CHANGES IN pH AT THE EQUIVALENCE POINT R. K. McALPINE University of Michigan, Ann Arbor, Michigan
IN NEUTRALIZATION titrations the four cases most commonly encountered are probably: (1) the titration of hydrochloric acid with sodium hydroxide; (2) the titration of potassium acid phthalate with sodium hydroxide; (3) the titration of acetio acid with sodium hydroxide; and (4) the titration of sodium carbonate with hydrochloric acid. In using approximately 0.1 N standard solutions it is customary to adjust the amount of the substance being titrated to approximately four milliequivalents so that a volume of 40 to 45 ml. of the reagent will be used. Then if the end point is within one drop of the equivalence point an accuracy of approximately one part in one thousand will be attained. In these titrations the equivalence point may be defined as the pH reached when exactly equivalent amounts of the two reagents have been brought together, while the end point is usually the color change of an indicator which has been added to the initial solution. The equivalence point may be calculated for any given case by use of the ionization constant of the acid and the water constant and by making reasonable assumptions as to the final volume of the solution, the number of milliequivalents being titrated, and the concentration of the standard solution. The necessary change in pH which produces the end point with a given indicator is best determined experimentally by preparing a series of buffered solutions differing from each other by small steps and extending somewhat beyond the range covered by the indicator. The volume should approximate that to be obtained a t the end of the titration and the same amount of indicator should be added to each that will be used in a regular titration. By comparison of the colors in the series it is a simple matter to fmd how much change in pH is necessary to produce a recognizable shift of color, or how closely one can judge the pH from the actual color a t the end point. In actual titrations the three indicators most widely used are phenolphthalein-colorless in acid solution, red in alkaline solutions and starting to show pink a t a pH of 8.4; methyl red modified by methylene blue-reddish purple in acid solution, green in alkaline solution and pale steel blue a t a pH of 5; and methyl orange-pink in acid solution, yellowin alkalme solution, and light orange or salmon colored a t a pH of 4. Since none of these coincide exactly with the pH a t the equivalence point in the titrations being considered, it is a matter of some interest to be assured that the end points in question can, if properly selected, indicate the equivalence points within an accuracy of one part in one thousand for the titrations. This involves the problem of determining the change in pH produced by the last
drop of reagent added in approaching the equivalence point in each of the different cases. This may be done without serious difficulty by first calculating the pH a t the equivalence point and then h d i n g the pH when one drop short of the equivalence point. Case I: The Titration of HC1 with NaOH. In this case the reaction involved is fundamentally: H+
+ OH-
=
H9O
If the two reagents are brought together in exactly equivalent amounts and the reaction reaches equilibrium slightly short of completeness, $he concentrations of H + and OH- will be equal. Since a t room temperature [H+] X [OH-] = 1 X 10-14, this means that a t the equivalence point the concentration of H + will be 1 X lo-' or the pH will be 7. To determine the pH when one drop short of the equivalence point, it may be assumed that the volume is approximately 100 ml., that the reagent being added is 0.1 N NaOH, that the volume of one drop is 0.05 ml., and that practically the whole of the OH-introduced by the last drop is used up by the reaction indicated. On this basis there is the equivalent of 0.05 ml. of 0.1 N HC1 still present in 100 ml. when the titration is one drop short of the equivalence point. In such a solution the concentration of H + would be 0.1 X 0.05/100 = 5 X M, or the pH would be 4.3. On that basis the change in pH brought about by the last drop of 0.1 N NaOH would be 7 - 4.3 or 2.7 pH units. S i c e the color change of methyl red-methylene blue lies within this range it is evident that this indicator may be used when the total volume of reagent added is approximately 40 ml. without introducing an error as great as one part in one thousand. Before leaving this case, however, it is necessaty to confirm the assumption that the reaction of this last drop went nearly 100 per cent to completi?n by comparing the concentration of H + when one drop short, 5 X lo-', with that at the equivalence point, 1 X lo-?. This shows that the extent of reaction of this last drop is 499 parts out of 500. If one wishes to consider the effect of one drop excess of 0.1 N NaOH in this titration, it will be noted that the OH- thus added accumulates in the solution and the total volume is still approximately 100 ml. so the concentration of OH- would be 5 X This gives a pOH of 4.3 or a pH of 9.7 (14-pOH). In other words, one drop excess of 0.1 N NaOH will change the pH from 7 to 9.7, or 2.7 pH units. Since phenolphthalein turns from colorless to light red within this range it can be seen that this indicatior may also be used for this titration.
