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Volume 71, 1942
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A Table of Debye-Hückel Functions for Use in Solution Chemistry.

[Read before the Auckland Institute, June 18, 1941; received by the Editor, April 24, 1941; issued separately, September, 1941.]

Since the appearance of the theory of Debye and Hückel (1923), the idea of interionic attraction, postulated by these authors, has been generally accepted as representing one important factor describing the deviation of electrolytic solutions from the laws of ideal or perfect solutions. Debye and Hückel derived an equation for the activity coefficient of an electrolyte which, for an aqueous solution at a temperature of 25°, takes the form:

[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

—log f = 0·5066 z1z2√I/(1+0·3288˚a√I) (1)

[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

where f is the rational activity coefficient, z1 and z2 are the valences of the ions of the electrolyte, I is the total ionic strength and å is a constant, characteristic of the electrolyte, designated as the distance of closest approach of the ions. The total ionic strength, I, is defined as ½ ∑ c1z12, where the summation is taken over all the kinds of ions present in solution and c1 denotes ionic concentration in gram-ions per litre. Thus for 1–1, 1–2, 2–2 and 1–3 electrolytes I = c, 3c, 4c and 6c respectively, c being the electrolyte concentration in mols per litre.

Equation (1) is of great importance in the theory of solutions and leads to a large amount of computation when values of log f are required over a range of c and å values. We have now constructed a table which facilitates these computations. The table is based on the following considerations. Rearranging equation (1) we obtain:

[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

—z1z2/log f=(1·974′/√I)0·649 å (2)

The table at the end of this paper contains, in the two right-hand columns, values of 0·649 å corresponding to a range of å values. The main body of the table contains values of 1·974/√I for a range of I values compiled, like a table of logarithms, so that values can be interpolated corresponding to four significant figures in I. It is important to note that the figures in the difference columns on the right are to be subtracted. The negative logarithm of the activity coefficient,—log f, is obtained for any value of å and c by finding the two values of 1·974/√I and 0·649 å from the table; the two values are added together and the reciprocal obtained from a table of reciprocals. This reciprocal is multiplied by the valence product, z1z2, to give −log f.

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The following examples will illustrate the use of the table as well as the extension to concentrations outside the range listed in the table:—

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Valence Type. Example. Z1Z2. c. I. å 1.974/√I 0.649 å Sum. Reciprocal. —log f.
1–1 KC1 1 0.2987 0.2987 5.0 3.612 3.245 6.857 0.1458 0.1458
1–1 1 0.002987 0.002987 5.0 36.12 3.245 39.365 0.0254 0.0254
1-1 1 0.02987 0.02987 5.0 11.42 3.245 14.665 0.0682 0.0682
1-1 1 0.02987 0.02987 4.0 11.42 2.596 14.016 0.0713 0.0713
1–2 BaCl2 2 0.5 1.5 4.0 1.612 2.596 4.208 0.2377 0.4754
1–3 LaCl3 3 0.00073 0.00438 5.0 29.83 3.245 33.075 0.0302 0.0906
2-2 ZnSO4 4 0.5 2.0 5.0 1.396 3.245 4.641 0.2155 0.8620

Reference.

Debye, P., and Hückel., E., 1923. Physikal. Z., 24, 185.

