The Journal of the American Chemical Society VOL. 53
JANUARY, 1931
[CONTRIBUTION FROM
THE
No. 1
ANIMALNUTRITION LABORATORY, UNIVERSITY OF ADELAIDE, SOUTH AUSTRALIA]
RECALCULATIONS OF THE FIRST DISSOCIATION CONSTANT OF PHOSPHORIC ACID IN AQUEOUS SOLUTION AT 1 8 O , AND INCIDENTAL DETERMINATION OF THE ACTIVITY COEFFICIENTS OF THE UNDISSOCIATED ACID MOLECULES BY JOSEPH W, H. LUGG RECEIVED JULY 1, 1930
PUBLISHED JANUARY 12, 1931
In 1907 Noyes and Eastman' measured the formula conductivities of phosphoric acid in aqueous solutions of low concentration a t No,and in 1909 Abbott and Bray2 used these data in conjunction with their own determination of the equivalent conductivity at infinite dilution of H + HzP04- ions, in calculations of the first stoichiometric dissociation constant. It appears that no reliable conductivity measurements of dilute phosphoric acid solutions have been made since then. Sherrill and Noyes* in 1926 recalculated the first constant from Noyes and Eastman's data and by way of correcting for inter-ionic electrical effects they made use of the fact noted by MacInnes,' namely, that mobilities of certain monatomic univalent ions (K+, Naf, C1-, etc.), up to about 0.1 N concentration, are independent of the oppositely charged ions associated with them provided these latter are monatomic and univalent. In their recalculations of the original data, Sendroy and HastingsKdid not correct for inter-ionic attraction forces. They calculated the ionic strengths of the solutions from the original phosphoric acid molarities instead of from the equivalent ion concentrations. Onsager'se admirable development of the Debye-Hiickel theory' furnishes means for correcting the measured conductivities of the phosphoric acid solutions for inter-ionic attraction forces. For completely dissociated
+
Noyes and Eastman, Carnegie Inst. Washington Publ. No. 63,1907, p. 262. 31, 729 (1909). Sherrill and Noyes, ibid., 48,1861 (1926). MacInnes, ibid.,43,1217 (1921). Sendroy and Hastings, J. B i d . Chem., 71, 783 (1927). 6 Onsager, Physik. Z., 28,277 (1927). Debye and Hiickel, Physik. Z.,24,305 (1923).
* Abbott and Bray, THISJOURNAL,
2
JOSEPH W. H. LUGG
VOl. 53
uni-univalent electrolytes in aqueous solution a t No,Onsager’s limiting equation is
+ O.159Am) V%
A = Am - (35.7
(1)
where A is the equivalent conductivity a t concentration c and A, is the equivalent conductivity a t infinite dilution. Practically all modern evidence weighs in favor of the contention that strong electrolytes such as NaC1, KC1, LiC1, HC1, KOH,*etc., and many of higher valence type, are completely dissociated in aqueous solution, even up to molar concentration. To conductivity data for the uni-univalent electrolytes, Debye and Hiickel* have applied equations of the form A = A,
-adz+ b2c
(2)
determining the constants a and b for each electrolyte by the method of least squares. It may be concluded from Onsager’s paper that a negative value for b connotes incomplete dissociation. The a values agree excellently with the Onsager theoretical coefficients, so the assumption that an equation of the form of (2) would apply to solutions containing only H + and H2P04- ions necessarily in equal concentration is well founded. With the value of A, a t 18’ known, a could be calculated from Equation 1, and b may be assigned the rough value f40, from a consideration of the positive Debye-Hiickel coefficients.
