The acidbase, reductantloxidant, and solidlsolute scales that are described here can he used to advantage as visual representations of equilibrium expressionssuch as (I), (2), and (3).There are other well known graphical representations of these expressions that have been expounded and popularized by Pourhaix,' Sillen,2 Waser3 and others. They used twodimensional diagrams in which two scales are set a t right aneles to each other. There are some advantaees, however, in setting two scales dong the same line. They &e easy to set up, involve onlv a minimum of calculation, and can he used as a starting point in the teaching ofequilibria. They provide an introduction to the interpretation of the more advanced EhpH, stability, and distribution diagrams. Figures 1to 4 establish the principles involved in setting up double scales for acidhase, reductantloxidant, and solid/ solute systems. Figures 5 to 8 provide one-dimensional alternatives to titration curve diagrams. Figure 1is an acidhase scale showing the positions of two acidhase couples. The units on the scale are pH units. The :midpoint of the huffer zone of each couple is at that pH value where pH = pK.. The buffer zone covers two tenfold changes in the [ACID]/[BASE] ratio. According to the relation pH
= pK. - log-
[BASE]
for Base + H+ = ACID
(1)
the [ACID]/[BASE] ratio is 1011 at pH = pK, - 1, hut it is 111 a t pH = pK. and is 1/10 a t pH = pK. 1. The acidhase huffer zone is therefore two pH units wide. At pH values to the left of the pK, value the acid is more dhundant than its hase. At pH values to the right of the pK. value the acid is less abundant than its hase. The acid is therefore written on the left of its pK, value, and the hase is written on the right. For every two couples on the scale the stronger acid belongs to the couple on the left, and the stronger hase belongs to the couple on the right. The spontaneous transfer of protons is always from left to right. The spontaneous reaction, involving a major transfer of protons between the two couples on the scale in Figure 1, occurs when the acid from the left hand couple is added to the base from the right hand couple, i.e.
+
covers two tenfold changes in the [RED]/[OX] ratio. According to the Nernst expression a t 25 "C 0.0592 [RED] E=Eo--log-forOX+ne-=RED (2) n LOXI the [RED]/[OX] ratio is 1011 at E = ED- 0.0592/n, i t is 111 at E = En. and is 1/10 at E = E Q 0.0592ln. The redox huffer zone is therefore 2 X 0.0592ln V wide. For the Fe3+/Fe2+ couole where n = 1the buffer zone is 2 X 0.0592 or 0.1184 V wide. For the Sn4+/Sn2+ couple where n = 2 the huffer zone is 0.0592 V wide. At E values to the left of the dovalue the reductant is more ahundant than its oxidant. At E values to the right of the ED value the reductant is less ahundant than its oxidant. The reductant is therefore written on the left of its E o value, and the oxidant is written on the right. For everv two counles on the scale the stroneer reductant belongs t o t h e coupie on the left, and the stronger oxidant belongs to the couple on the right. The spontaneous reaction, involving a major transfer of electrons between the two couples on the scale in Figure 2, occurs when the reductant from the left hand couple is added to the oxidant from the right hand couple, i.e.
+
-
Sn2++ 2Fe3+ 2Fe2++ Sn4+ Each small number on the upper side of the scale indicates the number of intervals of 0.04921n V from the nearer E n value, and indicates the power of ten in the value of [RED]/ [OX]. The approximate position of the equivalence point is
Figure 1. An acidlbase scale.
-
HF + NHa NHIC+ FEach small number on the uooer side of the scale indicates the number of pH units from tile nearerpK, value, and indicates the Dower often in the valueof IAClDlIlRASEI. . At -pH 6.1 the small number is 3 because, & that b ~the , value of [HF]/[F-] is 11103. The equivalence point (E.P.) is a t the pH value which is obtained when the pure salt is dissolved into water. The approximate position of the equivalence point of a solution of ammonium fluoride is at nH 6.2. where IF-l/IHFl . .. . = [NH4+]/[NH3]= 103. Fieure 2 is a reductantloxidant or redox scale showinn the positions of two redox couples. The intervnls on the scale are intewals of 0.060 V. The rnidwlnt of the huffer zone of each couple is a t that E value where E = E n . Each buffer zone 'Pourbaix, M. J. N., "Thermydynamics of Dilute Aqueous Solutions," Edward Arnold & Co., London, 1949. ?Wen, L. G., "Graphic Presentation of Equilibrium Data", in "Treatise on Analytical Chemistry," Part 1, Volume 2, (Editors: Kolthoff, 1. M., and Elving, P. J.) Interscience, New York, 1959. 3Waser, J., J. CHEM. EDUC., 44,275,1967. 24 I Jwmal of Chemical Education
Figure 2. A reduetantloxidam or redox scale.
