atkins & de paula: atkins’ physical chemistry 9e chapter 20: molecules in motion

Atkins & de Paula:

Atkins’ Physical Chemistry 9e

Chapter 20: Molecules in Motion


transport property, the ability of a substance to transfer matter, energy, or some other property from one place to another.

diffusion, the migration of matter down a concentration gradient. thermal conduction, the migration of energy down a temperature gradient. electric conduction, the migration of electric charge along an electrical potential

gradient. viscosity, the migration of linear momentum down a velocity gradient. effusion, the emergence of a gas from a container through a small hole.

MOLECULAR MOTION IN GASES

20.1 The kinetic model of gases kinetic model, a model of a gas in which the only contribution to the energy is from

the kinetic energies of the molecules. three assumption of kinetic model, The gas consists of molecules of mass m in ceaseless random motion. The size of the molecules is negligible; d << λ Elastic collision, a collision in which the total translational kinetic energy of the

molecules is conserved.


20.1(a) Pressure and molecular speeds pressure of a gas, root mean square speed, the square root of the mean of the squares of the speeds: c

= v21/2 = (3RT/M)1/2.

tAvVnN xA )/(# of molecules

xmv2Momentum change,

2/12

222

2

22

3

3

Tconstantatconstant,3

Pressure

Force;momentumofchangeofRate

22

changeMomentum

22222

M

RTc

V

nMcp

pVV

nMc

V

vnMp

V

nMv

V

nMAv

V

tnMAv

V

tAvnmNmv

V

tAvnN

nRTpV

vvvvcxx

x

xxAx

xA

zyx

231 nMcpV


distribution of speeds, the function f(v) which, through f(v)dv, gives the fraction of molecules that have speeds in the range v to v + dv.

Maxwell distribution of speeds,

2

1

)(

torangetheinFranction 21

v

vdvvf

vv

RTMvevRT

Mvf 2/2

2/32

24)(


RTMvdvvdvdvdv

zyxRTMv

zyxzyx

RTMvx

kNRMmN

adxex

kTmvxx

kTmvxzyx

kTmvkTmvkTmvkTmvmvmvkTE

zyxk

evRT

Mvf

dvvfdvdvdveRT

Mdvdvdvvfvfvf

eRT

Mvf

RT

M

kT

mK

m

kTKdveKdvvf

eKvfvfvfvff

eeKeKeKef

mvmvmvE

zyx

x

AA

axx

x

zyxzyxk

2/22/3

4

2/2/3

2/2/1

2/3,

2/3

2/13/1)/(2/3/1

2/3/1

2/2/2//)(/

2212

212

21

22

2

2

2/122

2

2222212

212

21

24)(

)(2

)()()(

2)(

22

211)(

)()()()(


mean speed, 475 ms-1 for N2 in air and 25oC.

most probable speed, relative mean speed,

2/122/32/1

0

2/32/3

82

2

1

24

24)( 0

2232

M

RT

M

RT

RT

Mdvev

RT

Mdvvvfc adxexRTMv

ax

BA

BArel mm

mmkTc

2/18

2/1* 2

0

M

RTc

v

f

McTc /1,

ccrel2/12

mmm BA


20.1(b) The collision frequency collision diameter, the distance of approach corresponding to a collision. collision frequency, z, the number of collisions made by a molecule in an interval

divided by the length of the interval; ~5×109 s-1 for N2 at 1 atm and 25oC. collision cross-section, σ, σ = πd2 .

20.1(c) The mean free path mean free path, λ, the average distance a molecule travels between collisions; ~70 nm

for N2 at 1 atm=103 molecular diameter.

