Principles of End Point Detection in Chelometric Titrations Using M eta I I oc hromic Indicators Characterization of End Point Sharpness CHARLES N. REILLEY and R. W. SCHMID University of Norfh Carolina, Chapel Hill, N. C.
b The optimal use of metallochromic indicators in chelometric titrations requires a thorough understanding of the factors involved. Unlike acid-base titrations, the interplay of a large number of equilibria must b e considered. From pM-pH or pY-pH diagrams the sharpness of an end point may be quantitatively characterized by only two end point indices and the end point sharpness under different conditions may be readily compared. Such diagrams are useful in predicting the possibility of selective titrations and the feasibility of activating an indicator response for the titration of one metal ion by the addition of a second. The principles outlined are helpful in the search for new chelometric methods and indicators.
C
titrations occupy an increasing role in the analysis of metal-ion mixtures. To n iden the scope of these applications, new indicators and titrants are being studied and proposed. In the evaluation of metallochromic indicators, two important but different characteristics of the end point must be distinguished and considered. First, a large degree of contrast must exist between the colors of the indicator before and after the end point. Second, this indicator color change must be sharp-i.e., produced by a minimum increment of titrant. In the literature, these two characteristics have often been confused. No quantitative scale has been set up for describing thme two characteristics of end point quality. This article proposes a numerical characterization of the second characteristic, end point sharpness. The relationship betneen such sharpness and the significant parameters involved in a titration are derived to create a method for selecting titration conditions from theoretical considerations. A quantitative characterization of the color contrast of an end point and the theoretical aspects of screening dyes will be the subject of a separate communication HELoJiETRIc
PH
Figure 1.
Titration of magnesium using Eriochrome Black T indicator leff. p M - p H diagram Righf. Corresponding pM-titration curves
The sharpness of the end point in any titration with an indicator may be illustrated by a plot of the color change of the indicator us. the equivalents of titrant. In acid-base titrations, the shape of such a curve depends on only a few equilibrium constants and characterization of the end point sharpness is relatively simple. In chelometric titrations, the number of pertinent equilibria is considerably larger. The end point sharpness depends on the stability constants of the metal chelonates and the metal-indicator complexes, on the concentrations of metal ion and indicator, and on pH, buffer, masking agents, and other cations present in solution. Color change curves have been calculated for a number of highly specified systems (3,4,16). Although the methods of calculation are of general applicability, separate calculations are required for each set of conditions. This procedure is tedious and the interplay of the various factors and equilibria is difficult to perceive. In this paper, phl-pH and pY-pH diagrams present the quantitative significance of the factors involved. From such diagrams two end point indices can be readily obtained whose numerical values allow immediate quantitative characterization of the color change in the titration, independent of the system being considered. Because such diagrams are directly related to color change curves, they allow rapid predic-
tion of optimum titration conditions. Other applications are in predicting the conditions necessary for selective titrations and in testing the feasibility of activating an indicator response for the titration of one metal by the addition of a second activating metal ion. pM-TITRATIONCURVES AND pM-pH DIAGRAMS
As in pH titration of acids and bases, metal ion titrations may be generally represented by pM titration curves. Examples of such curves are shown i n Figure 1, right, for the (ethylenedinitri1o)tetraacetic acid (EDTA) titration of magnesium at various pH values. The initial phI is characterized by point A and the phI after the titration by point C where the solution is 100% overtitrated. The shaded areas indicate the region of the color change of Eriochrome Black T indicator, B corresponding to the point of !joy0color change. Because the indicator forms a 1 t o 1 complex with the metal ion, the fractional color change from 0.1 to 0.9 covers about 2 pM units. If the total pM break is large, the situation is ideal when the 5oojb color change point, B, coincides with the inflection point of the phI curve because, at this point, a maximum color change is obtained by a minimum increment of titrant. The titration curve a t pH 10 is not too far from this ideal case. The curves a t pH 8 and 11 illustrate cases of dragging color changes; the end VOL. 31, NO. 5, MAY 1959
887
b -100
Figure 2. Color change curves various end point indices
for -eo
-eo
point in these cases tends to be early or late (pH 8 or 11). The pertinent points, A , B, and C are usually strongly pH-dependent under practical titration conditions. It is, therefore, advantageous to plot them as a function of p H in a pM-pH diagram (left side of Figure 1). This construction can be readily achieved with the help of the constants involved in the pertinent equilibria.
-40
- 20 -0
-100
-eo CONSTRUCTION
OF
pM-pH DIAGRAMS
Pertinent Equilibria. The construction of pM-pH diagrams requires the knowledge of the equilibrium constants for all the pertinent reactions. At this time these constants are known for only a few common metal-indicator reactions but are abundant for the various chelon-metal reactions (2). The determination of the metalindicator stability constants offers a fruitful field for future investigations. The most basic equilibria are given by M M
+Y + In
F? ~
MY MIn
(1)
(2)
For simplicity, the charges are omitted. The symbols are: hf, metal ion; Y, chelon; In, indicator. The corresponding equilibrium constants are:
- 60 - 40 -20
0 0
-0
T
-100
$
-eo
z a -60
0
- 40 a
3 0
0
-20
@ -0
(3)
-100
(4)
-eo
The concentrations, [Y] or [In], of the completely ionized forms are related to the stoichiometric concentrations [Y]’ or [In]’. These latter quantities correspond to the sum of the concentrations of the unmetallized species in all forms of ionization,
-60
-40 -20
+ [HYI + [HzYl + . . . [In]’ = [In] + [HIn] + [HzIn] + . . . [Yl’
=
VI
The concentration of the free form yU] or [In] ia related to the stoichiometric concentration of the unmetallated species by: [Y] = VI’ ~
ffY
(5)
where
E Q UIVALEN TS I
.96
888
ANALYTICAL CHEMISTRY
I
.97
I
.9e
I .99
I
LOO
I
1.01
I
102
8
1.03
where k , are the acidity constants of the chelating agent or indicator. The effective equilibria reactions may be represented: LI + H.Y MY aH+ (8)
+
1X
+ HJn
e MIn
+ bHf
(9)
whose effective stability constants are then given by:
The other pertinent equilibria and their effect are described in separate sections. Construction of Fundamental pMpH Diagram. A simple pM-pH diagram such as given in Figure l for the titration of magnesium with E D T A using Eriochrome Black T ( C I 203) as indicator, is readily constructed by the use of Equations 11, 12, and 13. The initial pbI, represented by line A is defined :
pMB value decreases with a slope, dpMe/bpH, of 1. At an even lower pH region, where the monohydrogen form (HIn) of the indicator converts into the dihydrogen form (HJn), a second break occurs in the pM-pH diagram, below which the slope, dpMB/dpH, becomes 2. I n Figure 1 this latter region is only partially visible. The breaks occur a t pH values corresponding to the pk, values of the indicator; for Eriochrome Black T, pkl = 6.3 and p k ~= 11.6 and the equation for line B is ~ M =B 7
- log (1 +
[H+]X 1O"J
pM-pH DIAGRAMS AND COLOR CHANGE CURVES
I n a pM-pH diagram, the distance between lines A and C represents, a t any pH, the extent of the total pM break. The occurrence of the color changeearly, a t the equivalence point, or lateis determined by the position of line B between A and C and may be characterized by the pM differences, A1 and A2 :
= log K M I-~ 01x1,
+
[H+] x 10'7.9)
At pH 6.8, lines B and A intersect; above this intersection the metal-indicator complex is stable, whereas below the hydrogen ions outcompete the metal for the indicator and the metal-indicator complex is unstable. The difference between lines B and A (or D)is actually a measure of the stability of the metalindicator complex and Figure 1 shows that the stability of the magnesiumEriochrome Black T complex increases with increasing pH but dissociates again above pH 12. Line C represents the ph4 value a t the point in the titration 100% past the end point (equimolar amounts of metal chelonate and chelon). The general equation for the pM of line C is
A2
+ log CM + log CM (14)
p h f ~- P&IA = log K&.
