TEMPERATURE VARIATION OF IONIZATION CONSTANTS I N AQUEOUS SOLUTIONS A. W. WALDE Department of Chemistry, Iowa State College, Ames, Iowa Received July 23, 1934
According to Harned and Embree (5), the temperature variation of the ionization constant can, as a first approximation, be represented by a quadratic equation of the type log K
- log K ,
=
-p(t
- el2
(1)
where K is the ionization constant, K , is the maximum value of the ionization constant, p is a general constant and has a value of 5.0 X deg.-2, t is the Centigrade temperature, and 0 is the temperature a t which K is equal to K,. For purposes of theoretical significance to be used later in connection with van’t Hoff’s isochor and the temperature variation of the heat of reaction, the temperature may be expressed in degrees Absolute without changing the value of p . Equation 1then becomes log K
- log K ,
= -P(T
-
(2)
e1)2
where T is the absolute temperature and is the absolute temperature a t which K is equal to K,. Thus, log K may be expressed by a quadratic equation log K = (log K,
- p e ; ) + 2 p e 1 ~-
p
~
2
(3)
in which K , and are constants characteristic of each substance, and the constant p is general. Likewise, equation 1 becomes log K = (log K ,
- pe2) + zpet
- pt2
(4)
By rearranging terms, equations 3 and 4 become, respectively, log K
+ pT2 = log K , - PO: + 2p&T
(5)
and log K
+ pt2 = log K , 477
- pe2
+ zpet
(6)
478
A. W. WALDE
It is the object of this paper to show that p is not a characteristic constant except under certain assumptions. The theoretical significance of p and its relation to d2log K/dT2will also be pointed out. By taking the first derivative of log K with respect to temperature in equation 1: d-log - K dT
-
- 2p(t - e)
(7)
Likewise, from equation 2: d log K -dT
- 2p(T - e,)
Rut from van’t Hoff’s isochor,
d-log - K dT
-
AH 2.303RT2
where AH is the heat of reaction, or in this case, the molal heat of ionization. By taking the second derivative of equation 1, d2 log K = d T2
- 21,
Likewise, by differentiating equation 9 : log K --dT2
d(AH)
d2
2AH
- 2.303RT2 d T - 2.303RT3
(1 1)
but
where AC, is the change in specific heat upon ionization. Therefore d2 AC, - 2AH --log K dT2 - 2.303RT2 2.303RT3
(13)
If the variation of log K with temperature is not originally expressed as a quadratic equation, and if AH and AC, are not calculated from this quadratic equation, the validity of the assumption, namely, that p is a general constant, may be tested by combining equations 10 and 13:
- 2p =
ACP - 2AH 2.303RT2 2.303RT3
and dividing by 2,
-
AC, = 4.606RT2
-
AH 2.303RT3
479
TEMPERATURE VARIATION O F IONIZATION CONSTANTS
Equation 15 may be tested in two ways. Since AH = 0 somewhere at or near the temperatures under consideration, one may determine the value of - p at A H = 0. This value will be the value of - p at log K, because d log K AH ___ = - and a t K,, A H = 0. Equation 15 then becomes dT RT2 4.606RT2 =
-
Table 1 shows the values of - p at K , found in the literature. It shows as was assumed by that - p does not have a constant value of 5.0 X Harned and Embree (5). In order to determine if - p remained constant for any particular acid through a range of temperatures, equation 15 was applied to the series of TABLE 1 The value of - p at K,,,
4.806RTZ
ACp X 10s
--p=
I
Forniic acid (4). ................................ Chloroacetic acid (9). .......................... Acetic acid (2). ................................ Propionic acid ( 3 ) , . ............................ Glycine (KA) (S)............................... Alanine ( K A ) (7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phosphoric acid (second hydrogen) (6). . . . . . . . . . Sulfuric acid (second hydrogen) (1) . . . . . . . . . . . . .
