Fogler_Ch10_Web.fm Page 1 Friday, December 11, 2015 12:37 PM
1
Chapter 10
Web 10.1 Rate Laws Derived from the
Pseudo-Steady-State Hypothesis
In Section 7.1 we discussed the PSSH where the net rate of formation of reactive intermediates was assumed to be zero. An alternative way to derive a catalytic rate law rather than setting
rAD
-------- 0
kA
is to assume that each species adsorbed on the surface is a reactive intermediate. Consequently, the net rate of formation of species i adsorbed on the surface will be zero:
The PSSH should
be used when
more than one
step is limiting
ri S 0
(10-54)
While this method works well for a single rate-limiting step, it also works well
when two or more steps are rate-limiting (e.g., adsorption and surface reaction). To illustrate how the rate laws are derived using the PSSH, we shall consider the isomerization of normal pentene to iso-pentene by the following
mechanism, shown as item 6 of Table 10-4:
kN
⎯⎯→ N S
N S ←⎯⎯
k
–N
kS
N S ⎯⎯→ I S
kI
⎯⎯→ I S
I S ←⎯⎯
k
–I
The rate law for the irreversible surface reaction is
rN rS k SCN S
The net rate
of the adsorbed
species (i.e., active
intermediates)
is zero.
(10-55)
The net rates of generation of N S sites and I S sites are
rN S k NPNCv k NCN S k SCN S 0
(10-56)
rI S k SCN S k ICI S k IPICv 0
(10-57)
Solving for CN S and CI S gives
k NPNCv
CN S -------------------k N k S
(10-58)
k SCN S k IPICv ⎛ k S k NPN
k I ⎞
CI S ----------------------------------------- ⎜ ------------------------------- ------P⎟ C
kI
k I I⎠ v
⎝ k I ( k N k S )
(10-59)
Fogler_Ch10_Web.fm Page 2 Tuesday, October 20, 2015 8:49 AM
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Chapter 10
and substituting for CN S in the surface reaction rate law gives
kN kS
-P C
rN -------------------k N k S N v
(10-60)
From a site balance we obtain
Ct CN S CI S Cv
(10-61)
After substituting for CN S and CI S , solving for Cv , which we then substitute
in the rate law, we find
⎛ k N k S Ct ⎞
PN
-⎟ ----------------------------------------------------------------------------------rN ⎜ -------------------⎝ k N k S⎠
⎛
kN
kS ⎞
k I
1 --------------------- ⎜ 1 ----- ⎟ PN ------- PI
k N k S ⎝
kI ⎠
kI
(10-62)
The adsorption constants are just the ratio of their respective rate constants:
k I
K I ------kI
and
kN
K N -------k N
k
⎫
⎪
⎬
⎪
⎭
Use the PSSH
method when
Some steps are
irreversible.
Two or more
steps are ratelimiting.
⎛ K N k S Ct ⎞
PN
-⎟ --------------------------------------------------------------------rN ⎜ ----------------------------1
⎝ 1 k S k N ⎠ 1 -------------------------- K NPN K IPI
1 k S k N
We have taken the surface reaction step to be rate-limiting; therefore, the surface specific reaction constant, kS , is much smaller than the rate constant for
the desorption of normal pentene, kN ; that is,
kS
1 » -------kN
and the rate law given by Equation (10-62) becomes
kPN
rA --------------------------------------1 K NPN K IPI
(10-63)
This rate law [Equation (10-63)] is identical to the one derived assuming that
rAD / kA 0 and rD / kD 0. However, this technique is preferred if two or more
steps are rate-limiting or if some of the steps are irreversible or if none of the
steps are rate-limiting.