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Table of Debye-hückel Functions.
I 0 1 2 3 4 5 6 7 8 9 Differences (to be subtracted, not added).
1 2 3 4 5 6 7 8 9
0.10 6.243 6.212 6.181 6.151 6.121 6.092 6.063 6.035 6.007 5.979 3 6 9 12 15 18 21 24 26
.11 5.952 5.923 5.800 5.873 5.847 5.821 5.796 5.771 5.747 5.723 3 5 8 10 13 15 18 20 23
.12 5.699 5.675 5.652 5.629 5.606 5.584 5.561 5.539 5.518 5.496 2 5 7 9 11 14 16 18 20
.13 5.475 5.454 5.434 5.413 5.393 5.373 5.353 5.333 5.314 5.295 2 4 6 8 10 12 14 16 18
.14 5.276 5.257 5.230 5.220 5.202 5.184 5.166 5.149 5.131 5.114 2 4 5 7 9 11 13 14 16
.15 5.097 5.080 5.064 5.047 5.030 5.014 4.998 4.982 4.966 4.951 2 3 5 7 8 10 11 13 15
.16 4.935 4.920 4.905 4.890 4.875 4.860 4.845 4.831 4.816 4.802 1 3 4 6 7 9 10 12 13
.17 4.788 4.774 4.760 4.746 4.733 4.719 4.706 4.692 4.679 4.666 1 3 4 5 7 8 9 11 12
.18 4.653 4.640 4.627 4.615 4.602 4.590 4.377 4.565 4.553 4.541 1 3 4 5 6 8 9 10 11
.19 4.529 4.517 4.505 4.494 4.482 4.470 4.459 4.448 4.436 4.425 1 2 3 5 6 7 8 9 10
.20 4.414 4.403 4.392 4.382 4.371 4.360 4.349 4.339 4.329 4.318 1 2 3 4 5 6 8 9 10
.21 4.308 4.298 4.288 4.277 4.267 4.257 4.248 4.238 4.228 4.218 1 2 3 4 5 6 7 8 9
.22 4.209 4.199 4.190 4.180 4.171 4.162 4.153 4.143 4.134 4.125 1 2 3 4 5 6 7 7 8
.23 4.116 4.107 4.099 4.090 4.081 4.072 4.064 4.055 4.047 4.038 1 2 3 3 4 5 6 7 8
.24 4.030 4.021 4.013 4.005 3.996 3.988 3.980 3.972 3.964 3.956 1 2 2 3 4 5 6 7 7
.25 3.948 3.940 3.932 3.925 3.917 3.009 3.902 3.894 3.886 3.879 1 2 2 3 4 5 5 6 7
.26 3.872 3.804 3.857 3.849 3.842 3.835 3.828 3.820 3.813 3.806 1 1 2 3 4 4 5 6 7
.27 3.799 3.792 3.785 3.778 3.771 3.764 3.758 3.751 3.744 3.737 1 1 2 3 3 4 5 6 6
.28 3.731 3.724 3.717 3.711 3.704 3.698 3.691 3.685 3.678 3.672 1 1 2 3 3 4 5 5 6
.29 3.666 3.660 3.653 3.647 3.641 3.635 3.628 3.622 3.616 3.610 1 1 2 2 3 4 4 5 6
.30 3.604 3.598 3.592 3.586 3.580 3.575 3.569 3.563 3.557 3.551 1 1 2 2 3 4 4 5 5
.31 3.546 3.540 3.534 3.529 3.523 3.517 3.512 3.500 3.501 3.495 1 1 2 2 3 3 4 4 5
.32 3.490 3.484 3.479 3.474 3.468 3.463 3.458 3.452 3.447 3.442 1 1 2 2 3 3 4 4 5
.33 3.436 3.431 3.426 3.421 3.416 3.411 3.406 3.401 3.396 3.391 1 1 2 2 3 3 4 4 5
.34 3.380 3.381 3.376 3.371 3.366 3.361 3.356 3.351 3.346 3.342 0 1 1 2 2 3 3 4 4
å 0.649 å
.35 3.337 3.332 3.327 3.323 3.318 3.313 3.309 3.304 3.299 3.295 0 1 1 2 2 3 3 4 4 2.0 1.298
.36 3.290 3.280 3.281 3.277 3.272 3.268 3.263 3.259 3.254 3.250 0 1 1 2 2 3 3 4 4 2.1 1.363
