V O L U M E 2 6 , NO. 5, M A Y 1 9 5 4
871
zene type. A summary of the aromatic type analyses by distillation fractions is shown in Tables VI1 and VIII. Often it is useful t o know not only the concentrations of the various aromatic types but also the concentration of actual aromatic carbon nuclei. This frequently gives a truer measure of aromaticity as far as the reactions and properties of carbon in aromatic ring structure are concerned. By a consideration of the above analyses and the structural formulas given in Tables IV and V it was possible to determine the average concentration of aromatic carbon nuclei for each of the six aromatic compound types found in the heavy cyclic oil and heavy gas oil. These values are shown in column 2 of Table IX as average weight per cent aromatic carbon per molecule. They represent weighted averages based on the molecular weight distribution of each aromatic compound type found in the cyclic oil and gas oil. Column 3 shows the weight per cent of each of the six types of substituted molecules and column 4 the weight per cent of the aromatic carbon nuclei in the total aromatic fractions for the six molecular types.
ACKNOWLEDGMENT
The authors are indebted to H. M. Tenney for the development and application of the infrared method, t o R. C. Cox for the distillations of the gas and cycle oils, and t o R. J. Vick for his valuable assistance on the ultraviolet spectroscopic and data correlations. LITERATURE CITED Clerc, R. J., Kincannon, C B., a n d \Vier, T. P., Jr., AN.IL. C H E Y . 2 2 , 8 6 4 (1950). Fink, D. F., Lewis, R. W., and Weiss, F. T., Ibid., 22, 858 (1950). Forziati, -4.F., Willingham, C. B , LIair, B. J., and Rossini, F. D . , J . Research, AratZ. Bur. Standards, 32, 11 (1944). Furby, i%. W., ANAL.C H E Y . ,2 2 , 8 7 6 (1950). Glasgow, .I.R., J r . , Willingham, C . B., Rossini, F. D., Ind. Eng. Chem.,4 1 , 2 2 9 2 (1949). l I a i r , B. J., Gabariault, d.L., and Rossini, F. D., Ibid.,39, 1672 (1947). Mair, R. J., S x e e t m a n , A. J., a n d Rossini, F. D., Ibid.,41, 2224 (1949).
RECEIVED for review June 10, 1953. Accepted February 17, 1954. Presented before the Division of Petroleum Chemistry a t the 119th Xeeting Of the .kMERrcAx CHEMICAL SOCIETY, Cleveland, Ohio, 1951.
Measurement of the Effect of Dilution upon pH ROGER G. BATES N a t i o n a l Bureau of Standards, Washington,
D. C.
A unit, the “dilution value,” ApH,/z, is proposed to meet the need for a numerical expression of the effect of dilution or concentration of the buffer substances upon the pH of the solution. The dilution value is defined as the increase of pH that results from diluting the solution with an equal volume of pure water. The dilution value and the Van Slyke buffer value, taken together, summarize completely the effectiveness of a solution for pH control. The factors causing a change of pH on dilution are discussed and equations for the dilution values of strong acids, strong bases, and buffer solutions are developed.
T
H E buffer value, defined by Van Slyke in 1922 ( 8 ) , has proved to be a useful unit for the measurement of the buffer capacity. However, the buffer value alone is not sufficient t o characterize the effectiveness of a buffer solution for the control of pH. If a buffer mixture is to regulate the acidity most completely, its p H should not only be little changed by the addition of m a l l amounts of strong acid or base but a190 relatively insensitive to changes in the concmtration of the buffer substance. At present, no generally accepted unit for the measurement of thip “dilution effect” exists. It is proposed that the dilution effect be expressed by a quantity designated ApH112, for which the name dilution value is suggested and which is defined by A P H I / ~= ( ~ H ) c , / 2- (PHI,,
(1)
I t is thus the increase of pH suffered by a solution of initial concentration c, upon dilution with an equal volume of pure waterthat is, to a final concentration of c,/2. The dilution value is positive when the pH increaqes on dilution and negative when it decreases.