-41.0 -48.9 -33.7 -37.1 -45.0 -23.2 -41.0 -74.5
298 269 295 293 328 321 316 269
5.06 7.38 4.23 4.72 4.57 2.46 4.50 11.25
acids in table 1. The constants used and the results calculated for these acids are given in table 2. Table 2 shows that p is not a constant, characteristic of any particular acid, unless the variation of log K with temperature is originally expressed as a quadratic equation. It was observed that p increased with temperature except where log K had been expressed as a quadratic function. Figure 1 shows that equation 13 will have a point of inflection at some temperature above that of the experiment. This fact cannot be true if log K is expressed as a general quadratic equation, as illustrated by lines I1 and 111 of figure 1. At the point of inflection,
‘ ZK = 0 and equation 13 becomes dT2 ACP
2.303RT2
-
2A H 2.303RTa
which, upon simplification, becomes AC, =
2AH
-
T
TABLE 2 Variation i n AH and ACD with temperature
Propionic acid 273 278 283 288 293 298 303 308 313 318 323 328 333
737 562 384 203 19 - 168 -358 -551 -746 -945 -1147 1351 - 1559
273 283 291 298 305 313
149 -593 -914 -1170 1402 -1639
283 288 293 298 303 308 313 318
1562 1439 1305 1159 1000 829 646 449
-
-
-
-34.7 -35.3 -35.9 -36.5 -37.1 -37.7 -38.3 -38.9 -39.4 -40.0 -40.6 -41.2 -41.8
-5.090 -4.993 -4.900 -4.811 -4.724 -4.652 -4.561 -4.483 -4.397 -4.324 -4.254 -4.187 -4.121
0.792 0.572 0.370 0.186 0.017 -0.139 -0.282 -0.412 -0.532 -0.643 -0.744 -0.837 -0.923
5.88 5.57 5.27 5.00 4.74 4.51 4.28 4.07 3.87 3.68 3.51 3.35 3.20
2.162 1.590 1.048 0.535 0.048 -0.415 -0.853 -1.270 -1.665 -2.043 -2.404 -2.746 -3.074
-46.83 -42.04 -38.21 -34.86 -31.51 -27.68
-6.869 -5.738 -4.933 -4.301 -3.703 -3.089
-0.160 -0.572 -0.811 -0.967 -1.080 -1.169
6.71 5.17 4.12 3.33 2.62 1.92
-0,368 -1.619 -2.360 -2.821 -3.295 -3.658
-23.4 -25.8 -28.2 -30.6 -33.0 -35.4 -37.8 -40.2
-3.194 -3.400 -3.591 -3.776 -3.930 -4.079 -4.218 -4.346
1.507 1.317 1.134 0.958 0 : 786 0.623 0.461 0.305
4.70 4.72 4.73 4.73 4.72 4.70 4.70 4.65
4.264 3.793 3.324 2,860 2.382 1.911 1.442 0.971
0.767 0.562 0.375 0.204 0.050 -0.093
5.48 5.16 4.86 4.58 4.32 4.09
2.095 1.562 1.062 0.588 0.145 -0.276
Acetic acid 273 278 283 288 293 298
714 552 389 223 57 -112
-32.15 -32.50 -32.85 -33.20 -33.55 -33.91
-4.716 -4.597 -4.484 -4,376 -4.272 -4.184
* AC, was calculated from Owen's equation, d(AH)/dt = 18.6 - 4%. It was observed that log K was expressed as a quadratic function. Calculation of AH and AC, from a quadratic equation in t must necessarily produce a constant value of p by this method, because the coefficient of the square term is p . Values from log KB for glycine were not computed because log RB had been expressed aa a quadratic function of temperature. 480
TABLE 2-Concluded
Acetic acid-Concluded
-282 -455
303 308 313 318 323 328 333
-628 -804 982 - 1161 - 1342
-34.26 -34.61 -34.96 -35.31 -35.66 -36.02 -36.37
273 278 283 288 293 298 303 308 313 318 323 328 333
931 755 573 384 189 - 13 -221 -436 -657 -884 -1118 -1358 - 1605
-34.6 -35.9 -37.1 -38.4 -39.7 -41.0 -42.3 -43.6 -44.8 -46.1 -47.4 -48.7 -50.0
-
'
-4.080 -3.988 -3.901 -3.817 -3.737 -3.660 . -3.586
-5.075 -5.078 -5.064 -5.061 -5.055 -5.059 -5.037 -5.024 -4.999 -4.988 -4.967 -4.949 -4.929
-0.222 -0.342 -0.448 -0.547 -0.637 -0.736 -0.795
3.86 3.65 3.45 3.27 3.10 2.92 2.79
-0.672 -1.049 -1.402 -1.738 -2.058 -2.360 -2.646
