Margarita Valero Juan Physical Chemistry Department Pharmacy Faculty Salamanca University ATHENS 2017

SALAMANCA, SPAIN

SALAMANCA MAIN SQUARE

SALAMANCA CATHEDRAL

TRANSPORT PHENOMENA 2.1.- Concept of Transport 2.2.- Diffusion 2.3.- Diffusion of Matter 2.3.1.- First Fick´s Law 2.3.2.- Second Fick´s Law 2.4.- Diffusion through Membranes 2.4.1.-Permeable Membranes 2.4.2.- Semi-Permeables Membranes 2.5.- Bibliography

2.1.- Transport J = f (X) Transport: FLUX (J): Transference of “some amount” of a physical property between two regions of a system. DRIVING FORCE (X)  SOME EFFECT: FLUX (J) FLUX (J): J = f (X) AMOUNT OF PHYSICAL MAGNITUD TRANSFERRED BY UNIT OF AREA AND TIME Physical Magnitud: Driving Force (X) Transport Phenomena * Energy: Heat Difference of Temperature Heat Transfer * Matter: a) fluid Difference in Concentration Osmosis b) particles Difference in Concentration Diffusion . * Electric Charge: Difference in Electric Potential. Electric Conductivity

2.2.- Diffusion <x>=0 <x>2 = 2Dt D = kT/f D = kT/6pr Definition: movement of molecules due to the thermal or kinetic energy. Brownian Movement: in the absence of concentration gradient “random walk”: by collision among particles D: Diffusion Coefficient I.S. m2/s t: time: seconds (s) <x>2: mean square distance: I.S.: m2 <x>=0 <x>2 = 2Dt f: frictional coefficient k: Boltzman´s Constant I.S. 1.3806504*10-23 J/K D: Diffusion Coefficient I.S. S.I. m2/s T: Temperature K (ºC + 273) D = kT/f Einstein´s Law: Stokes-Einstein´s Law : D = kT/6pr h: solvent viscosity I.S.: Pa*s ((N/m2)*s) r: particle radius (spherical particles) (rH= hydrodynamic radius): length

2.2.- Diffusion D = kT/6pr Stokes-Einstein´s Law : Diffusion Coefficient: a) increases as T increases: Arrhenius Equation: D=D0exp(-Ea/RT) b) decreases as r increases (size increases) c) decreases as viscosity ( ) increases Diffusion Coeffcient Determination: density measurments, NMR, Light Scattering

2.2.- Diffusion

2.2.- Diffusion <x>2 = 2Dt t = <x>2 / 2D EXAMPLE 1: The diffusion coefficient of glucose is 4.62*10-10 m2 s-1. Calculate the time required for a glucose molecule to diffuse through: a) 10000Å b) 0.1 m <x>2 = 2Dt t = <x>2 / 2D D: Diffusion Coefficient I.S. m2/s t: time: s <x>2: mean square distance: I.S. m2 <x>2 =(10000 Å*10-10m/Å)2=10-6m2 t=10-6m2/(2* 4.62*10-10m2s-1)=1.08*103s b) <x>2 =(0.1m)2=10-2m2 t=10-2m2/(2* 4.62*10-10 m2s-1)= 1.082 107 s ~ 116 days

2.2.- Diffusion D = kT/6pr r = kT/6pD Stokes-Einstein´s Law : EXAMPLE 2: Calculate the hydrodynamic radius of a sucrose molecule in water knowing that at 25ºC, Dsucrose= 69*10-9m2s-1 and H2O.=1.0*10-9 Ns/m2. Stokes-Einstein´s Law : D = kT/6pr r = kT/6pD h: solvent viscosity I.S.: Pa*s ((N/m2)*s) r: particle radius (spherical particles) (rH= hydrodynamic radius): length k: Boltzman´s Constant I.S. 1.3806504*10-23 J/K D: Diffusion Coefficient I.S. S.I. m2/s T: Absolute Temperature K r =(1.3806504*10-23 J/K)(25+273)K/ (6*3.1416*1.0*10-9 Ns/m2.* 69*10-9m2s-1)= = 3.16*10-10m = 3.16Å J=N*m

2.3.- Diffusion of Matter J = f (X) J = dn/A dt dC/dx: J = f (X) Flux: J : particles/ length 2 time Speed: v = dn/dt v: particles/ time dC/dx: Concentration Gradient: particles/ length 4 J = f (X) Fick´s laws Quantifying the Diffusion Process