theoretical considerations. If v (in liters per mole) is defined as l / c ( c is volume concentration; the difference between the molar and molal scales is small in dilute solutions and is unimportant for the present purpose, where concern is only with values of c less than O.l), one obtains for the interval from ci to ci/2
provided the variation of dpH/dv with concentration is nearly linear in the range cr to c,/2. I n order to relate dpH/dv and the dilution value to concentrations, dissociation constants, and activity coefficients, four important cases will be considered separately. For these derivations, p H will be defined formally by p H = -1Og
I t is possible not only to determine A p H l / ~experimentally according to Equation I but to calculate it approximately from
fcH
(3)
where aH is the hydrogen ion activity and f is the activity coefficient of a univalent ion. I t will be assumed that f can be calculated, when c does not exceed 0.1, by the Debye-Huckel equation in the form log f =
-A
di
1+ B ‘ d i
I/. is the ionic strength. I n Equation 4, B’ is defined by Ba,, where a, is the ion-size parameter, for which an average value of 4 will be taken; -4and B are constants of the Debye-
where
B’
=
Huckel theory (6). Solutions of Strong Acids and Bases. The change of p H on dilution of a solution of a strong acid or strong base with or a i t h out added strong salt is caused by changes in the concentrations of ions and changes in activity coefficients. The former are readily expressed in terms of two functions, g and D ( 7 ) g
C4LCULATION OF DILUTION VALUES
-log
BH
Ca
- cb; D
EGE
Ca
- COR
(5)
I n these expressions, c, and Cb are, respectively, the normal concentrations of strong acid and base. The changes in activity coefficimts are determined primarily by the alteration of the
ANALYTICAL CHEMISTRY
872 ionic strength, which is a function of the total electrolyte concentration. When only strong acids, bases, and salts are present
Differentiation of this equation with respect to v gives
_ dpH _= dv
where h, is the thermodynamic ioii-product constant for water. Hence, a n = 0.5fg
ia
o.8686c2(D c2 - D2
+
+ ( 2 2 + 1) p2 dlog
Equation 16 contains both dD/dv and dlogf/dp. obtained by differentiation of Equation 14:
ti,
there is obtained from Equa-
Equation 10 can be substituted for dlog f/dp. cated substitutions are made
dLH = fc' dv f(c2 - D 2 ) 2c(aH
+
Inawiuch as the numerical value of v is 1 g, dg/dz' = y*; however, the sign of the derivative is negative for positive g (acid solutions) and positive for negative g (alkaline solutions). Similarly, to obtain dlogfldv, 1' is set equal to 1 : p . Hence, from Equation 8
where the first term on the right has a positive sign when g is positive and a negative sign when g is negative. Equation 0 is applicable to s o l ~ t i o n sof strong acids, bases, and salts. Between p H 6 and 8, the terms containing K , exert a controlling influence. When the solution contains no added Ealt and the concentration of acid or Imqe is greater than 10-EJI, hoxever, 4K,/f2 is negligible n ith respect to g2. Furthermore, dlog , f / d p can be estimated by differentiation of Equation 4 with re-pect to ionic strength:
Under these special conditions, the general Equation 9 can be simplified to the folloi\-ing forms, valid a t 25' C.
lfonoacidic Strong Bases.
