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Surface chemistry equations

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In the forms given in Introduction to Colloid and Surface Chemistry by Duncan J. Shaw, publisher Butterworth-Heinemann. Data entry by Dienkuan Donn. Markup by meditor.

Brownian Motion x = \sqrt{2Dt}, Df = Tk, f = 6aηπ

Spreading Wetting S = \frac{-\Delta {G}_{s}}{A} = {\gamma }_{\mathrm{SG}}-\left({\gamma }_{\mathrm{LG}}+{\gamma }_{\mathrm{SL}}\right), Young's equation 0 = − γS + γSL + πSG + γLGcos(θ)

Adhesional Wetting, Dupre equation WA = \frac{-\Delta {G}_{a}}{a} = γLG + γSG − γSL, Young-Dupre equation Wa = {\gamma }_{\mathrm{LG}}\left(1+\mathrm{cos}(\theta )\right)

Immersional Wetting − ΔGi = γSG − γSL = {\gamma }_{\mathrm{LG}}\left(\mathrm{cos}(\theta )\right), Contact Angles p = \frac{2\gamma \mathrm{cos}(\theta )}{r}, {p}^{\prime } = \frac{2{\gamma }^{\prime }}{r}

Adsorption Isotherms, Langmuir Isotherm q = {K}_{a}{q}_{m}\frac{c}{1+{K}_{a}c} , Freundlich Isotherm q = {K}_{f}\sqrt[n]{c}, where q = \frac{x}{m}

Gouy-Chapman \frac{{d}^{2}\psi }{d{x}^{2}} = \frac{2e{n}_{0}z}{\epsilon }\mathrm{sinh}(\frac{e\psi z}{Tk}), \frac{d\psi }{dx} = \mathrm{sinh}(\frac{e\psi z}{2Tk})\sqrt{\frac{8Tk{n}_{0}}{\epsilon }}, ψ = \frac{2kt}{ez}\mathrm{ln}(\frac{1+\gamma {e}^{-\kappa x}}{1-\gamma {e}^{-\kappa x}}), where γ = \frac{-1+{e}^{\frac{e{\psi }_{0}z}{2Tk}}}{1+{e}^{\frac{e{\psi }_{0}z}{2Tk}}}, and κ = \sqrt{\frac{2{e}^{2}{n}_{0}{z}^{2}}{T\epsilon k}} = \sqrt{\frac{2{N}_{A}c{e}^{2}{z}^{2}}{T\epsilon k}} = \sqrt{\frac{2{F}^{2}c{z}^{2}}{T\epsilon k}}, in which F is coulombs/volts, at 25 C κ = 0.329e+10\sqrt[2]{\frac{c{z}^{2}}{\mathrm{mol}{\mathrm{dm}}^{-3}}}{m}^{-1}.σ0 = \mathrm{sinh}(\frac{e{\psi }_{0}z}{2Tk})\sqrt{8T\epsilon k{n}_{0}}

Debye-Huckel Approximation, When \frac{e{\psi }_{0}z}{2Tk} is far less than 1 then {e}^{\frac{e{\psi }_{0}z}{2Tk}} ~= 1+\frac{e{\psi }_{0}z}{2Tk}, ψ = ψ0e − κx, σ0 = εκψ0

Stern Layer, As above substitute ψ0 with ψd.σ1 = \frac{{\sigma }_{m}}{1+{e}^{\frac{\phi +e{\psi }_{d}z}{Tk}}\left(\frac{{N}_{A}}{{n}_{0}{v}_{m}}\right)}

Capacitances C1 = \frac{{\sigma }_{0}}{{\psi }_{0}-{\psi }_{d}} = \frac{{\epsilon }^{\prime }}{\delta }, C2 = \frac{{\sigma }_{0}}{{\psi }_{d}}, ψd = \frac{{C}_{1}{\psi }_{0}}{{C}_{1}+{C}_{2}}, at low potentials, 25 C C2 = εκ = 2.28\sqrt[2]{\frac{c{z}^{2}}{\mathrm{mol}{\mathrm{dm}}^{-3}}}F{m}^{-2}

Surface Potentials \frac{d\zeta }{d\mathrm{pAg}}(\mathrm{\zeta ->0}) = \frac{d{\psi }_{0}}{d\mathrm{pAg}}\frac{d{\psi }_{d}}{d{\psi }_{0}}(\mathrm{\zeta ->0}) = -59\left(\frac{{C}_{1}}{{C}_{1}+{C}_{2}}\right)\mathrm{mV}

Huckel Equation uE = \frac{{v}_{E}}{E} = \frac{\epsilon \zeta }{\mathrm{1.5}\eta }, Smoluchowski Equation uE = \frac{{v}_{E}}{E} = \frac{\epsilon \zeta }{\eta }, Electro-Osmosis \frac{d{V}_{\mathrm{EO}}}{dt} = AVEO = \frac{AE\epsilon \zeta }{\eta }

Colloidal Stability V = VA + VR where VR = {e}^{-H\kappa }\left(\frac{32{T}^{2}a\epsilon {\gamma }^{2}{k}^{2}\pi }{{e}^{2}{z}^{2}}\right), and VA = \left(\frac{1}{2x}\right)\left(-\frac{A}{12}\right) = \frac{-Aa}{12H}

A132 = \left(\sqrt[2]{{A}_{11}}-\sqrt[2]{{A}_{33}}\right)\left(\sqrt[2]{{A}_{22}}-\sqrt[2]{{A}_{33}}\right), A131 = {\left(\sqrt[2]{{A}_{11}}-\sqrt[2]{{A}_{33}}\right)}^{2}

Coagulation Kinetics -\frac{dn}{dt} = k2n2, \frac{1}{n}-\frac{1}{{n}_{0}} = k2t, k2 = \frac{4Tk}{3\eta }, W = \frac{{k}_{2}}{{k}_{2}}

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