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Chapter 5 Collection and Analysis of Rate Data
Two common types of reactors for obtaining kinetics data: Batch & Differential reactors
Seven methods for data analyses: (1) Differential method; (2) Integration method; (3) Half-life
method; (4) Initial-rate method; (5) Linear regression method; (6) Non-linear regression
method; (7) Excess method.
§ Batch Reactors
<1> Differential method (A) For constant V
A → P, AAAA
AAA Ckdt
dCdt
dCCkr lnlnln αα +=
−⇒−==−
α)( AP
P
A
A Cdt
dC
k−
=
ln(dt
dCA−
ln(CA)
S =α
ln(dt
dCA−
ln(CA) CAP
P
A
dtdC
−
Remark: Let AAA
A krdt
dCC =−=−⇒= 0ln (intercept)
To obtain dtdCA− graphical, numerical, or polynomial fit:
(a) Graphical method
CA -- -- -- -- -- --
t -- -- -- -- -- --
The area under ∆CA/∆t is the same as that under the curve.
Read estimates of dtdCA from the curve
(b) Numerical method
CA CA0 CA1 CA2 CA3 CA4 …
t t0 t1 t2 t3 t4 …
By three-point differentiation formulas
Initial: )43(21
2100
AAAt
A CCCtdt
dC−+−
∆=
CA
t
Interior: )(21
11 −+ −∆
= AiAit
A CCtdt
dC
i
Final: )34(21
12 AfAfAft
A CCCtdt
dC
f
+−∆
= −−
(c) Polynomial fit
nna tatataaC ++++= LL2
210
⇒ Least-square fitting ⇒ a0, a1, a2, …, an can be obtained!
12321 32 −+++=∴ n
nA tnatataa
dtdC
LL
Remarks: (i) The order of polynomial have to be carefully chosen. (See Levenspiel, p.65, ex3-2 &
p.259-260)
(ii) Excess + Differential method.
βαBAA
A CkCrdt
dC=−=−
ββαβα
ααββα
BBABAA
AABA
AABBAA
BBAB
CkCkCCkCr
constantCCCCbCkCkCCkCr
constantCCCCa
''0
000
'0
000
)(
)(
===−⇒
=≈⇒>>===−⇒
=≈⇒>>
From (a) & (b) ⇒ α, β can be obtained ⇒ obtain k!
(B) For total pressure-time data
For gas-phase reaction ))()(1(0
0
0 TT
PP
XVV
Aε+=
At V = constant, and isothermal operation
)(1
)(,
)(
)1(
00
0
00
0
0
0
0
000
0
00
AkCdtdP
RTr
yNN
dtdP
PyRTP
dtdP
PRTP
PPP
dtd
RTP
dtdXC
dtdCrSince
PPPXXPP
AA
AT
A
A
AA
AAA
AA
AA
LLLα
δ
δδεδε
ε
εε
==−∴
====
−==−=−
−=⇒+=⇒
)260.,1.5.()(),(
]/)[()1()1( 00
0
000
pExSeePfdtdPA
RTPPP
PPP
RTPXCC AA
AAA
=⇒
−−=
−−=−=
代入式
又δ
εQ
(C) For constant pressure operation
αα
ααα
εε
εε
ε
ε
εε
ε
ε
+−
=+
⇒
+=
+−
==−
+=
−+−
=−
=−∴
+−
=+−
==⇒
+=⇒
+=≠
−
A
AA
A
A
A
A
A
A
AAAA
A
A
A
AA
A
AA
A
AA
A
AAAA
A
A
XXkC
dtdX
X
dtdX
XC
XXkCkCrSince
dtdX
XC
dtXdN
XVdtdN
Vr
XXC
XVXN
VNC
XVVisothermal&PconstantAtTT
PPXVVConstantV
11
)1(1
)1(11
)1(
)1()1(
11)1(
)1()1()1(
)1(
))()(1(,
10
00
0
0
0
00
0
0
0
00Q
t
XA
dtdX ASlope =
kCX
Xdt
dXX A
A
AA
A
lnln)1(11ln
)1(1ln 0 +−+
+−
=
+
⇒ αε
αε
XA -- -- -- -- -- --
t -- -- -- -- -- --
Plot XA vs. t
dtdX A -- -- -- -- -- -- dt
dXX
A
Aε+11
-- -- -- -- -- --
XA -- -- -- -- -- --
A
A
XXε+−
11 -- -- -- -- -- --
Slope = α
)11ln(
A
A
XXε+−
)1
1ln(dt
dXX
A
Aε+
Find slope = α, From intercept (α-1) ln(CA0)+ln(k) k obtained!!