dv
=
Buffer Solutions Composed of a Weak Acid or Base and Its Strong Salt in Equal Molar Concentrations. The ionic equilibrium in equiniolal buffer solutions of both the arid and baqe types can be formulated in a general manner as follow: HBz+l
Bz
+ H+
(13)
\%here2 is the charge (with its proper sign) borne by the base, B, conjugate to the weak acid, HB It is assumed that the stoichiometric concentrations of both acid and base are c and that Equation 13 and the ionization of water are the only equilibria of concern. The thermodynamic equilibrium contant for 13 nil1 be designated K,, and it ie again convenient to employ the quantity D , defined by
- COH
;
= - (a,
-
2)
From the maw law expression for Equilibrium -13, taking into account the effect of H B and R on the ionization of xater
When the indi-
+ a()=) [0.8686D -
K i t h the appropriate choice of i and p i c ! 1:quation 18 is valid for buffer systeme composed of a weak Imse and its completely dissociated salt as well as for those contaiiiing a weak acid and its salt. Because of the approximate nature of the expression for the activity coefficient, however, it is limited to ionic strengths less than about 0.2. The values of r and p i c for buffer systems of five valence types (Tyithout ntldcd neutral qalt) are given in Table I.
Table I.
\-allies of z and p c for ,icid-Base Buffer Systems
Acid
L!
Base
,
c
Certain simplificatione can often be made in Equation I8 for the particular syst,em under consideration. If the weak acid 1 = - 1; if it is a univalent cation (the conis uncharged, 2 2 1 = 1. For both cases, p/c = 1. jugate base is uncharged), 22 When D is less than c/10, c2 - 0 2 is practically equal to c2. Furt,herniore, either a a or aoH n-ill he negligible except in the immediate vicinity of the neutral point. At 25' C., A = 0.51 and R' = 1.3, inasmuch as an average ion-size parameter of 4 has been chosen for t,he calculations. Buffer Solutions Containing the Acid Salt of a Dibasic Acid. The p H of a c molar solution of the acid salt, MHA, of the dibasic acid €I2,\ i. giyen approximately by the following equation ( 1 , 3):
+
Monobasic Strong Acids. dPH __ dv
Ca
The former
+ d0.25f"g2 + K,
By differentiation with respect to tion 7
D
(16)
~
pH =
+
1
- 2 log KlK2
c + D . + 21 log xXC - D + 2logf -
(19)
Because of its form, the concriitration term in the exact expression, namely (CH,A D)/cH*A,is not sensitive to errors in the estimated CH,A when the latter is considerably larger than D For this reason, i t ie permissible, when D is small, to evaluate CH,4 from the equilibrium constant for the reaction
+
2HA- G H2.4
+ A-
neglecting the small hydrolysis of HA-. The approximate lowel concentration limit for the validity of Equation 19 is c = 0.02 a t D = &0.001; between p H 4.5 and 9.5 the equation holds a t lower concentrations. I n Equation 19, K 1 and KS are the two thermodynamic dissociation constants of HA, D is given by Equation 14, and x is defined by (20)
V O L U M E 2 6 , N O 5, M A Y 1 9 5 4
073 DILUTION
4 , 0(
PHENOMENA
I n Figure 1, the values of the dilution effect, ApH{/*, for solutions of hydrochloric acid and for four buffer solutions -0.2 composed of equal concentrations ( M ) of a monobasic acid and its strong salt (buffer type HB, B - ) are plotted as a function of pH. T h e dilution effects were computed by Equation 2 with the aid of Equations 9 (strong acid) and 18 (buffer solutions). The influence of the strength of the weak acid upon the dilution effect is shown by the position of the dotted lines, 3 5 7 9 w h i c h are l a b e l e d PH with the correspondFigure 1. Dilution Values of Hydrochloric Acid and Weak Acid Buffer Solutions ing values of K a . Buffer type HB, B The figure shows that the change of the DH of solutions