2Definitions 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.
3Transport Properties Defined Examples (cont’d).Viscosity - transport of linear momentum along a velocity gradient.Electrical conductivity - transport of charge along a potential gradient.
4Migration 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.
5Transport Properties in an Ideal Gas Transport of matter.Transport of energy.T -thermal conductivity coefficient.D - diffusion coefficient.Transport of momentum.=viscosity coefficient.
6Diffusion Consider the following system. Z=0 +Z -Z z Nd Nd(-) Nd(z=0)
7Number Densities and Fluxes The number densities and the fluxes of the molecules are proportional to the positions of the molecules.
8The Net FluxThe net (or total) flux is the sum of the J(LR) and the J(RL).
9The Diffusion Coefficient To a first approximation.
10The 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
11The Final EquationTaking into account of the number of molecules that do not reach the wall.
12Thermal ConductivityConsider the following system.Z=0+Z-Z
13Number 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)(+)
14The Net FluxThe net (or total) flux is the sum of the J(LR) and the J(RL).
15The Thermal Conductivity Coefficient To a first approximation.
16The Final EquationTaking into account of the number of molecules that do not reach the wall.
17ViscosityConsider the following system.Z=0+Z-ZDirection of flow
18Number Densities and Fluxes Molecules traveling L R transport linear momentum (mvx()) to the new layer at z = 0!zmvxmvx(-)mvx(z=0)mvx(+)
19The Net FluxThe net (or total) flux is again the sum of the J(LR) and the J(RL).
20The Viscosity Coefficient To a first approximation.
21The Final EquationTaking into account of the number of molecules that do not reach the wall.
22Viscosities 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
23Transport in Condensed Phases Discussions of transport properties have taken place without including a potential energy term.Condensed phases - the potential energy contribution is important.
24Viscosities in Liquids Liquid layers flowing past one another experience significant attractive interactions.Z=0+Z-ZDirection of flow
25The Viscosity Equation For liquid systemsE*a,vis= activation energy for viscous flowA = pre-exponential factor
26Conductivities 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.
27Resistance 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)
28Charge 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.
29Drift SpeedConsider the following system.+-12Length = l
30Forces 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
31The 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.
32Connection Between Mobility and Conductivity Consider the following system.+-d+=s+td-=s-t-ZZ=0+Z
33Ion Fluxes For the cations J+ = + cJ NA s+ += Number of cations cJ = electrolyte concentrationS+ = Cation drift speed
34Ion Flux (Cont’d) Flux of anions J- = - cJ NA s- - = Number of cationscJ = electrolyte concentrationS- = anion drift speed
35Ion 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
36The Conductivity Equation. Ohm’s lawI = Jcharge AThe conductivity is related to the mobility as followsF = Faraday’s constant = C/mole
37Measurement 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.
38The 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!
39Molar Conductivities Molar conductivity M = 1000 / cJNote c in mole/lMolar conductivity - extensive propertyTwo casesStrong electrolytesWeak electrolytes
40Ionic ContributionsThe molar conductivity can be assumed to be due to the mobilities of the individual ions.
41Molar Conductivities (Cont’d) Molar conductivities as a function of electrolyte concentration.mC1/2Strong electrolytesWeak electrolytes
42Strong Electrolyte Case Kohlrausch’s lawom = molar conductivity of the electrolyte at infinite dilutionA = molar conductivity slope - depends on electrolyte type.
43Weak Electrolytes The Ostwald dilution law. K = equilibrium constant for dissociation reaction in solution.
44Law of Independent Migration Attributed to Kohlrausch.Ions move independently of one another in dilute enough solution.Table of o values for ions in textbook.
45Conductivity and Ion Diffusion Connection between the mobility and conductivities of ions.DoJ = ionic diffusion coefficient at infinite dilution.
46Ionic Diffusion (Cont’d) For an electrolyte.Essentially, a restatement of the law of independent migration.ONLY VALID NEAR INFINITE DOLUTION.
47Transport NumbersFraction of charge carried by the ions – transport numbers.t+ = fraction of charge carried by cations.t- = fraction of charge carried by anions.
48Transport Numbers and Mobilities Transport numbers can also be determined from the ionic mobilities.u+ = cation mobility.u- = anion mobility.