ch 24 pages 619-625
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Lecture 6 – Diffusion. Ch 24 pages 619-625. Summar y of lecture 5. - PowerPoint PPT PresentationTRANSCRIPT
Ch 24pages 619-625
Lecture 6 – Diffusion
We have introduced the general problem of random walk and provided the remarkable simple result relating the mean square displacement and therefore the root mean square displacements with the number of steps N taken and the length of each step
Summary of lecture 5
For the random diffusion of molecules in a gas, the mean square displacement of each molecule can be expressed in terms of number of collisions and mean free path:
22 zlr
lNr 2/12/12 22 Nlr
The probability of having a certain displacement x can then be expressed in terms of the step length l:
Summary of lecture 5
The frequency (if the number of hops or steps per unit time is N’, then the number of hops N=N’t
NlxeNl
xW22 2/
22
1)(
WN t
eN t
ex l N x l N t
1
2
1
22
2
2
22 2 2 2
/ /
We have introduced the concept of diffusion from a microscopic perspective when discussing the motion of molecules in a gas
The diffusion coefficient has been defined by Einstein as a measure of the distance traveled over time (on average) by a particle undergoing diffusion
Microscopic Diffusion
2 2 6r r Dt
An equivalent description can be provided in macroscopic terms when we consider the concentration C(x,t) of a solute in a solvent system (e.g. a protein in water)
When a solution is at equilibrium the concentration of solute is uniform throughout. If the solute concentration is not uniform, a solute concentration exits that must be reduced to zero if the system is to attain equilibrium
Diffusion is the process whereby concentration gradients in a solution are reduced spontaneously until a uniform homogeneous solution is obtained
Diffusion occurs whenever there is a concentration difference
As a consequence of diffusion, an equilibrium state of uniform concentration (or heat, if it heat diffusion) is reached
Macroscopic Diffusion
Let us think of molecules in solution as they move across a certain surface as they ‘diffuse’. Flux is the amount of matter (heat, charge, etc.) that crosses an area per unit time in a direction perpendicular to the surface
The flux of the solute J2 is related to its concentration C2 (how
much solute there is) and its transport velocity v2 (how fast it
moves) by:
Macroscopic Diffusion
units are mol/s cm2J v C2 2 2
If there is no difference in concentration (concentration gradient), there will be no flux; if the concentration is higher on the right, solute will go from right to left to equalize the concentration and reduce the gradient
There is a net transport of material in the direction opposite the concentration gradient; the steeper the concentration gradient, the larger the flux
These considerations lead to Fick’s First Law of diffusion:
Macroscopic Diffusion
dx
dcDxJ 2
2 )(
D is a phenomenological property called the diffusion coefficient; the units of D are cm2/s
Fick’s First Law of diffusion:
Macroscopic Diffusion
dx
dcDxJ 2
2 )(
This equation expresses the fact that, as diffusion occurs, the gradient of concentration decreases, reducing flux until, at equilibrium, the next flux ceases and diffusion stops
We shall now generalize the concept of diffusion
If a temperature gradient exists in a material, heat will be conducted through the material from the region of higher temperature to a region of lower temperature; this process is called heat conduction
The thermal conductivity is the rate at which heat is transferred through a material per unit temperature gradient
We can define an analogous of the diffusion coefficient (thermal conductivity) as the constant of proportionality that relates the heat flux h (Joules per m2 per second) to the thermal gradient dT/dx (degrees K per meter)
Heat Conduction
The thermal conductivity is the constant of proportionality that relates the heat flux h (Joules per m2 per second) to the thermal gradient dT/dx (degrees K per meter)
We have a relationship analogous to Fick’s First Law of Diffusion which relates the solute mass flux J2 (kg per m2 per second) to the
concentration gradient dC(x)/dx (kg/ m3 per m):
Heat Conduction
22 2
dC x dT xJ D h
dx dx
From Fick’s First Law, D2 must have units of m2/s
From the heat flux equation it is clear that must have units of
Heat Conduction
22 2
dC x dT xJ D h
dx dx
Because of their units, fluxes like h and J2 are sometimes called
current densities, because they measure the amount of a quantity that passes through a unit area per unit time
1 1 1J K m s
Generally, we think of a force acting on an object as inducing movement; since we observe flow, we can think that there must be a ‘force’ that induces the solute to ‘flow’
A force occurs when a potential difference exits
In the case of an electric charge, the potential difference is electrostatic (measured as a voltage difference)
In the case of heat flow, a thermal gradient induces heat transfer
In the case of solute transport the potential difference results from a concentration gradient
Chemical Potential
In analogy to the classical concept of force as of a potential gradient, we can introduce a chemical potential to express differences in free energy that induce flux
If the concentration of solute C2 is a function of x, the chemical
potential has the general form:
Chemical Potential
A difference in chemical potential exercises a force on the solute molecules; the force that induces solute flow is related to the chemical potential by the equation
FdG x
dxRTd C x
dx
RT
C x
dC x
dxext 2 2
2
2( ) ln ( )
( )
( )
)(ln)( 2022 xCRTGxG
Consider a molecule in solution. If an external force F is applied to the particle, the particle obviously accelerates according to F=ma
The particle will not accelerate for long. After a brief period, the velocity becomes constant as a result of resistance from the surrounding fluid. This velocity is called the steady state velocity vT and fulfills the condition:
Friction
fv FT
fv is the frictional force exerted by the surrounding fluid on the particle and f is the frictional coefficient of the particle
The fictional coefficient depends on the size and shape of the particle but not on its mass. For a spherical particle with radius R
Friction
fv FT
f R 6
Where is the viscosity of the fluid (Stoke’s Law)
The force acting on each single solute particle is:
Friction
The diffusion coefficient is related to the temperature and to the frictional coefficient f that depends on the solvent property (viscosity) and on the molecular property of the solute (size, shape and hydration)
F
Nv f
RT
N C x
dC x
dx
k T
C x
dC x
dxext B
02
0 2
2
2
2 ( )
( )
( )
( )
J v C xk T
f
dC x
dxB
2 2 22 ( )( )
If the solute flux J2(x) into a volume V=Ax is not equal to the flux
out of the volume J2(x+x), then the solute concentration in the
volume must change by the same amount (matter is conserved).
