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1. Chemistry 232 Transport Properties

2. 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.

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. Nd(-) Nd(z=0) Nd Nd(+) -Z +Z Z=0 z Diffusion • Consider the following system.

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

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. Collision Ao

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

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

13. (-) (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.

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

15. The Thermal Conductivity Coefficient • To a first approximation.

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

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

18. 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!

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

20. The Viscosity Coefficient • To a first approximation.

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

22. 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

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. Direction of flow -Z Z=0 +Z

25. The Viscosity Equation • For liquid systems E*a,vis= activation energy for viscous flow A = 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 Speed • Consider the following system. 1 2 Length = l

30. 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

31. 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.

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

33. Ion Fluxes • For the cations • J+ = + cJ NA s+ • += Number of cations • cJ = electrolyte concentration • S+ = Cation drift speed

34. Ion Flux (Cont’d) • Flux of anions • J- = - cJ NA s- • - = Number of cations • cJ = electrolyte concentration • S- = anion drift speed

35. 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

36. 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

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*cellR • Standard solutions - KCl (aq) of various concentrations!

39. Molar Conductivities • Molar conductivity M = 1000  / cJ Note c in mole/l • Molar conductivity - extensive property • Two cases • Strong electrolytes • Weak electrolytes

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

41. Molar Conductivities (Cont’d) • Molar conductivities as a function of electrolyte concentration. Strong electrolytes m Weak electrolytes C1/2

42. Strong Electrolyte Case • Kohlrausch’s law om = molar conductivity of the electrolyte at infinite dilution A = molar conductivity slope - depends on electrolyte type.

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

44. 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.

45. Conductivity and Ion Diffusion • Connection between the mobility and conductivities of ions. DoJ = ionic diffusion coefficient at infinite dilution.

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

47. 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.

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