Transport phenomena 1/16 Transport (kinetic) phenomena: diffusion, electric conductivity, viscosity, heat conduction... Flux of mass, charge, momentum, heat,...... J = amount (of quantity) transported per unit area (perpendicular to the vector of flux) within time unit Units: energy/heat flux: J m 2 s 1 = W m 2, current density: A m 2 Cause = (generalized, thermodynamic) force F = gradient of a potential (chemical potential/concentration, electric potential, temperature) Small forces linearity J = const F In gases we use the kinetic theory: molecules (simplest: hard spheres) fly through space and sometimes collide
Diffusion macroscopic view 2/16 First Fick Law: Flux J i of compound i (units: mol m 2 s 1 ) J i = D i c i is proportional to the concentration gradient ( c i = grad c i = x, y, ) ( ci c z i = x, c i y, c ) i z D i = diffusion coefficient (diffusivity) of molecules i, unit: m 2 s 1 Example. A U-shaped pipe of lenght l = 20 cm and cross section A = 0.3 cm 2. One end is in Coca-Cola (11 wt.% of sugar), other end in pure water. How much sugar is transported by diffusion in one day? D sucrose (25 C) = 5.2 10 6 cm 2 s 1. 0.74 mg For mass concentration in kg m 3, the flux is in kg m 2 s 1 110 g sucrose in 1 L: cw = 110 g dm 3 = 110 kg m 3 grad cw = cw/l = 550 kg m 4 D = 5.2 10 6 cm 2 s 1 = 5.2 10 10 m 2 s 1 J = D grad cw = 2.56 10 7 kg m 2 s 1 m = JAt = 2.56 10 7 kg m 2 s 1 0.3 10 4 m 2 24 60 2 s = 7.4 10 7 kg
Diffusion microscopic view Flux is given by the mean velocity of molecules v i : 3/16 J i = v i c i Thermodynamic force = grad of the chemical potential: ( ) µi F i = = k BT c N A c i i Difference of chemical potentials = reversible work needed to move a particle (mole) from one state to another where formula µ i = µ i + RT ln(c i/c st ) for infinity dillution was used. Friction force acting against molecule moving by velocity v i through a medium is: F fr i = f i v i where f i is the friction coeficient. Both forces are in equilibrium: F i fr + F i = 0 tj. F i fr = f i v i = f i J i c i = F i = k BT c i c i On comparing with J i = D i c i we get the Einstein equation: D i = k BT f i (also Einstein Smoluchowski equation)
Einstein Stokes equation [blend -g che/sucrose] 4/16 Colloid particles or large spherical molecules of radius R i in a solvent of viscosity η it holds (Stokes formula) f i Arrhenius law for F i = 6πηR i v i viscosity (decreases Einstein Stokes equation: with increasing T), diffusivity (increases D i = k BT D f i = k BT with increasing T i 6πηR i Opposite reasoning hydrodynamic (Stokes) radius defined as: R i = k BT 6πηD i effective molecule size (incl. solvation shell) Example. Estimate the size of the sucrose molecule. Water viscosity is 0.891 10 3 m 1 kg s 1 at 25 C. R = 0.47 nm
Second Fick Law Non-stationary phenomenon (c changes with time). The amount of substance increases within time dτ in volume dv = dxdydz: [J x (x) J x (x + dx)]dydz = dv J x,y,z J = D c c i τ = D i 2 c i (equation of heat conduction) [plot/cukr.sh] 5/16 Example. Coca-Cola in a cylinder (height 10 cm) + pure water (10 cm). What time is needed until the surface concentration = half of bottom concentration? { c τ = c c0 x < l/2 D 2 x 2 c(x, 0) = 0 x > l/2 c(x, τ) = c 0 2 + 2c 0 π 1 ( ) ( ) 3πx 3 cos exp 32 π 2 l l 2 Dτ [ ( ( πx ) cos exp l π2 l 2 Dτ + 1 ( ) ( 5πx 5 cos exp l ) 52 π 2 l 2 Dτ 4 months ) ]
Diffusion and the Brownian motion Instead of for c( r, τ), let us solve the 2nd Fick law for the probability of finding a particle, provided that it sits at origin for τ = 0. We get the Gaussian distribution with half-width τ 1/2 1D: c(x, τ) = (4πDτ) 1/2 exp 3D: c( r, τ) = (4πDτ) 3/2 exp ( x2 4Dτ ) ( ) r2 4Dτ c(x,t) 0.2 0.1 [traj/brown.sh] 6/16 0.5 t=0 t=1 0.4 t=2 t=3 0.3 t=4 t=5 0.0-3 -2-1 0 1 2 3 x 1D: x 2 = 2Dτ 200 The previous example: τ x 2 /2D = 4 months (x = 0.1 m) y y 0 3D: r 2 = 6Dτ 0 0 0 x x
Brownian motion as a random walk (Smoluchowski, Einstein) within time τ, a particle moves randomly by x with probability 1/2 by x with probability 1/2 [show/galton.sh] + 7/16 Using the central limit theorem: in one step: Var x = x 2 = x 2 in n steps (in time τ = n τ): Var x = n x 2 Gaussian normal distribution with σ = n x 2 = τ/ τ x: 1 e x2 /2σ 2 = 1 τ 2πσ 2πτ x exp which is for 2D = x 2 / τ the same as c(x, τ) [ x2 2τ NB: Var x def. = (x x ) 2, for x = 0, then Var x = x 2 τ x 2 Example. Calculate Var u, where u is a random number from interval ( 1, 1) ] 1/6
