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Chemistry 232 Transport Properties

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Definitions Transport property. 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.

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Transport Properties Defined Examples (cont’d). Viscosity - transport of linear momentum along a velocity gradient. Electrical conductivity - transport of charge along a potential gradient.

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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.

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Transport Properties in an Ideal Gas Transport of matter. D - diffusion coefficient. =viscosity coefficient. T -thermal conductivity coefficient. Transport of energy. Transport of momentum.

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Diffusion Consider the following system. Z=0 +Z-Z z NdNd N d (- ) N d (z=0) N d (+ )

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Number Densities and Fluxes The number densities and the fluxes of the molecules are proportional to the positions of the molecules.

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The Net Flux The net (or total) flux is the sum of the J(L R) and the J(R L).

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The Diffusion Coefficient To a first approximation.

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The Complication of Long Trajectories Not all molecules will reach the imaginary wall at z=0! AoAo Collision 2/3 of all molecules will make it to the wall in a given time interval t.

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The Final Equation Taking into account of the number of molecules that do not reach the wall.

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Thermal Conductivity Consider the following system. Z=0 +Z -Z

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Number Densities and Fluxes Assume each molecule carries an average energy, = k B T. =3/2 for a monatomic gas. =5/2 for a diatomic gas, etc. z (- ) (z=0) (+ )

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The Net Flux The net (or total) flux is the sum of the J(L R) and the J(R L).

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The Thermal Conductivity Coefficient To a first approximation.

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The Final Equation Taking into account of the number of molecules that do not reach the wall.

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Viscosity Consider the following system. Z=0+Z-Z Direction of flow

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Number Densities and Fluxes Molecules traveling L R transport linear momentum (mv x ( )) to the new layer at z = 0! z mv x mv x (- ) mv x (z=0) mv x (+ )

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The Net Flux The net (or total) flux is again the sum of the J(L R) and the J(R L).

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The Viscosity Coefficient To a first approximation.

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The Final Equation Taking into account of the number of molecules that do not reach the wall.

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Viscosities Using Poiseuille’s Law Poiseuille’s law Relates the rate of volume flow in a tube of length l to Pressure differential across the tube Viscosity of the fluid Radius of the tube

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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.

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Viscosities in Liquids Liquid layers flowing past one another experience significant attractive interactions. Z=0+Z-Z Direction of flow

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The Viscosity Equation For liquid systems E * a,vis = activation energy for viscous flow A = pre-exponential factor

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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.

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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)

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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.

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Drift Speed Consider the following system. Length = l 11 2

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Forces on Ions Accelerating force Due to electric field, E f = ( 2 - 1 ) / l Retarding force Due to frictional resistance, F`= f s S = drift speed F = frictional factor - estimated from Stokes law

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The Drift Speed The drift speed is written as follows z J = charge of ion o = solvent viscosity e = electronic charge =1.602 x C a J = solvated radius of ion In water, a J = hydrodynamic radius.

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Connection Between Mobility and Conductivity Consider the following system. Z=0 +Z -Z d + =s + t d - =s - t

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Ion Fluxes For the cations J + = + c J N A s + + = Number of cations c J = electrolyte concentration S + = Cation drift speed

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Ion Flux (Cont’d) Flux of anions J - = - c J N A s - - = Number of cations c J = electrolyte concentration S - = anion drift speed

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Ion Flux and Charge Flux Total ion flux J ion = J + + J - = S c J N A Note = Total charge flux J charge = J ion z e = (S c J N A ) z e = ( c J N A ) z e u E f

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The Conductivity Equation. Ohm’s law I = J charge A The conductivity is related to the mobility as follows F = Faraday’s constant = C/mole

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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.

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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 R Standard solutions - KCl (aq) of various concentrations!

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Molar Conductivities Molar conductivity M = 1000 / c J Note c in mole/l Molar conductivity - extensive property Two cases Strong electrolytes Weak electrolytes

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Ionic Contributions The molar conductivity can be assumed to be due to the mobilities of the individual ions.

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Molar Conductivities (Cont’d) Molar conductivities as a function of electrolyte concentration. mm C 1/2 Strong electrolytes Weak electrolytes

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Strong Electrolyte Case Kohlrausch’s law o m = molar conductivity of the electrolyte at infinite dilution A = molar conductivity slope - depends on electrolyte type.

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Weak Electrolytes The Ostwald dilution law. K = equilibrium constant for dissociation reaction in solution.

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Law of Independent Migration Attributed to Kohlrausch. Ions move independently of one another in dilute enough solution. Table of o values for ions in textbook.

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Conductivity and Ion Diffusion Connection between the mobility and conductivities of ions. D o J = ionic diffusion coefficient at infinite dilution.

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Ionic Diffusion (Cont’d) For an electrolyte. Essentially, a restatement of the law of independent migration. ONLY VALID NEAR INFINITE DOLUTION.

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Transport Numbers Fraction of charge carried by the ions – transport numbers. t + = fraction of charge carried by cations. t - = fraction of charge carried by anions.

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Transport Numbers and Mobilities Transport numbers can also be determined from the ionic mobilities. u + = cation mobility. u - = anion mobility.

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