cre lecture 15
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8/16/2019 CRE Lecture 15
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Lecture 15
Example – Finding theRate Law
2
t(min) 0 1 2 3
C A(mol/L) 1 0.7 0.5 0.35
0.3 0.2 0.15t
C A
t
C A
.1
.2
.3
t1 2 3
Areas equal for both
sides of the histogram
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Find f(t) of using equal area differentiationt
C A
C A 1 0.7 0.5 0.35
-dC A/dt 0.35 0.25 0.175 0.12
Plot (–dC A/dt) as a function of C A
ln
C A
Slope = α
dC A/dt
ln
3
Example – Finding theRate Law
Choose a point, p, and find the concentration and
derivative at that point to determine k.
4
Example – Finding theRate Law
Ap
p
A
C
dt
dC
k
ln
C A
Slope =α
dC A/dt
ln
p
A
dt
dC
p AC
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Linear Least-Square Analysis
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In the case where the rate law is dependent onthe concentration of more than one reactants andmethod of excess is not applicable.
B A A
A C kC r dt
dC
0000)(
B A A
A C kC r dt
dC
000lnlnln)(ln
B A
A C C k dt
dC
Linear Least-Square Analysis
6
1002010
,ln,ln,ln,)/ln( ak aC X C X dt dC Y B A A
22110 X a X aaY
000 lnlnln)(ln
B A
A C C k dt
dC
N
j j
N
j j j
N
j j
N
j j j
N
j j j
N
j j
N
j j
N
j j j
N
j j
N
j j
N
j j
X a X X a X aY X
X X a X a X aY X
X a X a NaY
1
2
221
2111
201
2
1 212
1
2
111
101
1
1 22
1 110
1
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Non-Linear Least-Square Analysis
7
We want to find the parameter values (α, k, E) forwhich the sum of the squares of the differences, the
measured rate (Cm), and the calculated rate (Cc) is
a minimum.
That is, we want to be a minimum. 2
n
i
icim
K N
S
K N
C C
1
22
2
8
For concentration-time data, we can combine the
mole balance equation for to obtain:
Rearranging to obtain the calculated concentration
as a function of time, we obtain:
r A kC A
00
A A
A A
dC kC
dt
t C C
1 1
0 (1 )
A AC C kt
1 1/(1 )
0[ (1 ) ] Ac A A
C C C kt
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9
Now we could use Polymath or MATLAB to find the values of α and kthat would minimize the sum of squares of differences between themeasured (C Am) and calculated (C Ac) concentrations.
That is, for N data points,
Similarly one can calculate the time at a specified concentration, tc
and compare it with the measured time, tm, at that same concentration.
That is, we find the values of k and α that minimize:
Example
10
Find reaction order and specific reaction rate by nonlinear least square analysis .
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1 1
M ethod of Init ial Rates
1 2
A series of experiments are carried out at different
initial concentrations, C A0 and initial reaction rate
–r A0 is determined for each run.
The initial rate can be found by differentiating the
data and extrapolating to zero time.
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Example
1 3
Solution
1 4
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M e thod of Ha lf-Lives
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The half-live of a reaction is the time it takes for theconcentrationofthereactanttofall tohalf of itsinitial value.
By determining the half-life as a function of the initialconcentration, the reaction order and constant can bedetermined.
If two reactants are involved, use the method of excess inconjunctionwiththemethodofhalf-livestoarrangetherateoftheform:
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Differential Reactors
2 0
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Using very little catalyst and large volumetric flow rates:
CA0-CAecan be made quite small
Use method of initial rates to find reaction order and specificreaction rate.
Example
2 2
T=500 F
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2 3
Solution
2 4
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