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CHAPTER 19 – Molecules in Motion I. Transport in Gases.

A. Fick’s Law of Diffusion.

1. Matter transport is driven by concentration gradient.

Flux (matter) = Jm= −DdN

dz

N = number density of particles Flux Jm is measured in moles or molecules per unit area per unit

time D = diffusion coefficient (m2/s)

Particles are in random motion and moving between cells passively. Flux Jm ∝ net number moving in forward direction ≠0 if N is different between cells Jm > 0 if N is less on the right Jm < 0 if N is less on the left

Nature tries to fill every void, a consequence of entropy increase, i.e., increase in the dispersal of matter.

2. Thermal energy transport is driven by temperature gradient.

Flux (energy) = JE= −κ

dTdz

κ = coefficient of thermal conductivity = J/m2-K = W/m-K

3. Transfer of momentum is driven by velocity gradient.

Flux (momentum) = Jp= −η

dvx

dz

η = viscosity = kg/m-s 1 Poise (P) ≡ 0.1 kg m-1s-1

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B. The transport parameters D, κ and η.

1. Diffusion coefficient D.

Use the Maxwell-Boltzmann distribution of molecular velocities. Can calculate one-way flux of molecules passing through boundary

between two adjacent “cells”, going both directions.

J(net flux L to R) = J L → R( ) − J L → R( )

= −13λvmean

dNdz

#

$%%

&

'((L,R boundary

So D= −13λvmean

λ = mean free path = kBT/σp

vmean = mean speed = 8RTπM

"

#$$

%

&''

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D ↑ as T ↑ D ↓ as σ ↑ D ↓ as p ↑

2. Thermal conductivity κ. Works similarly. Molecules passing between cells with slightly

different T’s carry their average thermal energy with them across boundary.

J(net flux L to R) = J L → R( ) − J L → R( ) On average each molecule carries across boundary an average energy νkBT, where ν = 3/2 for atoms and ν = more for molecules, with more degrees of freedom.

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J(net energy flux) = −13νvmean

λNkB

dTdz

$

%&&

'

())L,R boundary

κ =13νvmean

λNkB

κ ↑ as λ ↑ but λ =kBT

σp=1

σN

κ ↓ as σ ↑ or N ↑ Since constant a volume heat capacity of ideal gas is:

CV,m

= Navo

∂νkBT

∂T

#

$%%

&

'((V

the flux equation can be re-written such that

κ =13vmean

λ particles#$

%&CV,m

κ ↑ as CV,m ↑ and [ ] is molar concentration

3. Viscosity η.

η =13vmean

λmN

m = mass of one particle

η =pMDRT

or MD[particles]

M = molar mass η of gas is independent of p

η of gas ↑ as T ↑ (opposite behavior of liquids) C. Effusion. Rate of randomly hitting a “hole” in the wall of area A.

Rate effusion = ZwA =

pANavo

2πMRT( )12

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II. Molecular Motion in Liquids.

A. Ion Transport.

1. Conductance of electrolyte solutions.

a. The conductance properties of electrolytes are due to migration

of ions and obeys Ohm’s Law: current I = V/R (voltage/resistance)

(amperes) = (volts v/ohms Ω) b. Fits standard “linear response theory” format: “flow” = “conductance coefficient” x “driving force” I = GV G = 1/R = conductance (Atkins uses G instead of traditional L) G in Ω-1 = “Siemans” unit c. G depends on cross-sect area A of conductor, the length

traversed and intrinsic properties of the ions (which we are most interested in).

G =κA

A = electrode area (depends on cell geometry)= length between electrodes (depends on cell geometry)κ=conductivity (intrinsic property of ions here)

cell constant defined as k = / A . We want to factor this out.

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If type 1 and type 2 are identical:

Z12 → Z11 =

€

v πd2

2

N12

where d is diam of 1

€

v is 8kBT

πm1

1/2

II. Molecular Motion in Liquids.

A. Ion Transport.

1. Conductance of electrolyte solutions.

a. The conductance properties of electrolytes is due to migration of ions and

obeys Ohm’s Law: current I = V/R (voltage/resistance)

(amperes) = (volts v/ohms Ω) b. Fits standard “linear response theory” format: “flow” = “conductance coefficient” x “driving force” I = GV G = 1/R = conductance (Atkins uses G instead of traditional L) G in Ω-1 = “Siemans” unit c. G depends of cross-sect area A of conductor, the length traversed

€

and intrinsic properties of the material (which we are most interested in).

€

G =κA

A = electrode area (depends on cell geometry)

= length between electrodes (depends on cell geometry)

κ = conductivity (intrinsic property of ions here)

cell constant defined as

€

k = / A . We want to factor this out.

