Diffusion. Spring Quarter 2004 Instructor: Richard Roberts. Reading Assignment: Ch 6: Tinoco; Ch 16: Levine; Ch 15: Eisenberg&Crothers

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1 Chemistry 24b Spring Quarter 2004 Instructor: Richard Roberts Lecture 2 RWR Reading Assignment: Ch 6: Tinoco; Ch 16: Leine; Ch 15: Eisenberg&Crothers Diffusion Real processes, such as those that go on continually in biological systems are always attended by a free energy decrease for the system. One of the simplest ways to describe this approach of the system towards a state of lower free energy is by way of the concept of the thermodynamic force. We can illustrate the power of this approach first by deriing Fick s Law of Diffusion. You might recall from elementary mechanics in physics that a particle in a mechanical or graitational field always moes from a point of high potential energy to one of lower potential energy. Another way of stating this fact is that there is a force on the particle gien by ( ) = V( r ) F r or in one dimension F x = dv or V x Howeer, in contrast to motion in free space or a acuum, motion of molecules in a gas or liquid inoles resistance or frictional forces. A particle moing in such a medium must cause displacement and separation of these interacting molecules. Force is required and the net result is an expenditure of energy. When the path of a particle inoles the separation of two or more molecules, work is done by the particle on these two or more molecules. This work or energy expenditure by the particle is not recoered by the particle when the two or more molecules again return to their aerage intermolecular distances. Instead, this work of separation is dissipated as heat characteristic of irreersible processes. In fact, the rate of energy dissipation by the moing particle is proportional to its elocity. One frequently speaks of this rate of energy dissipation as a frictional force. This frictional force is not only a property of the particular medium through which the particle is moing,

2 Chemistry 24b Lecture 2 2 but is also dependent on the particle size and shape. The larger the size, the greater the resistance encountered. The inherent property of the medium which determines the friction force is called the iscosity η, and is related, in part, to the degree of molecular interaction and molecular size and shape. In 1850, Stokes showed that for a sphere where F f = 6πrη F f = frictional force r = radius of sphere η = iscosity = elocity of motion of sphere F f r Figure 1-1 The quantity 6πrη is often referred to as the frictional coefficient, i.e., f = 6πrη Since the frictional force encountered by a particle moing in a iscous medium is proportional to the elocity of the particle, it is eident that a driing force acting on the particle cannot lead to its continual acceleration. The particle will reach a elocity such that the frictional force opposing the motion becomes equal to the driing force. Gien a constant driing force, the elocity of motion of a particle in a iscous medium will be a constant, and is obtained by equating F f = F d where F d driing force on particle or F d = 6πrη = 1 F f d = λf d λ = f 1 mobility elocity of particle assumed when unit driing force is acting on it

3 Chemistry 24b Lecture 2 3 Concept of a Flux Now consider a large number of molecules k moing with an aerage elocity k as a result of this driing force F d. F d Figure 1-2 Let us calculate the number of such molecules crossing 1 cm 2 of this imaginary boundary flow per unit time or sec. This quantity is called the flux. J k = C k k where C k concentration of moles / cm 3 Substituting for k, J k = λ k C k F dk flux moles / cm 2 / sec Deriation of Fick s Law of Diffusion (First Law) The aboe discussion assumes that the particles are subjected to graitational potential. Howeer, F d need not be mechanical in origin. F d could be the thermodynamic force. For neutral solutes, we can define a thermodynamic force or in 1-D, F dk = µ k ( r ) 1 ( F dk ) = µ k(x) x x 1 Now µ k (x ) = µ k 0 + RT lnc k (x)

4 Chemistry 24b Lecture 2 4 Therefore, ( F dk ) = 1 RT d ln C k (x) = k x B T 1 C k dc k (x) From this result, we see that a concentration gradient proides a thermodynamic force for the particles or molecule to diffuse. Also, particles diffuse from higher µ k to lower µ k, or from higher C k to lower C k. dµ k < 0, F d = dµ k > 0 µ k x Figure 1-3 With this result, we can obtain the flux: J k = λ k C k F dk J k = λ k C k k B T 1 C k dc k (x) J k = λ k k B T dc k (x) = D k dc k Fick' s Law (First Law) where D k diffusion coefficient of species k = k B T f = k B Tλ k (cm 2 / sec)

5 Chemistry 24b Lecture 2 5 Table 1-1. Typical Diffusion Coefficients at 20 C in H 2 O (cm 2 /sec) Glycine Leu Gly Gly Ribonuclease Human Serum Albumin Human Hb Tobacco Mosaic Virus H Phospholipids in Bilayer Membrane 10 7 to 10 8 Membrane Proteins or lower Diffusion of Electrolytes For charged species k (as opposed to neutral solutes discussed aboe) F dk = µ k ( r ), µ = electrochemical potential where µ k (x) = µ k 0 (T, P) + RT ln C k (x) + z k F Φ (x) From this, 1 J k = λ k C k dµ k 1 = λ k C k RT dc k + z k F dφ ( x) C k = λ k k B T dc k ( x) + z k e C k k B T dφ( x) which shows that the flux is controlled by both the concentration and the electrical potential gradient. Now dφ may exist purely as a result of diffusion of the electrolyte (or free diffusion of ions), or it may be imposed on the system from an external source, with the use of appropriate electrodes. For example:

6 Chemistry 24b Lecture 2 6 (1) Free Diffusion You cannot hae macroscopic separation of charges in space. Accordingly, there is no net charge flow z C + z Aν ν + x free diffusion of ions Figure 1-4 Therefore z k J k = 0 for all alues of k k For simplicity, consider a binary electrolyte: Conditions for Free Diffusion: 0 = z + λ + dc + (x) Since for binary electrolytes C k = ν k C where ν k stoichiometric number of ions We obtain dφ = RT F + λ 2 +z + F RT C dφ(x) dc + + z λ (x) + λ 2 z F RT z + ν + λ + + z ν λ z 2 + ν + λ + + z 2 ν λ d ln C(x) C dφ(x) This result shows that in general one expects some electrical potential gradient to be associated with the simple diffusion of an electrolyte. dφ depends on: (a) z + ν + λ + + z ν λ (b) d ln C(x)

7 Chemistry 24b Lecture 2 7 Only if z + ν + λ + + z ν λ = 0 would dφ = 0 in the presence of d ln C. For a simple unialent-unialent electrolyte, where z + = 1, z = 1, ν + = 1, ν =1 dφ = RT F ( ) ( ) λ + λ λ + + λ d ln C Thus, if the mobility of the cation λ + is larger than that of the anion λ, then the resulting electrical potential gradient has the opposite sign to the concentration gradient, or in other words, opposes the motion of cations. C( x 0 + x) x 0 + C( x 0 + x) concentration gradient If λ is greater than λ +, then dφ Figure 1-5 dc has the same sign as, and tends to oppose the motion of anions. In effect, what exists in a diffusion system is a slight displacement of charge; the more mobile charge leads the less mobile one by a slight amount, just enough to gie rise to dφ obsered. The diffusion of cations and anions are coupled together in this fashion by their electrical attraction for each other. (2) Electrophoresis Discuss later.