Figure 3. A solidlsolute scale
Figwe 4. A generalized scale for two couples xlY and Zlw.
at E = +0.357 V, where [Sn"+]/[Sn2+] = [Fe2+]/[Feq+] = 107.02. -Figure 3 is a solid/solute scale showing the positions of two solid/solute couples, Ag2Cr04/Cr042- and AgBrBr-. The units on the lower side of the scale arepAgt units, where pAg+ = -log[Ag+]. The Ag+ ion is the ion that is common to the two couples. The pAg+ value at the midpoint of the Ag2CrO41 Cr042- huffer zone is obtained by suhstituting pCr042- = 0 into the expression ThepAg+ value at the midpoint of the AgBrBr- huffer zone is ohtained hy suhstituting the value of pBr- = 0 into the expression pKsl = 12.3 = pAg+
+ pBr-
(4)
The two pAg+ values a t the limits of the AgzCr04/Cr042buffer zone are ohtained by suhstitutingpC10~2-= -1, and pCr0d2- = + I , respectively, into expression (3). The two pAg+ values a t the limits of the AgBrIBr- huffer zone are ohtained by suhstitutingpBr- = -1 and pBr- = +I, respectively, into expression (4). The small numbers on the upper side of the scale indicate either pCr042- or pBr- values. At pAg+ = 4.30 on the scale, the small number, 3, is the pCr042- value that is ohtained by suhstituting pAg+ = 4.30 into expression (3). At pAg+ = 9.3
the nBr- value is 3. because toeether these two values satisfv expiession (4). A sb~utioncontaining Ag+ and Br- ions is saturated solution with respect t o AgBr when itspAg+ value is a t the same point on the scale as itspBr-value. A solution is a supersaturated solution with respect to AeBr when its pAg' &r is at a point further 14, the left o f ' i t s p ~ r -value. A snlltti,m that is undersaturated with respwt to ArBr has a pAg+ value that is further to the right o n t h e scali than its pBr- value. Because solid AgBr is favored to form when the pAg+ value shifts to the left, the solid is written over the left hand side of the huffer zone and the solute is written over the rieht hand side. The solid/solute couple that is further to the left on the scale is the couple that has the more soluble solid. In Figure 3 solid Ag2Cr04 is more soluble than solid AgBr. A major transfer of Ag+ ions from one of these solid phases to the other occurs spontaneously when the solid from the left hand couple is added to the solute from the right hand couple, i.e.
a
Ag&rOd,,,
+ 2Br-
-
2AgBrl,,
+ Cr04Z-
The general principle is illustrated from Figure 4. Spontaneous transfer of common ion, proton, or electron is always from left to right. The spontaneous reaction involving a major transfer of common ion, proton, or electron occurs when X is added to W
x+w-Y+Z When Y is added to Z, however, a minor transfer occurs which results in the formation of small amounts of X and W. The same mixture of X W + Y Z is ohtained a t equlihrium, whether X is added to W or Y is added to 2. The greater the distance between the twopK values on the scale the smaller will the quantities of X and W be a t equilibrium.
+
+
Titration Theory
Figure 5. An acidlbase scale for the titration of an acid, HB, with standard NaeOH- solution.
Figure 5 sets out the data involved in the titration of a sample of a weak acid (pK. = 4.7) that is neutralized by standard sodium hydroxide from a buret. The indicator is phenolphthalein whosepK,value is 9.6. The titration reaction is Na'OHHR standard 'sample
Figure 6. A redox scale for the titfation of Fez+ with standard HCrO12- mluti.
Figure 7. A solilsolutescale f a m e timtion of sample CIT(0.10M) w h standard A~+NO~-.
Figure 8. A complexomehic titration of a sample of Ca2+(0.1M) with standard EDTA.
-
Na+B-
+ Hz0
At thc rquivalenc.~point of the titration (K.P.) H sulutinn of Na'H- is ijhtsined u,hose ~.quivalenceapprnximation is IHH] = [OH-\. 'l'l~epH ~1'8.85ur E.P. on the scale is I hr pH oiatl 0.1 Af soluti