of I n the range of validity of Equation 19, D 2is always negligible hydrochloric acid with dilution is much larger than that of buffer with respect to x V . Hence, by differentiation of Equation 19 solutions of comparable concentration. As p H i is approached, with respect to v and combination with Equation 17, one obtains the dilution value of the strong acid falls off to zero. At' high p H values; the dilution effect of weak acid buffer solutions, positive a t low pH, becomes negative and the p H falls with dilution of the solution. It is further evident that an invewion of the concentration-lpH1/ relationship occurs near p H 4 : \)elow this point the pH of the dilute buffer solutions increases more markedly on dilution than does that of the more concentrated solutions. The calculation of d o g f,'dp can be made, as before, by EquaFigure 2 is a similar plot of the dilution effect for solutions tion.10. The ionic strength of a solution of the acid qalt 3IHA of sodium hydroxide and four buffer solutions composed of equal i j given approximately Ir)r(1, 3 ) concentrations (121) of a monoacid base and its strong d t (buffer type HB+, B). T h e values were computed by Equation3 p = c(1 0.5.~) 1.50 (22) 2, 9, and 18. T h e dotted lines relate the dilution effect to the from whirti dissociation constant of the weak base, Kb = K,/K,. The curves of Figures 1 and 2 are evidently of identical shape u = (1 0 . 5 ~ ) 1.5; (23) and are arranged symmetrically with respect to p H 7 and a buffer value of 0. The different disposition of the correFponding curves on the two plots ip caused by the change of sign of the quanThe practical upper limit for z i? satisfactorily represented by 1 (buffer tity g (strong acid and hase, Equation 9) and of 22 potassium hydrogen phthalate ( K l / K z = 290, 2 = appros. 0.12) and the lower limit hy potassium hydrogen tartrate ( K , / K , = systems, Equation 18). The p H of solut'ions of strong bases evidently fall:: rapidly 22, 3' = approx. 0.33). Furthermore, the term 1.5 D/c may vary with dilution. For buffer systems containing weak /)aws of from 0 to a maximum of +O.Oi for the limiting condition, c = charge 0 or + 1 , the inversion of the concentration dependence 0.02 and D = +0.001. Thcl mean value of p/c-namely 1.11 &O.l-wilI therefore apply to the acid salts of all dihacic weak acids for which K,/K? is grttater than 20 and less than 300. From the definition of x (Equation 20) Table 11. \-slues of dlnxldv from c = 0.1 to c = 0.001 HCI
+
+
+
+
+
C
0.1
It is evident from Table 11, Ivhqre the relative magnitudes of dlnxldv and c are given for K I / K ~= 20 and K I / K z = 200, t h a t dlnxldv is so much smaller than c in dilution solutions t h a t its contribution can be ignored in Equation 21. An approximate correction can be made b y means of Table I1 for concentrations u p to 0 1M.
0.05 0.01 0.005 0 001
KI/K? 20 200 20
200 20 200
20
200
20 200
dlnz/dr
dlnz/dtl ( % of c )
-6.1 -8 1 -2.7 -3 5 -0.33 - 0 42 -0.13 -0.16
6.1 8.1
-0.013 -0.017
1.3 1.7
x
loa
5.4
7.0
3.3
4.2 2 6 3.2
ANALYTICAL CHEMISTRY
874
water (an acid and a base) upon the buffer 0.1( equilibria, causing a shift toward neutrality. The second is the effect of dilution on the activity coefficients. At pH values from 5 to 9, the acid-base effect of the water is often neg( ligible, and the change of pH on dilution is the N \ r e s u l t o n l y of t h e changes in activity coI a efficients. In the regions where Equations 4 and 10 are valid, dlog f / d p is neg-0,1( ative. The changes in i n t e r i o n i c forces on dilutionof the solutions OaO025M thus tend to increase the activity coefficient NoOH and to raise the pH of strong bases, weak acid buffer solutions -0.2( (22 1 negative), and 3 5 9 I1 acid salts. On