AI
=
log KAY - log KLln = log K X Y - log K M I n log CWY log a I n (15)
+
AI is a measure of the tightness of the metal-indicator complex as described in the previous section and Az expresses the extent to which the chelon displaces the indicator from the metal-indicator complex. Most important is the fact that AI and Az, which henceforth are termed = -log C M (11) end point indices, define completely the nhere CM is the total concentration of sharpness of an indicator end point, if the metal ion. Equation 11 is valid only indicator concentration is relatively where no precipitating or complexing small. The sharpness of an end point agents are present, or, if present, are not may, therefore, be characterized simply reactive with the metal ion being conby two numbers-3.4 and 3.0, meaning sidered. Thus for the titration of A, = 3.4 and A2 = 3.0. that 0.01M magnesium illustrated in Figure The example of Figure 1 shows that 1, pMA has a constant value of 2 in the early end points are obtained if A1 is pH range up to pH 10.2, the pH a t which small and Az is large (pH 8), but late end formation of magnesium hydroxide compoints are obtained if A, is small and AI mences in a 0.01M solution. At pH valis large (pH 11). The sharpest end ues greater than 10.2, the phlg increases points will be obtained with both A1 n-ith increasing pH with a slope, and Az large. These end point indices, At high p H values ( > l o ) where the pM/pH, of 2, as illustrated by line D. obtained from a pM-pH diagram, allow The hydrolysis constant, log K M ~ ( O B ) ,chelon, EDTA, exists in the free base form ([Y] = V I ' ) and (YY equals .l, a simple but quantitative comparison of = 9.8, as obtained from potential-pH the pMc equals log KMY (= 8.8) as 11the end point sharpness under varying measurements with the third class eleclustrated in Figure 1. As the pH is deconditions of pH, type of indicator and trode system: creased, regions of pH occur where the chelon, buffer, concentration of metal Hg I Hg-EGTA-2, Mg-EGTA-2, Mg +2 various acid forms (HY, HZY) become ion, and related parameters. This relationship is obtained by compredominant and the value of pR4C drops [EGTA = ethylene glycol bis(0-amiof the in a manner analogous to that bining Equations 14 and 15 with 16. noethyl ether)-N,N'-tetraacetic acid]. PMB curve, except that the breaks occur Equation 16, derived by Fortuin, KarThe electrode system works as pMg eleca t different pH values because pkl = sten, and Kies (4) and later by Flaschka trode within certain pH ranges (16). 2.0, pkz = 2.76, pka = 6.16, pka = and Khalafalla (S), represents the titraLine B corresponds to the pM value 10.26. The equation for line C (for tion curve (color change us. equivalents a t which the indicator is present 50% in the magnesium-EDTA complex) is given of chelon added) of a direct chelometric free dye form and 50% in metal-indicatitration using a metallochromic indicaby tor complex form. For a 1 to 1 metal-intor. dicator complex, the pM a t the 50% p?vfc = 8.8 - log (1 [H+] X 1O10.'6 color change point, B, is equal to the [H+I2 x 1016.42 ,..) logarithm of the effective stability constant of the metal-indicator complex: The terms omitted in the above equation are negligible in the pH region above 5. Line C intersects line A a t a pH someThus a t high p H values (>12 )where what below 5; at pH values below this the indicator exists in the free base form intersection the magnesium-EDTA com([In] = [In]') and (YI, equals 1, the plex is unstable, and a t pH values above, where a corresponds t o the equivalents pM8 equals the logarithm of the staof chelon added; y, the fractional color the complex is stable. The distance bebility constant, log KMI, = 7.0 (magtween lines C and A (or D)is actually a change (fraction of free indicator) ; Car, nesium-Eriochrome Black T complex), measure of its stability. EDTA will total metal ion concentration (titrated as illustrated in Figure 1. As the pH is dissolve magnesium hydroxide in the p H plus untitrated) ; CI,, total indicator decreased, a pH region occurs where the region from 10 to 13 and this is readily concentration; [MIn], concentratjon of free indicator (In) converts into the predicted from Figure 1 where line C is the metal indicator complex; K M Y ef, monohydrogen form (HIn) and the above D in this pH region. fective stability constant of metal
+
+
+
VOL. 31, NO. 5, MAY 1959
889
chelonate, KbIn,effective stability constant of metal indicator complex. Thus in terms of the end point indices, the equation is: a =
1-
1-y 1 ( 7 lOa, ) +
Because the product CMKLy, which is equal to 10A1+*Z,has to be a t least lo4 to 106to obtain a usable end point break, the last term of Equation 17 may be neglected without introducing a n error in a of more than 0.1%. Furthermore, in the fourth term, [MIn] may be neglected compared to 10A2[MIn]. Thus, after rearranging the term involving the indicator concentration, a becomes a = l 1-y
1
(7 10a,+) (&) lOA*CM
[l
1
3% -
+ 10A2(1- y)]
(18)
The color change during the course of the titration depends only on the parameters AI, A2 and the ratio CI,/CM. Furthermore, if the ratio of the concentration of indicator to the ion being titrated is or less, the last term of Equation 18 contributes less than 0.1% to a and may be neglected. Under this condition, the only parameters which govern the color change curve are A1 and A2.
