F. J. C. Rossotti and I Potentiometric Titrations

Hazel Rossotti University of Oxford I Using Gran Plots

England I A texfbookomission

A number of recent textbooks (1-10) of analytical chemistry discuss potentiometric acid-base titrations, but none of these mentions Gran's graphical method (11) of end point determination. Gran's method seems to us to be the best yet suggested, and has been successfully used for many years in Stockholm and elsewhere for the precise analysis of acids and bases.

The method is very simple. A volume V of acid of initial concentration H is titrated with a volume u of strong base of concentration B in the galvanic cell (I)

electrode reversible t,o I titration / I reference - ) hydrogen ions solot~on half-cell (+)

The potential of cell (I) is given by

where Eo' includes the potential of the reference half- cell and the standard potential of the probe half-cell, and E, is the liquid junction potential. A cell of type (I) is also used in commercial pH meters, which, when suitably calibrated, give direct readings of pH.

Titrotion of Strong Acid with Strong Base

For any value of v before the equivalence point US,

If pH is measured directly, we may make use of a func- tion defined as

+ = ( V + u)[H+]rH = (V + u)l0-nA (3)

From equations (2) and (3)

+ = (us - VWYH

If, instead of pH, the potential E is determined, we may

use a similar function defined by + = (V + u)10-EF/l.a03KT (5)

Fromequations (I), (2), and (5)

After the equivalence point,

where K,, is the stoichiometric ionic product of water. For use in the alkaline range, functions analogous to + and J. may he defined in terms of the experimental quantities pH and E respectively, viz.

From equations (7) and (8)

+' = ( V - ua)B/K,yr (10)

and from equations (I), (7), and (9)

If YH, 4, and K , remain constant throughout the titration, then from equations (4) and (10) the quanti- ties + and 4' are linear functions of u, such that +(u), +'(u) and the abscissa all intersect a t the point 0, US.

Similarly, equations (6) and (11) show that, for constant m, Ej, and K,, the functions J.(v) and J.'(u) are also linear, and both functions again cut the abscissa a t the point 0, u,. Thus the value of u, may he ohtained by plotting any one of the quantities +, +', J., or J.' against u, and extrapolating the line ohtained to cut the abscissa -

a t the point u = v,. The method may be illustrated by the following

Volume 42, Number 7, July 1965 / 375

Figvre 1. Gran plots d v l and Q ' M for the titration of o strong ocid with a strong bose. Doto for the filled points are given in the table.

example which is suitable for inclusion in an under- graduate course on pH. A mixture of 5 mlO.l M HC1 and 50 ml water was titrated with -0.1 M NaOH, using a cell containing glass, and saturated calomel, electrodes. The calculations1 are shown in the table, and the resulting points plotted as full circles in Figure 1.

Titration of o Mixture of 5.00 ml 0.1 000 M HCI $ 50.0 ml H z 0 with -0.1 M N a O H Using a

Commercial oH Meter

~ ~ . . . . . 5.00 60.0 9.828 ... 6.73 X 101 OW2 6.00 61.0 11.268 ... 185.2 X 10' 0.056 7.00 62.0 11.549 . . . 354.0 X 109 0.109

From Figure 1, ma = 4.97 ml, whence B = 0.1006 M.

Deviations from linearitu may be due to one or more of the following factors: (1) Curvature of any of the Gran plots a t values of v remote from v, suggests that the quantity E,' = E, - RTF-' In 7~ is not negligible a t the extreme values of [H+] or [OH-] used. If the plots are linear nearer the equivalence point, the value of v, must be obtained from this region alone (Fig. 2). For the junction H20/Sat:KCI, linear functions are

1 From equations (4), (6), (lo), and ( l l ) , the point of inter- section 0, u, of the plots for acid solution either with those for alkaline solution or with the abscissa, is unaffected by changes in the slopes of the lines. The various quantities JI, O, JI,' and $' may therefore be multiplied by any constants which make plotting more convenient. The terms 10"F'P.aD8"T and 10-pH may, if required, be replaced by 10(C-E)"P.~omT and 1 0 ( C ~ u H ~ where C is any constant. A value of 'pH' = pH - C may be obtained merely by using an unstrtndardized commercial pH meter.