This means the change in solute concentration per unit time equals the flux gradient (see diagram below)
The Diffusion Equation
C
t
J x J x x
x2 2 2
( ) ( )
The relationship between the concentration change and the flux gradient is combined with Fick’s First Law to Produce the Diffusion Equation, which is also called Fick’s Second Law and describes how the concentration gradient changes with time:
C
t
J x J x x
x
dC
dt
dJ x
dx
k T
f
d C
dxB2 2 2 2 2
22
2
( ) ( ) ( )
To be formally correct, it should be expressed in terms of partial derivatives since the concentration depends both on time and space:
22
2
222
22
x
CD
x
C
f
Tk
t
C b
The Diffusion Equation
D is again the diffusion coefficient
The Diffusion Equation has the following general solution:
22
2
222
22
x
CD
x
C
f
Tk
t
C b
tDxetD
CtxC 2
2 4/
2
02
4),(
The Diffusion Equation
The Diffusion Equation has the following general solution:
00.050.10.150.20.250.3
-20 -10 0 10 20
x
Dt=1
Dt=4
Dt=16
tDxetD
CtxC 2
2 4/
2
02
4),(
The Diffusion Equation
The function C2(x,t) has the form of a bell-shaped (i.e.
Gaussian) curve; it can be used to calculate the average displacement of a solute particle:
04
),( 22 4/
2
0 tDxetD
xCdxtxxCx
The mean squared displacement and root mean square displacement are not 0:
tDdxtxCxx 222 2),(
tDxx rms 222
tDxetD
CtxC 2
2 4/
2
02
4),(
The Diffusion Equation
Notice the similarity with the random walk problem
tDdxtxCxx 222 2),(
tDxx rms 222
2 2 6r r Dt
The diffusion coefficient is half of the mean square displacement per unit time. Because the solution to the diffusion equation has gaussian shape, the diffusion coefficient is related to the width of the gaussian at half height
The Diffusion Equation
The diffusion coefficient is half of the mean square displacement per unit time. Because the solution to the diffusion equation has gaussian shape, the diffusion coefficient is related to the width of the gaussian at half height
00.050.10.150.20.250.3
-20 -10 0 10 20
x
Dt=1
Dt=4
Dt=16
The Diffusion Equation
The concentration c is obviously maximum at t=0; as time increases, it spreads, while the concentration at half maximum is
00.050.10.150.20.250.3
-20 -10 0 10 20
x
Dt=1
Dt=4
Dt=16
The width at half maximum is:
2ln2 Dt
2ln4 Dt
The Diffusion Equation
00.050.10.150.20.250.3
-20 -10 0 10 20
x
Dt=1
Dt=4
Dt=16
One can measure the diffusion coefficient by monitoring how a concentration gradient disappears (non equilibrium experiment)
Because the diffusion coefficient is also related to the random motion of molecules, it can be measured under equilibrium conditions by measuring the random motion of molecules directly
One such method is laser light scattering
2ln4 Dt
The Diffusion Equation
00.050.10.150.20.250.3
-20 -10 0 10 20
x
Dt=1
Dt=4
Dt=16
Laser light scattering: The method monitors the scattering of highly monochromatic light as it travels through the solution
Because each particle moves at a slightly different speed, scattering will induce a distribution of Doppler shifts that will be reflected in a broadening of the monochromatic light beam
The width at half height of the signal is proportional to D, when the molecules are much smaller than the wavelength of the laser beam (you need a laser because it is highly monochromatic)
The Diffusion Equation