6 Transport A

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6. Transport phenomena

Basel, 2008



6. Transport phenomena

References :

•P. Atkins, J. de Paula, “Atkins‘ PhysicalChemistry”, Oxford Univ. Press, Oxford, 8thed., 2006, Chapter 21.• I. Tinoco, K. Sauer, J.C. Wang, J.D. Puglisi“Physical Chemistry, Principles andapplications in biological sciences”, Prentice-Hall, New Jersey, 4th ed., 2002, Chapter 6.

Supplementary material :German version for thischapter (Prof. Huber lecture

from 2007).

1. Introduction

2. Phenomenological equations for transport properties

3. Transport properties of a perfect gas

4. Diffusion in a fluid

5. Measurement of the diffusion coefficients, viscosity and thermalconductivity



6.1 Introduction

Transport properties of a substance: the ability of the substance to transfermatter, energy or some other properties, from one place to another.

Diffusion : migration of matter down a concentration gradient.

Thermal conduction : migration of energy down a temperature gradient

Electric conduction : migration of electric charges along an electricpotential gradient

Viscosity : migration of linear momentum down to a velocity gradient.

Transport properties are expressed by phenomenological equations(empiric equations obtained from a summary of experiments andobservations)



6.2 Phenomenological equations

Flux, J represents the quantity of a property passing through a given areain a given interval of time/(area and duration of time).

Types of flux:

- Matter flux: if the matter is flowing through the area in the interval oftime > number of molecules/(m 2 s).

- Energy flux: if energy is flowing through the area in the interval of time> J/(m 2s).

Experimental observations on transport properties show that the flux of aproperty is proportional to the first derivative of a related property(gradient of the property).





Fick first law of diffusion

J > 0 flux towards +Oz direction

J < 0 flux towards -Oz direction

Fick first law of diffusion for the molecules of aperfect gas:

dz dN

Dmatter J −=)(

D – diffusion coefficient; [D] = m 2 /s

(6.2)





6.2.3 Viscosity

Hypothesis: A Newtonian fluid is formed by a series of layers moving past

one another, in a tube/container.- the layer next to the wall is stationary

- the velocity of the succesive layers depends on the distance from the wall.

Molecules move between layers and bring a x-component of linearmomentum they have in their original layer (initial layer) to the layer inwhich they move (final layer).

The final layer is accelerated orretarded, dependng on the linearmomentul of the molecule.



Viscosity

Viscosity: net retarding effect of molecules to different layers.It depends on the transfer of x-component of linear momentum (flux in theOz direction) into the layer of interest.

( )dz

d momentumcomp x J xv∝−−

Similar to the diffusion relation, the flux of energy due to the thermalmotion is :

( )dz

d momentumcomp x J xv

η −=−−η = coefficient of viscosity (orviscosity)

[η ] = kg/(m s) or Poise (P)

1P = 10 -1 kg/m s

(6.6)

(6.5)



Thermal conductivity and viscosity



6.3 Transport properties of a perfect gas

Model: simple kinetic theory of gases

Diffusion coefficient, D:

c D λ 31

= cλ - mean free path of particles

- mean speed of the particles in agas

λ decreases as the pressure in the gas is increasing > the diffusioncoefficient is decreasing. Gas molecules diffuse more slowlythan the pressure is high.

increases with the temperature > diffusion coefficient is increasing withthe temperature. Molecules in a hot sample diffuse more quicklythan in the case of a cool sample.

c

D is greater for small molecules, than for large molecules (because themean speed of the particles is inverse proportional to the mass of the

particles).

(6.7)



6.3.1 Thermal conductivity – perfect gas

(6.8)

Coefficient of thermal conductivity, k :

[ ] AC ck mv,31

λ =

cλ - mean free path of particles

- mean speed of the particles in agas

[ A] - molar concentration of the gasmolecules

Cv,m – molar heat capacity at Vconstant.

λ decreases as the pressure in the gas is increasing

λ decreases as the molar concentration of the gas is increasingk ≠ f(pressure)

k is higher for gases with a high heat capacity because a gradient oftemperature corresponds to a higher variation of energy.



6.3.2 Viscosity – perfect gas

Viscosity, η :

[ ] AcM λ η 31

=

cλ - mean free path of molecules

- mean speed of molecules in agas

[ A] - molar concentration of the gasmolecules

M – molar mass of molecules

increases when the temperature increases (T 1/2) viscosity isincreasing as temperature increases (for gases).c

λ decreases as the pressure in the gas is increasing

[ A] increases as the molar concentration of the gas isincreasing

η ≠ f(pressure)

(6.9)



6.3.3 Transport properties for gases

c(m s -1)

k

η Cvk



Transport properties for gases

Units

m2 s-1

J K-1

m-1

s-1

kg m -1 s-1



Diffusion coefficients for various media

P=1bar

H2 in H2

Xe in Xe

H2 in airI2 in air

NaCl in H 2O Na +

NaCl in H 2O Cl -

ethanol in H 2O

Ag in Cu (6.55 mol%)

Cu in Zn (75 mol%)

Substance and

diffusion medium

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