CHAPTER 19 Molecules in Motion. 1. Matter transport is driven by concentration gradient.

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1 CHAPTER 19 Molecules in Motion I. Transport in Gases. A. Fick s Law of Diffusion. 1. Matter transport is driven by concentration gradient. Flux (matter) = J m = D dn dz N = number density of particles Flux J m is measured in moles or molecules per unit area per unit time D = diffusion coefficient (m 2 /s) Particles are in random motion and moving between cells passively. Flux J m net number moving in forward direction 0 if N is different between cells J m > 0 if N is less on the right J m < 0 if N is less on the left Nature tries to fill every void, a consequence of entropy increase, i.e., increase in the dispersal of matter. 2. Thermal energy transport is driven by temperature gradient. Flux (energy) = J E = κ dt dz κ = coefficient of thermal conductivity = J/m 2 -K = W/m-K 3. Transfer of momentum is driven by velocity gradient. Flux (momentum) = J p = η dv x dz η = viscosity = kg/m-s 1 Poise (P) 0.1 kg m -1 s -1 1

2 B. The transport parameters D, κ and η. 1. Diffusion coefficient D. Use the Maxwell-Boltzmann distribution of molecular velocities. Can calculate one-way flux of molecules passing through boundary between two adjacent cells, going both directions. J(net flux L to R) = J( L R) J ( L R ) = 1 3 λv # dn& mean % ( $ dz ' L,R boundary So D= 1 3 λv mean λ = mean free path = k B T/σp " v mean = mean speed = 8RT % $ ' # πm & D as T D as σ D as p 2. Thermal conductivity κ. Works similarly. Molecules passing between cells with slightly different T s carry their average thermal energy with them across boundary. J(net flux L to R) = J L R 1 2 ( ) J ( L R ) On average each molecule carries across boundary an average energy νk B T, where ν = 3/2 for atoms and ν = more for molecules, with more degrees of freedom. 2

3 J(net energy flux) = 1 3 νv λnk $ dt ' mean B & ) % dz ( L,R boundary κ = 1 3 νv mean λnk B κ as λ but λ = k B T σp = 1 σn κ as σ or N Since constant a volume heat capacity of ideal gas is: # νk C V,m = N B T & avo % $ T ( ' V the flux equation can be re-written such that κ = 1 3 v λ # particles % mean $ & C V,m κ as C V,m and [ ] is molar concentration 3. Viscosity η. η = 1 3 v mean λmn m = mass of one particle η = pmd or MD[particles] RT M = molar mass η of gas is independent of p η of gas as T (opposite behavior of liquids) C. Effusion. Rate of randomly hitting a hole in the wall of area A. Rate effusion = Z w A = pan avo ( 2πMRT) 1 2 3

4 II. Molecular Motion in Liquids. A. Ion Transport. 1. Conductance of electrolyte solutions. a. The conductance properties of electrolytes are due to migration of ions and obeys Ohm s Law: current I = V/R (voltage/resistance) (amperes) = (volts v/ohms Ω) b. Fits standard linear response theory format: flow = conductance coefficient x driving force I = GV G = 1/R = conductance (Atkins uses G instead of traditional L) G in Ω -1 = Siemans unit c. G depends on cross-sect area A of conductor, the length traversed and intrinsic properties of the ions (which we are most interested in). G = κa A = electrode area (depends on cell geometry) = length between electrodes (depends on cell geometry) κ=conductivity (intrinsic property of ions here) conductivity (intrin cell constant defined as k = / A. We want to factor this out. 4

