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<span class='text_page_counter'>(1)</span>Physical Chemistry of Minerals and Solutions. 2008/2009. Advection and Diffusion Physical Chemistry of Minerals and Aquous Solutions D.M. Sherman, University of Bristol. Diffusion-Limited Reactions •Reactions for which the elementary chemical steps are very fast will be diffusion controlled. •Diffusion constants of ions in water range from 3 x 10-6 to 2 x 10-4 cm2/s. •If diffusion is fast enough, the rate of the reaction will be advection controlled.. Page ‹#›.
<span class='text_page_counter'>(2)</span> Physical Chemistry of Minerals and Solutions. 2008/2009. Diffusion at the Mineral-water interface. Diffusion: Ficks First Law The flux (J, in moles/cm2-s) of a chemical species is given by the concentration gradient:. J = "D. !. #C #x. Diffusion constants of ions in water range from 10-6 to 10-4 cm2/s.. Page ‹#›.
<span class='text_page_counter'>(3)</span> Physical Chemistry of Minerals and Solutions. 2008/2009. Diffusion: Ficks Second Law The concentration as a function of t and x is found By solving the differential equation (Fick’s 2nd law):. "C " 2C =D 2 "t "x With the appropriate boundary conditions to define the problem... !. Diffusion: Analytical Solutions For a simple constant-source problem, the boundary condition is C(0,t) = C0. The solution to Fick’s Law is:. C(x,t) =. 1 x C0 (1" erf ) 2 2 Dt. !. Page ‹#›.
<span class='text_page_counter'>(4)</span> Physical Chemistry of Minerals and Solutions. 2008/2009. Diffusion: A Cheap Solution... A general approximation for any geometry is that. C(x,t) ". 1 C0 4. when x = Dt. !. Numerical Diffusion Simulations Divide system into cells of length l. Allow components to diffuse from one cell to another.. Page ‹#›.
<span class='text_page_counter'>(5)</span> Physical Chemistry of Minerals and Solutions. 2008/2009. Diffusion in PHREEQC TITLE diffusion simulation SOLUTION 1 units mol/kgw Na 0.001 Cl 0.001 SOLUTION 2-50 TRANSPORT -cells 50 -time_step 1000 -diffusion_coefficient 1.0e-4 -flow_direction diffusion_only -shifts 50 -punch_frequency 125 SELECTED_OUTPUT -file nacl.txt -totals Na Cl END. Diffusion in PHREEQC. Page ‹#›.
<span class='text_page_counter'>(6)</span> Physical Chemistry of Minerals and Solutions. 2008/2009. The Advection-Diffusion Equation:. "Ci " 2Ci "C = D 2 #v i "t "x "x. !. Isotherm-Based Reactive Transport. Si = f (Ci ). !. "Ci " 2Ci "Ci Ri = Di # v "t "x "x 2 Where Ri is the retardation factor:. !. Ri = 1+. "b $Si # $Ci. ! Page ‹#›.
<span class='text_page_counter'>(7)</span> Physical Chemistry of Minerals and Solutions. 2008/2009. Problems with the Kd approach. "Si = Kd "Ci. is not constant!. Moreover, the retardation factor approach will fail for pptn-dissolution reactions since :. "Si "Ci. !. is undefined.. !. Advection Simulations Divide system into cells. Transfer solution from cell n-1 to cell n at each step. Cell 0 is always full. The flow velocity = (cell length)/(time step).. Page ‹#›.
<span class='text_page_counter'>(8)</span> Physical Chemistry of Minerals and Solutions. 2008/2009. Advection Rainwater; pH 5.6, pCO2 = 3.5 Buried DU 2 g Schoepite; pCO2 = 3.5 With FeOOH 1g FeOOH; pCO2 = 3.5. 1g FeOOH; pCO2 = 3.5. No FeOOH. 1g FeOOH; pCO2 = 3.5. 1g FeOOH; pCO2 = 3.5. Sorption by FeOOH retards transport. The Advection-Diffusion with Kinetics: If a dissolved species is being produced (consumed) at a rate k(Cs-C) where Cs is the saturation concentration, then the concentration C will obey. "C " 2C "C = D 2 #v + k(Cs # C) "t "x "x Ignoring diffusion, the concentration will have a steady State profile when. k(Cs " C) = v. !. #C #x. ! Page ‹#›.
<span class='text_page_counter'>(9)</span> Physical Chemistry of Minerals and Solutions. 2008/2009. Calcite Dissolution: Kinetics + Advection TITLE Kinetic dissolution of calcite SOLUTION 0 pH 7.00 temp 25.00 EQUILIBRIUM_PHASES 0 CO2(g) -3.5 1000. SOLUTION 1-20 pH 7.00 temp 25.00 KINETICS 1-20 Calcite -m0 1.0 -parms 5 0.3 TRANSPORT -shifts 200 in m -cells 20 -lengths 1 in seconds to give v= 1m/300s -time_step 300 -punch_frequency 25. Calcite Dissolution: Kinetics + Advection. [Ca] = 0.44 mmol/l at sat’n 0.003 m/s 0.01 m/s 0.1 m/s. Resulting steady state profiles. Page ‹#›.
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