(19) Because A, and A2 occur in Equation 19 in additive terms, their influence on the color change curve is easily separated and interpreted. The term containing A1 occurs with a negative sign and thus determines the sharpness of the color change prior to the equivalence point. A2 determines the sharpness of the color change after the equivalence point. Or, dragging before the equivalence point is caused by insufficient stability of the metal-indicator complex and dragging after this point by insufficient displacement power of the chelon ton-ards the metal-indicator complex. By using Equation 19, the color change curves were calculated for various combinations of AI and A2 and the results are given in Figure 2. An important fact is that these color change curves are perfectly general and, therefore, completely independent of the particular system being studied. hccordingly, if the end point indices of any system are known-e.g., from a pM-pH diagram-the color change expected during the titration can be predicted 890
ANALYTICAL CHEMISTRY
from the corresponding curves in Figure I 3-3
n
z. SHARPNESS
OF
INDICATOR END POINT
The sharpness of the indicator end point is readily evaluated from Figure 2, where the color change curves for end point indices, AI and A*, are plotted (assuming the indicator concentration negligible). The larger the A-values, the sharper the end point. Tl'ith both end point indices equal to 4, the fractional color change from 0.2 to 0.8 occurs within less than 0.1% of added titrant, which is about the maximum precision expected from most volumetric procedures. Indices larger than 4 give no practical improvement. On the other hand, for values less than 2, the end point drags appreciably. However, an end point with a low index on one side may be improved with a high index on the other side. Because A1 decreases nith dilution, initially high AI indicates the possibility of titrations in very dilute solutions. The color generally need not undergo a change as wide as from 0.2 to 0.8 to be easily detected visually. The exact range of color Change which the eye detects best is a complex function of the tristimulus values of the two colors involved and varies according to the metal and the indicator. With most unscreened indicators it is convenient for the observer to follow the disappearance of the color of the metal-indicator complex. The disappearance of the last tinge of this color detectable visually occurs within a certain fixed range of the color change. I n using Eriochrome Black T, for example, the color becomes (to the eye) pure blue a t a color change of about 0.70 to 0.85. In these instances it is usually desirable to have an asymmetrical color change curve, especially if the total pM break is small (A1 A2 = 5 to 6 p M units). With a very large total pM break, asymmetry causes little improvement because the end point is already sharp. The significance of skewed color change curves has been too frequently neglected. The sensitivity or steepness of the color change curve, dylda, as a function of the value y of the color change may be obtained by differentiation of Equation 19.
I .2
Figure 3. change
.4
.6
COLOR
CHANGE
Sensitivity
.e
1.0
y
of
color
Negligible indicator concentration Equivalence points (short veriical lines)
80
u w
5
60
S
5
40
0
# 20
0 95
97
96
OB 99 EQUIVALENTS
1.00
101
Figure 4. Influence of indicator concentration on color change curve
A, = 3.5,
A2
= 3.5
+
.2
.4
.6
,8
1.0
Y
Figure 5. Sensitivity of color change with 1 % relative indicator concentration Equivalence point (short vertical lines)
Figure 3 shows the sensitivity of the color change for various asymmetries of color change curve for which the sum of the indices is 6 and the color change value a t the equivalence point is indicated by a short vertical line. Only with the symmetrical curve do the equivalence point and the point of maximum sensitivity coincide. For the asymmetrical curves, a small error occurs whose
value is 0.16 and 0.26% for indices of 2.53.5 and 2-4, respectively. Consider a n indicator where the value of 80% color change (dotted line in Figure 3) is best detected by the operator. In this case, maximum sensitivity occurs with the asymmetrical distribution 2.5 and 3.5 of the end point indices. For a symmetrical color change curve (3.0
I
I I I I
6
7
8
9
1
0
small length of light path, unusually high concentrations of indicator may have been employed. The influence of the indicator concentration on the sensitivity of the color change is illustrated in Figure 5. The top curve shows the sensitivity with negligible indicator concentration for the symmetrical case 3 and 3 (as Figure 3). The bottom family of curves represents the sensitivity for various end point indices in the presence of 1% of indicator (CI, = 0.01CM) and illustrates the leveling of the sensitivity caused by the presence of high indicator concentration. Also the difference between the point of maximum color change and the equivalence point (vertical lines in the diagram) is larger when the amount of indicator is high. The differences are 0.50, 0.47, and 0.43 for the cases 3-3, 2.53.5, and 2-4,respectively. TITRATION CURVES IN PRESENCE OF PRECIPITATE
and 3.0), the sensitivity is much lower than when the indices are 2.0 and 4.0. The maximum absolute-e.g., to a spectrophotometer-sensitivity is obtained with a symmetrical color change (AI = A2) and a t a color change of 0.5. k'ith small end point indicLs (small total phI break) this point is often difficult to detect visually because of the gradual color change. However, when the indicator is perfectly screened, the color change occurs between complementary colors and because the color is gray a t the 50% color change point, the equivalence point is accurately and sensitively observed. Because the stability constants under varying titration and ionic strength conditions differ by several tenths of PI< unit from the literature values, the color change curves derived from A-values are approximations for the color change curves encountered in an actual titration. INFLUENCE OF INDICATOR CONCENTRATION
In very dilute metal ion solutions the concentration of the indicator may actually total as much as a few per cent of the metal ion concentration in order that a sufficient color intensity be obtained. In such cases, several drops of the titrant may actually be required to titrate the metal-indicator complex and the sharpness of the end point is appreciably decreased. For calculation of the color change curve in such cases, the last term of Equation 18 has to be considered. For practical con-