<2> Integral Method Key concepts: (1) The integral method is used most often when the reaction order is known.
(2) It is desired to evaluate the specific reaction rate constants at different temperatures to
determine Ea.
(3) We are looking for the appropriate function of concentrations corresponding to a particular rate
law that is linear with time.
Slope = -k
t
CA
(A) Constant-V Batch Reactors
∫ =−⇒=−∴
=−=−
=−
A
A
C
CA
A
A
A
AAA
A
ktCf
dCkdtCf
dC
Cfkdt
dCdt
dNV
r
0 )()(
)(1
Remark: The integral method uses a trial-and-error
procedure to find the reaction order!
For PA k→ (i) zero-order reactions:
ktCCkdt
dCkr
AAA
A
−=⇒−=∴
=−
0
Slope= k
∫−A
A
C
CA
A
CfdC
0 )(
t
Slope = k
t
A
A
CC 0ln
S = k
t
AX−11ln
(ii) first-order reactions:
kt
CC
kdtCdCkC
dtdC
kCr
A
A
A
AA
A
AA
=⇒
=−⇒=−∴
=−
0ln
)14,132.(1
1ln
)1()1(
)1(
)1()1(
0
00
00
−=−
⇒
=−
⇒=−
∴
−===−⇒
−=−
==
∫
exPseektX
ktX
dXkdtX
dX
XkCkCdt
dXCdt
dC
XCV
XNVNCSince
A
X
A
A
A
A
AAAA
AA
AAAAA
A
A
S = k
t
AC1
0
1
AC
Slope= CA0 k
t
A
A
XX−1
(iii) second-order reactions:
ktCC
kdtCdCkC
dtdC
kCr
AA
C
CA
AA
A
AA
A
A
=−⇒
−=⇒=−∴
=−
∫
0
22
2
11
0
A
A
AA
AAA
XX
XktC
ktCXC
−=−
−=⇒
=−−
⇔
11
11
1)1(
1
0
00
For A + B → P, -rA = kCACB
0
00
0
),(
)()(
A
BBABA
ABAB
CCXC
XabCCconstantViv
=−=
−=∴=
θθ
θQ
ktCdXX
dXX
tobelongifwhereBBA
BABA
XXBAXBA
XB
XA
XXLet
ktCXX
XXkCdt
dXCdt
dCr
AA
X
AB
BA
X
A
B
BB
BB
ABA
AB
ABAABA
A
X
ABA
ABAAA
AA
A
AA
A
000
00
200
11
11
1
iii,1;1,1
11
0
))(1()()(
)()1())(1(1
))(1(1
))(1(
=−−
+−−
⇒
><−=≠
−=
−=⇒
=+=+
⇒
−−+−+
=−
+−
=−−
=−−
⇒
−−==−=−∴
∫∫
∫
θθθ
θθθθ
θθ
θθ
θ
θ
ktCXX
ktCXX
ABAB
AB
AB
AB
BA
B
0
0
)1()1(
ln
)ln(1
1)1ln(1
1
−=−−
⇒
=−
−+−
−∴
θθθ
θθ
θθ
(v) Reactions of shifting order
A → R , A
AAA Ck
Ckdt
dCr2
1
1+=−=− (1)
At high CA → 21 / kkrA =− (zero order)
At low CA→ AA Ckr 1=− (1st order)
Integrating equation (1), tkCCkC
CAA
A
A10,2
0, )(ln =−+
AA
AA
CCCC−0,
0, /ln
AAAA
AA
CCtkk
CCCC
−+−=
−∴
0,
12
0,
0,
)(/ln
AA CCt−0,
☆ The other reaction mechanisms for integral methods, see P.41~P.63, Levenspiel.
