Appendix IV

Debye-Hückel Equation and Kielland Table For Ion-Activity Coefficients

Debye-Hückel expression for individual ion-activities:

$log$ $\gamma = \frac{-Az^2\sqrt{I}}{1 + \r{a}B\sqrt{I}}$

Values of the Ion-Size Parameter å for Common Ions Encountered in Natural Water:

å × 10⁸	Ion
2.5	$\ce{NH^+_4}$
3.0	$\ce{K+}$ , $\ce{Cl-}$ , $\ce{NO^-_3}$
3.5	$\ce{OH-}$ , $\ce{HS-}$ , $\ce{MnO^-_4}$ , $\ce{F-}$
4.0	$\ce{SO_4^{2-}}$ , $\ce{PO_4^{3-}}$ , $\ce{HPO_4^{2-}}$
4.0–4.5	$\ce{Na+}$ , $\ce{HCO^-_3}$ , $\ce{H2PO^-_4}$ , $\ce{HSO^-_3}$
4.5	$\ce{CO3^{2-}}$ , $\ce{SO3^{2-}}$
5	$\ce{Sr^{2+}}$ , $\ce{Be^{2+}}$ , $\ce{S^{2-}}$
6	$\ce{Ca^{2+}}$ , $\ce{Fe^{2+}}$ , $\ce{Mn^{2+}}$
8	$\ce{Mg^{2+}}$
9	$\ce{H+}$ , $\ce{Al^{3+}}$ , $\ce{Fe^{3+}}$

Parameters A and B at 1 Bar

Temperature (°C)	A	B (× 10^–8)
0	0.4883	0.3241
5	0.4921	0.3249
10	0.4960	0.3258
15	0.5000	0.3262
20	0.5042	0.3273
25	0.5085	0.3281
30	0.5130	0.3290
35	0.5175	0.3297
40	0.5221	0.3305
50	0.5319	0.3321
60	0.5425	0.3338

Kielland Table of Ion-Activity Coefficients at 25º Celcius Arranged by the Size of Ions (Based on Debye-Hückel Equation)

Charge	*Size, å**	Ions	l = 0.0005	0.001	0.0025	0.005	0.01	0.025	0.05	0.1
1	2.5	$\ce{Rb+}$ , $\ce{Cs+}$ , $\ce{Ag+}$ , $\ce{NH^+_4}$ , $\ce{Tl+}$	0.975	0.964	0.945	0.924	0.898	0.85	0.80	0.75
	3	$\ce{K+}$ , $\ce{Cl-}$ , $\ce{Br-}$ , $\ce{I-}$ , $\ce{CN-}$ , $\ce{NO^-_3}$ , $\ce{NO^-_2}$ , $\ce{OH-}$ , $\ce{F-}$ , $\ce{ClO^-_4}$	0.975	0.964	0.945	0.925	0.899	0.85	0.805	0.755
	4	$\ce{Na+}$ , $\ce{IO^-_3}$ , $\ce{HCO^-_3}$ , $\ce{HSO^-_3}$ , $\ce{H2PO^-_4}$ , $\ce{CLO^-_2}$ , $\ce{C2H3O^-_2}$	0.975	0.964	0.947	0.928	0.902	0.86	0.82	0.775
	6	$\ce{Li+}$ , $\ce{C6H3O^-_2}$	0.975	0.965	0.948	0.929	0.907	0.87	0.835	0.80
	9	$\ce{H+}$	0.975	0.967	0.950	0.933	0.914	0.88	0.86	0.83
2	4.5	$\ce{Pb^{2+}}$ , $\ce{Hg2^{2+}}$ , $\ce{SO4^{2-}}$ , $\ce{CrO4^{2-}}$ , $\ce{CO3^{2-}}$ , $\ce{SO3^{2-}}$ , $\ce{C2O4^{2-}}$ , $\ce{S2O3^{2-}}$ , H citrate^2–	0.903	0.867	0.805	0.742	0.665	0.55	0.455	0.37
	5	$\ce{Sr^{2+}}$ , $\ce{Ba^{2+}}$ , $\ce{Cd^{2+}}$ , $\ce{Hg^{2+}}$ , $\ce{S^{2-}}$ , $\ce{WO4^{2-}}$	0.903	0.868	0.805	0.744	0.67	0.555	0.465	0.38
	6	$\ce{Ca^{2+}}$ , $\ce{Cu^{2+}}$ , $\ce{Zn^{2+}}$ , $\ce{Sn^{2+}}$ , $\ce{Mn^{2+}}$ , $\ce{Fe^{2+}}$ , $\ce{Ni^{2+}}$ , $\ce{Co^{2+}}$ , phthalate^2–	0.905	0.870	0.809	0.749	0.675	0.57	0.485	0.405
	8	$\ce{Mg^{2+}}$ , $\ce{Be^{2+}}$	0.906	0.872	0.813	0.755	0.69	0.595	0.52	0.45
3	4	$\ce{PO4^{3-}}$ , $\ce{PO4^{3-}}$	0.796	0.725	0.612	0.505	0.395	0.25	0.16	0.095
3	9	$\ce{Al^{3+}}$ , $\ce{Fe^{3+}}$ , $\ce{Cr^{3+}}$ , $\ce{Sc^{3+}}$ , $\ce{In^{3+}}$ , and rare earths	0.802	0.738	0.632	0.54	0.445	0.325	0.245	0.18

*Note that these sizes are rounded values for the effective size in water solution and are not the size of the simple ions, unhydrated. For a more detailed discussion, see the original paper from which these values were taken [J. Kielland, J. Amer. Chem. Soc., 59, 1675 (1973)].

References

BERNER, R. A. 1971. Principles of Chemical Sedimentology. McGraw-Hill, New York, 240 pp.

KLOTZ, I. M. 1950. Chemical Thermodynamics. Prentice-Hall, Eaglewood Cliffs, N.J., 369 pp.

MANOV, G. G., R. G. BATES, W. J HAMER, and S. F. ACREE. 1943. Values of the constants in the Debye-Hückel equation for activity coefficients. J. Amer. Chem. Soc., 65, pp. 1765–1767.

HWB

Hydrogeologists Without Borders

Appendix IV