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DECEMBER. 1948
Case 11: The Titration of KHCsHI04 with NaOH. Potassium acid phthalate is essentially completely ionized into K+ and HCsOa04-, but the latter is a moderately weak acid with an ionization constant of approximately 4 X On this basis the net reaction involved in the titration may be represented by the equation: HCsH&i
+ OH-
=
+
CaH404- HIO
"Completing the square": 12.25 X 10-'0 - 7 X
+ z2 = 7.25 X 10-"'
3 . 5 X 10-5 - z = 4 7 . 2 5 X 10-'O = 2.7 X z = 3.5 X 10-6 - 2.7 X lo-' = 0.8 X 10-a
From this:
If one assumes that the HCs6O4- and OH- are brought together in exactly equivalent amounts then when equilibrium is reached the concentration of HCJLOIand OH- will be equal. If the final volume is taken as 100 ml., and 4 milliequivalents of the KHCsHaOnwere used and the net reaction goes practically to completion, the h a 1 concentration of CsH404-will be approximately 0.04. On this basis the pH a t the equivalencepoint may be calculated.
Substituting 0.04 for [CsH40~=land 1 X 10-14/[H+l for [HC&04-] one obtains the equation:
Rearranging:
[OH-] = 1 X l o w - z = 2 X 10-8 [H+l = 5 X 10-8, pH = 8.3
Thus the last drop of NaOH added changes the pH from 8.3 to 9.0 or 0.7 unit. IIowever, since phenolphthalein starts to develop a pink color a t a pH of 8.4 and is a definite light red a t a pH of 8.8, it is evident that with this indicator the titration may be carried out with the desired degree of accuracy. Further, it is the only one of the three which would be suitable for this titration. Incidentally, it is interesting to calculate how com~ l e t e l vthe last droo of NaOH would enter into reaction in this case. If 5 X lo-' - x = the amount of HCr HnOl-used in reaction by the last drop, and x = 0.8 X lo", this decrease in HCsH404- equals the amount of OH- used up. 4.2 X = amount of OH- reacting out of 5 X added. Thus the extent of the reaction = 4.2/5 or 84 wer cent. Case 111: The itr ration of HCzH302 with NaOH. This case is quite similar to the previous one in that a moderately weak acid is being titrated by a strong base. The net reaction is:
In calculating the pH when one drop short of the HCn1408 OH- = CzHaOn- HIO equivalence point it will be noted that the last drop introduces a concentration of OH- of 5 X M into the solution, but it cannot be assumed that practically On the basis of the same general assumptions as were all of this enters into the reaction given since a signifi- made in the earlier case, a t the equivalence point the cant fraction must accumulate in the solution to give a concentration of C2H302-will be 0.04, and the concenh a 1 concentration of OH- of 1 X In this case it trations of HC2HaO2and of OH- will be equal. becomes convenient to call the amount of OH- that accumulates equal to x and the amount which enters into the reaction equal to 5 X lo-' - x. On this basis, Substituting 0.04 for [C2H302-]and 1 X lO-I4/ [H+l for when the reaction is one drop short of the equivalence [HC2H301]the equation becomes: point [CsH40,=]E 0.04, [HC&04-] = 1 X (5 X 10-5 - x) (amount a t the equivalence point plus the amount which reacted when the last drop of NaOH was added), and [OH-] = 1 X 10W5- x (the amount a t the equivalence point minus the amount which ac- From this: cumulated as the last drop was added). This permits [H+IP= 4.5 X 10-18 setting up the equations to represent conditions when [H+] = 2.1 X lo-* pH = 8.67 one drop short of the equivalence point For the case of one drop short of the equivalence point, it may again he assumed that the last drop of '0.1 N NaOH introduces a concentration of OH- of 5 X From these equations two expressions for H + may he that x = the amount of this which accumulates, and 5 X 10-5 - x = the amount that is used in the reacobtained and set equal to each other: tion. Thus a t this point [C2H302-] = 0.04. [OH-] = (4.7 X 10-6 - x), and [HC2HaO2]= 4.7 X (5 X 10-6 - x) or 5.47 X 10" - x. From the ionization constant of HC2H302and from the water constant rearranging:
+
+
+
+
JOURNAL OF CHEMICAL EDUCATION
696
two expressions for the concentration of H + may be obtained and placed equal:
rearranging :
But [H+] = [HCOs-] and [H2COz] = 0.02. Therefore:
This leads to the equation: 2.5709 X 10-10 - 5.94 X 10-92
+ xP = 2.2 X 10-1'
[H+] = 4 3 X lo-' X 0.02 = 7.7 X 10-6
This may be changed to: (2.97X
- z)l = 6.47 X
lo-'@
- z = 2.54 X
lo-&
2.97 X
z
=
0.43 X 10-6
From this: [OH-] = 4.7 X
figure is used in calculations involving equilibrium constants. Since one assumes that exactly equivalent amounts of H + and GO3- are brought together in the titration, at the equivalence point any slight incompleteness of reaction will leave [H+] = [HCOS-I. Thus from the ionization constant of H2COa one may calculate the pH at the equivalence point.
