Chapter 1
When a particle is pulled with a constant force f, it eventually comes to a constant velocity,
called the drift velocity. For small particles that move not too fast, the drift velocity is
proportional to the applied force:
f
υd =
ζ
ζ = the friction or drag coefficient. This can be calculated for spherical objects by Stokes’
law:
ζ =6 πηrr
= the viscosity of the fluid medium (water is about 10-3 N m-2 s)
r = the radius of the particle
The amount of material that is transported per unit of time depends not only on the drift
velocity, but also on the concentration c. Flux is the quantity of the transported amount.
J=c υ d
Heavy particles will settle to the bottom of the vessel under influence of gravity. This is
called sedimentation.
The net force on a particle settling in a fluid medium is the force of gravity minus the lift
force due to the displaced fluid:
f =mg− ρf Vg=∆ ρVg
m = the mass of the particle
g = 9.81 m/s2
f = the density of the surrounding fluid
V = is the volume of the particle
∆ = the difference between the particle and the fluid
∆ ρVg
υd =
ζ
V = 4/3πrr3
When ∆ > 0 the particles move downward. When ∆ < 0 the particles move upward.
The process by which the particles spread out spontaneously is called diffusion.
Fick’s law:
∂c
J=−D
∂x
, 𝛛c/𝛛x = the slope of the concentration profile; the steeper the slope, the larger the
diffusive flux.
The total flux of molecules is the sum of the drift flux and the diffusive flux:
∂c
J tot =J drift +J dif =υ d c−D
∂x
When an equilibrium is reached, nothing changes anymore:
mg ∂c
J tot = c −D =0
ζ ∂x
Concentration c as a function of height x.
−mgx
ζD
c ( x )=c ( 0 ) e
The barometric formula:
−mgx
c ( x )=c ( 0 ) e KBT
Combining these formula gives: 𝛇D = kBT or:
kBT
D=
ζ
The Stokes-Einstein relation:
kBT
D=
6 πηrr
A salt solution in which we create an electric field 𝞮, for example by placing two parallel
plates in the solution across which we apply a voltage Ecell. If the electrodes are a distance 𝓵
apart, we know that the electric field is:
Ecell
E=
l
In such a field, ions of a charge ze (z is the valence and e = 1.60E-19 C) experience a force
fE=ze E
The ions will drift with a net speed:
When a particle is pulled with a constant force f, it eventually comes to a constant velocity,
called the drift velocity. For small particles that move not too fast, the drift velocity is
proportional to the applied force:
f
υd =
ζ
ζ = the friction or drag coefficient. This can be calculated for spherical objects by Stokes’
law:
ζ =6 πηrr
= the viscosity of the fluid medium (water is about 10-3 N m-2 s)
r = the radius of the particle
The amount of material that is transported per unit of time depends not only on the drift
velocity, but also on the concentration c. Flux is the quantity of the transported amount.
J=c υ d
Heavy particles will settle to the bottom of the vessel under influence of gravity. This is
called sedimentation.
The net force on a particle settling in a fluid medium is the force of gravity minus the lift
force due to the displaced fluid:
f =mg− ρf Vg=∆ ρVg
m = the mass of the particle
g = 9.81 m/s2
f = the density of the surrounding fluid
V = is the volume of the particle
∆ = the difference between the particle and the fluid
∆ ρVg
υd =
ζ
V = 4/3πrr3
When ∆ > 0 the particles move downward. When ∆ < 0 the particles move upward.
The process by which the particles spread out spontaneously is called diffusion.
Fick’s law:
∂c
J=−D
∂x
, 𝛛c/𝛛x = the slope of the concentration profile; the steeper the slope, the larger the
diffusive flux.
The total flux of molecules is the sum of the drift flux and the diffusive flux:
∂c
J tot =J drift +J dif =υ d c−D
∂x
When an equilibrium is reached, nothing changes anymore:
mg ∂c
J tot = c −D =0
ζ ∂x
Concentration c as a function of height x.
−mgx
ζD
c ( x )=c ( 0 ) e
The barometric formula:
−mgx
c ( x )=c ( 0 ) e KBT
Combining these formula gives: 𝛇D = kBT or:
kBT
D=
ζ
The Stokes-Einstein relation:
kBT
D=
6 πηrr
A salt solution in which we create an electric field 𝞮, for example by placing two parallel
plates in the solution across which we apply a voltage Ecell. If the electrodes are a distance 𝓵
apart, we know that the electric field is:
Ecell
E=
l
In such a field, ions of a charge ze (z is the valence and e = 1.60E-19 C) experience a force
fE=ze E
The ions will drift with a net speed:
