Correction to the Debye-Hückel Limiting Law
Author's Data: Housam Binous
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Acknowledgement :
This problem was presented in the excellent book by S. I. Sandler, Chemical and Engineering Thermodynamics, 3rd Edition, Wiley 1999 (Illustration 9.1-8 page 654).

Off[General::"spell1"]

Activity coefficients of HCl in aqueous hydrochloric acid solutions versus HCl molality at 25°C.

tbl1 = {{0.0005, 0.975}, {0.001, 0.965}, {0.005, 0.928}, {0.01, 
            0.904}, {0.05, 0.83}, {0.1, 0.796}, {0.5, 0.757}, {1.0, 
            0.809}, {3. , 1.316}, {5. , 2.38}, {8. , 5.90}, {10. , 
            10.44}, {12. , 17.25}, {14. , 27.3}, {16. , 42.4}};

tbl2 = 
      Transpose[{Map[Sqrt[#[[1]]] &,  tbl1], 
          Map[Log[#[[2]]] &,  tbl1]}]

plt1 = 
      ListPlot[tbl2, 
        PlotStyle ->  {PointSize[0.02], RGBColor[1, 0, 0]}, 
        DisplayFunction ->  Identity]

Values of the parameters (α, β and a) appearing in the Debye-Hückel limiting law and its corrections.

α = 1.178; a = 4; β = 0.3291;

Corrections to the Debye-Hückel limiting law. We find δ=0.278995 and  βa=1.77879.

\(\(LogGamTheo = 
        Map[\(-α\)\ \@#/\((1 + βa\ \@#)\) + δ\ # &, \(Transpose[tbl1]\)[\([1]\)]];\)\)

sol1 = 
      FindMinimum[
        Sum[(tbl2[[i, 2]] - LogGamTheo[[i]])^2, {i, 1, 
            15}], {δ, 0.5}, {βa, 1.5}]

βa obtained using experimental data is equal to 1.77879. This value is close to 1.3164 obtained from Tabulated values of β and for an average radius of hydration a=4 (Overscript[A, o]).

β a

plt2 = 
      Plot[-α x/(1 + βa x) + δ x^2 /. 
          sol1[[2]], {x, 0, 4}, PlotStyle ->  RGBColor[0, 0, 1], 
        DisplayFunction ->  Identity]

Debye-Hückel limiting law

plt3 = 
      Plot[-α x, {x, 0, 4}, 
        PlotStyle ->  RGBColor[0, 1, 0], 
        DisplayFunction ->  Identity]

Plots of experimental data of activities versus HCl molality and the various theoretical models. We see that the Debye-Hückel limiting law gives accurate prediction of activity coefficients only below 0.01 molal. Significant deviations from this law are observed at higher ionic strengths.

Show[plt1, plt2, plt3, 
      DisplayFunction -> $DisplayFunction]