- 4.3 X
lo-' = 0.4 X 10-1
pH = 4.11
To calculate the pH when one drop of 0.1 N HCl less than an equivalent amount has been added it may be noted that the one drop by itself in 100 ml. would increase the concentration of H + by 5 X However only a fraction of the H + accumulates as such, the rest entering into the reaction. If x = the actual accumulation of H + from the. last drop the concentration of H+ when one drop short will be 7.7 X - x, and the concentration of HCOa- a t this point will be (5 x 10-5 - X) or 12.1 x 1 0 4 - X. 7.7 x Using these values in the ionization constant of HCO,-:
+
From these figures it appears that the- last drop of 0.1 N NaOH in reaching the equivalence point in such a titration changes the pH from 7.6 to 8.67, or 1.07 units. Since phenolphthalein shows a definite pink color a t a pH of 8.5 it is evident that if the titration is carried out with this as the end point the error will be well within the limit set. So far as the extent to which the last drop of 0.1 N OH- enters into reaction is concerned, this may be and calculated easily. The value of x is 0.43 X thus the amount of OH- used up would be 5 X 0.43 x 10-5 or 4.57 x 10-5.
On this basis, when one drop short of the equivalence point the concentration of H+ would be 7.7 X 2.06 X or 5.64 X and the pH = 4.25. From these calcnlations it appears that the last drop of 0.1 N HC1 would change the pH from 4.25 to 4.11 or only 0.14 unit. Further, of the last drop added only 2.94/5 or 59.8 per cent enters into the reaction. Thus 91.4 per cent of the last drop is used in convertThese figures show that the last drop added in aping HC2H30zto CaH302-. proaching the equivalence point produces an unusually Case IV: The Titratwn of NmCOa with HC1. I n this small change in pH, while the equivalence point itself is case the salt of a weak acid which hydrolyzes to give a slightly above the neutral point of the indicator, basic solution is being treated with a strong acid. The methyl orange (at neutral point pH = 4). If one acover-all reaction may be written as follows: tually titrates to the neutral point with methyl orange one will be slightly past the equivalence point. Calculations based on the assumption that the concentration However this reaction takes place in two stages and the of H2C08 remains 0.02 show that approximately 4/5 first one is practically complete when little more than drop of 0.1 N HCl would be sufficient to reach this neuhalf the fuU amount of HCl has been added so only the tral point. However, it is to be noted that at the equivasecond reaction is involved when approaching the lence point the solution is highly supersaturated with equivalence point. These two reactions are: COzfor the pressure of COz in ordinary air, therefore if much swirling is indulged in, combined with slow tiCOa- f H+ = HCOZ(1) tration, a considerable loss of COZwill occur, so the acHCOaH' = HnCO1 (2) tual concentration of HzC03 at the end of a titration For the over-all reaction the equivalent weight of may be much lower than 0.02. Under these conditions NaC03 is half its molecular weight, therefore, if one themethyl orange end point may involve an overtitration by more nearly 2 drops of the 0.1 N HC1, introducstarts with approximately 4 milliequivalents of N* C03 the concentration of H&03 at the equivalence ing an error approaching two parts per thousand. To demonstrate the changes in color of indicators point (in 100 ml.) will be 0.04 N o r 0.02 M. The latter
-
+
DECEMBER, 1948