Conductivity Ohm Law (here: U = voltage, U = φ 2 φ 1 ): 8/16 R = U I I = 1 R U 1/R = conductivity, [1/R] = 1/Ω = S = Siemens (Specific) conductivity (conductance) κ is 1/resistance of a unit cube 1 R = κ A l Vector notation: j = κ E = κ φ A = area, l = layer thickness, [κ] = S m 1 j = el. current density, j = I/A, E = el. field intensity, E = U/l substance κ/(s m 1 ) silver 63 10 6 sea water 5 tap water 0.005 to 0.05 Si 1.6 10 3 distilled water (contains CO 2 ) 7.5 10 5 deionized water 5.5 10 6 glass 1 10 15 1 10 11
Molar conductivity 9/16 Strong electrolytes: conductivity proportional to concentration. Molar conductivity λ: λ = κ c Units: [κ] = S m 1, [λ] = S m 2 mol 1. Watch units best convert c to mol m 3! Example. Conductivity of a 0.1 M solution of HCl o is 4 S m 1. Calculate the molar conductivity. 0.04 S m 2 mol 1
Mobility and molar conductivity Mobility of an ion = averaged velocity in a unit electric field: 10/16 u i = v i E = U/l = el. intensity, U = voltage E Charges z i e of velocity v i and concentration c i cause the current density κ i j i = v i c i z i F = u i Ec i z i F! = λ i c i E λ i = u i z i F = molar conductivity of ion i Ions (in dilute solutions) migrate independently (Kohlrausch law), for electrolyte K z + ν A z A ν : here we define z A > 0 Mathematically j = j A + j K = (λ A c A + λ K c K )E = (λ A ν A + λ K ν K )ce λ = κ c = i ν i λ i Deprecated (λ e i = equivalent conductivity) λ = i ν i z i λ e i λ e i = u if = λ i z i
Nothing is ideal [mz pic/grotthuss.gif] 11/16 Limiting molar conductivity = molar conductivity at infinite dilution λ i = lim λ i c 0 Departure from the limiting linear behavior (cf. Debye Hückel theory): Typical values: λ = λ(c) = λ const c nebo λ = λ const I c cation λ /(S m 2 mol 1 ) anion λ /(S m 2 mol 1 ) H + 0.035 OH 0.020 Na + 0.0050 Cl 0.0076 Ca 2+ 0.012 SO 4 2 0.016 Mobility and molar conductivity decreases with the ion size (Cl is slow), solvation (small Li + + 4 H 2 O is slow) H +, OH are fast movie credit: Matt K. Petersen, Wikipedia
Conductivity of weak electrolytes 12/16 We count ions only, not unionized acid In the limiting concentration: κ = λ c ions = λ αc def. = λ exptl c 0.04 HCl α = λexptl λ 0.03 Ostwald s dilution law: K = c c st α 2 1 α = c c st (λ exptl ) 2 λ (λ λ exptl ) λ / S m 2 /mol 0.02 NaOH 0.01 AgNO 3 CuSO 4 CH 3 COOH 0 0 0.2 0.4 0.6 0.8 1 1.2 c 1/2 / (mol/l) 1/2
Conductivity and the diffusion coefficient 13/16 Einstein (Nernst Einstein) equation: D i = k BT f i = k BT F i /v i = k B T z i ee/(u i E) = k BT = RTu i z i e/u i z i F z i FD i = RTu i λ i = u i z i F = z2 i F2 RT D i microscopically: u i = z ie k B T D i here z i is with sign diffusion: caused by a gradient of concentration/chemical potential migration: caused by el. field D J i = D i c i = c i i µ RT i z j i = c i FD i i µ RT i = c i u i µ i j i = κ i φ = c i λ i φ = c i u i z i F φ Let us define the electrochemical potential µ i = µ i + z i Fφ, then j i = c i u i µ i = c i D i z i F RT µ i = c i λ i z i F µ i
Transference numbers [pic/nernstovavrstva.sh] 14/16 Transference number (transport number) of an ion is the fraction of the total current that is carried by that ion during electrolysis. t = I I = I + I Different ions carry different fractions of the current because different ions move at different speeds under the same potential gradient. For K z + ν A z ν (electroneutrality: z c = z c ): here z > 0 I t = v c z v c z + v c z = v v + v = u u + u = z D z D + z D = ν λ ν λ + ν λ Properties: t + t = 1 t t = u u Example. Electrolysis of CuSO 4 : t Cu 2+ = 40 %, t 2 SO4 = 60 %. cathode Nernst layer bulk total Cu 2+ 100 %I diffusion Cu 2+ 60 %I Cu diffusion SO 2 60 %I 4 migration Cu 2+ 40 %I migration SO 2 60 %I 4
Examples 15/16 Example. Calculate the specific conductivity of a uni-univalent electrolyte MA of concentration 0.01 mol dm 3, assuming that both M and A are of the same size as the sucrose molecule (D = 5.2 10 6 cm 2 s 1 at 25 C)? κ = 0.04 S m 1 (λ± = 0.002 S m 2 mol 1 ) Note: 0.01 M of KCl has κ = 0.14 S m 1 > 0.04 S m 1, because the ions are smaller than sucrose Example. Calculate the migration speeds of ions M +, A between electrodes 1 cm apart with applied voltage 2 V. vm + = va = 4 10 6 m s 1 = 15 mm h 1
Conductivity measurements 16/16 Usage: determination of ion concentrations (usually small): solubility, dissociation constants, conductometric titration... Resistance constant of conductance cell (probe) C (dimension = m 1 ): 1 R = κ A Rκ = l l A = C C determined from a solution of known conductivity (e.g., KCl), C = R κ.