t= V_e\). In the case of monoprotic acids, there is only one inflection in the pH range of interest. Therefore, an effective method to find the equivalence volume is to plot the titration curve and find the inflection point. Alternately, plot the first derivative of the titration plot and look for a maximum.

Gran Plot¶

Another method to find the ANC of an unknown solution is the Gran plot technique. When an ANC determination is being made, titration with a strong acid is used to “cancel” the initial ANC so that at the equivalence point the sample ANC is zero. The Gran plot technique is based on the fact that further titration will result in an increase in the number of moles of \(H^+\) equal to the number of moles of \(H^+\) added. Thus after the equivalence point has been attained, the number of moles of \(H^+\) added equals the number of moles of \(H^+\) in solution. An equation describing this mass balance is provided as:

(35)¶\[ N_{t} \left(V_{t} -V_{e} \right)=\left(V_{s} +V_{t} \right)\left[H^{+} \right]\]

Solving for the hydrogen ion concentration:

(36)¶\[ \left[H^{+} \right]=\frac{N_{t} \left(V_{t} -V_{e} \right)}{\left(V_{s} +V_{t} \right)}\]

Equation (36) can also be solved directly for the equivalent volume.

(37)¶\[ V_{e} =V_{t} -\frac{\left[H^{+} \right]\left(V_{s} +V_{t} \right)}{N_{t} }\]

Equation (37) is valid if enough titrant has been added to neutralize the ANC. A better measure of the equivalent volume can be obtained by rearranging equation (37) so that linear regression on multiple titrant volume - pH data pairs can be used.

(38)¶\[\frac{\left(V_{s} +V_{t} \right)}{V_{s} } \left[H^{+} \right]=\frac{N_{t} V_{t} }{V_{s} } -\frac{N_{t} V_{e} }{V_{s} }\]

We define \(F_1\) (First Gran function) as:

(39)¶\[ F_1 = \frac{V_s +V_t }{V_s } {[H}^+ {]}\]

If \(F_1\) is plotted as a function of \(V_t\) the result is a straight line with slope = \(\frac{N_{t} }{V_{s} }\) and abscissa intercept of \(V_e\) (Fig. 7).

The ANC is readily obtained given the equivalent volume. At the equivalence point:

(40)¶\[ V_s ANC= V_e N_t\]

Equation (40) can be rearranged to obtain ANC as a function of the equivalent volume.

(41)¶\[ ANC=\frac{V_e N_t }{V_s }\]

Fig. 7 Gran plot from titration of a weak base with 0.05 N acid. \(C_T\) = 0.001 moles of carbonate and sample volume is 48 mL. The equivalent volume was 4.8 mL. From equation (41) the ANC was 5 meq/L.

pH Measurements¶

The pH can be measured either as activity \(\mathrm{\{}H^+\mathrm{\}}\) as measured approximately by pH meter) or molar concentration ([H^+]). The choice only affects the slope of F1 since \([H^+] = \mathrm{\{}H^+\mathrm{\}/\gamma}\).

(42)¶\[F_1 =\frac{V_s +V_t }{V_s} [H^+] = \frac{V_s + V_t}{V_s} \frac{\{ H^+ \} }{\gamma} = N _t \frac{V_t - V_e}{V_s}\]

where \(\gamma\) is the activity correction factor and the slope is \(N_t/V_0\). If \([H^+]\) concentration is used then

(43)¶\[F_1 = \frac{V_s +V_t }{V_s } { \{ H}^+ {\}} = \gamma N_t \frac{V_t - V_e}{V_s}\]

where the slope is \(\frac{\gamma \cdot N_t}{V_s}\).

This analysis assumes that the activity correction factor doesn’t change appreciably during the titration).

There are many other Gran functions that can be derived. For example, one can be derived for Acidity or the concentration of a single weak or strong acid or base.

To facilitate data generation and subsequent Gran plot construction and analysis pH versus titrant volume can be read directly into a computer, that can be programmed to analyze the data using the Gran plot theory. The program generates the Gran function for all data and then systematically eliminates data until the Gran function (plot) is as linear as possible. The line is then extrapolated to the abscissa to find the equivalent volume.

ANC Determination for Samples with pH < 4¶

After the equivalence point has been reached (adding more acid than ANC = 0) the only significant terms in equation (33) are \(\left[{H}^{+} \right]\) and ANC.

(44)¶\[ \left[{H}^{+} \right]>>{\; }\left[{HCO}_{{3}}^{{-}} \right]+{\; 2}\left[{CO}_{{3}}^{{-2}} \right]+\left[{OH}^{{-}} \right]{\; }\]

When the pH is 2 pH units or more below the lowest \(pK_a\) of the bases in the system the only species contributing significantly to ANC is the hydrogen ion (equation (44)) and thus the ANC is simply

(45)¶\[ ANC= - [H^+]\]

For a sample containing only carbonates, if the pH is below 4 the ANC is approximately equal to -[\(H^+\)] and no titration is necessary.

Titration Techniques¶

Operationally, the first few titrant volumes can be relatively large increments since the important data lies at pH values less than that of the equivalence point (approximately pH = 4.5 for an Alkalinity titration). As the pH is lowered by addition of acid the ionic strength of the solution increases and the activity of the hydrogen ion deviates from the hydrogen ion concentration. This effect is significant below pH 3 and thus the effective linear range is generally between pH 4.5 and pH 3.0. The maximum incremental titrant volume (\(\mathrm{\Delta}V_a\)) that will yield n points in this linear region is obtained as follows.

If \(V_s\) >> \(V_t\) then equation (35) reduces to

(46)¶\[{N}_{{t}} {\; \; \; }\frac{(V_{t} -V_{e} )}{V_{s} } \cong {\; [H}^{+} {]}\]

Let \([H^+]_e\) be the concentration of hydrogen ions at the equivalence point and \([H^+]_f\) be the final concentration of hydrogen ions at the end of the titration.

(47)¶\[N_t \frac{(V_e - V_e)-(V_f - V_e)}{V_s} =[H^+]_e [H^+]_f\]

Thus the volume of acid added to go from \([H^+]_e\) to \([H^+]_f\) is

(48)¶\[ V_f - V_e =\frac{V_s \left([H^+]_f -[H^+]_e \right)}{N_t}\]

To obtain n data points between \([H^+]_e\) - \([H^+]_f\) requires the incremental titrant volume (\(\mathrm{\Delta} V_t\)) be 1/n times the volume of acid added between the equivalence point and the final titrant volume. Thus by substituting \(n\mathrm{\Delta}V_t\), and typical hydrogen ion concentrations of \([H^+]_e = 10^{-4.5}\) and \([H^+]_f = 10^{-3.0}\) into equation (48) the maximum incremental titrant volume is obtained.

(49)¶\[\Delta V_t\cong \frac{(0.001-0.00003)V_s }{n\; N_t} \cong \frac{0.001V_s}{n\; N_t}\]