+
Determination of A m (H+ HzP04-)at 18’ The quantity p which occurs frequently in this paper is not quite the same as the “ionic strength” as defined by Lewis and Randall,Qbut is defined in the identity, p = l / Z z 1 ( Czz), where C is the concentration of an ion in moles per liter, and z is its valence. “p” defined in this way has greater significance in the Onsager and Debye-Hiickel equations. In arriving a t a value for A, (H+ HzP04-), the conductivity data of Abbott and Bray2 for the salt NaH2P04 were employed, but were first corrected for dissociation of the HzP04- ions. The corrections were considerably smaller than those calculated by Abbott and Bray. A generalized equation for the second stoichiometric dissociation constant of phosphoric acid and rough values for the mobilities of the ions H+, H2P04and HP04-, used in calculating the corrections to be applied to A NaH2PO4 as measured, appear below
+
pKz
7.16
-
2 f i
1
+ 1.5 Jp
(3)
A,H+ = 314; A,HzPOd- = 27; I ~ , ~ / ~ H P O = ~53. - Decreases in these values with increase in p are calculable from the Onsager equations. Equation 3 has been deduced by the author from his own data for the Debye and Hiickel, Physik. Z., 24, 305 (1923). Lewis and Randall, “Thermodynamics and the Free Energy of Chemical Substances,” McGraw-Hill Book Co., Inc., New York, 1923. 8
9
Jan., 1931
FIRST DISSOCIATION CONSTANT OF PHOSPHORIC ACID
3
second constantlo and from the data in W. Mansfield Clark’s 1928 edition of “The Determination of Hydrogen Ions.” From these sources i t is (i. e., pK2 p Y H + ) = 7.16 found that in presence of most salts, 1.5 dp/l a dz)a t Bo,where p signifies negative common logarithm, YH+ is the activity coefficient of H f ion, and a is a constant determined by the ionic environment. Replacing prH+from Equation 12 of this paper, one obtains for ~ K azt Bo, the approximate equation pKz = 7.16 2 dE/(l a 4;). The most likely value for a in NaH2P04 solutions seems to be about 1.5, but the uncertainty will not materially affect the calculations. Where d is the degree of dissociation of the H2P04- ion, calculated with the help of Equation 3, we have
+
+
+
A,,,,.
NaHtPOa = ANaH2POd
- dAH+ - 2dh1/2HP04‘ + dAH2P04-
Table I contains the data of Abbott and Bray for NaH2P04 conductivity; corrections to be applied for secondary dissociation of the H2P04ions; corrected values of ANaH2P04; values of p ; and values of dii. TABLEI EQUIVALENT CONDUCTIVITY OF NaH2P04 AT 18O Molarity of NaHzPO4
0 0.001 .003 .005 ,010 .020
.ox .loo
.200 a
Approx.
4;
0 0.032 .055 .071
.loo ,141 ,229 .316 .447
Molar conductivity (measured), ANaHaPO4
Corrn.
Corr. molar or equivalent conductivity, Awrr. NaHqPOd
71.68 ? 69.23 67.58 65.53 61.43 ? 58.26 54..43 49.74
-3.8 -2.4 -1.8 -1.1 -0.7 - .5 - .3 - .2
70.1” 67.9 66.8 65.8 64.4 61.7 57.8 54.1 49.5
Extrapolated.
Corrected values of A were plotted against corresponding values of
v’E, and the curve drawn was extrapolated from the points for lowest concentrations almost linearly with d,L to the point, fi = 0, Acorr, = 70.1. Slopes of tangents to the curve a t low fi values corresponded very well with the slope of the theoretical limiting straight line, A = 70.1 - (35.7 + 0.159 X 7O.1)fipJ divergences being in the direction expected for a completely dissociated salt. For A,NaCI, the value 108.9 determined by Debye and Hiicke18 from the data of Kohlrausch and Maltby was accepted. A value 379.3 for A,HCl was obtained from the data of Goodwin and Haskelll* by extralo
Not yet published. Goodwin and Haskell, Phys. Rev., 19,369 (1904).