2.3.1- First Fick´s Law J = f (X) J =-D dC/dx J = dn/A dt = -D dC/dx Flux of particles J =-D dC/dx D: Diffusion Coefficient dC/dx: Concentration Gradient UNITS: * dC/dx: particles/length4 (c=particles/length4) * dn/dt: particles/ time * D: length2/time A: length2 I.S: length: m; time: seconds J = dn/A dt = -D dC/dx v = dn/dt = -D A dC/dx

2.3.1- First Fick´s Law Steady State Conditions: J =cte and dC/dx= cte along x x1 x2 x3 J1 J2 J3 J1=J2=J3 J = dn/A dt = -D dC/dx v = dn/dt = -D A dC/dx dX1=dX2 dC1=dC2 C1≠C2 ≠C3 J = -D dC/dx J= -D (DC/Dx)

2.3.1- First Fick´s Law Steady State Conditions: J =cte and dC/dx= cte EXAMPLE 3: In one container there is a wall that separates two regions through a circular disc of 6 mm of diameter and 5 mm in thickness. In the compatmet 1, there is an 0.2M aqueous urea solution; whereas compartment 2 has only water. How many grams of urea passes from compartment 1 to 2 in 1s?, Durea= 9.37*10-10m2s-1 and Murea=60g/mol. Steady State Conditions: J =cte and dC/dx= cte 0.2M Urea H2O J = Dn/A t H2O J = -D (DC/Dx) Dn/t= -DA (DC/Dx) 5mm D = 9.37*10-10m2s-1 A= pr2 = 3.1416*(3 mm*10-3m/mm)2=2.83*10-6 m2 DC=-0.2M DX=5 mm*10-3m/mm=5*10-3m Dn/t=-9.37*10-10m2s-1*2.83*10-6 m2 *(-2*10-4 mol m-3/5*10-3m)= 1.05*10-12 mol/s In t=1s: Dn=1.05*10-12 mol/s x 1s=1.05*10-12 mol 1.05*10-12 mol * 60g/mol= 63.17*10-12g= 63.17 pg

2.3.2- Second Fick´s Law Non Steady State Flux: J ≠ cte and dC/dx ≠ cte along x x1 x2 x3 J1 J2 J3 J1≠J2 ≠ J3 Particles Flux dX1=dX2 dC1 ≠ dC2 C1≠C2 ≠C3 J = f (X) ∂C/∂t = D (∂/∂x(∂C/∂x))= D(∂2C/∂x2) J =-D dn/dx D:Diffusion Coefficient dC/dx: Concentration Gradient

2.3.2- Second Fick´s Law t<0: C=0 from –x to x x1 x2 x3 J1 J2 J3 t = 0: x=0 Nmolecules/m2 Boundary Conditions Unidirectional Transport

2.3.3- Some Complications It is assumed that: Fickian Diffusion Coefficient Equals to Stokes-Einstein relation: D=kT/6phr A: ARISING FROM THE DIFFUSING MATTER (1) High concentration of the solute: then a=gC : then D*( 1+dlng/dlnC) b) Changes the viscosity (h) of the medium (2) Ionic solutes: D is an average of ion, ion-pair and molecular species

2.3.3- Some Complications It is assumed that: Fickian Diffusion Coefficient Equals to Stokes-Einstein relation: D=kT/6phr A: ARISING FROM THE MEDIUM WHERE THE PARTICLE DIFFUSES Complex Systems: The effective diffusion coefficient D*, should be considered

2.4.- Diffusion Process through Membranes 2.4.1. Permeable Membranes Steady State Conditions:J=cte and dC/dx =cte along X C1 C2 x1 x2 l C1*P C2*P C1 C2 x1 x2 l C1*P C2*P J= -D (DC/Dx) C1 C2 x1 x2 l P= Cm/C P= Cm/C J= -D (DC/Dx) PERMEABILITY: D/Dx J= - PDC

2.4.- Diffusion Process through Membranes 2.4.2. Semi-Permeable Membranes DIALYSIS: diffusion of a permeable solute OSMOSIS: diffusion of solvent molecules

2.5.- Bibliography Physical Chemistry with Applications to Biological Systems. Chapter 5. Raymond Chang. Collier Macmillan Canadá, Ltd. 1977.ISBN:0-02-321020-6 Physical Chemistry of Foods. Chapter 5. Pieter Walstra. Marcel Decker Inc. New York.2003.ISBN:0-8247-9355-2