5 d. Find k for your cell by studying a standard solution of known κ M KCl at 25 C has κ = Ω -1 m -1 G KCl = /k measure G (= 1/R) solve for k for our cell e. Now, you ve eliminated geometric aspects of the cell, can convert all subsequent R readings to κ. κ depends on concentrations of ions and their intrinsic mobilities. a sheer measure of number a more interesting microscopic property f. For electrolyte of concentration c, of stoichiometry A ν+ B ν ν + A Z+ + ν B Z with fractional ionization α (0 α 1) κ = 1000αcνF(u + + u - ) Here 1000αc is conc of A + and/or B - in mol-m -3 ν = ν + Z + = ν- Z- = 1 for KCl, KAc, HCl, etc. F is Faraday C/mol, charge of 1 mole of protons And (u + + u - ) are the mobilities of ions g. Let s define a new quantity which factors out the concentration aspect. Λ m = κ/1000νc Λ m called Molar conductance or equivalent conductance or molar conductivity. Λ m = αf(u + + u - ) α = 1 for strong electrolytes KCl, HCl, KAc α < 1 for HAc weak acid, etc. 5

h. Attraction between ions lowers the mobilities: Λ m = Λ m (1 - β C) -general observation for strong electrolytes, (α =1) (Kohlrausch s Law) where term Λ m is for ions

limiting molar conductances of individual species λ o + = o Fu+ λ o - = o Fuu o +, u o - = mobilities at dilution or for general stoichiometry: Λ m = ν + λ + - o + ν + λ o

6 h. Attraction between ions lowers the mobilities: Λ m = Λ m (1 - β C) -general observation for strong electrolytes, (α =1) (Kohlrausch s Law) where term Λ m is for ions acting completely independently, called the limiting molar conductance of the electrolyte. Λ m = λ o + + λ o -! limiting molar conductances of individual species λ o + = o Fu+ λ o - = o Fuu o +, u o - = mobilities at dilution or for general stoichiometry: Λ m = ν + λ + - o + ν + λ o Note: ions independently contribute to conductance - law of independent migration of ions (infinitely dilute limit). i. Λ m not linear with c for HAc (weak electrolyte) due to α being < 1 and concentration-dependent. Hard to get Λ m. 6

7 It still can be determined by combining other quantities to obtain: Λ m (HAc) = λ o H+ + λ o Acu o Ac- = λ o Ac- /F Example: Λ o (HAc) = Λ o (HCl) + Λ o (KAc) - Λ o (KCl) all are strong electrolytes which can extrapolated to -dilution be measured and Proof: Λ o (HAc) = (λ o H+ + λ o Cl- ) + (λ o K+ + λ o Ac- ) - (λ o K+ + λ o Cl- ) = (λ o H+ + λ o Ac- ) j. α, degree of ionization can be obtained for a weak acid HAc: α = Λ m /Λ m measure estimate by combining strong electrolyte data as above True: since Λ m = αf(u + + u - ) and Λ m = F(u + + u - ) k. Another method of obtaining Λ m and K a of a weak acid - Ostwald s dilution law Can derive by starting with α = Λ m /Λ m and substituting in an expression for α in terms of K a to obtain: 1 Λ m = 1 Λ m o + Λ m c K a (Λ m o ) 2 7

8 2. What determines ion mobilities u +, u - themselves? a. Diffusive motion of ions in electric field: Drift speed s determines overall conductivity, hence mobility. s determined by balance between driving force of the field E and the opposing frictional resistance force: force of field = zee = charge of ion x potential field frictional resistance force = - fs Set equal: zee = -(-fs) zee = fs f = friction coefficient s = speed of drift s = zee/f = drift speed of ion. b. Friction coefficient of a sphere of radius r in a fluid: Stokes Law f = 6πηr η is viscosity; liquid H 2 O η 1 centipoise c. s relation to E s = ue 8

9 where: u = ze/f = ionic mobility (as in previous section) Can measure u from conductance experiment. ze known, can calculate f. Then can calculate r, effective radius of ion, called the hydrodynamic radius. d. Hydrodynamic radius may not equal actual physical radius. (vdw radius) e. Anomalously high speed of conduction of hydrogen ions in water - the Grotthus mechanism: 9

10 Notes: 10

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