4
VOl. 53
JOSEPH W. H. LUGG
polating to 4; = 0 a curve connecting A with dill just as in the case of NaHzP04. These values of Am for the three electrolytes, NaH2P04, NaCl and HC1, are for 18', and A m (H+ f HzP04-) a t 18' is readily obtained from 379.3 70.1 - 108.9, giving 340.5. Thus the equation one would expect to apply to aqueous solutions a t 18' containing only H + and HzP04- ions is
+
A
=
+
A, -(35.7 f 0.159 X 340.5) 45 40 X 2e, that is A = A, - 127.1 fi 80e
+
(4)
where e is the equivalent ion concentration and A refers to a dilution l/e. Although the second term (80e) is not negligible, it is obviously small compared with the first (127.1 v'z) up to e = 0.04, so the uncertainty attached to the coefficient 80 will be found scarcely to affect the calculations below that value of e. Calculation of the First Stoichiometric Dissociation Constant.-In aqueous solution phosphoric acid dissociates as follows H3P01 eH+
+ H2POa-i
HzPO4-
eHf
+ HPOI';
HPOa'
H+
+ POI'
For any but almost infinitely dilute solutions of the free acid, the magnitudes of the three dissociation constants are such as to allow the production of only H + and HzPOs- ions in any appreciable quantity. OH- ion concentrations, of course, are negligibly small in consequence of the hydrogen ionization of the acid. Thus the conductivity of phosphoric acid solutions down to a molarity of, say, 0.0001 may be considered due entirely to the presence of H + and HzPOd- ions in equal concentration. Hence where A is the molar conductivity at molarity c of a solution of phosphoric acid a t 18', and d is the degree of the first dissociation, we have, neglecting viscosity changes which are small enough at low concentrations A/d = A,(Hf
+ HaP'Od-)
- A dz + Bdc
(5)
where A is 127.1 and B is 80. Transposition, squaring of both sides and subsequent rearrangement lead to
+ (2BA,
(B2~2)d4
- AZC)d8-
(2BAc
- A&)d' -
(2AA,)d
4-A'
= 0
(6)
Certain roots of this biquadratic in d would have to be excluded and it must be recalled that only the positive extraction of 6in Equation 5 is to be used. Defining A,,,,, in the relation A,,,,/A, = d, and substituting in Equation 5 , one obtains AooFr. = A
+ A d .\/z - Bdzc
(7)
and -Acorr. is A corrected for inter-ionic attraction forces. From Equation 7 A,,,, may be obtained by a series of successive approximations and then d may be calculated from the definition. Two such approximations were found sufficient to give values of A,,,, unaltered in the first decimal place by successive approximations, and values of d appearing in
Jan., 1931
FIRST DISSOCIATION CONSTANT OF PHOSPHORIC ACID
5
Table I1 were derived from these instead of from laborious solution of Equation 6. By the law of mass action we have for the first stoichiometric dissociation constant, square brackets signifying molarities
In logarithmic form, where p signifies negative common logarithm, Equation 8 becomes PKi = P[H+l P[HzPO41 - $[Hap041 = 2Pd PC - P(1 - d) (9) Table I1 contains in the order of the columns, the data of Noyes and Eastman, corrections to A, A,,,., d, p , fi,K1 and ~ K I .
+
+
TABLE I1 Original H:P?4 molanty, C
A
Corm.
Am=.
0
0.0002 .0020 .OlOO .0125 .0500
330.8' 1.8 332.6 283.1 4.3 287.4 203 5.9 208.9 191.2 6.0 197.2 122.7 6.0 128.7 .os00 104 6.0 110.0 .loo0 96.5 5.9 102.4
d
I
Am
1 .0000 0.9769 .8441 .6136 .5791 .3780 .3231 .3008
4;
p=e=dc
0
K1 =
0
0.0001954 0.0140 .001688 ,0411 .006136 .0783 .007239 .OS51 .01890 * 1375 .02585 .1608 ,03008 ,1734
d9c
(1 - d )
0 0.00826 .00915 .00974 .00997 .01149 .01234 ,01294
fiK1
2 . 090°
2.083 2.039 2.011 2.001 1.940 1.909 1.888
Extr.