ditions where A 2 2 2, the correction term for the indicator concentration may be simplied to: Y =
b - U &CIn
The (negative!) correction term y is added linearly to the right side of Equation 19 and is a simple function of the ratio of the indicator to metal ion concentration. The correction is illustrated in Figure 4 for the case of AI = A, = 3.5. The influence of an indicator concentration C I = ~ 10-3CM is practically nondetectable in a volumetric procedure which has a precision of 0.1%. The lowest concentration of indicator which may be used in a titration is limited by its visibility, generally about 0.5 to 2 X lO+M in a 250-ml. Erlenmeyer flask. The exact minimum concentration depends on factors such as absorptivity, tristimulus value, eye perception of the operator, light source, length of light path through the titration vessel, and colored impurities. In a 10-ZM solution of sample the indicator correction may be neglected, but becomes increasingly important in more dilute solutions. In the titration of a 10-4M solution of metal ion, an end point type which is very sharp in 10-2M solution drags considerably because CI, is as large as 0.1 CM. Under such circumstances the end point sharpness depends very little on the end point indices. The indicator concentration is also a significant factor in the titration of small volumes because, to compensate for the
From equilibrium considerations, the chelometric titration of a precipitated metal ion is possible in many cases. Although equilibrium is usually reached slowly in a titration which involves a precipitate, and such procedures are generally avoided, it is of interest to discuss briefly the color change curves in such titrations. In alkaline solution, most metal ions precipitate as hydroxides, M
+ zOH
~
M(0H)
-2
(21)
For a solution in equilibrium with R hydroxide precipitate a lon-er limit is imposed on the pM according to P ~ T D= log KY(oH),- ZpK, -t- zpH (23)
In pX-pH diagrams this p b l ~limit is represented by a line D (Figure 1). During the titration the pM value remains constant until all the hydroxide is dissolved, then the pM follom the titration curve which would have been obtained had no hydrolysis taken place. This is illustrated in Figure 1 for the titration of magnesium where line D is determined by the formation of magnesium hydroxide. At pH 11, the initial pM is equal to 3.9 and magnesium hydroxide is precipitated (provided the solution is more concentrated than 10-3.9M). It follows from Figure 1 (right), that a reasonable color change still may be obtained if the titration is carried out slowly enough to ensure equilibrium. At a pH of 12.5, however, the initial phi would be 7 and the magnesium is practically completely masked. The color change of the indicator remains a t a constant value during titration (until the precipitate disappears) as VOL. 31, NO. 5, MAY 1959
891
may be calculated from Equation 4 after inserting the solubility product of the precipitate. Howei er, in practice the dye frequently forms a lake with the precipitate and the total color will be different. EFFECT OF BUFFER AND COMPLEXING AGENTS
Buffers are employed in chelometric titrations, not only to keep the pH constant, but also to prevent hydrolysis and subsequent precipitation of the metal ion. In cases where the buffer does not prevent precipitation, an additional auxiliary complexing agent is added to the solution (addition of tartrate for the titration of lead in ammonia buffer). An ideal buffer should have a complexing power just sufficient to keep the metal ion in solution, thereby necessitating only a minimum decrease in the initial metal ion activity. The complexing properties of buffers may affect the pM values to a considerable extent, not only a t the beginning of the titration, but also a t the color change and past the end point. For a theoretical estimation of the end point sharpness under these circumstances, the pM-pH plot must be constructed not from Equations 11, 12, and 13 but from Equations 27, 30, and 33 which take into account the various effects of the buffer. Effect on pMA. Metal ions in the presence of a complexing species, X, undergo the general reaction
M
+ mX e MX,
(24)
the corresponding equilibrium constant being
The initial concentration of the free (hydrated) metal ion in the presence of the complexing species, X, may be represented by
where PMX is a factor depending on pH, acidity constants of species X, the complex constant KMX,, and concentration of the complexing species. In the absence of complexing agents, j&x is equal to unity. In the presence of a buffer the initial pM is: phfs = -log CM
+ log BMX
(27)
Effect of pMB. I n certain cases, a mixed complex between metal ion, indicator, and buffer may form M In X MInX (28)
+ +
and
a compound of this type is ZnInNHs, with In = Eriochrome Black T (CI 203) 892
ANALYTICAL CHEMISTRY
or Eriochrome Blue Black R (CI 202). In such a case the color change occurs a t a higher pM value
+ log [XI
~ M = B log- KMx,x ff1nax
where ax takes the acid-base properties of the buffer into account and is defined by Equation 7 . Effect of pMc. Mixed complexes between metal ion, chelon, and another complexing ligand have been observed in several cases. RI
+ Y + X e MYX
(31)
The stability of such mixed complexes is larger than the stability of the simple metal chelonate and as a result a larger p M break is obtained. The pM a t lo070 past the end point then becomes pMc
+
+
log K M Y X log X log m y m x (33)
titration may be evaluated by calculating the piLI difference between the appropriate curves of Figure 6. A t pH 10 the following values for the end point indices are obtained: Ammonia Concentration (JI) 0 0.01 0.1 1.0
A, - A2 6.0-7.0
6.8-5.2 4.54.1 1.6-3.1
The data show that excessive concentrations of ammonia cause a strong decrease of the end point sharpness. On the other hand, line D indicates that a t least about 0.1M ammonia should be present to prevent hydroxide precipitation. Although the titration with less ammonia buffer would give higher end point indices, this condition is not desirable because of the presence of precpitate. If no mixed indicator complex would form, 0.1M ammonia would ha1re g'ivena poor end point (Al = 1.6) and 1M ammonia would actually have displaced the zinc from the indicator.