Figure 2. Gron plot +lvl or JIIvl for a system in which Ej' = IEj-RTF-' In ?HI varies with [H*]. 35.0 ml of 0.210 M HNOs war titrated with -2 M LiOH in 50% v / v aqueous dioxone containing 0.5 M (Li', Hi) NCJ-.

obtained for [H+] 5 8 X M. A method for cal- culating E,' from Gran plots has been described else- where (12) together with a similar method for obtaining y, for HC1 from data obtained during titration of HCI in a Harned type cell without liquid junction. (2) If the acid solution contains a metal ion which starts to

Figure 3. Gron plot Q O or +lvI for the itandardirotion of o strong acid containing hydrolyzable metal ions. 40.0 ml of -0.01 M HClOr containing -50 mM VOICIOth was titrated with 0.021 1 8 M NoHCOs in a 3 M (No)- C104medivm.

376 / Journal o f Chemical Education

hydrolyze a t a hydrogen ion concentration in the re- gion H > [H+] > M the functions $(u) and + ( u ) may become curved as the acidity is decreased. If the plots are linear a t the highest acidities used, the equiv- alence point of the strong acid may again be deter- mined by extrapolation to $ = 0 or + = 0 (Fig. 3). (3) If the strong base is contaminated with carbonate, the functions @'(u) and $'(u) are curved in the region of v,.

The value of u, = VH/B (where B = ([OH-] + 2[COa2-1) is the total concentration of base) may be found using only measurements in the acid region u < u, (Fig. 4). If the functions +'(u) and $'(u) are linear over an appreciable range of v, they may be extrapolated to cut the abscissa a t the point 0.' = VH/B1 (where B' is the total concentration of hydroxyl ions in the solu- tion of base). An estimate of the carbonate ion con- centration may therefore be obtained.

Titration of Weak Acid with Strong Base

If an acid HA is not fully dissociated in solution, the free hydrogen ion concentration is no longer given by equation (2) but by the relationship

[H+l = K,[HAlI[A-I (12 )

where K. is the stoichiometric acid dissociation constant of HA. If the titrant is the monacidic base MOH, then

[A-] = [M+] + [H+] - [OH-] = vB

V + u ---- + [Hf1 - [OH-] (13)

and VH - vR

[HA] = C A - [A-I = - - - [Ht] + [OH-] ( 1 4 ) V + u

where CA is the total concentration of HA in the solu- tion. If

>> [Ht] - [OH-] V + v (15)

and VH - vB

V + v >> [Ht] - [OH-] (16 )

then from equations (12), (13), and (14) K.(VH - oB) - K.(v. - u ) [H'] = -

vB v (17 )

By analogy with equations (3), (4), (5), and (6) we may define the functions2

r = v[H+lm = vlO-pH = Kdv. - (18 )

After the end point, v > v, and equations (7) to (11) are again valid.

Since, for a weak acid, [H+] is low even a t the be- ginning of a titration, E, will probably be negligible. If, moreover, the initial solution of HA is not too con- centrated, the ionic strength will not vary grossly throughout the titration, and the terms K,, K,, and y, will remain approximately constant. Then T and 0

Figure 4. Portions of Gron plots +lvl or H v l ond +'(*I or $'lvl for the titro- lion of a strong acid with o solution containing both carbonate ond hy- droxide.

are linear functions of u, such that r(u) intersects +'(u) and 0(u) intersects $'(v) a t the point 0, v, (Fig. 5) .

Deviations from linearity may be due to one or more of the following factors: (1) At the beginning of the titra- tion, condition (15) may not be fulfilled, especially if the acid is only moderately weak. The value of v, must then be found using only that part of the function ~ ( u ) or 0(u) which is found to be linear (Fig. 5). (2) As u approaches u,, condition (16) may not be fulfilled if the

The remarks in footnote 1 apply to the quantity r, and also to +, +',and T below.

Figure 5. Gron plots r lv l or B(vl and +7vI or $'(v) for the standordim- tion of a weak ocid with a strong bare. 20.0 mi of -0.05 M acetic ocid war titrated with 0.1 06 M NaOH.

Volume 42, Number 7, July 1965 / 377

acid is very weak. The equivalence point may then be found by extrapolation of the linear region of ~ ( v ) or B(u) to 7 = 0 or B = 0. (3) If the acid is contaminated with a metal ion which forms complexes with A-, equations (13) and (14) will no longer be valid. De- viation of r(u) and B(u) from linearity will increase as u approaches 8,. (4) Curvature in the alkaline region will again be observed if the base contains carbonate (see p. 377).

Titration of Strong Base with Strong Acid

For the titration of a volume Y of strong base with a volume y of strong acid, the functions analogous to Q and Q' [equations (4) and (lo)] are

a = ( Y + y ) [ H t 1 m = ( Y + ~ ) 1 0 - ' ~ = (Y - y.)Hrx (20 )

in the acidic and alkaline regions re~pectively.~ Here ye is the equivalence point. Similar functions, * and W, analogous to (6) and (11) may be obtained from measurements of B.