The Constant at Infinite Dilution. Activity Coefficients of H3P04.By definition and the mass law, the first activity dissociation constant of phosphoric acid K: is given in Equation 10 and is an exact constant. ax signifies activity of X, and ax = rx [XI, where TX is the activity coefficient, expresses its relation to the molarity. The chosen standard state in which Yx = 1 (i. e., ax = [XI) for H+, H2POc, and HaPOs, is infinite dilution in each case. Hence replacing ax by YX [XIand expressing in logarithmic form PKi = P W + l PPM"41 - P[&POcl Pya+ P ~ H S P O-~PYHIPO4 Substitution from Equation 9 followed by rearrangement, gives PKi = PK: - PYH+ - P Y ~ P o I prmo4 (11) For variation of the activity coefficient of an ion with p , the DebyeHtickel'* equation has been found to apply admirably up t o p = 0.5 or more
+
+
+
+
l2
Debye-Hiickel, Phyysik. Z., 24, 186,(1923).
6
JOSEPH W. H. LUGG
Vol. 53
where zi is the valence of the ion and aiis the mean limit of approach of ions to ions of kind “i.”13 In place of 3.3 X 107aione may use the coefficientsa for H + and b for HzP04-ions, in solutions containing H + and HzPO4- ions. * Then -fir,+ -PYHlpo4- may be put in the form and provided a and b be of the same order of magnitude the expression may be reduced for practical purposes to l+a4/5
where a now has some mean value. The expression approaches -4ji as V’G +0. Debye and McAu1ayl4 have investigated variation of the activity coefficient of a non-electrolyte in presence of an electrolyte, and‘have deduced a theoretical equation which may be expressed P-fnon-elsot.
=
-.%
(13)
where S is a constant depending upon the mean effective ionic radius, among other things. Experimentally, it has been shown by these authors and by Randall and Failey16 that the variation depends almost entirely upon the concentration of electrolyte and scarcely a t all upon the concentration of non-electrolyte up to moderate concentrations, and is, moreover, in agreement with the general equation. One may therefore replace pHaPO4 in Equation 11 by -SF, where S is a constant. Equation 11 becomes PK1 = PK: - 1 +4 apfi - sg (14)
+
dpKl/d fi = - 1/(1 a dP)z- 2s dF, giving slopes of tangents to the curve of Equation 14 and showing that the slope is - 1 when 1 / F = 0. Values of pK1 appearing in Table I1 were plotted against corresponding values of v5,but practically no weight was placed upon the point corresponding with lowest phosphoric acid concentration, because A, and Acorr. cannot be known exactly and for this point it requires an error of but *O.l in either to effect an error of *0.01 in pK1. A smooth curve was drawn with tangential slopes continually approaching -1 as 1/T; deF = 0 (i.e., &) was found t o be 2.09. The creased to zero.16 pK1 a t d values obtained by Sherrill and Noyes and by Sendroy and Hastings, are 2.08 and 2.11, respectively. 1s The additional “salting out term,” ( - Cg), deduced by Hiickel [Physik.Z., 26, 931 (1925)], where C is small and positive, may well be omitted here since its effects would be inappreciable up to fi = 0.2. A very small increase in the value of ai compensates for the omission up t o much higher values of Debye and McAulay, ibid., 26, 22 (1925). 16 Randall and Failey, Chem. Reviews, 4, 271 (1927). 16 When S is positive, Equation 14 permits such a smooth extrapolation only provided that S > a. The condition is satisfied in this case.
a.
