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##### Chemistry 232

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**Chemistry 232**Transport Properties**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.**Transport Properties Defined**• Examples (cont’d). • Viscosity - transport of linear momentum along a velocity gradient. • Electrical conductivity - transport of charge along a potential gradient.**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.**Transport Properties in an Ideal Gas**• Transport of matter. • Transport of energy. T -thermal conductivity coefficient. D - diffusion coefficient. • Transport of momentum. =viscosity coefficient.**Nd(-)**Nd(z=0) Nd Nd(+) -Z +Z Z=0 z Diffusion • Consider the following system.**Number Densities and Fluxes**• The number densities and the fluxes of the molecules are proportional to the positions of the molecules.**The Net Flux**• The net (or total) flux is the sum of the J(LR) and the J(RL).**The Diffusion Coefficient**• To a first approximation.**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. Collision Ao**The Final Equation**• Taking into account of the number of molecules that do not reach the wall.**-Z**+Z Z=0 Thermal Conductivity • Consider the following system.**(-)**(z=0) (+) z 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.**The Net Flux**• The net (or total) flux is the sum of the J(LR) and the J(RL).**The Thermal Conductivity Coefficient**• To a first approximation.**The Final Equation**• Taking into account of the number of molecules that do not reach the wall.**Direction of flow**-Z Z=0 +Z Viscosity • Consider the following system.**mvx(-)**mvx(z=0) mvx mvx(+) z Number Densities and Fluxes • Molecules traveling L R transport linear momentum (mvx()) to the new layer at z = 0!**The Net Flux**• The net (or total) flux is again the sum of the J(LR) and the J(RL).**The Viscosity Coefficient**• To a first approximation.**The Final Equation**• Taking into account of the number of molecules that do not reach the wall.**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**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.**Viscosities in Liquids**• Liquid layers flowing past one another experience significant attractive interactions. Direction of flow -Z Z=0 +Z**The Viscosity Equation**• For liquid systems E*a,vis= activation energy for viscous flow A = pre-exponential factor**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.**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)**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.**-**+ + - - + - + + - + - - + + - Drift Speed • Consider the following system. 1 2 Length = l**Forces on Ions**• Accelerating force • Due to electric field, Ef = (2 - 1) / l • Retarding force • Due to frictional resistance, F`= f s • S = drift speed • F = frictional factor - estimated from Stokes law**The Drift Speed**• The drift speed is written as follows zJ = charge of ion o = solvent viscosity e = electronic charge =1.602 x 10-19 C aJ = solvated radius of ion In water, aJ = hydrodynamic radius.**-**+ + - - + - + + - + - - + + - Connection Between Mobility and Conductivity • Consider the following system. d+=s+t d-=s-t -Z +Z Z=0**Ion Fluxes**• For the cations • J+ = + cJ NA s+ • += Number of cations • cJ = electrolyte concentration • S+ = Cation drift speed**Ion Flux (Cont’d)**• Flux of anions • J- = - cJ NA s- • - = Number of cations • cJ = electrolyte concentration • S- = anion drift speed**Ion Flux and Charge Flux**• Total ion flux Jion = J+ + J- = S cJ NA Note = + + - • Total charge flux Jcharge = Jion z e = (S cJ NA) z e = ( cJ NA) z e u Ef**The Conductivity Equation.**• Ohm’s law I = Jcharge A • The conductivity is related to the mobility as follows F = Faraday’s constant = 96486 C/mole**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.**The Cell Constant**• The cell constant, C*cell = * R* • * - literature value for conductivity of standard solution. • R* - measured resistance of standard solution. • Conductivity - = C*cellR • Standard solutions - KCl (aq) of various concentrations!**Molar Conductivities**• Molar conductivity M = 1000 / cJ Note c in mole/l • Molar conductivity - extensive property • Two cases • Strong electrolytes • Weak electrolytes**Ionic Contributions**• The molar conductivity can be assumed to be due to the mobilities of the individual ions.**Molar Conductivities (Cont’d)**• Molar conductivities as a function of electrolyte concentration. Strong electrolytes m Weak electrolytes C1/2**Strong Electrolyte Case**• Kohlrausch’s law om = molar conductivity of the electrolyte at infinite dilution A = molar conductivity slope - depends on electrolyte type.**Weak Electrolytes**• The Ostwald dilution law. K = equilibrium constant for dissociation reaction in solution.**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.**Conductivity and Ion Diffusion**• Connection between the mobility and conductivities of ions. DoJ = ionic diffusion coefficient at infinite dilution.**Ionic Diffusion (Cont’d)**• For an electrolyte. Essentially, a restatement of the law of independent migration. ONLY VALID NEAR INFINITE DOLUTION.**Transport Numbers**• Fraction of charge carried by the ions – transport numbers. t- = fraction of charge carried by anions. t+ = fraction of charge carried by cations.**Transport Numbers and Mobilities**• Transport numbers can also be determined from the ionic mobilities. u- = anion mobility. u+ = cation mobility.