(B) For total pressure-time data (V = V0, T = T0)
-k2
S=k1
RT
PadP
ac
RT
PPadP
XadCC
RT
PacP
ac
RT
PPacP
XacCC
RT
PabP
ab
RT
PPabP
XabCC
Similarly
CRT
PPPyRT
PPPP
PPRTP
PPP
RTPXCC
PfCfCkfrConceptRT
PPPPPRT
CdtdP
RTrand
PPPXXPP
DD
ADAD
CC
ACAC
BB
ABAB
AA
AAAAAA
AAA
AAA
AA
δ
εθδθ
δ
εθδθ
δ
εθδθ
δε
δε
εε
ε
δε
δδ
εε
+−=
−+
=+=
+−=
−+
=+=
−+=
−−
=−=
=−+
=−+
=
−+=
−−=−=
=⇒=−
−+=
−−==−
−=⇒+=
00
0
0
00
0
0
00
0
0
0
00
00
0
00
0
000
2
000
0
00
)()()(
)()()(
)()()(
,
)1(])1[()1()1()1(
))()(),(:(
)1()(1,1
)1(Q
(See ex. 5.2, P269)
Slope= kδRT
t
∫P
P PfdP
0 )(2
Slope= k
t
)1ln(0A
A XC εε
+
tvs
PfdPPlotiii
RTtkdtRTkPf
dPii
PkfdtdP
RTriSteps
P
P
tP
P
A
.)(
)(
)()(
)(1)(:
0
0
2
02
2
∫
∫∫ ==
==−
δδ
δ
(C) Constant P, T, V ≠ Constant, Batch Reactors
ktXCX
dXC
kdt
dXX
Ckrreactionsorderzeroi
dtdX
XCr
XXCCXVV
AA
X
A
AA
A
A
A
A
A
A
AA
A
AAAA
A =+=+
⇒
=+
=−−><+
=−
+−
=+=
∫ )1ln(1
1
:1
11);1(
000
0
0
00
εεε
ε
ε
εε
ktXXXkC
dtdX
XC
kCrreactionsorderfirstii
AA
AA
A
A
A
AA
−=−⇒+−
=+
=−−><
)1ln(11
1
:
00
εε
tkCXX
X
tkCdXXX
XXkC
dtdX
XC
kCrreactionsordersecondiii
AAA
A
AA
X
A
A
A
AA
A
A
A
AA
A
0
00 2
22
00
2
)1ln(1
)1(
)1(1
11
1
:
=−+−
+⇒
=−+
⇒
+−
=+
=−−><
∫
εε
εεε
<3> Method of Initial Rates
00
00
lnln)ln( AA
nAA
nAA
CnkrkCrinitalkCrif
+=−⇒=−⇒=−
Steps: (i) A series of experiments is carried out at different CA0.
(ii) Plot CA vs. t
(iii) Determine –rA0 by differentiating the data and extrapolating to t = 0.
(iv) Plot ln(–rA0) vs. ln(CA0).
S = n
kln
)ln( 0Ar−
0ln ACt
0AC
01AC02AC03AC04AC
00
AC
A rdt
dCslopeA
==
Remark: When a significant reverse reaction is present, the use of the differential method for data analysis to
determine reaction orders and specific reaction rates is unsuitable.
⇔ Initial rate method! See Ex. 5-4, p. 2778.
<4> Method of Half-Lives Definition t1/2 : half-life of a reaction is defined as the time it takes from the concentration of the
reactant to fall to half of its initial value.
(A) Constant V
)(, VConstantkCdt
dCr AA
Aα=−=−
S = 1-α
0ln AC)1(12ln
1
−−−
α
α
k
1/2tln
A0
1
2/11A0
1
/1
1A0
1
2/1A02/1
1
A
A01
A01
A01
A
1A0
1A
C)1()1(12lnln,
C1
)1(1,
C1
)1(12C
21,
1CC
)1(C1
C1
C1
)1(1
,)1(CC
ααα
α
αα
α
α
α
α
α
α
α
ααα
αα
−+−−
=∴−−
=
−−
=⇒=
−
−
=
−
−=⇒
−=−⇒
−
−
−
−
−
−
−−−
−−
kt
kntSimilarly
ktCtat
kkt
kt
n
A
Steps: (i) Conduct CA vs. t, (different CA0)
(ii) Get table.
A0C -- -- -- -- -- --
2/1t -- -- -- -- -- --
(iii) Plot 2/1ln t vs. A0Cln .
(iv) from slope, S = 1 - α, α = 1 - S; and from intercept, I = )1(12ln
1
−−−
α
α
k , k.
Remark: If two reactants are involved in the chemical reaction, use the method of excess
in conjunction with the method of half lives.
