Ch 24 pages 619-625

1. Lecture 6 – Diffusion Ch 24 pages 619-625

2. Summary of lecture 5 • 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 • 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:

3. Summary of lecture 5 • The probability of having a certain displacement x can then be expressed in terms of the step length l: • The frequency (if the number of hops or steps per unit time is N’, then the number of hops N=N’t

4. Microscopic Diffusion 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 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)

5. Macroscopic Diffusion 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

6. 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: units are mol/s cm2

7. Macroscopic Diffusion 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: D is a phenomenological property called the diffusion coefficient; the units of D are cm2/s

8. Macroscopic Diffusion Fick’s First Law of diffusion: 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

9. Heat Conduction 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)

10. 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):

11. Heat Conduction From Fick’s First Law, D2 must have units of m2/s From the heat flux equation it is clear that l must have units of 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

12. Chemical Potential 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

13. 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: 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

14. Friction 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:

15. Friction 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 Where h is the viscosity of the fluid (Stoke’s Law)

16. Friction The force acting on each single solute particle is: 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)

17. The Diffusion Equation If the solute flux J2(x) into a volume V=ADx is not equal to the flux out of the volume J2(x+Dx), 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)

18. The Diffusion Equation 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: To be formally correct, it should be expressed in terms of partial derivatives since the concentration depends both on time and space:

19. The Diffusion Equation D is again the diffusion coefficient The Diffusion Equation has the following general solution:

20. The Diffusion Equation The Diffusion Equation has the following general solution:

21. 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: The mean squared displacement and root mean square displacement are not 0:

22. The Diffusion Equation Notice the similarity with the random walk problem 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

23. 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

24. The Diffusion Equation The concentration c is obviously maximum at t=0; as time increases, it spreads, while the concentration at half maximum is The width at half maximum is:

25. The Diffusion Equation 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

26. The Diffusion Equation 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)