σ = πd2

λ

pkTzctc

ckTpctz

pckTttckTp

kTpVNtc

relrel

relrel

relrel

rel

//

//1

//1/

///1/1densitynumber

tubeofvolume

N

VT constatonindepentis

Tconstatpz),V(const constat NTz


20.2 Collisions with walls and surfaces collision flux, ZW, the number of collisions with an area in a given time interval

divided by the area and the duration of the interval, ~3×1023 cm-2 s-1 for O2 at 1 bar and 300 K.

collision frequency, the collision flux multiplied by the area of the region of interest.

tAv xN

# of molecules = N×volume = # of collisions

2/1//

2/1

2/12/1

0

2/2/1

0

0

0

)2(4

1

2

22)(

)(

)(collisionsof#

0When

0

22

mkT

pc

m

kTZ

m

kTdvev

RT

Mdxvfv

dxvfvZ

dvvfvtA

v

kTpVnNW

adxxex

kTmvxxx

xxW

xxx

x

A

axx

NNN

N

N

2/18

M

RTc


20.3 The rate of effusion effusion, the emergence of a gas from a container through a small hole. Graham’s law of effusion: the rate of effusion is inversely proportional to the square

root of the molar mass.

2/10

2/10

0 )2()2(effusionofRate

MRT

NpA

mkT

pAAZ A

W

Knudsen method, a method for the determination of the vapour pressures of liquids and solids.

tA

m

M

RTp

tmA

mZ

tmAZm

MRT

pN

mkT

pZ

W

W

AW

0

2/1)2()2(

0

0

2

loss,mass

2/12/1

See Example 20.2Effusion of a gas


20.4 Transport properties of a perfect gas flux, the quantity of a property passing through a given area in a given time interval

divided by the area and the duration of the interval. matter flux, the flux of matter, J(matter) dN/dz [m-2s-1]. energy flux, the flux of energy, J(energy) dT/dz [Jm-2s-1]. Fick’s first law of diffusion: the flux of matter is proportional to the concentration

gradient, J(matter) = –DdN/dz; D: diffusion coefficient.

cD 3

1

See Further information 20.1

]A[m,31

VCc coefficient of thermal conductivity, κ, the coefficient κ in J(energy) = –κdT/dz.


momentum flux, J(momentum) dv/dz. Newtonian (laminar) flow, flow that occurs by a series of layers moving past one

another. coefficient of viscosity, η, the coefficient η in J(momentum) = –ηdvx/dz.

See Further information 20.1

Nmc3

1

][3

1AcM


][31 AcM

D; λ 1/p D 1/p, T D T, λ 1/σ D 1/molecular dimension κ; λ 1/p, [A] p κ is independent on p, κ CV,m η; λ 1/p, [A] p η is independent on p, T η T

c

c

pkT /

2/18

M

RTc


MOLECULAR MOTION IN LIQUIDS

20.5 Experimental results NMR, EPR, inelastic neutron scattering, viscosity measurements, study on the

molecular motion in liquids. viscosity measurements, η eEa/RT (mobility of the particles e-Ea/RT )


20.6 The conductivities of electrolyte solutions conductance, G, the inverse of resistance; [G]=Ω-1 or S. conductivity, the constant κ in G = κA/l; [κ]=Sm-1. molar conductivity, Λm = κ/c. strong electrolyte, an electrolyte with a molar conductivity that varies only slightly

with concentration. weak electrolyte, an electrolyte with a molar conductivity that is normal at

concentrations close to zero, but falls sharply to low values as the concentration increases.

Kohlrausch’s law, for the concentration dependence of the molar conductivity of a strong electrolyte, Λm = Λm – Kc1/2.

limiting molar conductivity, Λm, the molar conductivity at zero concentration. law of the independent migration of ions, Λm = v+λ+ + v–λ–; λ+ and λ– are the limiting

molar conductivity of cations and anions, respectively, v+ and v– are the numbers of cations and anions per formula unit of electrolyte (v+ = v– = 1 for HCl, CuSO4, v+ = 1 and v– = 2 for MgCl2).

Chapter 20: Molecules in Motion20.7 The mobilities of ions20.7(a) The drift speed drift speed, s, the terminal speed when an accelerating force is balanced by the viscous

drag. mobility of an ion, the coefficient u in the expression s = uE; u = ze/6πηa. hydrodynamic radius (Stokes radius), the effective radius of a particle in solution. Grotthuss mechanism, a mechanism for the conduction of protons in solution in which

neighbouring H2O molecules transfer a proton.

EE

FF

F

EFE

uf

zes

affsl

zeze

l

fric

fric

whenzeroforcenet

relation)s(Stokes'6

Grotthuss mechanism; high u of H+

Table 20.5

1.5 ps


20.7(b) Mobility and conductivity ionic conductivity, the contribution of ions of one type to the molar conductivity: λ =

zuF.