1.000 0.768 0.553 0.352 0.164 -0.011 -0.174 -0.328 -0.469 -0.601 -0.725 -0.841 -0.950
6.08 5.85 5.62 5.41 5.22 5.05 4.86 4.70 4.53 4.39 4.24 4.11 3.98
2.731 2.136 1.564 1.012 0.481 -0.032 -0.526 -1.005 -1.466 -1.911 -2.343 -2.760 -3.165
10.99 IO.62 10.26 9.94 9.68 9.38 9.04 8.84 8.60 8.38 8.18 8.00 7.81
-0.428 -1.508 -2.550 -3.561 -4.435 -5.501 -6.411 -7.311 -8.179 -9.029 -9.858 10.668 -11,458
5.86 5.22 4.60 4.00 3.39 2.80
2,325 1.777 1.281 0.851 0.480 0.171
Sulfuric acid (second hydrogen) 273 278 283 288 293 298 303 308 313 318 323 328 333
-146 -533 -934 -1351 1782 -2229 -2692 -3172 -3665 -4176 -4704 -5249 -5811
-
-76.0 -78.9 -81.8 -84.8 -87.9 -90.9 -94.1 -97.3 -100.6 -103.7 -107.2 -110.7 -114.1
-11.148 -11.161 -11.165 -11.177 -11.173 -11.216 -11.205 -11.213 -11.226 -11.210 -11.233 -11.249 -11.249
-0.157 -0.542 -0.901 -1.237 -1.514 -1.842 -2.165 -2.374 -2.613 -2.833 -3.052 -3.255 -3.441
-
dl-Alanine ( K A ) ~ 293 298 303 308 313 318
913 720 538 369 215 79
-39.8 -37.5 -35.1 -32.3 -29.0
-25.4
-5.068 -4.627 -4.180 -3.722 -3,236 -2.746
0.794 0,595 0.423 0.276 0.153 0.054
t A mistake in AC9 was observed in the original data. $ ACp was calculated by approximate methods from A H .
48 1
482
0
A. W. WALDE
IO
20
30 40 TEMPERATURE O C .
50
6C
FIG.1. VARIATIONOF SECOND DERIVATIVE OF IONIZATION CONSTANTWITH TEMPERATURE
TEMPERATURE VARIATION OF IONIZATION CONSTANTS
483
FIQ.2. VARIATIONOF FIRSTDERIVATIVE OF IONIZATION CONSTANTWITH T ~ M P E R A T U R E
484
A. W. WALDE
A tangent to any point on a curve in figure 1 then represents
3d3 log K dT3 '
which can be determined by differentiating equation 13:
d3-log K d T3
- 4AC, 2.303RT3
'
6AH 2.303RT4
'
1 d(ACP) 2.303RT2 dT
(19)
This equation likewise expresses the rate at which AH/T is approaching
Ac,.
Another method of testing whether or not log K is a quadratic function
of temperature is to plot log against temperature. Straight lines dT having the same slope should be obtained. The expression d log K/dT may be obtained by combining equations 8 and 9 :
AH = -2p(T 2.303R T2
-
01)
=
d log K
dT
AH obtained in table 2 have been plotted in figure 2; the 2.303RT2 values do not fall on straight lines, showing that d log K/dT is not a linear function of temperature. The slopes of the lines are not the same for all acids. Glycine is found to plot as a straight line. This fact must be true, however, since examination of the original article (8) shows that AH and AC, apparently had been calculated under the assumption that log K was a quadratic function of temperature. The data for
'
CONCLUSION
1. Log K is not a quadratic function of the temperature. . 2. The theoretical significance of p and its relation to d log K/dT and d2 log K/dT2 have been pointed out. 3. The significance of the third derivative has been pointed out for the first time. REFEREKCES
(1) (2) (3) (4) (5) (6) (7) (8) (9)
HAMER,W. J.: J. Am. Chem SOC.66,860 (1934). HARNED, H. S., AND EHLERS, R. W.: J. Am. Chem. SOC.66,652 (1933). HARNED, H. S.,AND EHLERS, R. W.: J. Am. Chem. SOC.66,2379 (1933). HARNED, H. S.,AND EMBREE, N. D.: J. Am. Chem. SOC.66, 1042 (1934). HARNED, H. S., AND EMBREE, N. D.: J. Am. Chem. SOC.66, 1050 (1934). NIMS,L. I?.: J. Am. Chem. SOC.66,1946 (1933). NIMS,L. F., AND SMITH,P. K.: J. Biol. Chem. 101,401 (1933). OWEN,B. B . : J. Am. Chem. Soo. 66, 24 (1934). WRIGHT, D. D.: J. Am. Chem. SOC.66, 314 (1934).
