Transport Properties

Transcription

1 EDR = Engel, Drobny, Reid, Physical Chemistry for the Life Sciences text Transport Properties EDR Chapter 24 "I have said that the Brownian movement is explained...by the incessant movement of the molecules of the fluid, which striking unceasingly the observed particles, drive about these particles irregularly through the fluid except in the case where these impacts exactly counterbalance one another. Ithas, to be sure, been long recognised, especially in explanation of the facts of diffusion, and of the transformation of motion into heat, not only that substances in spite of their homogeneous appearance, have a discontinuous structure and are composed of separate molecules, but also that these molecules are in incessant agitation, which increases with the temperature and only ceases at absolute zero." Jean Baptiste Perrin, 909, Nobel Prize in Physics in 926 "for his work on the discontinuous structure of matter, and especially for his discovery of sedimentation equilibrium". Living systems exploit transport phenomena: systhesis of DNA/proteins; movement of nutrients and waste materials across cell walls/membranes; ions transported between neurons in the Introduction to Transport (24.) brain and body; O 2 /CO 2 exchanged in the lungs of animals, gills of fish, and stomata of plants. In CHEM 340 and so far this semester, you have examined equilibrium properties of a system. In this chapter and the following chapters on kinetics we are dealing with nonequilibrium processes which occur at nonzero rates. Asystem can be out of equilibrium if matter and/or energy are being transported between the system and its surroundings or between different parts of the system. Properties that depend on rates of movement of matter and/or energy are known as transport properties. The property which is transferred is one of the extensive conserved quantities of the system such as mass, momentum, energy, or charge and is transferred from an area of higher amount to an area of lower amount. This spatial imbalance is a gradient of an intensive state variable of the system such as number density (N/V ), velocity, temperature, or electric potential and is the driving "force" for the transfer. Particles, molecules, ions, atoms, electrons, etc, move or flow in response to this spatial gradient in an attempt to diminish the gradient (shades of Le Châtelier s Principle) and restore equilibrium. The number of particles moving per unit time is the flow (remember Z from last week?). The flux of particles, the number of particles per unit time per unit area is proportional to the flow and is some function of the gradient. While the functional form can be quite complex, in many cases an accurate approximation can be made by assuming that the flows and fluxes are linearly proportional to the negative of these gradients The flow through a plane is the collision frequency Z from last week. flow = flux area = α f (gradient ) (24.) where α is the proportionality constant which is called the transport coefficient of the material. To take an example consider the flux J x of gas particles in opposition to the gradient in particle number density N/V shown in the figure on the left (EDR Ñ = N/V ). For this diffusion process the flux and gradient are related by Area A of wall J x is flow of concentration through area perpendicular to x J x = A dn dt = D d(n/v ) = A Below are some transport processes with their corresponding fluxes and gradients (coefficients evaluated for hard sphere gases). Z W = N/V Table I. Some Transport Properties transport process property transported flux gradient coefficient* molecular mass J = dn = D d(n/v ) D = A dt 3 λ v diffusion Fick s st law (3π /6) m 2 /s viscosity linear J = dp x = η dv x η = A dt dz 3 λ v ρ momentum Newton s law (5π /32) kg/(m s) thermal energy J = dq = κ dt κ = A dt 3 λ v C V,m ρ/m conductivity (as heat q) Fourier s law (25π /64) J/(K m s) *numbers in parentheses derived from more accurate kinetic theory