Fuzvuzvvv

zuFvczvcuF

l

AG

RI

l

zvcuFAFAzvcuJAI

FzvcuzvcsFsvcNzezeionsJJ

svcNtA

tAsvcNionsJ

tAsvcNtAs

vcN

vc

m

c

l

usA

AA

A

A

m

)(

/

)()charge(

)(

inionsof#

density#

ionoftypeeachofconc.molar

0

/

/

E

E

E

E

Kohlrausch’s law, Λm = Λm –K c1/2

ion–ion interactions


20.7(c) ion–ion interactions relaxation effect, the reduction of an ion’s mobility due to distortion of the ionic

atmosphere. electrophoretic effect, the enhanced viscous drag due to the counter current of

oppositely charged ions. Debye–Hückel–Onsager theory, a theory of the concentration dependence of the

molar conductivity of a strong electrolyte, K = A + BΛm.

retardation of an ion’s mobility

No E

E

Λm = Λm – Kc1/2


I20.2 Ion channel passive transport, the tendency for a species to move spontaneously down a concentration

or potential gradient. active transport, transport that must be driven by an exergonic process. channel former, a protein that creates a hydrophilic pore in a membrane. ion channel, a protein that effects the movement of a specific ion down a potential gradient. ion pump, proteins that effect the active transport of ions. patch clamp technique, for studying ion transport across biological membranes.

patch clamp technique

K+ channel


DIFFUSION

20.8 The thermodynamic view The diffusion equation thermodynamic force, dw = dμ = (μ/x)p,T dx, dw = -F dx F = –(μ/x)p,T.

Fick’s first law of diffusion μ = μo+RTlna F =-RT(lna/x)p,T, for ideal solution F =-RT/c(c/x)p,T

Einstein relation, D = uRT/zF. J=-Ddc/dx, J=sc sc=-Ddc/dx s=-D/c dc/dx=DF /RT For electrolyte solutions; s=uE, F=NAezE= zFE uE= zFE D/RT D = uRT/zF

Stokes–Einstein equation, u=ez/f D = kT/f = kT/6πηa.

No charge term!!It can apply to neutral molecules in solution.


20.9 The diffusion equation diffusion equation (Fick’s second law of diffusion), the relation between the rate of change of concentration at a point and the spatial variation of the concentration at that point: c/t = D2c/x2.

2

2

Net;

Outflow;

Inflow;

x

cDll

x

cc

xD

x

cD

x

cD

x

cDJJ

l

JJ

t

cl

J

Aldt

AdtJ

t

cl

J

Aldt

JAdt

t

c

Nature abhors a wrinkle!!


20.9(a) Diffusion with convection convection, the transport of particles arising from the motion of a streaming fluid. convective flux, the amount of substance passing through an area in a given interval

by convection divided by the area and the length of the interval; J = cv.

generalized diffusion equation, the diffusion equation including convection

x

cv

l

vl

x

ccc

l

JJ

t

ccv

tA

tcAvJ

)(

20.9(b) Solutions of the diffusion equation

DtxeDtA

ntxc

x

cD

t

c 4/2/1

02

22

)(),(

x

cv

x

cD

t

c

2

2

DtreDt

ntrc 4/

2/30

2

)(8),(;diffusioinradialFor


20.10 Diffusion probabilities average distance travelled in diffusion, x = 2(Dt/π)1/2. root mean square distance travelled in diffusion, x21/2 = (2Dt)1/2.

2/1

0

4/2/10

0

2)(

1 2

Dt

dxxeDt

dxN

xcANx DtxA

dx

Adx dx # of particles=cANAdxProbability that any of the N0=n0NA particle in the slab =cANAdx/N0

Diffusion is a very slow process!


Impact on Nanotechnology; DLA diffusion limited aggregation (DLA)

Formation of nanoporous membranes through DLA process

S. W. Han et al., J. Mater. Chem. 2008, 18, 2208.

atkins & de paula: atkins’ physical chemistry 9e chapter 20: molecules in motion

Documents

migration of energy

assumption of kinetic

migration of matter

molecules of mass

kinetic energies

motion slide

velocity gradient

concentration gradient