Molecular motion in liquids

(2015) Chemical Kinetics by M Lim 1

Intermolecular PE ~ KE

20.5 Experimental results• NMR, ESR: relaxation time the mobilities of the molecules• Inelastic neutron scattering: motion of particles,

internal dynamics of macromolecules

• Viscosity,

Consider the motion of an ion in liquid, and see that the information that motion provides can be used to infer the behavior of uncharged species, too

cf. the structure of liquids (section 16.6)the radial distribution function, g(r): the mathematical description of the average location of the particles

~ aE RTeη

(2015) Chemical Kinetics by M Lim 2

Electric resistance, resistivityResistance, (unit: ohm, )VR

I= Ω

Resistivity: specific electric resistance

VA

Ω =

( , resistivity, )lA

mρ ρ= Ω

(2015) Chemical Kinetics by M Lim 3

20.6 The conductivities of electrolyte solutions-11Conductance, (unit: mho, , siemens)1G S

R−Ω ≡=

Λm varies with C# of ions in solutions might not be proportional to C.

1 A CSV Vs

= = =Ω

Strong electrolyte, (experimentally)

(Kohlrausch's law)

: limiting molar conductivity is found to depend more on the stoichiometry of the

om m

om

K C

K

Λ = Λ −

Λelectrolyte

( , conductivity, )Al

Sm

κκ=

[ ]1 2

3

of a solution

Molar conductivity, m

S m S mmol m moC l

κ

κ−

−

∝

=

≡

Λ

(2015) Chemical Kinetics by M Lim 4

The limiting molar conductivity of BaCl2 in water at 298K.

om ν λ ν λ+ + − −Λ ≡ +

( ) 2 1 2 112.72 2 7.63 27.98 om mS m mol mS m mol− −Λ = + × =

The law of the independent migration of ions (by Kohlrausch)

: limiting molar conductivity of the cations : limiting molar conductivity of the anions

λλ+

−

20.6 The conductivities of electrolyte solutions-2

(2015) Chemical Kinetics by M Lim 5

20.7 The mobilities of ions(a) Drift speeds

An ion ( ) in an electric field ( )

Frictional force, , 6 (frictional coeff.)fric

ze El

F zeEF fs f a

φ

πη

∆=

== =

Illus. Cs+ (a=170 pm) in 1.0 cP.8 2 1 1

4

5 10For 1 electric field, 5 an ion passing about 10 solvent molecules per second.

u m V sV cm s m sµ

− − −≈ ×=

: hydrodynamic radius (or Stokes radius)

Li Cs

Li Cs

a

a a

u u+ +

+ +

>

∴ <

When , the ions reach a terminal speed, the drift spee

d,

,

( : mobility of the ion)

6

fricF F s

z zeEsf

ze zeuf a

eE fs

us uEπη

=

=

= =∴≡

=∴

(2015) Chemical Kinetics by M Lim 6

20.7 The mobilities of ionsGrotthuss mechanism

2

The proton, although it is very small, has a very high =349.6 / at 298Instead of a highly solvated proton moving,rearrangement of bonds in a group of water molec

Scm mol Kλ

λ

ules (Grotthuss mechanism)

(2015) Chemical Kinetics by M Lim 7

Review 20-3Conductance, ( , conductivity, )

Molar conductivity,

For strong elctrolyte, (Kohlrausch's law)

m

om m

SAGl

CK C

mκκ

κ

=

Λ ≡

Λ = Λ −

( )Mobility of the 66

ion : ze zeu f af a

πηπη

= = =

Drift speed: s uE=

Limiting molar conductivity: om ν λ ν λ+ + − −Λ ≡ +

(2015) Chemical Kinetics by M Lim 8

20.7 (b) mobility and conductivityzuFλ =

( )