2 -2- Mass Transport: Diffusion and Its Time Variation ( ) Mass, momentum, and energy can all be transported by diffusion. Diffusion is a macroscopic motion of components of a system in response to a concentration difference. Diffusion can be described phenomenologically (in much the same spirit as the first law of thermodynamics is introduced, there is no "proof"), it can have a phenomenologcal statistical description with a microscopic basis, and it can be described by kinetic theory. Phenomenological Description: Fick s Laws, 855 (24.3) The relationship between the flux in the number of particles and the gradient of the number density examples of diffusion in three states of matter Gas Liquid Solid J x = A dn dt = D d(n/v ) (24.5) is Fick s first law of diffusion. As Table II shows diffusion is applicable to gases, liquids, and solids. The diffusion coefficent can vary strongly and generally increases with T and decreases with molar mass. Table II. Some Diffusion Coefficents, D (cm 2 /s) gas pair H 2 -O 2 He-Ar O 2 -CO 2 CO-C 2 H 4 D, 0 o C, atm infinite dilution NaCl sucrose myoglobin hemoglobin (20 o C) 0 5 D,water, 25 o C, atm infinite dilution in Cu Ni Cu (025 o C) Ni Cu (630 o C) Al Cu (20 o C) D, solid, atm self diffusion H 2 O 2 CO 2 Xe D, gas, 0 o C, atm self diffusion H 2 O Hg C 2 H 5 OH n-c 3 H 7 OH 0 5 D,liquid, 25 o C, atm Fick also derived anequation for the rate of change of number density as a result of diffusion. The layer of fluid between x and x + in the figure below will have material entering from one side and leaving from the other. Since the two planes at x and x + have the same area, the increase in the number density with time, (N/V )/ t, will equal the excess of molecules diffusing into the region over those diffusing out per unit volume or the difference between the flux in and the flux out per unit distance, W Remember frequency Z from last week: Z dn W = Now ) the flux: J W = W dn dt and 2) V = A A dt A (N/V ) t = [J(x) J(x + )] where (a partial is used since Ñ = N/V depends upon x and t) J(x + ) = J(x) + J x So substituting in for J using Fick s first law (N/V ) t = J x = x D (N/V ) x J(x) J(x + ) = D 2 (N/V ) THE DIFFUSION EQUATION (24.9) x 2 where the last line is Fick s second law of diffusion and follows if D is independent of x. This secondorder linear differential equation can be solved (with thin film boundary conditions) yielding

3 -3- N(x = 0, t = 0) Ñ(x, t) = e x2 /4Dt (24.20) (4π Dt) /2 Note that for any t, Ñ(x, t) = N(x = 0, t = 0) so mass is conserved. In this one-dimensional mathematical problem, Ñ is given asnumber of mole- cules per unit distance. If we consider a real (threedimensional) system then the right hand side of the above mathematical solution should be divided by the cross-sectional area A. Ñ would then become the unsual particle density, N/V. Onthe right is the particle density profile at various times for molecules diffusing from the origin. At t = 0the molecules are all located on a plane of unit area at x = 0. Since Eq. (24.20) is normalized to N you might remember from last week that we can define a probability. Here the probability P(x) that a particle will diffuse a distance x in any direction in time t is P(x) = p(x) = Ñ(x, t) N = (4π Dt) /2 e x2 /4Dt Now we can determine some properties. First the average distance that a particle which diffuses according to Eq. (24.20) travels in time t is obviously zero. We can arrive at this conclusion three ways. First the distribution is symmetrical about x = 0,just like avelocity component, so every move to x has its counterpart to x.then, mathematically, x = xp(x) = x Ñ(x, t) N = (4π Dt) /2 xe x2/4dt where we can immediately see that the integral vanishes as the integrand is a product of an odd function x and an even function, the Gaussian. You can also use the kinetic theory integrals to arrive at the same conclusion (integral 4 in Levine Table 4.). Again, taking a clue from velocity, what about the mean square displacement? To evaluate, use integrals and 3 in Levine Table 4.; n =, a = /4Dt. x rms = x 2 = (4π Dt) /2 x2 e x2/4dt = 2Dt (24.2) Simple and very useful. In two dimensions r rms = 4Dt and r rms = 6Dt in 3D for distance r. EX. Take typical values of D of 0,0 5,and 0 20 cm 2 s,for a gas, liquid, and solid, respectively. Determine the typical rms x displacement for each in one minute. concentration /2 0 distance While diffusion in liquids is slow on a macroscopic scale, it is fairly rapid on the scale of biological cell distances (see EX 2.).

4 -4- Statistical Description (24.4) In 905 Einstein gave the first satisfactory theory of Brownian motion - the zig-zag motion of suspended particles in a liquid - where he theorized that the particles behave like molecules in solution whose motion follows the diffusion equation. One of his derivations was phenomenological with a microscopic basis (Stokes-Einstein equation, 24.37) while the other relied upon a statistical theory (a random walk) and was independently arrived atbymarian von Smoluchowski using a different approach (Einstein-Smoluchowski equation, 24.30). Einstein s theory was verified by the careful experimental work of Perrin. Perrin s experiments finally convinced the world of skeptics that molecules were real. Equation (24.20) gives the probability of finding a particle at a distance x from the origin after a certain amount of time. This equation was found phenomenologically by solving Fick s second law of diffusion. The same formula can be derived from microscopic data using a random walk model. random walk in one dimension Question: What is the probability that a particle has moved X integral steps from the origin after taking N random steps? The probability is the total number of ways the particle can be found at X after taking N steps (number of paths which end at X) divided by all the possible paths the particle can take. Since each step has two paths, either to the left or to the right, the number of possible paths is 2 N. Onthe right the number of ways the particle can arrive at different locations for, 2, and 3 steps is enumerated. Looking at the number of possible paths for a given number of steps we recognize that the enumeration is just the coefficients in the binomial expansion of (x + y) N given bypascal s triangle. So for atotal number of N steps with N + steps to the right and N steps to the left the number of paths W is W = N! N +!N! = N! N +!(N N + )! (24.23) Since each step is of unit length, the final position is X = N + N = N + (N N + ) = 2N + N or N + = ½(N + X) and since N = N + + N then N = N N + = ½(N X). Therefore the probability is P(N, X) = W 2 N = N! 2 N [½(N + X)]![½(N X)]! Using Stirling s approximation for the factorials, ln n! = (n + ½) ln n n + ln(2π ) /2 becomes P(N, X) = /2 2 e X 2 /2N π N Brownian movement of a gamboge particle in water. The points indicate the successive positions of the particle over 30-second intervals. The observations were performed by J. Perrin under a microscope at a magnification of approximately 3,000. Pascal's Triangle number of steps the probability which has the same Gaussian form as Eq. (24.20) and is equivalent when it is realized that Eq. (24.20) is normalized to while P(N, X) isnormalized to 2. Therefore <x 2 >* 3 2 for random coil on p. 7

5 π N /2 2 X /2N e = 4π Dt /2 e x2 /4Dt As the random walk statistical theory and the Fick s second law description need to converge to the same result, the exponents in the exponentials must be equivalent. If each random step covers a distance x 0 between collisions, so that x = Xx 0,and if the time between steps (collisions) is τ,sothat N = t/τ,then X 2 2N = x2 τ 2x 2 0 t = x2 4Dt => D = x2 0 2τ (24.30) where the last equality is the Einstein-Smoluchowski equation. If one were to associate the distance x 0 with λ, the mean free path, and τ with the average time between collisions, since x 0 = v τ then for gases D = λ v (24.3) 2 EX 2. The diffusion coefficient of the protein lysozyme is cm 2 /s. How long does it take this protein to diffuse an rms distance of a) 0 cm, b) 0 µm, typical diameter of a eukaryotic cell (one with a nucleus)? Motion on molecular scale can be observed by single molecule imaging via fluorescence microscopy. Protein of interest Organic dyes XFP s Quantum dots Antibody Natural or synthetic ligand Organic dye or or or Fluorescent protein Biocompatible quantum dot Trajectory length Short (< 0s) Very short (< s) Long (up to minutes) Probe accessibility Good to excellent Excellent Good Examples of trajectories (Membrane receptors on live neurons) 250 nm 250 nm 250 nm Single Molecule Tracking techniques in live cells based on fluorescent probes: left, glutamate receptors tracked in neuron plasma membrane; middle, glutamate receptor fused to variant of YFP; and right, AMPA receptor tracked in neuron plasma membrane with antibody functionalized quantum dots. Laurent Cognet, Cécile Leduc, Brahim Lounis, Advances in live-cell single-particle tracking and dynamic super-resolution imaging, Current Opinion in Chemical Biology, Elsevier, 204, 20C, p.78.

6 -6- Actual diffusion experiment and the experimental diffusion images and data. Note Gaussian Sliding and hopping diffusions of a protein (green) on an elongated DNA molecule (orange). Typical experimental trajectories on DNA could have sliding and hopping motions convoluted. (a) Image of a protein molecule, GFP-LacI, diffusing along DNA. Two large dots at the DNA ends are LacI-lacO 256 sites and the green segment on the DNA (arrow) is the trace of the diffusing GFP-LacI. (b) Image series of the diffusing protein (arrow) corresponding to green dots in (c), displacement versus t curve of the protein. (d) Fluorescence time trace of the diffusing GFP-LacI. (e) Gaussian distribution of x i x i-. The scale bar is 0.5 mm. Y.M. Wang and R.H. Austin, Single-Molecule Imaging of LacI Diffusing Along Nonspecific DNA, Biophysics of DNA-Protein Interactions from Single Molecules to Biological Systems, M.C. Williams, L.J. Maher III (eds.), Springer, 20, p. 9. Diffusion in Liquids and Viscosity of Liquids (24.5.) Diffusion results from the random thermal motion of molecules. Considering Brownian motion of a particle immersed in a solvent, the particle experiences a time-varying force due to random collisions with the solvent molecules. In addition the particle experiences a frictional force which opposes its motion due to the solvent s viscosity. The property that characterizes a particle s resistance to flow is its viscosity η (Greek eta). Newton s second law for the particle s motion must account for both of these forces. Einstein did this and found that the particle s mean square displacement increased with time as x rms = 2kTt/ f where f is the frictional force. If the diffusing particle is spherical Stokes law gives the frictional force as f = 6πηr where r is the radius of the sphere. Substituting these results into the diffusion coefficient given ineq. (24.2) yields D = x2 2t = kt f = kt 6πηr (24.37) the latter result is the Stokes-Einstein equation which allows a molecular parameter, r, to be determined. The radius determined by this equation is the hydrodynamic radius as it includes solvent molecules which are strongly associated with the particle. The units of viscosity are the poise (P) or 0. kg m s. Liquid viscosities are generally given incentipoise cp (0 2 )P. EX 3. The diffusion coefficient of ribonuclease from bovine pancreas, an enzyme that digests RNA, has been measured in a dilute buffer at 20 o C. The value is D = cm 2 s. Assuming that the protein molecule is a sphere, calculate its hydrodynamic radius.

7 -7- Diffusion Coefficients of Random Coils (24.6) In the previous section the diffusing particle was assumed to be reasonably rigid.. However, many large biological molecules, macro-- molecules, are different. In particular, they are flexible. As the DNA moves through a solution, a large amount of water is strongly associated with it and moves along with the DNA. Hydrodynamically, the DNAmolecule behaves as a highly hydrated sphere. The radius of this sphere is proportional to the three-dimensional size of the molecule. The simplest model to describe the solution structure of biomolecules is a radom coil. In this description DNAwould be considered a macromolecule of many rigid subunits of length l connected by bonds which can rotate. The direction of rotation (and the bond between subunits) is random. The over-all shape of the molecule follows a random walk with step sizes of length l. A measure of the shape is given by the distance r between the beginning and end of the molecule (the end-to-end distance). The mean-square end-to-end distance in a random coil macromolcule consisting of n subunits or segments of length l is r 2 = nl 2 (24.49) On p. 4 of these notes r 2 is enumerated for three random walk steps where r rms = r 2 is the average distance between the beginning and end of a path of n random steps each of length l. There r 2 is shown to be equal to n for unit step lengths, l =. The size of the macromolecule is approximated by the volume described by an n-step random walk, modeled as a sphere whose radius is given by r rms = l n. So the size is proportional to r rms which is proportional to n. However the number of steps n in the random walk is just the number of subunits in the macromolecule which is proportional to the molecular weight of the macromolecule, M. Thus r rms M if the macromolecule is flexible. From Eq. (24.37) the diffusion coefficient is inversely proportional to the radius of the sphere, D /r rms,sothe diffusion coefficient is also inversely proportional to M.