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d. Find k for your cell by studying a standard solution of known κ. 0.02000M KCl at 25° C has κ = 0.27653 Ω-1m-1 GKCl0.02000= 0.27653/k measure G (= 1/R) solve for k for our cell e. Now, you’ve eliminated geometric aspects of the cell, can

convert all subsequent R readings to κ. κ depends on concentrations of ions and their intrinsic mobilities. ↑ ↑ a sheer measure of number a more interesting microscopic property f. For electrolyte of concentration c, of stoichiometry

Aν+

Bν−

→ ν+AZ+ + ν

−BZ−

with fractional ionization α (0 ≤ α ≤ 1) κ = 1000αcνF(u+ + u-) Here 1000αc is conc of A+ and/or B- in mol-m-3 ν = ν+Z+ = ν-|Z-| = 1 for KCl, KAc, HCl, etc. F is Faraday 96485 C/mol, charge of 1 mole of protons And (u+ + u-) are the mobilities of ions g. Let’s define a new quantity which factors out the concentration aspect. Λm = κ/1000νc Λm called Molar conductance or equivalent conductance or molar conductivity. Λm = αF(u+ + u-) α = 1 for strong electrolytes KCl, HCl, KAc α < 1 for HAc weak acid, etc.

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h. Attraction between ions lowers the mobilities:

Λm = Λm°(1 - β√C) -general observation for strong electrolytes, (α =1) (Kohlrausch’s Law) where term Λm° is for ions acting completely independently,

called the limiting molar conductance of the electrolyte. Λm° = λo

+ + λo- ! limiting molar conductances of individual

species λo + = Fu+

o λo - = Fu-

o u+

o , u-o = mobilities at ∞ dilution

or for general stoichiometry: Λm° = ν+λo

+ + ν+λo-

Note: ions independently contribute to conductance - law of

independent migration of ions (infinitely dilute limit). i. Λm not linear with √c for HAc (weak electrolyte) due to α being <

1 and concentration-dependent. Hard to get Λm°.

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It still can be determined by combining other quantities to obtain:

Λm° (HAc) = λo

H+ + λoAc-

uo

Ac- = λoAc-/F

Example: Λo(HAc) = Λo(HCl) + Λo(KAc) - Λo(KCl) all are strong electrolytes which can be measured and

extrapolated to ∞-dilution Proof: Λo(HAc) = (λo

H+ + λoCl-) + (λo

K+ + λoAc-) - (λo

K+ + λoCl-)

= (λo

H+ + λoAc-)

j. α , degree of ionization can be obtained for a weak acid HAc: α = Λm/Λm° ↑ ↑ measure estimate by combining strong electrolyte data as above True: since Λm = αF(u+ + u-) and Λm° = F(u+ + u-) k. Another method of obtaining Λm° and Ka of a weak acid -

Ostwald’s dilution law Can derive by starting with α = Λm/Λm° and substituting in an expression for α in terms of Ka to obtain:

1Λm

=1Λmo+

Λmc

Ka(Λmo)2

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2. What determines ion mobilities u+, u- themselves?

a. Diffusive motion of ions in electric field:

Drift speed s determines overall conductivity, hence mobility. s determined by balance between driving force of the field E and

the opposing frictional resistance force: force of field = zeE = charge of ion x potential field frictional resistance force = - fs f = friction coefficient s = speed of drift Set equal: zeE = -(-fs) zeE = fs s = zeE/f = drift speed of ion. b. Friction coefficient of a sphere of radius r in a fluid: Stokes’ Law

f = 6πηr η is viscosity; liquid H2O η ≈ 1 centiPoise c. s relation to E s = uE

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2. What determines ion mobilities u+, u- themselves? a. Diffusive motion of ions in electric field:

Drift speed s determines overall conductivity, hence mobility. s determined by balance between driving force of the field E and the opposing

frictional resistance force: force of field = zeE = charge of ion x potential field frictional resistance force = - fs f = friction coefficient s = speed of drift Set equal: zeE = -(-fs) zeE = fs s = zeE/f = drift speed of ion. b. Friction coefficient of a sphere of radius r in a fluid: Stokes’ Law

f = 6πηr η is viscosity; liquid H2O η ≈ 1 centiPoise c. s relation to E s = uE where:

u = ze/f = ionic mobility (as in previous section) Can measure u from conductance experiment. ze known, can calculate f. Then can calculate r, effective radius of ion, called the hydrodynamic radius.

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where: u = ze/f = ionic mobility (as in previous section) Can measure u from conductance experiment. ze known, can calculate f. Then can calculate r, effective radius of ion, called the

hydrodynamic radius. d. Hydrodynamic radius may not equal actual physical radius. (vdW

radius)

e. Anomalously high speed of conduction of hydrogen ions in water

- the Grotthus mechanism:

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Notes: