Atkins & de Paula:
Atkins’ Physical Chemistry 9e
Chapter 20: Molecules in Motion
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.
Chapter 20: Molecules in Motion
20.1(a) Pressure and molecular speeds
 pressure of a gas, pV  13 nMc2
 root mean square speed, the square root of the mean of the squares of the speeds: c
= v21/2 = (3RT/M)1/2.
Momentum change, 2m vx
nN A Avx t
nm NA Avx2 t nMAvx2 t
Moment umchange 
 2m vx 

2V
V
V
nMAvx2
Rat eof changeof moment um
; Force
V
nMvx2
nM  v x2  c 2  v 2   v x2    v 2y    v z2  nMc2
P ressure 
p
     

V
V
3V
pV  const antat
, const antT
1/ 2
nMc2
 3RT 
pV  nRT
p


 c  

3V
M


# of molecules
(nNA / V )  Avx t
Chapter 20: Molecules in Motion
 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,
3/ 2
 M  2 Mv 2 / 2 RT
f (v)  4 
 ve
 2RT 
Franctionin therangev1 tov2 

v2
v1
f (v)dv
Chapter 20: Molecules in Motion
Ek  12 m vx2  12 m vy2  12 m vz2
f  Ke
 Ek / kT
 Ke
 ( 12 mv x2  12 mv 2y  12 mv z2 ) / kT
 Ke
2
 mv x2 / 2 kT  mv y / 2 kT  mv z2 / 2 kT
e
e
1 / 3  mv x2 / 2 kT
f  f (v x ) f (v y ) f (v z )  f (v x )  K e




f (v x )dvx  1  1  K 1/ 3  e
 m 
K  

 2kT 

3/ 2
 mv x2
/ a)
/ 2 kT
1 / 3  2kT 
dvx     K 

m


 ax 2
dx  (
 e
 M 
 M , R  N Ak
mN
A

  

 2RT 
1/ 2
1/ 2
3/ 2
1/ 2
 M   Mv x2 / 2 RT
 f (v x )  
 e
2

RT


 M 
f (v x ) f (v y ) f (v z )dvx dvy dvz  

2

RT


 M 
    f (v)  4 

2

RT


dv x dv y dv z  4v 2 dv
3/ 2
e  Mv
2
/ 2 RT
3/ 2
2  Mv 2 / 2 RT
ve
dvx dvy dvz  f (v)dv
Chapter 20: Molecules in Motion
c  T , c  1/ M
 mean speed, 475 ms-1 for N2 in air and 25oC.
 M 
c   vf (v)dv  4 


 2RT 
3/ 2
 M 
v
e
dv







4



0
 2RT 
1/ 2
most probable speed, f  0  c*   2RT 
v
 M 

 relative mean speed,
crel

2
 3 ax
dx 1/ 2 a 2
0 x e
3  Mv 2 / 2 RT
 8kT
 
 
1/ 2




m A mB
m A  mB
mA  mB  m
crel  21/ 2 c
3/ 2
2
1/ 2
1  2RT   8RT 
 
 

2  M   M 
Chapter 20: Molecules in Motion
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.
volume of tube  
λ
number density  1 /   1 / crel t  N / V  p / kT
p / kT  1 / crel t  t  kT / crel p
z  1 / t  crel p / kT  crel N
z  T at constV(const N ), z  p at constT
  crel t  crel / z  kT / p
σ=
πd2
 is indepent onT at const V
Chapter 20: Molecules in Motion
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.
When vx  0

# of collisions NAt  vx f (vx )dvx
0

ZW  N  vx f (vx )dx
0


0
1/ 2
 M 
vx f (vx )dx  

 2RT 
1/ 2
 kT 
ZW  N 

 2m 



0
vx e
1/ 2
 8RT 
c 

 M 
1/ 2
 kT 
dvx    

 2m 
2

ax
dx 1 / 2 a
0 xe
1
p
V  p / kT
c N N nN
A /


4
(2m kT)1/ 2
# of molecules = N×volume = # of collisions
NAvx t
 mv x2 / 2 kT
Chapter 20: Molecules in Motion
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.
Rate of effusion  ZW A0 
pA0
pA0 N A

(2m kT)1/ 2 (2MRT )1/ 2
 Knudsen method, a method for the determination of the vapour pressures of liquids
and solids.
See Example 20.2
Effusion
of a gas
mass loss, m  ZW A0 mt
p
pN A
ZW 

m
 2RT 
( 2mkT )1 / 2 ( 2MRT )1 / 2
ZW 
      
 p  

A0 mt
M


1/ 2
m
A0 t
Chapter 20: Molecules in Motion
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.
1
D  c
3
See Further information 20.1
  13 c CV ,m[A]
 coefficient of thermal conductivity, κ, the coefficient κ in J(energy) = –κdT/dz.
Chapter 20: Molecules in Motion
 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.
1
3
  c mN
1
3
  Mc [ A]
See Further information 20.1
Chapter 20: Molecules in Motion
  kT / p
1/ 2
 8RT 
c 

 M 
 13 Mc[ A]
D; λ  1/p  D  1/p, c  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, c  T  η  T
Chapter 20: Molecules in Motion
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 )
Chapter 20: Molecules in Motion
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 Motion
20.7 The mobilities of ions
20.7(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.
E
F fric

ze
 F  zeE 
l
l
 fs f  6a (Stokes's relation)
net force zero when F  F fric  s 
zeE
 uE
f
1.5 ps
Grotthuss mechanism; high u of H+
Table 20.5
Chapter 20: Molecules in Motion
20.7(b) Mobility and conductivity
 ionic conductivity, the contribution of ions of one type to the molar conductivity: λ =
zuF.
molar conc. of each type of ion  vc
# density  vcN A
# of ionsin stA  vcN A stA
vcN A stA
 vcN A s
tA
uE
J (charge)  J (ions)  ze  ze  vcN A s  zvcsF s
 zvcuEF
J (ions) 
  / l
I  JA  zvcuEFA E

zvcuFA
l

A
 G 
R
l
m  / c
  zvcuF 

  / vc    zuF
I
0m  v   v   (v z  u  v z u ) F
 Kohlrausch’s law, Λm = Λm –K c1/2
ion–ion interactions
Chapter 20: Molecules in Motion
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.
No E
Λm = Λm – Kc1/2
E
retardation of an
ion’s mobility
Chapter 20: Molecules in Motion
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.
K+ channel
patch clamp technique
Chapter 20: Molecules in Motion
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.
Chapter 20: Molecules in Motion
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 = D2c/x2.
c JAdt J


t Aldt l
c
J Adt
J
Outflow;  

t
Aldt
l
c J  J 
Net; 
t
l
c
c
c
   c  
 2c
J  J   D  D
  D  D c   l   Dl 2
x
x
x
x   x  
x
Inflow;
Nature abhors a wrinkle!!
Chapter 20: Molecules in Motion
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.
J
cAvt
c J  J   
c   v
c
 cv  
 c  c  ( )l    v
At
t
l
x   l
x
 
 generalized diffusion equation, the diffusion equation including convection
c
 2c
c
 D 2 v
t
x
x
20.9(b) Solutions of the diffusion equation
n0
c
 2c
 x 2 / 4 Dt
 D 2  c( x, t ) 
e
t
x
A(Dt)1/ 2
For radial diffusioin; c(r , t ) 
n0
 r 2 / 4 Dt
e
8(Dt)3 / 2
Chapter 20: Molecules in Motion
20.10 Diffusion probabilities
 average distance travelled in diffusion, x = 2(Dt/π)1/2.
 root mean square distance travelled in diffusion, x21/2 = (2Dt)1/2.
Adx
# of particles=cANAdx
Probability that any of the N0=n0NA particle in the slab =cANAdx/N0
dx
 x  

0
xcANA
1
dx 
N0
(Dt)1/ 2


0
1/ 2
xe
 x 2 / 4 Dt
 Dt 
dx  2 
 
dx
Diffusion is a very slow process!
Chapter 20: Molecules in Motion
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.