Page 627 - Bird R.B. Transport phenomena

Problems 607

1 Fig. 19B.4. Oxidation of silicon.
Sit)
SiO ?
4 V •:.•'•••:••.•
v /
Si Si + O 2 -> SiO 2

19B.4. Oxidation of silicon (Fig. 19B.4). 1 A slab of silicon is exposed to gaseous oxygen (species A)
at pressure p, producing a layer of silicon dioxide (species B). The layer extends from the sur-
face z = 0, where the oxygen dissolves with concentration c A0 = Kp, to the surface at z = 5(0,
where the oxygen and silicon undergo a first-order reaction with rate coefficient /c". The thick-
ness 5(0 of the growing oxide layer is to be predicted. A quasi-steady-state method is useful
here, inasmuch as the advancement of the reaction front is very slow.
(a) First solve the diffusion equation of Eq. 19.1-18, with the term dc /dt neglected, and apply
A
the boundary conditions to obtain

in which the concentration c at the reaction plane is as yet unknown.
A8
(b) Next use an unsteady-state molar O balance on the region 0 < z < 5(0 to obtain, with the
2
aid of the Leibniz formula of §C3,
(19B.4-2)
at az
(c) Now write an unsteady-state molar balance on SiO in the same region to obtain
2
+k" c = ±j. (19B.4-3)
} AS
(d) In Eq. 19B.4-2, evaluate dS/dt from Eq. 19B.4-3 and dcjdz from Eq. 19B.4-1. This will
yield an equation for c \
M

~®A~B~ \ %я) М = Сл ° (19В.4-4)
Inserting numerical values into Eq. 19B.4-4 shows that the quadratic term can safely be
neglected. 1
(e) Combine Eqs. 19B.4-3 and 19B.4-4 (without the quadratic term) to get a differential equa-
tion for 5(0. Show that this leads to
Jr- + £ = VW (19B.4-5)

1
which agrees with experimental data. Interpret the result.
19B.5. The Maxwell-Stefan equations for multicomponent gas mixtures. In Eq. 17.9-1 the
Maxwell-Stefan equations for the mass fluxes in a multicomponent gas system are given. Show
that these equations simplify for a binary system to Fick's first law, as given in Eq. 17.1-5.
19B.6. Diffusion and chemical reaction in a liquid.
(a) A solid sphere of substance A is suspended in a liquid В in which it is slightly soluble,
and with which it undergoes a first-order chemical reaction with rate constant k". At steady

1
R. Ghez, A Primer of Diffusion Problems, Wiley-Interscience, New York (1988), pp. 46-55; this book
discusses a number of problems that arise in the microelectronics field.