.37 3.245 3.241 3.237 3.232 3.228 3.224 3.219 3.215 3.211 3.207 0 1 1 2 2 3 3 3 4 2.2 1.428
.38 3.202 3.198 3.194 3.190 3.180 3.182 3.177 3.173 3.169 3.165 0 1 1 2 2 2 3 3 4 2.3 1.493
.39 3.161 3.157 3.153 3.149 3.145 3.141 3.137 3.133 3.129 3.125 0 1 1 2 2 2 3 3 4 2.4 1.558
2.5 1.623
.40 3.121 3.117 3.114 3.110 3.106 3.102 3.098 3.094 3.091 3.087 0 1 1 1 2 2 3 3 3 2.6 1.688
.41 3.083 3.079 3.076 3.072 3.068 3.064 3.061 3.057 3.053 3.050 0 1 1 1 2 2 2 3 3 2.7 1.752
.42 3.046 3.043 3.039 3.035 3.032 3.028 3.025 3.021 3.017 3.014 0 1 1 1 2 2 2 3 3 2.8 1.817
.43 3.011 3.007 3.003 3.000 2.997 2.993 2.990 2.980 2.983 2.979 0 1 1 1 2 2 2 3 3 2.9 1.882
.44 2.976 2.973 2.969 2.966 2.963 2.959 2.956 2.953 2.949 2.946 0 1 1 1 2 2 2 3 3 3.0 1.947
3.1 2.012
.45 2.943 2.940 2.930 2.933 2.930 2.927 2.923 2.920 2.917 2.914 0 1 1 1 2 2 2 3 3 3.2 2.077
.46 2.911 2.907 2.904 2.901 2.898 2.895 2.892 2.889 2.886 2.883 0 1 1 1 2 2 2 2 3 3.3 2.142
.47 2.880 2.877 2.873 2.870 2.867 2.864 2.861 2.858 2.855 2.852 0 1 1 1 2 2 2 2 3 3.4 2.207
.48 2.849 2.846 2.844 2.841 2.838 2.835 2.832 2.829 2.826 2.823 0 1 1 1 1 2 2 2 3 3.5 2.272
.49 2.820 2.817 2.814 2.812 2.809 2.806 2.803 2.800 2.797 2.795 0 1 1 1 1 2 2 2 3 3.6 2.337
3.7 2.402
.50 2.792 2.789 2.786 2.783 2.781 2.778 2.775 2.772 2.770 2.767 0 1 1 1 1 2 2 2 2 3.8 2.466
.51 2.764 2.762 2.759 2.756 2.754 2.751 2.748 2.746 2.743 2.740 0 1 1 1 1 2 2 2 2 3.9 2.531
.52 2.738 2.735 2.732 2.730 2.727 2.725 2.722 2.719 2.717 2.714 0 1 1 1 1 2 2 2 2 4.0 2.596
.53 2.712 2.709 2.707 2.704 2.701 2.699 2.696 2.694 2.691 2.689 0 1 1 1 1 2 2 2 2 4.1 2.661
.54 2.686 2.684 2.681 2.679 2.677 2.674 2.672 2.669 2.667 2.664 0 0 1 1 1 1 2 2 2 4.2 2.726
4.3 2.791
.55 2.662 2.659 2.657 2.655 2.652 2.650 2.647 2.645 2.643 2.640 0 0 1 1 1 1 2 2 2 4.4 2.856
.50 2.638 2.636 2.633 2.631 2.629 2.626 2.624 2.622 2.619 2.617 0 0 1 1 1 1 2 2 2 4.5 2.921
.57 2.615 2.613 2.610 2.608 2.606 2.603 2.601 2.599 2.597 2.594 0 0 1 1 1 1 2 2 2 4.6 2.986
.58 2.592 2.590 2.588 2.585 2.583 2.581 2.579 2.577 2.574 2.572 0 0 1 1 1 1 2 2 2 4.7 3.051
.59 2.570 2.568 2.566 2.564 2.561 2.559 2.557 2.555 2.553 2.551 0 0 1 1 1 1 1 2 2 4.8 3.116
4.9 3.180
5.0 3.245
0.6 2.549 2.528 2.507 2.487 2.468 2.449 2.430 2.412 2.394 2.377 2 4 6 8 10 11 13 15 17 5.1 3.310
0.7 2.359 2.343 2.327 2.311 2.295 2.280 2.265 2.250 2.235 2.221 2 3 5 6 8 9 11 12 14 5.2 3.375
0.8 2.207 2.193 2.180 2.167 2.154 2.141 2.129 2.116 2.104 2.093 1 3 4 5 6 8 9 10 12 5.3 3.440