produced by small changes in pH it is convenient to use commercially available buffer tablets to make up a series of buffered solutions, each of 100 ml. volume in 250ml. Erlenmeyer flasks, to cover the desired range. For phenolphthalein this might conveniently include four solutions a t pH = 8.4, 8.6, 8.8, and 9.0, respectively. Using a 1 per cent solution of the indicator in alcohol, two drops are added to each of the flasks. At a pH of 8.4 a faint pink may he seen, the color deepening in each successive flask to a bright red a t a pH of 9.0. After observing this series with just two drops of the indicator, it is instructive to add 3 4 drops more to each of the flasks and note the increases in depth of color obtained. This shows that with a one-color indicator such as phenolphthalein the depth of color depends on the amount of indicator used and the volume of the solution as well as the pH. For methyl red plus methylene blue, the indicator solution is prepared by using 1.25 g. of the methyl red and 0.825 g. of methylene blue, dissolving in alcohol and diluting with 90 per cent alcohol to 1 liter. Four buffered solutions are prepared, of pH 4.6,4.8, 5.0, and 5.2, respectively. To each is added 3 drops of the indicator. Thesolution with pH of 4.6 shows a dilute permanganate color, the one with pH 5.2 has a definite green color. The intermediate solutions show a violet color a t pH of 4.8 and a faint steel blue a t pH of 5.0. In the case of methyl orange, the indicator solution is prepared by dissolving 1g. in 1 liter of water. The buffered solutions should give pH's of 3.6, 3.8,4.0, and 4.2 (4.4 might also be included). On treating with 2 drops of the indicator the solutions show small changes of color ranging from an orange (reddish tinge) to a moderate yellow. The distinctions between adjacent solutions are rather slight, but when solutions are compared having a difference of 0.4 pH unit the two colors can he distinguished without dificulty.
SUMMARY
(1) It has been calculated that in the titration of HCI with 0.1 N NaOH, if the volume a t the end point is 100 ml. the change in pH produced by the last drop of reagent is 2.7 units. The extent of neutralization of this last drop is 99.8 per cent. (2) In the similar titration of 4 milliequivalents of potassium acid phthalate the change in pH produced by the last drop of 0.1 N NaOH (in a vol. of 100 ml.) is 0.7 pH unit. In this case the last drop of NaOH would he 84 per cent neutralized. (3) In the titration of 4 milliequivalents of acetic acid the change in pH produced by the last drop of 0.1 N NaOH in 100 ml. volume is 1.07 pH units. The extent of neutralization of this last drop is 91.4 per cent. (4) In the titration of 4 milliequivalents of NazCO, by 0.1 N HC1 the change in pH produced by the last drop of 0.1 N HCl in 100 ml. volume of solution is 0.14 pH unit. The extent of neutralization of the last drop of HC1 is 59 per cent. (5) By noting the colors produced in buffered solutions by the common indicators it may be shown that the change from violet to green in the case of methyl red-methylene blue, or the change from colorless to a moderate pink in the case of phenolphthalein may be taken as a satisfactory end point in the titration of a strong acid with 0.1 N NaOH. The change from colorless to a light red in the case of phenolphthalein may he taken as a satisfactory end point in the titration of either potassium aciQ phthalate or acetic acid with 0.1 N NaOH. The change in pH by the last drop of 0.1 N HC1 used in the titration of NazCOs is too slight to give a satisfactory color change with methyl orange. Therefore the direct titration of NazCOp with HC1 is inherently less accurate than the other titrations which have been discussed.
ERRATA Owing to the author's absence abroad while the article, "A Modification of the Periodic Table," Vol. 24, No. 12, Dec., 1947, was in press, the following corrections were omitted from the proofs. On?. 588, column 2, the equation should resd:
,'Z
=
+ + 2(y + 1) + 1.5 + 1 . 5 ( - l ) J Y ;
on p. 590, column 2, the equations should read as follows:
T y=
'/rl~l
+ '/n
- L/2(-1)V11;
On p. 591, column 2, line 9 should read: to anisotopic groupings, subsidiary tables or medid
