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
Transport:
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
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
<x>=0
<x>2 = 2Dt
Einstein´s Law: D = kT/f
Stokes-Einstein´s Law :
D: Diffusion Coefficient I.S. m2/s
t: time: seconds (s)
<x>2: mean square distance: I.S.: m2
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/6pr
: solvent viscosity I.S.: Pa*s ((N/m2)*s)
r: particle radius (spherical particles) (rH= hydrodynamic radius): length
2.2.- Diffusion
Stokes-Einstein´s Law :
D = kT/6pr
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
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
a) <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
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
: 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
a) 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)
Flux:
Speed:
J = dn/A dt J : particles/ length 2 time
v = dn/dt v: particles/ time
Concentration Gradient:
J = f (X) Fick´s laws
dC/dx: particles/ length 4
Quantifying the Diffusion Process
2.3.1- First Fick´s Law
Flux of particles
J = f (X)
J =-D dC/dx
D: Diffusion Coefficient
dC/dx: Concentration Gradient
J = dn/A dt = -D dC/dx
v = dn/dt = -D A dC/dx
UNITS:
* dC/dx: particles/length4
(c=particles/length4)
* dn/dt: particles/ time
* D: length2/time
•A: length2
I.S: length: m; time: seconds
2.3.1- First Fick´s Law
Steady State Conditions:
J =cte and dC/dx= cte along x
J1=J2=J3
J1 J2 J3
J = dn/A dt = -D dC/dx
v = dn/dt = -D A dC/dx
x1 x2 x3
C1≠C2 ≠C3
dX1=dX2
J = -D dC/dx
dC1=dC2
J= -D (DC/Dx)
2.3.1- First Fick´s Law
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
H2O
5mm
J = -D (DC/Dx)
J = Dn/A t
Dn/t= -DA (DC/Dx)
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
J1≠J2 ≠ J3
J1 J2 J3
Particles Flux
x1 x2 x3
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
dX1=dX2
dC1 ≠ dC2
2.3.2- Second Fick´s Law
t<0: C=0 from –x to x
t = 0: x=0 Nmolecules/m2
J1≠J2 ≠ J3
J1 J2 J3
Boundary Conditions
x1 x2 x3
Unidirectional Transport
2.3.3- Some Complications
It is assumed that: Fickian Diffusion Coefficient Equals to StokesEinstein relation: D=kT/6pr
A: ARISING FROM THE DIFFUSING MATTER
(1) High concentration of the solute:
a) then a=gC : then D*( 1+dlng/dlnC)
b) Changes the viscosity () 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 StokesEinstein relation: D=kT/6pr
A: ARISING FROM THE MEDIUM WHERE THE PARTICLE
DIFFUSES
(1) 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
J= -D (DC/Dx)
C1
x2
x1
x2
x1
C1*P
C2*P
C1
C1
C2
C1*P
C2
C2*P
C2
l
l
l
x2
x1
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:0-02321020-6
-Physical Chemistry of Foods. Chapter 5. Pieter Walstra. Marcel Decker
Inc. New York.2003.ISBN:0-8247-9355-2