Titration of Weak Base with Strong Acid

If the base MOH is incompletely dissociated, equa- tion (7) must be replaced by

where K, = [ I f+] [OH-]/ IMOH]. Now

and

Y B - yH - [OH-] + [Ht1 ( 2 4 ) [MOH] = Cnr - lM+l = - - - Y + Y

where Cnr is the total concentration of base in the solu- tion. If

y H A Y + Y ) >> [OH-] - [Ht1 (25 )

and

( Y B - yH)I (Y + >> [OH-] - lH+l ( 2 6 )

then from equations (22), (23), and (24)

Thus, in the alkaline range, a functionZ

T = y/lHt1m = yIODH = K d y . - y ) / K , ~ x ( 2 8 )

can be plotted against y to give a straight line such that T = 0 when y = ye. A similar function can be defined for use with measurements of E. In the acid range, equation (20) is again valid, and y, can be located using plots of *' or Q' against y. Even when the plot in the alkaline region is nonlinear owing to the presence of carbonate in the base, or to the fact that conditions (25) and (26) are not fulfilled, a precise value of ye may be obtained using only the measurements obtained after the end point.

Advantages of Gran's Method

Simplicity of Measurement. Potentiometric readings can be taken after regular increments in u throughout the whole titration; unlike differential plots of d E / d u or

dpH/dv against u, Grau plots do not require large numbers of readings, corresponding to very small changes in u, in the region of the equivalence point.

Simplicity of Calculation. The calculations are quick and easy. One point may be computed and plotted in the one or two minutes which elapse between the addition of base and the taking of a steady potentio- metric reading (see the table).

Versatility. The method may be used when only part of the pH range is accessible to measurement, e.g., when the acid contains metal ions which hydrolyze in the region pH-2.6 (see Fig. 3). It would be very difficult to determine the concentration of strong acid in a solution of this type by conventional methods. Gcan's method has the further advantage that the presence of carbonate in the alkali can readily be detected.

The application of the method to other types of po- tentiometric data (e.g., those obtained in titrations of polybasic acids and in redox, complex formation, and precipitation reactions) is discussed in Gran's original paper (11).

-

Precision. The end points obtained by a linear Gran extranolation are much more ~recise than those ob- tain& by the differential meihod, especially if the titration curve is not symmetrical.

The value of v, obtained from a Gran plot may, if required, be refined to give an even more precise value. If the plot of #(u) or +(v) is linear, then E,' = 0 and YE = 1, whence from equations (1) and (2)

If either H or B is accurately known, values of Eo' may be calculated from (29) using several different values of u, within the small range permitted by the Gran ex- trapolation. The most precise value of u, is obhined by trial and error as that which gives the most nearly constant value of Eol. A value of y, may be refined similarly.

Literature Cited

( 1 ) LINGANE, J . J., "Electroanalytieal Chemistry,'' 2nd ed., Intmcience Publishers, New Yark, 1958.

( 2 ) KOLTHOFF, I. M., AND ELVING, P. .I., Editors, "Treatise an Analytical Chemistry, Part I, Vol. I, Interscience Publishers, New York, 1959.

( 3 ) BELCHER, R., AND NUTPEN, A. J., "Quantitative Inorgitnic Analysis," 2nd ed., Butterworths, London, 1960.

( 4 ) EWING, G. W., "Instrumental Methods of Chemical Analvsis." McGraw-Hill Book Co.. New York. 1960. . .

(5) LAITINEN, H. A,, "Chemical Analysis," ~e~rak- ill Book Co., New York, 1960.

( 6 ) VOOEL, A. I., "Quantitative Inorganic Analysis," 3rd ed., Longmans, London, 1961.

( 7 ) STROUTS, C. R. N., WILSON, H. N., AND PARRY-JONES. R. T., Editors, "Chemical Analysis," Val. 11. Oxford University Press, 1962.

(8) MEITES, L. Editor, "Handbook of Analytical Chemistry," MrGraw-Hill Book Co.. New York. 1963.

(9) S ~ o o o , D. A,, AND WEET, D. M., "Fundamentals of An- slyticsl Chemistry," Holt, Rinehart, and Kinstcn, New York 1Qfi.Z - ..~. ,

(10 ) BUTLER, J . N., "Ionic Eqnilibriom," Addison-Wesley Pub- lishing Co., Reading, Mass., 1964.

( 1 1 ) GHAN, G., Analyst, 77, 661 (1952). ( 1 2 ) Rossor~r, F. J . C., AND ROSSOTTI, H. S., J. Phys. Chem., 68,

3773 (1964).