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.

Collisions With Walls and Surfaces Rate at which molecules collide with a wall of area A

Effusion Rate at which molecules pass through a small hole of area A o, r

Effusion (Cont’d) Effusion. A gas under pressure goes (escapes) from one compartment of a container to another by passing through a small opening.

Effusion

The Effusion Equation Graham’s Law - estimate the ratio of the effusion rates for two different gases. Effusion rate of gas 1  r 1.

Effusion Equation (Cont’d) Effusion rate of gas 2  r 2.

Effusion Ratio Ratio of effusion rates.

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 momentum. Transport of energy. D - diffusion coefficient.  =viscosity coefficient.  T -thermal conductivity coefficient.

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

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(L  R) and the J(R  L).

The Diffusion Coefficient To a first approximation.

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.

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

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

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)  (+ )

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

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.

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

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 (+ )

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

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. Z=0+Z-Z Direction of flow

The Viscosity Equation For liquid systems E * a,vix = 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. Length = l 11 2

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

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.

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

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

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

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

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

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

Molar Conductivities Molar conductivity  M = 1000  / c J 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. mm C 1/2 Strong electrolytes Weak electrolytes

Strong Electrolyte Case Kohlrausch’s law  o m = 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. D o J = 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 cations. t - = fraction of charge carried by anions.

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