( )

arg

number density

Current,

,

A

Aions A

ch e ions A

CNs tA CNJ s CN

A tJ zeJ z s C zs C z uE CF

I JA z uE CFA zu CFAE zu CFAl

l AI IR RA l

zu CF

z FC C

e N F

u

νν ν

ν ν ν

φν ν ν

φ κ φκ

κ νκ κλ νλν

∆ ⋅= =

∆= = = =

∆= = = =

∆ ∆= = → =

∴ =

= = Λ = =

4

-

-8 2 1 1

2 1

For CuSO ( 2, 2, 1,

~ 5 10 )

~ 19 om

z z

u m V s

mS m mol

ν ν+

+

− − −

−

= == =

×

Λ

( )om

om

z u z u Fν λ ν λ

ν ν+ + + − − −

+ + − −

Λ = +

Λ ≡ +

(Faraday constant, )AF N e=

(2015) Chemical Kinetics by M Lim 9

20.7 (c) ion-ion interactionsKohlrausch's law: o

m m K CΛ = Λ −C1/2 dependency of the conductivity in

• Relaxation effect: the reduction of the ions’ mobility due to the opposite charge

• Electrophoretic effect: a enhanced viscous drag due to the ionic atmosphere moving in an opposite direction to the central ion.

• Debye-Huckel-Onsager theory: attempt to obtain quantitative expression

Diffusion

(2015) Chemical Kinetics by M Lim 10

20.8 The thermodynamic viewIonic motion to migration of neutral molecules

max,,

ep T

ddw d dx dxdxµµ = = = −

F

,

,

, ,

concentration gr

For a concentration gradient, and if ln ln 1

(force a adientrising from a )1p T p T p T

p T

a ca c c

x x c

cRTc x

x

≈

∂ ∂ ∂ = =

∂

∂ ∂ ∂

= −

∂∴

F

,

,

Thermodynamic force,

ln ( ln )

p T

o

p T

ddx

aRT RT ax

µ

µ µ

≡ −

∂ = − = + ∂

F

(2015) Chemical Kinetics by M Lim 11

20.8 (b) Einstein relation (D and u)Fick's first law of diffusion

cJ Dx∂

= −∂

0

In case of an ion in an electric field, , for 1 mole

(Einstein relation)

Es uE qE zeN E zFE

D D zFDs uE zFE u

uRTD

RT RT R

F

T

z

= = = =

= =

=

= ∴ =

FF

mole density

moles

cc As tJ sc

A t⋅ ∆

= =∆

D c D c Dsc x c RT RT∂ = − = − − = ∂

F F

c As tJ scA t⋅ ∆

= =∆

,

,

1p T

p T

cRTc x

c cx RT

∂ = − ∂

∂ ∴ = − ∂

F

F

(2015) Chemical Kinetics by M Lim 12

20.8 (c) The Stokes-Einstein equation

( )

2 2

20 2 2

Recall

m

z F DzuFRT

F z D z DRT

λ λ

ν λ ν λ ν ν+ + − − + + + − − −

= ∴ =

Λ = + = +

Ex 20.4 D, λ, a of SO42- in water at 298K when u=8.29×10-8 m2 s-1V-1.

89 2 1

8 2 1

2310

9 2 1

8.29 10 8.314 298 1.1 102 96485

2 8.29 10 96485 0.016 1 1.38 10 298 2.2 10 220

6 6 6 0.000891 1.1 10

uRTD m szF

zuF S m molf kTa m pm

D m s

λ

πη πη π

−− −

− −

−−

− −

× ⋅ ⋅= = = ×

⋅= = ⋅ × ⋅ =

× ⋅= = = = × =

⋅ ⋅ ×

Bond length 144 pm:small degree of solvation

Recall ,

6

ez zFD zeDu uf R

kT kTDf

T kT

aπη=

= =

∴ =

=

(2015) Chemical Kinetics by M Lim 13

Review 20-4Limiting molar conductivity: o

m ν λ ν λ+ + − −Λ ≡ +

( )Mobility of the ion : 66

ze zeu f af a

πηπη

= = =

zuFλ =

(Stokes-Einstein relation)

, (Einstein relation)

kTDf

zFD uRTu DRT zF

=

= =