2 Definitions Transport property. Examples. The ability of a substance to transport matter, energy, or some other property along a gradient.Examples.Diffusion - transport of matter along a concentration gradient.Thermal conductivity - transport of thermal energy along a temperature gradient.
3 Transport Properties Defined Examples (cont’d).Viscosity - transport of linear momentum along a velocity gradient.Electrical conductivity - transport of charge along a potential gradient.
4 Migration Down Gradients Rate of migration of a property is measured by a flux J.Flux (J) - the quantity of that property passing through a unit area/unit time.
5 Transport Properties in an Ideal Gas Transport of matter.Transport of energy.T -thermal conductivity coefficient.D - diffusion coefficient.Transport of momentum.=viscosity coefficient.
6 Diffusion Consider the following system. Z=0 +Z -Z z Nd Nd(-) Nd(z=0)
7 Number Densities and Fluxes The number densities and the fluxes of the molecules are proportional to the positions of the molecules.
8 The Net FluxThe net (or total) flux is the sum of the J(LR) and the J(RL).
9 The Diffusion Coefficient To a first approximation.
10 The Complication of Long Trajectories Not all molecules will reach the imaginary wall at z=0! 2/3 of all molecules will make it to the wall in a given time interval t.CollisionAo
11 The Final EquationTaking into account of the number of molecules that do not reach the wall.
12 Thermal ConductivityConsider the following system.Z=0+Z-Z
13 Number Densities and Fluxes Assume each molecule carries an average energy, = kBT.=3/2 for a monatomic gas.=5/2 for a diatomic gas, etc.z(-)(z=0)(+)
14 The Net FluxThe net (or total) flux is the sum of the J(LR) and the J(RL).
15 The Thermal Conductivity Coefficient To a first approximation.
16 The Final EquationTaking into account of the number of molecules that do not reach the wall.
17 ViscosityConsider the following system.Z=0+Z-ZDirection of flow
18 Number Densities and Fluxes Molecules traveling L R transport linear momentum (mvx()) to the new layer at z = 0!zmvxmvx(-)mvx(z=0)mvx(+)
19 The Net FluxThe net (or total) flux is again the sum of the J(LR) and the J(RL).
20 The Viscosity Coefficient To a first approximation.
21 The Final EquationTaking into account of the number of molecules that do not reach the wall.
22 Viscosities Using Poiseuille’s Law Relates the rate of volume flow in a tube of length l toPressure differential across the tubeViscosity of the fluidRadius of the tube
23 Transport in Condensed Phases Discussions of transport properties have taken place without including a potential energy term.Condensed phases - the potential energy contribution is important.
24 Viscosities in Liquids Liquid layers flowing past one another experience significant attractive interactions.Z=0+Z-ZDirection of flow
25 The Viscosity Equation For liquid systemsE*a,vis= activation energy for viscous flowA = pre-exponential factor
26 Conductivities in Electrolyte Solutions Fundamental measurement of the mobilities of ions in solutions electrical resistance of solution.Experimentally - measure AC resistance.Conductance - G = 1/R.R = AC resistance of solution.
27 Resistance Measurements Resistance of sample depends on its length and cross-sectional area = resistivity of the solution. = conductivity of the solution.Units of conductivity = S/m = 1/( m)
28 Charge Transport by Ions Interpreting charge transport.Amount of charge transported by ions.The speed with which individual ions move.The moving ions reach a terminal speed (drift speed).Force of acceleration due to potential gradient balances out frictional retarding force.
29 Drift SpeedConsider the following system.+-12Length = l
30 Forces on Ions Accelerating force Retarding force Due to electric field, Ef = (2 - 1) / lRetarding forceDue to frictional resistance, F`= f sS = drift speedF = frictional factor - estimated from Stokes law
31 The Drift Speed The drift speed is written as follows zJ = charge of iono = solvent viscositye = electronic charge=1.602 x CaJ = solvated radius of ionIn water, aJ = hydrodynamic radius.
32 Connection Between Mobility and Conductivity Consider the following system.+-d+=s+td-=s-t-ZZ=0+Z
33 Ion Fluxes For the cations J+ = + cJ NA s+ += Number of cations cJ = electrolyte concentrationS+ = Cation drift speed
34 Ion Flux (Cont’d) Flux of anions J- = - cJ NA s- - = Number of cationscJ = electrolyte concentrationS- = anion drift speed
35 Ion Flux and Charge Flux Total ion fluxJion = J+ + J-= S cJ NANote = + + -Total charge fluxJcharge = Jion z e= (S cJ NA) z e= ( cJ NA) z e u Ef
36 The Conductivity Equation. Ohm’s lawI = Jcharge AThe conductivity is related to the mobility as followsF = Faraday’s constant = C/mole
37 Measurement of Conductivity Problem - accurate measurements of conductivity require a knowledge of l/A.Solution - compare the resistance of the solution of interest with respect to a standard solution in the same cell.
38 The Cell Constant The cell constant, C*cell = * R* * - literature value for conductivity of standard solution.R* - measured resistance of standard solution.Conductivity - = C*cell RStandard solutions - KCl (aq) of various concentrations!