Jan., 1931
c
F I F S T DISSOCIATION CONSTANT O F PHOSPHORIC ACID
(
In Fig. 1 are shi3vn the curve connecting pK1 with fi and passing through the point. f i K ~= 2.09, dF = 0, the limiting straight line pK1 = 2.09 - dp,and :A cr.i.rvepK1 = 2.09 - dP/(l 1.0 4;). The choice of a value 1.0 for a in Equation 14 is suggested from an examination of scattered data for d i ' k e n t electrolytes, and it is considered unlikely to be in error by more than 4-0.5or -0.3. 'From the divergence of the two curves in the figure, S in Equation 14 is readily calculated to be 1.8. An error
+
1.80
0 '
2
1.90
c)
d p., x
% 2.00
2.10
0
0.02 0.04
0.06
0.08 0.10
0.12
0.14
0.16
0.18
0.20
4;. Fig. 1.
of *0.3 in the chosen value for a, would result in an error of about *0.2 in the calculated value of 5'. With the assumed value 1.0 for a, therefore, the equation connecting PK1 for pure phosphoric acid in dilute aqueous with calculated from the H + and HzP04- ions, is solution a t Bo, = 2.09
-
1
+v1.0' v7 - 1 . 8 ~
(15)
making the equation for the activity coefficient of undissociated phosphoric acid in presence of H + and HzP04- ions in aqueous solution a t 18'. PYH~Po= ~ -1.8~
(16)
In conclusion the author wishes to acknowledge his indebtedness to Dr. S. W. Pennycuick of the Adelaide University for reading the manuscript and suggesting alterations, and making the necessary arrangements for publication. Summary 1. From data in the literature it has been found that A, NaHaPOd is 70.1 a t 18', and from this and the values 108.9 and 379.3 for A, NaCl and
8
HERBERT S. HARNED AND GEORGE M. MURPHY
VOl. 53
L H C 1 , respectively, A, (H+ 4-H~POI-)has been calculated to be 340.5 a t that same temperature. 2. Values for the first stoichiometric dissociation constant of phosphoric acid alone in aqueous solution a t 18' have been calculated from the data of Noyes and Eastman. PIC; (i. e., $JKIa t infinite dilution) has been assigned the value 2.09. An equation well fitting the data is @K1= 2.09 dZ/(l 1.0 fi)- 1.8~. 3. An approximate equation for the activity coefiicient of the undissociated fraction in pure dilute aqueous solutions of phosphoric acid at18' is pYHIP0,= - 1 . 8 where ~ ~ P is calculated from th eequivalent ion concentration.
+
[CONTRIBUTION
FROM THE
DEPARTMENT O F CHEMISTRY, YALE UNIVERSITY ]
THE TEMPERATURE COEFFICIENT OF DISSOCIATION OF ACETIC ACID IN POTASSIUM AND SODIUM CHLORIDE SOLUTI0N S BY HERBERTS. HARNED AND GEORGE M. MURPHY RECEIVED JULY 15, 1930
PUBLISHED JANUARY 12, 1931
It has been shown by Harned and Robinson, and Harned and Owen2 that the ionization constant, hydrogen- or hydroxide-ion concentrations, and ionic activity coefficientsof weak acids and bases may be determined from measurements of cells without liquid junction. Their measurements were all made a t 25'. Since it is a matter of considerable importance to study the effect of temperature on the dissociation of a weak electrolyte both in a pure solvent and in salt solutions, the present study was undertaken. To this end accurate measurements of the cells HZ 1 HC1 (0.01), MCl (m)I AgCl Ha I HAC (0.2), MC1 (m) I AgCl
I
Ag, and
I Ag
(1) (11)
at 20, 25 and 30' have been made, in which HAC is acetic acid, and MCl sodium and potassium chlorides. Experimental Procedure and Results The cells employed were of the usual H-type, and were equipped with a three-way stopcock through which the hydrogen entered the cell. By the proper manipulation of this stopcock, the cell could easily be transferred from a thermostat maintained a t one temperature to one at another temperature without danger of the hydrogen electrode coming into contact with air. Equilibrium in the cell was considered established when 1 The present investigation represents part of the thesis presented by George M. Murphy in partial fulfilment of the requirements for the degree of Doctor of Philosophy, Yale University, 1930. 2 Harned and Robinson, THISJOURNAL, 50, 3157 (1928); Harned and Owen, ibid., 52, 5079 (1930).