Mixed complexes of this type are freCOMPARISON OF END POINTS WITH VARIOUS quently formed by the EDTA complexes CHELONS AND INDICATORS of mercury(I1) and cobalt(II1)HgYNH,, H ~ Y ~ \ " ~ C H Z C H ~ N H ~The , quality of an end point in the tiHgYPy (Py = pyridine), HgYCl, tration of a given metal ion depends both on the indicator and the chelon. A CoYCNS (10, 15,1'7, 18). Titration of Zinc in Ammonia Buffer. given indicator may yield excellent reThe influence of various concentrasults with one titrant but poor results tions of ammonia on the titration of with another. Yet a second indicator zinc using Eriochrome Blue Black R may give good results with both titrants. By a pM-pH diagram, the properties (CI 202) is shown in Figure 6. of an end point with different chelons In the initial solution zinc is comand indicators may be readily complexed by ammonia and a mixture of pared a priori. In Figure 7 the characnine ammine complexes Zn(NH3)++, teristic lines A , B, C, and D are given for Zn(KHJ2++, Zn(NH3)3++, and Znthe titration of zinc in triethanolamine (NH3)4++ exists. The pM may be obbuffer with the chelons-EDTA, tetratained from Equation 27 with
where K1 (102.2), KI (102.1), and K I (102J)are the stability constants of stepwise complex formation ( I ) , k., is the acidity constant of ammonia, and Cx the total ammonia concentration. The graph shows that the initial PMAincreases greatly with increasing ammonia concentration (lines AI, A2, and AS). The pMB for 50% color change of the indicator Eriochrome Blue Black R, log K z n ~ = a 12.5 (6),in the absence of ammonia is shown as a dotted line in Figure 6. In the presence of ammonia the indicator forms the complex , 16.4 (6),and ZnInNHa, log I f Z n x n ~ E = the corresponding ~ M is B given by Equation 30. The lines BI, B2,and B, represent the 50% color change a t various ammonia concentrations. Because the zinc-EDTA complex does not react with ammonia, line C remains unaltered and Equation 13 applies. The effect of the ammonia concentration on the end point sharpness of this
ethylencpentamine (tetren) [log K Z ~=Y 15.4 (9), pka = 7.9, pkq = 9.1, pk6 = 9.9 (7') ] and triethylenetetramine (trien). H 15.4 ) ~ for calculating The log R z ~ ( o = line D was obtained from potential pH measurements with the third class electrode system : Hg IHgEGTA -2, ZnEGTA-2, Zn+?. The lines of 50% color change ( B ) apply to the indicators Eriochrome Black T (log ICZnIn = 13.0), Eriochrome Blue Black R, and zincon. For a pH of 7.8, the optimal buffer region of triethenolamine, the end point indices read from the diagram are as listed below. From a consideration of the end point indices, the behavior noted in Table I was expected. Experimental experience is in agreement with the end point properties predicted from the diagram. TITRATION OF MIXTURE OF CHELONS
Curves for the titration of mixtures of chelons with a given metal ion titrant,
XI, and the optimum conditions for selective titrations may be readily evaluated from pM-pH diagrams. Because the metal ion is used as a titrant, the pM value would be infinity a t the beginning of the titration and would then drop as the titration proceeds. Consider the titration of a mixture of EDTA and tetren with zinc ions, the course of the titration being followed by a pZn electrode. The principles involved are illustrated by Figure 7. At pH 10, for example, the pM starting a t infinity would drop as the titration proceeds and a pM buffering region would occur a t the pM values in the region of the lines C (pRI 15 to 3 6) for EDTA and tetren w-ith no clear pM break between them. Finally, after the titration of the sum of EDTA and tetren, a distinct pM break of about 7 units would occur. At pH 6, however, EDTA could be titrated selectively in the presence of tetren with a pM break of about 6 units. This theoretical consideration shows that a selective visual titration of EDTA a t p H 6 would be feasible if an indicator of suitable stability with zinc (color change a t pM 9) could be found. pY-pH DIAGRAMS
pM-pH diagrams are useful in evaluating the effect of factors such as pH, buffer, indicator, and chelon on the titration curve of a single metal ion. However, where more than one metal ion is present (such as titration of multicomponent mixtures, substitution titrations, sensitization of an indicator for the titration of one metal ion by addition of a second activating metal ion, and masking of metal ions) information concerning the relative influence of the competitive equilibria may be more clearly elucidated from a pY-pH plot. The reason for this is general. The titration curve in any titration may be represented by a plot of p(titrate) or of p(titrant) us. the titrant volume. For instance, the titration of acetate with acid may be plotted with p(acetate) or p H as the ordinate. Although there are no basic differences or theoretical advantages betm-een the two presentations, a p H plot is commonly made because of its more general applicability and ease of measurement. In the titration of mixtures, the situation is more complicated and the choice of presentation is important. In a plot of p(titrate), the number of breaks appearing on the graph depends on which presentation is employed and also, for the p(titrate) plot, on which titrated species is selccted as the master variable. For example, in the titration of a mixture of acetate ion and ammonia with acid as titrant, one break will be obtained in a p(acetate) plot whereas two will be obtained in a p(ammonia) or p H plot. The full number of breaks i s always obtained
Table
Indicator ErioT
ErioR Zincon
I.
Prediction of End Point Sharpness for Chelometric Titration of Zinc (pH 7.8)
Titrant EDTA tetren
Ai
Az
5.4 5.4
4.8 1.7
trien EDTA tetren trien EDTA
5.4 3.0 3.0 3.0 1.9
-0.8 7.2 4.1 1.5 8,4
tetren
1.9
5.3
trien
1.9
2.7
in a plot of the titrated species which reacts first with the titrant; this is actually the basis of the use of a mercury electrode in chelometric titrations (12). However, a p(titrant) plot is much more general because this type function exhibits the corresponding breaks. This technique is generally adopted in acid and base titrations because, in the titration of mixtures of acids with alkali as titrant, a pH plot is equivalent to a p(titrant) plot because the p(titrant), pOH, is equal to pK, - pH. The titration curve of a single metal ion may be suitably represented in a pM plot; in contrast, for a titration involving more than one metal ion, the breaks and their correct sequence are best represented in a p(titrant)-pYplot. The reverse case, a titration of several chelating agents with one metal ion, is best illustrated by a pR? plot which corresponds in this case to a p(titrant) plot, an example of which has been described. Because the titrant species may exist in several forms-i.e., complexed, hydrated-depending on the solution conditions, the use of the ultimate (or effective) titrant as the master variable is more general and offers the real basis for a unified treatment of the titration curve. By the ultimate titrant is meant that ingredient which actually limits the conversion of titrate species from the form present before to the form present after titration. Because the ultimate titrant species is usually present in a combined form, a measure of its true activity usually is not possible and consequently some practical parameter for its measurement is substituted. CONSTRUCTION
OF
pY-pH DIAGRAMS
Line A cannot, as in the case of pM graphs, be represented by the initial condition of the sample because the pY here is infinity. The system prior to the end point is therefore characterized by the pY a t half-titration: py4 = log K Y Y - log PMX
(34)
Comments and Expected Behavior Very sharp Dragging after end point, titrate until first red tinge appears No end point Sharp; titrate until last blue tinge disappeare Sharp; titrate until last blue tinge disappears Poor; dragging before and after end point Dragging before end point; titrate until pure yellow Dragging before end point, titrate until pure yellow Poor; dragging before and after end point
