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Chemistry 232 Transport Properties

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**Definitions 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.

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

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

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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(LR) and the J(RL).

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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! 2/3 of all molecules will make it to the wall in a given time interval t. Collision Ao

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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, = kBT. =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(LR) and the J(RL).

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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 (mvx()) to the new layer at z = 0! z mvx mvx(-) mvx(z=0) mvx(+)

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

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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**

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. + - 1 2 Length = l

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**Forces on Ions Accelerating force Retarding 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

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**The Drift Speed The drift speed is written as follows**

zJ = charge of ion o = solvent viscosity e = electronic charge =1.602 x C aJ = solvated radius of ion In water, aJ = hydrodynamic radius.

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**Connection Between Mobility and Conductivity**

Consider the following system. + - d+=s+t d-=s-t -Z Z=0 +Z

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**Ion Fluxes For the cations J+ = + cJ NA s+ += Number of cations**

cJ = electrolyte concentration S+ = Cation drift speed

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**Ion Flux (Cont’d) Flux of anions J- = - cJ NA s-**

- = Number of cations cJ = electrolyte concentration S- = anion drift speed

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**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

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**The Conductivity Equation.**

Ohm’s law I = Jcharge 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 / cJ 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. m C1/2 Strong electrolytes Weak electrolytes

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**Strong Electrolyte Case**

Kohlrausch’s law om = 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. DoJ = 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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