t h e PH other hand, the pH of Figure 2. Dilution Values of Sodium Hydroxide and Weak Base Buffer Solutions s o l u t i o n s of s t r o n g acids and of buffer soluBuffer t y p e HB +,B tions composed of a weak base and its salt (22 1 positive) tends to decrease withdilution. The net result of these two effects is to cause the pH of weak Table 111. Dilution T’alues (in pH Units) for Four Types of Buffer Solutions between pH 4.5 and 9.5 acid buffer systems x i t h pH greater than 7 to pass through a maxBase Acid Acid Acid imum a t some low concentration. In the same tvay, the pH Type Type Salt of buffer solutions composed of a weak base and its salt may C B2TeB HB, B HB-, BMHB 0.059 -0.028 0.028 0 105 pass through a minimum at a pH lees than 7 . A s a result of the 0 1 0,050 0.05 0 096 -0.023 0.023 relative changes in the two terms that make up the numera0.039 0.025 0 082 -0,019 0,019 0 079 0.037 0 02 -0.017 0.011 of Equation 21, the dilution value of solutions of acid salts tor 0.028 -0.013 0.013 0 065 0 01 0.022 0 005 -0.010 0.010 0 047 may pass through a minimum and increase in dilute solutione. 0.016 0 0025 -0.007 0.007 0 040 The dilution value for solutions of potassium hydrogen phthalate reaches its minimum value near a concentration of 0.02-W. Table IV. Observed and Calculated Dilution Values, The calculated dilution effects for solutions of hydrochloric ApHl/z, for Hydrochloric iicid and Buffer Solutions acid and for phthalate and phosphate buffer solutions are comAlolar Concn. of Each Solute Species Solution 0.1 0 05 0.02 0 01 0 005 pared in Table I V with the observed changes of pH on dilution. Hydrochloric acid The “observed” dilution effects for hydrochloric acid were com0.29 0.29 0.29 0.28 0.28 Obsd. puted from the activity coefficients of the acid ( 5 )on the assump0.31 0.31 0.32 Calcd. 0.30 0.30 tion that the activity of hydrogen ions equals the mean activity of hydrochloric acid, The pH values of the phthalate and phosphate buffer solutions were taken from the literature (2, 4 ) .
a
y
\
+
+
gen phosphate
Obsd. Calcd.
0,101 0,105
0.088
0.096
0.073 0.079
0.059 0.065
0,046
0.047
ACKNOWLEDGMEST
The author is indebted to J. E. Ricci for valuable suggestions. of the dilution effect occurs a t about pH 10. The pH of buffer solutions of these types decreases upon dilution when the pH exceeds about 5 but increases a t lower pH values. Both figures show that the dilution effect for weak buffer systems is independent of the strength of the weak acid or base in the region of intermediate pH, changing only n.ith concentration and type of buffer. The dilution values, ApH1/zr of buffer solutions of four types between p H 4.5 and 9.5 are listed in Table I11 for seven values of e , the molar concentration of each component. From the fundamental point of view, there are two effects of dilution of the pH values of buffer solutions, and these may profitably be considered separately. The first is the effect of the
LITER4TURE CITED (1) Bates. R. G., J. A m . Chem. Soc., 70, 1579 (1948). (2) Rates. R. G., and ricree, S.F., J . Research S a t l . Bur. Standards, _?4 . , _272 . l 1144.5) ,----,. (3) Bates, R. G., et al., Ihid., 47, 433 (1951). (4) Hamer, W. J., Pinching, G. D., and Acree, S . F.. I t i d . , 36,47 (1946). (5) Harned, H . S.,and Owen, B. B., “The Physical Chemistry of Electrolvtic Solutions,” 2nd e d . , appendix A, New York, Reinhold Publishing Corp., 1950. (6) Alanov, G. G., Bates, R . G.: Hamer, TV. J., and Acree, S. F., J. Am. Chem. Soc., 6 5 , 1765 (1943). (7) Ricci, J. E., “Hydrogen Ion Concentration,” p. 16, Princeton, N. J., Princeton University Press, 1952. ( 8 ) Van Slyke, D. D., J . Bid. Chem., 52, 525 (1922). RECEIVED for review August 27, 1953. Accepted February 11, 1954.