The pYB a t 50% color change point of the indicator is obtained by combination of Equations 3 and 12. The term [MY] is set equal to Cad because, under practical titration conditions, the color change should occur very near to this stoichiometric point. log
PYB
KMY
+
log CYIn log C M - log KMIn (35)
Line C represents the point in the titration 100% past the end point. Here [Y] is equal to CM and pYc
=
log
oly
- log Cu
(36)
The end point indices, AI and A2, for characterization of the color change sharpness may be obtained in the same way from pY-pH diagrams as from pll-pH plots. AI =
~ MB MA
A, = pMc
= PYA - p l i ~(37) - ~ M B PYB - pYc (38)
In a pY-pH diagram, only line A is independent of the metal ion concentration, lines B and C depending on the metal ion concentration CM. This dependency is just opposite to that existing in pM-pH diagrams. SELECTIVE TITRATIONS
Selectivity in the chelometric titration of a mixture of metal ions is of considerable importance and pY-pH diagrams offer an excellent unified basis for judging the degree of selectivity and for elucidating the influence of the various factors which govern it. One usual method of selective titrations employs the inherent differences between the stability constants of the various metal chelonates. When the difference is sufficient, the metal ion which forms the more stable chelonate will be titrated first, and a pY and pM break occur before the second metal ion reacts to any practical extent with the titrant. Such end points often are easily detected with physical methods such as potentiometry, amperometry, and spectrophotometry. VOL. 31, NO. 5, MAY 1959
e
893
For visual end point indication using metallochromic indicators, certain additional requirements must be fulfilled. First, the stability of the indicator complex of the metal ion to be titrated must be of a rather specified nature. Second, the indicator should not form, under the titration conditions employed, a complex with the other metal ions. These conditions are outlined on the pY diagram, Figure 8, for the titration of zinc in the presence of calcium or magnesium with EDTA using pyridine buffer pH 6. Figure 8, left, shows the characteristic lines A , B , and C as defined in the previous section. The pyridine zinc complex is so weak that line A remains practically unaltered; the curvature of line A near pH 8 is caused by the hydrolysis of the zinc ion. The indicator, 7-(4-sulfo-1-naphthylazo) -8-quinolinol= 6.9, pkl 5-sulfonic acid [log = 3.0, pkz = 7.0 (I??)], forms a complex with zinc above pH 3.5 (the intersection of lines A and B ) but no appreciable complex formation occurs with the alkaline earth ions a t pH 6 ( I S ) . The metalEDTA complexesare stable a t pH values above the intersection points of the lines A and C. The pY titration curve for the titration of a zinc solution (pH 6) in the absence of alkaline earth ions is shown as curve I on the right side of Figure 8 with the color change indicated by the shaded area. The end point indices obtained from the pY-pH diagram are Ai = P Y A ,
Zn
-PYB
= 2.9
and
indicating a satisfactory titration condition according to Figure 2. The end point indices a t other pH Values can be ascertained from Figure 8 and the expected color change curves obtained from Figure 2. For the titration of zinc alone, the sharpest color change curve should be obtained near pH 7 . In the presence of calcium, pH 6 gives better results. The pY titration curve in the presence of an equal amount of calcium (pH 6) is illustrated as line I1 on the right side of Figure 8. The end point break for the titration of the zinc occurs between the lines A for zinc and A for calcium. After the titration of calcium only a very small pY break occurs, between line A for calcium and line C. The end point index AI, in the zinc titration, does not change by the addition of calcium. The second end point index AZ is now obtained as the difference between pYB and pYa,c. and is e,qual to 2.9, considerably lower than the A2 obtained in the absence of calcium ion. The titration of zinc in the presence of calcium ion is nevertheless possible although the end point will not be so sharp (see Figure 2). 894
ANALYTICAL CHEMISTRY
s
4
5
0
7
1
pw
I
2 EPUIVILCNTS
or I
Figure 8. Selective titration of zinc ( 1 O-3M) in presence of equal amount of calcium Pyridine buffer, EDTA titrant, quinolinol-5-sulfonic acid
indicator: 7-(4-sulfo-l -naphthylazo)-8-
From the pY-pH diagram the choice of pH 6 was made to yield a symmetrical color change curve (Al = Az) because the Az) is maximum over-all break (A, constant over the pH range from 4.5 to 7. Line C indicates pY a t loo'% past the end point in a titration of zinc alone while the lines Ca and Mg indicate the pY a t 50% past the end point in a titration in the presence of an equimolar amount of one of these alkaline earth ions. The sharpness of the end point depends on the concentration of the alkaline earth ions present in solution. This effect also may be evaluated from a pY-pH diagram. In the presence of a molar concentration of magnesium ion, tenfold greater than that of zinc, line Mg in Figure 8 is shifted upwards by one pY unit. A hundredfold excess would shift the Mg line up two units and the sharpness of the zinc titration in this case (Al = 2.9, A2 = 2.8) would be comparable to that for zinc in the presence of an equimolar quantity of calcium (A, = 2.9, A2 = 2.9). Concentrations of calcium much greater than that of zinc produce drawn-out color changes beyond the end point-e.g., AZis small.
+
MASKING
The selective titration of a metal ion,
M, in a mixture is possible (disregarding rate effects) only when the effective stabilities of the chelonates of the other metal ions are sufficiently lower than that of M. By adding suitable quantities of appropriate masking agents, the required difference in the effective stability constants, and thus selectivity, can often be achieved. Generally, a given masking agent is effective for the desired purpose only under specified conditions of pH and then only with a given metal indicator and a given titrant. The best choice of these variables has usually been evaluated from many trial-and-error experiments. However, the behavior of a
masking agent under various conditions may be predicted from a theoretical consideration of the equilibria involved and other applications become apparent. This theoretical approach is greatly simplified and the results more easily visualized when the various effective equilibria are plotted in a pY-pH diagram. The optimal set of conditions (such as choice of indicator, titrant, pH, buffer) may then be read from the diagram as well as the advantages and limitations of a masking agent. As an example, consider the application of tetraethylenepentamine (tetren) as a masking agent. Figure 9 shows the pY-pH diagram for titrations with EDTA in the presence of 0.1M tetren. The effect of the masking agent is perfectly analogous to the complex effect and is treated identically. PYA, the half-titration point, is expressed by Equation 34, the factor PMXtaking into account the complexing effect of the masking agent X. Tetren forms 1 to 1 complexes with metal ions (9) and the term log pMx,which must be subtracted from the horizontal lines representing the stability (log KMx)of EDTA complexes of the metal, is equal to log KMXlog cyX log Cx. The diagram indicates that mercury(II), copper, nickel, and zinc are masked in the alkaline region (pH lo), because their curves (dashed lines) drop below pYc, the pY value beyond the end point. Cadmium also is masked, but less effectively. At this pH, lead, calcium, and magnesium may be titrated in the presence of those other metal ions. The diagram also reveals the masking properties of tetren a t other pH values. In acid solution (pH 5) only copper and mercury are masked, and a titration of lead is possible in the presence of copper, mercury, magnesium, and small quantities of calcium. The titration of zinc, nickel, or cadmium in the presence of copper and mercury a t pH 5 is less favorable. Triethylenetetramine (trien) is superior to tetraethylenepentamine as a
+
Ha
Figure 9.
