Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...
Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ... Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...
5.3 Transport processes in gases 111 I The general transport equation: We consider the net flux of Γ through an area at = 0 that is transported by gas molecules coming from an area above at = 0 + and from below from an area above at = 0 − (where isthemeanfreepathlength). Denoting the respective transport quantity per molecule as Γ, these partial fluxes are given by the number of gas kinetic collisions hitting on ( = 0 ) times the Γ carried by each gas molecule (number density ): • Using the gas kinetic wall collision frequency wall = 1 (Eq. 5.92), we obtain 4 Γ + = 1 µ Γ µΓ 4 × × 0 + (5.102) and • Net effect: Γ − = 1 4 × × µΓ 0 − µ Γ (5.103) Γ = Γ + − Γ − = 1 µ Γ 2 × × (5.104) • Net flux (with − sign to account for the fact that the flux is directed along the downhill gradient of Γ): −→ Γ = Γ µ Γ = −1 2 with given by (see the derivation of wall above) = µ 8 (5.105) 12 (5.106) and, for only one type of gas molecules A (i.e., A - A collisons), = 1 √ 2 1 2 × (5.107) • If also varies along (diffusion), Eq. 5.105 may be recast by defining the overall transport quantity Γ 0 from the transport quantity per molecule Γ as Γ 0 = Γ (5.108) into the form −→ Γ = − 1 2 Ã ! Γ 0 (5.109) In the following, ³ we shall apply Eqs. 5.105 or 5.109 to determine the self-diffusion coefficient Γ =1 Γ 0 = ´, the heat conductivity ¡ Γ = ¢ ,andtheviscosity ¡ ¢ Γ = of gases.
5.3 Transport processes in gases 112 5.3.2 Diffusion I Figure 5.9: The diffusion in a mixture A + B of one sort of molecules A against a time-independent concentration gradient is described by Fick’s first law. • Fick’s first law: −→ = µ = − (5.110) • In order to derive an expression for the self-diffusion coefficient (diffusion of an isotope of A in normal A), we make the ansatz Γ =1 Γ 0 = (5.111) y • Result: −→ = − 1 µ 2 (5.112) = 1 (5.113) 2 • Fick’s second law (time-dependent concentration gradient): • Einstein’s diffusion law in 1D: • Einstein’s diffusion law in 3D: = 2 2 (5.114) ∆ 2 =2 (5.115) ∆ 2 =6 (5.116)
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5.3 Transport processes in gases 112<br />
5.3.2 Diffusion<br />
I<br />
Figure 5.9: The diffusion in a mixture A + B of one sort of molecules A against a<br />
time-independent concentration gradient <br />
<br />
is described by Fick’s first law.<br />
• Fick’s first law:<br />
−→ = µ <br />
<br />
= − <br />
<br />
(5.110)<br />
• In order to derive an expression for the self-diffusion coefficient (diffusion of an<br />
isotope of A in normal A), we make the ansatz<br />
Γ =1 Γ 0 = (5.111)<br />
y<br />
• Result:<br />
−→ = − 1 µ <br />
2 <br />
<br />
(5.112)<br />
= 1 (5.113)<br />
2<br />
• Fick’s second law (time-dependent concentration gradient):<br />
• Einstein’s diffusion law in 1D:<br />
• Einstein’s diffusion law in 3D:<br />
<br />
<br />
= 2 <br />
2 (5.114)<br />
∆ 2 =2 (5.115)<br />
∆ 2 =6 (5.116)
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