(B) For total pressure-time data
ktRTP
PPP
RTkt
PPP
RTkt
PP
RTktdt
RTk
PPdP
RTPPkkC
dtdP
RTr
RTPPC
P
P
tP
P
AAA
)1(1)1(
)()1(1
)1(1
)()1(1
11
)()(])1[(
)1(1)1(
1
0
1
0
0
1
1
0
1
0
1
1
0
1010
00
0
0
−
=−
−+
⇒
−=
−
−+
⇒
=
−+−
==−+
⇒
−+===−⇒
−+=
−−
−
−−
−
−
−− ∫∫
αδε
εε
δα
εε
δεα
δδε
δε
δδε
αα
α
αα
α
α
ααα
αα
Q
''0
'0 2/12/1
2;21 ttPPorttPPAt =⇔==⇔=
1
11
0''
1
11
0'
101
11
'
1
0'1
)1(
)(11
lnln)1(ln,
)1(
)(112
2
lnln)1(ln
)1(
)(112
2
)1(112
2
2/1
2/1
2/1
2/1
−
−−
−
−−
−−
−−
−−
−
−
++−=
−
−
++−=⇒
−
−
+=⇒
−=−
+⇒
α
αα
α
αα
αα
αα
αα
εα
δεε
α
εα
δεε
α
εα
δεε
δεα
εε
k
RTPtSimilarly
k
RTPt
Pk
RTt
RTPkt
Steps: (i)
P -- -- -- -- -- P0 -- -- -- -- --
T -- -- -- -- -- '2/1
t -- -- -- -- --
(ii) Plot ln(t1/2’) vs. ln(P0), k, α obtained!
t
0P
01P02P03P04P
'2/1
t
Slope= 1-α
'2/1
ln t
k
0ln P
<5> Method of Excess A + B → P
βαBA
A CCkrdt
dCA =−=− (Constant V)
(i) CB,0 >> CA,0 => CB ≈ CB,0 = constant
∴ααββαAABBAA CkCkCCkCr ′===− 0, , where
ββ0,BB kCkCk ==′
Applying “differential, integral, initial or half-lines methods” to obtain α
(ii) CA,0 >> CB,0 => CA ≈ CA,0 = constant
ββαβαBBABAA CkCkCCkCr ′′===−∴ 0, , where
αα0,AA kCkCk ≈=′′
β can be determined !
§ Differential Reactors (initial rate) A differential reactor consists of a tube containing a very small amount of catalyst usually arranged
in the form of a thin wafer or disk.
Properties: (i) Conversion is extremely small
(ii) Reactant concentration through the reactor is essentially constant
(iii) No concentration gradient
(iv) Heat release is small => isothermal
(v) No by pass or channeling
(vi) No catalyst decay !
Mass balance 0,0, =⋅′+− wFF AeAA γ
FA, FA,e
∆L catalyst
wXF
wVCCV
r AAeAAA
0,,0,0' =−
=−∴
If A → P (stoichiometric ratio = 1), wF
wXF pAA =⇒ 0,
(Fp: product flow rate)
For constant volumetric flow, V0 = Ve = V, ( )
wCV
wCCV
r peAAA
0,0,0 =−
=′−
As w → 0, V0 → ∞ ⇒ CA,0 - CA,e → 0
⇒ 2,0,
.eAA
bACC
C+
= or CA,b ≈ CA,0 , where CA,b: bed concentration (see Ex 5.5, p.284)
§ Least-Square Analysis
For a constant-volume batch reactor, βαγ BAA
A CkCdt
dC=−=−
22110
0,0,0
0,0,0
lnlnlnln
XaXaaY
CCkdt
dC
CkCdt
dC
BAt
A
BAtA
++=⇒
++=
−⇒
−⇒
=
=
βα
βα
=>Using multiple linear regression (利用統計)
(see Ex.5-5 P. 251)
for the nonlinear least-squares analysis
γc : rxn rate estimated from calculation
γm : rxn rate measured
( )∑ −= 2ii mCL γγ ;
KNL
KNs
−=
−=
22σ
Minimize L/N-K (see Ex. 5.6 P. 255) (先看 p. 253 Table 5-2)
N: # of runs , K: # of parameters
Remarks:
(i) We can compare the residual plots for model checking
(是否缺少某一參數,比較不同 model之適切性…)
(ii) The model with physical meanings is preferred!
(iii) It is also possible to use the nonlinear regression to determine the rate law parameters from
conc.-t data!