Masking with tetraethylenepentamine
pY-pH diagram for EDTA titration of 1 0 - 3 M metal ions in presence of 0.1M tetraethylenepentamine Effect of masking agent (formation of hydrogen complexes neglected) Color change of magnesium-Eriochrome T complex (Horizontal) values of log KH-EDTA
20
pCu
N$
\, \
\ \
CPb
Zn. C d ~
\&
the latter is masked with tetren, is not possible using Eriochrome Black T. With metalphthalein, however, the titration is possible. The sharpness of an indicator end point in the presence of a masking agent may be evaluated from the pY-pH dia-
--_ ... -
TLe polyamine serves also as a buff& in these cases. The titrations were carried out with 0.01M EDTA. The amount of nickel, copper, and cobalt masked by polyamines is limited in a practical sense because the intense color of the corresponding polyamine complexes obscures the color change of the indicator.
PH
2oL/ 24
22
2 Cd
20
18
16
k 14
12
10
8
4
E
6
IO
PH
Figure 10. tetramine
Masking with triethylene-
pY-pH diagram for titration of 1 O - W metal ions in presence of 0.1 M triethylenetetramine Effect of masking agent (formation of hydrogen complexes taken into account)
6t 2
3
4
5
6
7
PH
Figure 1 1 . Activation of indicator response b y addition of second metal ion
---
EDTA titrations of 1 O-*M metal ion solutions in 0.1M acetate buffer using PAN indicator 611O-4M copper present
masking agent for this purpose (Figure 10). Although a metal ion is masked in respect t o the titrant, it is not necessarily masked in respect to the indicator. In selective titrations where metallochromic indicators are employed, the indicator must combine only with the metal ion to be titrated. This point may be checked ,by comparing the effective stability constants of the metal-indicator complexes in the presence of the masking agent or, as is often better, by a quick experiment.
~ ~ - 13 ~0 681 0 -'M
The existence of mixed complexes NInX (where X is the masking agent) complicates the situation. For example, the red nickel-Eriochrome Black T complex (which does not react with EDTA) turns more violet upon the addition of tetren. This mixed complex also does not react with EDTA and the indicator is blocked by nickel even in the presence of tetren. Thus the titration of magnesium in the presence of nickel, where
The horizontal solid lines in Figures 9 and 10 represent the log KxY values of the metal-EDTA chelonates. The actual pYA values are equal to these log K Z f Y values only when the effects of dissociation of MY by acid, formation of metal-chelonate derivatives, hydrolysis of metal ion, and complexation of metal ion by buffer or masking agents are negligible. For example, the pH effect causes the P Y A to increase and blend into line C at lower pH values, but has no effect on PYA a t p H values above those where PYA and C intersect. Because pY-pH diagrams predict the reaction sequence of metal ions with titrant, they can be utilized for establishing conditions for selective photometric titrations. At pH 10, calcium (also manganese, lead, and magnesium) can be selectively titrated with EDTA in the presence of nickel, mercury, and copper, provided tetren is added as a partial masking agent. After the calcium reaction, nickel (but not copper or mercury) will react. At pH 7 , lead (or manganese) may be selectively titrated in the presence of copper, calcium, mercury, and nickel. The calcium and nickel will cotitrate in the next phase of the titration without interference from copper or mercury. A number of other permutations using different partial masking agents, pH values, and chelon titrants extend the scope of this approach. SENSITIZATION OF METALLOCHROMIC INDICATOR WITH ANOTHER METAL I O N
The titration of a metal ion, &I1, which forms too weak a complex with an indicator may often be achieved, with a VOL. 31, NO. 5, MAY 1959
895
Table II. Masking with Polyamines Ion Condition, Ion Masked, Taken, Found, Titrated Indicator PH Mmole Mmole Mmole Tetraethylenepentamine, approx. 0.1M Mg(I1) Erio T NHa, 10 0 . 1 Hg(I1) 0.1005 0.1008 1.0 0.1003 0.1 Zn(I1) 0.1004 . . 1.0 0.1001 0 . 1 Cu(I1) 0.1008 0 . 1 Cd(11) 0.1001 0 . 1 Cd(I1) 0.1007 0 . 1 Ni(I1) a 0 . 1 Co(I1) a Mg(I1) Metalphthalein NH3,IO 0 . 1 K(I1) 0.1005 0.102b 0.102b 0 . 1 Co( 11) 0.103b Ba(I1) Methyl thymol NaOH, 12 0.1 Cd(I1) 0.0985 0.0983 blue 0 . 1 Zn( 11) 0.0983 1.0 0.98" 0.1 Co(I1) 0.098C Pb(I1) Methyl thymol NH3, 12 tartrate 0.1 Ni(I1) 0.0540 0.0545 blue 0.1 Hg( 11) 0.0952 0.0954 0.1 Zn( 11) 0.0953 Triethylenetetramine, approx. 0.1M Pb(11) Xylenol orange Hexamethylenetetramine, 0 . 1 Hg(I1) 0,0952 0,0950 5 0.05 Cu( 11) 0.0540 0.0545 Zn(I1) Xylenol orange Hexamethylenetetramine, 0 . 1 Hg(I1) 0.1004 0,1008 5
+
b
Indicator is blocked. Poor color change. Indistinct end point.