0.9 2.081 2.069 2.058 2.047 2.036 2.025 2.015 2.004 1.994 1.984 1 2 3 4 5 6 8 9 10 5.4 3.505
1.0 1.974 1.964 1.955 1.945 1.936 1.927 1.917 1.908 1.900 1.891 1 2 3 4 5 6 7 7 8 5.5 3.570
5.6 3.635
1.1 1.882 1.874 1.865 1.857 1.849 1.841 1.833 1.825 1.817 1.810 1 2 2 3 4 5 6 6 7 5.7 3.700
1.2 1.802 1.795 1.787 1.780 1.773 1.766 1.759 1.752 1.745 1.738 1 1 2 3 4 4 5 6 6 5.8 3.765
1.3 1.731 1.725 1.718 1.712 1.705 1.699 1.693 1.687 1.681 1.674 1 1 2 3 3 4 4 5 6 5.9 3.830
1.4 1.668 1.663 1.657 1.651 1.645 1.639 1.634 1.628 1.623 1.617 1 1 2 2 3 3 4 5 5 6.0 3.894
1.5 1.612 1.607 1.601 1.596 1.591 1.586 1.581 1.576 1.571 1.566 1 1 2 2 3 3 4 4 5 6.1 3.950
6.2 4.024
6.3 4.089
1.6 1.561 1.556 1.551 1.546 1.542 1.537 1.532 1.528 1.523 1.519 0 1 1 2 2 3 3 4 4 6.4 4.154
1.7 1.514 1.510 1.505 1.501 1.497 1.492 1.488 1.484 1.480 1.476 0 1 1 2 2 3 3 3 4 6.5 4.219
1.8 1.471 1.467 1.463 1.459 1.455 1.451 1.448 1.444 1.440 1.436 0 1 1 2 2 2 3 3 4 6.6 4.284
1.9 1.432 1.428 1.425 1.421 1.417 1.414 1.410 1.407 1.403 1.399 0 1 1 1 2 2 3 3 3 6.7 4.349
2.0 1.396 1.392 1.389 1.386 1.382 1.379 1.375 1.372 1.369 1.366 0 1 1 1 2 2 2 3 3 6.8 4.414
2.1 1.362 1.359 1.356 1.353 1.350 1.346 1.343 1.340 1.337 1.334 0 1 1 1 2 2 2 3 3
6.9 4.479
2.2 1.331 1.328 1.325 1.322 1.319 1.316 1.313 1.310 1.307 1.305 0 1 1 1 1 2 2 2 3 7.0 4.543
2.3 1.302 1.299 1.296 1.293 1.291 1.288 1.285 1.282 1.280 1.277 0 1 1 1 1 2 2 2 2
2.4 1.274 1.272 1.269 1.266 1.264 1.261 1.259 1.256 1.254 1.251 0 1 1 1 1 2 2 2 2
2.5 1.249 1.246 1.244 1.241 1.239 1.236 1.234 1.231 1.229 1.227 0 0 1 1 1 1 2 2 2
2.6 1.224 1.222 1.220 1.217 1.215 1.213 1.210 1.208 1.206 1 204 0 0 1 1 1 1 2 2 2
2.7 1.201 1.199 1.197 1.195 1.193 1.190 1.188 1.186 1.184 1.182 0 0 1 1 1 1 2 2 2
2.8 1.180 1.178 1.176 1.174 1.171 1.169 1.167 1.165 1.163 1.161 0 0 1 1 1 1 1 2 2
2.9 1.159 1.157 1.155 1.153 1.151 1.149 1.147 1.146 1.144 1.142 0 0 1 1 1 1 1 2 2
3 1.140 1.121 1.104 1.087 1.071 1.055 1.040 1.026 1.013 1.000 2 3 5 6 8 10 11 13 14
4 0.987 0.975 0.963 0.952 0.941 0.931 0.921 0.911 0.901 0.892 1 2 3 4 5 6 8 9 10
5 0.882 0.874 0.866 0.858 0.850 0.842 0.834 0.827 0.820 0.813 1 2 2 3 4 5 5 6 7
6 0.806 0.799 0.793 0.787 0.780 0.774 0.768 0.763 0.757 0.752 1 1 2 2 3 4 4 5 5
7 0.746 0.741 0.736 0.731 0.726 0.721 0.716 0.711 0.707 0.702 0 1 1 2 2 3 3 4 4
8 0.698 0.694 0.689 0.685 0.681 0.677 0.673 0.669 0.665 0.662 0 1 1 2 2 2 3 3 4
9 0.658 0.654 0.651 0.647 0.644 0.640 0.637 0.634 0.631 0.627 0 1 1 1 2 2 2 3 3