EDTA
EDTA
EETA
DTPA
PH
Figure 12. Effect of chelon on activated indicator response AI = 2.4,A2 =3.0 A1 = 1.6, As = 4.1 C. Ai = 2.8, A2 = 3.4 D. Ai 2.6, Az = 4.0 A. E.
very good end point, by adding a small amount of a second metal ion M t (or its chelonate). The metal ion MZ is one which by itself gives a good color change in a titration with this indicator. Examples are the titration of heavy metal ions with PAN [l-(%-pyridylazo)-% naphthol] or 1-naphthyl azoxine in the presence of a few drops of copper-EDTA solution and the titration of calcium with Eriochrome Black T after addition of a few per cent of magnesium-EDTA. A basic requirement is that the reaction
896
ANALYTICAL CHEMISTRY
takes place extensively in the direction as written. This reaction may occur in the desired direction even if the chelonate MzY is more stable than MIY because the free energy change of the over-all Reaction 39 involves also the formation of the stable indicator complex, M J n . At the end point the indicator is liberated from MJn as Reaction 39 is reversed. The quality of the end point depends strongly on the exact ease with which the reaction is reversed. This in turn depends on the concentration of the activating metal ion M2 (or its chelonate), on the stability constants involved, and the pH. These factors are discussed
using tlie copper-PAN and and calciumPalatine Fast Blue GGNA (Pr 144) systems as examples. The same principles may be applied for the selection and development of new indicator systems of this type (11). Figure 11 shows the PYA curves corresponding to the half-titration point for a number of metal ions in acetate buffer [log K H ~ ( O A= ~ ) *8.51. The dotted line represents copper. The lines B,, Bz, and B3 designate pYB for 50% color change of the copper-PAN indicator system for three concentrations of coploW3,and respectively). per For PAN, log K C ~ P=A16.0 ~ (20% dioxane), pkl = 1.9 and pkn = 12.2 according to Pease and Williams (8). The end point sharpness in the titration of a 10-?M solution of copper in 0.1M acetate a t different pH values may be evaluated from the curves, Cu, Ba, and C exactly as explained in the previous section. At pH 5 , the end point indices Fj.9 and 3.5 indicate a very sharp color change which occurs in practice. The other metal ions listed in Figure 11 give with PAN no or such weak complexes that the copper-PAN complex is formed preferentially (Reaction 39). The characterization of the end point sharpness for these metal ions using copper-PAS is more complicated than for the titration of copper alone and the determination and use of the end point index A1 involve certain reservations. As a first approximation, AI may be chosen as the differedce between the lower PYA line of the metal ions involved and the indicator line, pYB, corresponding t o the particular concentration of the activating metal ion (copper). For titration of 0.01M lead in the presence of 10-4M copper, A1 is taken as the pY difference between the lines Pb and B1. For an analogous titration of indium, AI is taken as the pY difference between the lines Cu and B1. The error introduced by this choice depends on the relative concentrations and relative effective chelonate stability constants of the titrated and the activating metal ion. A rigorous treatment would be more complicated than desirable for the scope of this publication. However, an end point index AI so determined represents the least sharpness to be expected and the actual end point sharpness may be slightly better. Effect of pH. Figure 11 shows that for the titration of nickel in the presence of lO-4iM copper, sharp end points will be obtained a t p H values of 3.5 (end point indices 3.1 and 4.0) and above. At lower p H values the end point will become sluggish. For zinc, the minimum p H under the same conditions is about 5 (end point indices 1.9 and 5.5). Effect of Concentration of Activating Metal Ion. The concentration of the activating metal ion-with constant
concentration of the indicator-niny have a pronounced effect on thc end point. When the copper concentration is employed of lO-3M (instead of (line B2) for the titration of 10-2M zinc, the end point is considerably improved. At p H 5 this corresponds to a change in the end point indices from 1.9 and 5.5 to 2.9 and 4.5. At pH 4.5, the end point would be sluggish with IO-‘M copper present (1.6 and 5.0); with lO-3M copper, the end point is improved (2.6 and 4.0). A further tenfold increase in the copper concentration (10+M) would not yield any worthwhile improvement (3.6 and 3.0). In the titration of nickel, an increase of the copper concentration from 10-4 to lO-3M, does not improve the end point. These considerations show that the optimum amount of copper-EDTA to be added to the sample depends upon several factors and that the use of a prepared copperPAN complex where the amount of copper is fixed is not recommended. Effect of Chelon. At p H 12 the indicator Palatine Fast Blue GGNA (Prototype KO. 144) gives an exccptionally sharp color change in the titration of calcium Ivith E D T A (sharpness 4.6 and 4.1 in 0.01M solution). For this indicator, log K c n ~ n= 7.5, pkl = 8.9, pkz = 12.9 (6). A pY-pH diagram for this system is given in Figure 12B. This end point is sharper than that of the magnesium-Eriochronie Black T system (Figure 12, A ) (3.4 and 3.0 in 0.01M in solution). It might appear that the calcium-Palatine Fast Blue system offers advantages for end point indication in the titration of barium over the customary titration of barium employing an equimolar amount of magnesiumEDTA and Eriochrome Black T (at pH 10). However, very poor end points are obtained. This paradoxical behavior is readily interpreted using Figure 12, A and B. The sharpness of the barium titration with the magnesium-Eriochrome Black T system is 2.4 and 3.0, and with the calcium-Palatine Fast Blue system 1.6 and 4.1. In the second case the color change drags before the end point although the metal indicator complex employed is stronger. Application of a different chelon represents a frequently overlooked but important parameter which will often greatly improve the end point. Either or both of two conditions must exist. In
the barium titration, the barium complex of the selected chelon should be more stable than that of EDTA (line Ba rises) or the calcium complex of the selected chelon should be n-eaker (lines Ca and Ca-Pal drop). With both effects present, the end point will improve even further. Figure 12, C and D,show the diagrams for the chelons p,pdiamino diethyl ether-N,N‘-tetraacetic acid (EETA) and diethylenetriaminepentaacetic acid (DTPA) where the end point sharpness is 2.8 and 3.4 and 2.6 and 4.0, respectively. A successful titration method for barium (and for sulfate indirectly) was developed from these considerations (6). For DPTA, log K C ~ Y = 10.6, log K B ~= Y 8.6, pk6 = 10.5 (1.4,19). The shaded areas on Figure 12 indicate the region where Palatine Fast Blue turns pink. Because the calcium complex is also pink, a useful visual end point cannot be observed a t high pH although the values for the end point index are good.
At half-titration, the PHAis equal to the pK. of the titrated base PHA = pK.
(40)
The indicator color change occurs a t ~ H =B ~ K I . (41) a t 100% past the end point, the characteristic point is pHc
=
-log C
(42)
where C is the total concentration of the titrated species. Thus: AI = ~ H A pHo = PI
