lecture 3elements/6e/powerpoints/2013lectures/lec3... · 2019. 8. 7. · lecture 3 review of...
TRANSCRIPT
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Chemical Reaction Engineering (CRE) is the field that studies the rates and mechanisms of
chemical reactions and the design of the reactors in which they take place.
Lecture 3
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Lecture 3
Review of Lectures 1 and 2
Building Block 1
Mole Balances (Review)
Size CSTRs and PFRs given –rA= f(X)
Conversion for Reactors in Series
Building Block 2
Rate Laws
Reaction Orders
Arrhenius Equation
Activation Energy
Effect of Temperature
2
Chapter 3
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Reactor Differential Algebraic Integral
V FA 0 FA
rA
CSTR
Vrdt
dNA
A
0
A
A
N
N A
A
Vr
dNtBatch
NA
t
dFA
dV rA
A
A
F
F A
A
dr
dFV
0
PFR
FA
V
dFA
dW r A
A
A
F
F A
A
r
dFW
0
PBRFA
W3
The GMBE applied to the four major reactor
types (and the general reaction AB)
Building Block 1: Mole Balance
Review Chapter 1
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fedA moles
reactedA moles X
D a
d C
a
c B
a
b A
ncalculatio of basis asA reactant limiting Choose
D d C c B b A a
4
Review Chapter 2
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Reactor Differential Algebraic Integral
A
A
r
XFV
0CSTR
AA rdV
dXF 0
X
A
Ar
dXFV
0
0PFR
Vrdt
dXN AA 0
0
0
X
A
AVr
dXNtBatch
X
t
AA rdW
dXF 0
X
A
Ar
dXFW
0
0PBR
X
W5
Building Block 1: Mole BalanceIn terms of Conversion
Review Chapter 2
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Levenspiel Plots
6
Review Chapter 2
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Reactors in Series
7
reactorfirst tofedA of moles
ipoint toup reactedA of molesXi
Only valid if there are no side streams
Review Chapter 2
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Reactors in Series
8
Review Chapter 2
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Building Block 2: Rate Laws
9
Power Law Model:
BAA CkCr
βα
β
α
OrderRection Overall
Bin order
Ain order
Chapter 3
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Building Block 2: Rate Laws
10
C3BA2
A reactor follows an elementary rate law if the
reaction orders just happens to agree with the
stoichiometric coefficients for the reaction as written.
e.g. If the above reaction follows an elementary rate
law
2nd order in A, 1st order in B, overall third order
BAAA CCkr 2
Chapter 3
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Building Block 2: Rate Laws
11
Rate Laws are found from Experiments
Rate Laws could be non-elementary. For
example, reaction could be:
› Second Order in A
› Zero Order in B
› Overall Second Order
2
A A Ar k C
2
B B Ar k C
2
C C Ar k C
2A+B3C
Chapter 3
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Relative Rates of Reaction
12
dDcCbBaA
d
r
c
r
b
r
a
r DCBA
Da
dC
a
cB
a
bA
Chapter 3
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Relative Rates of Reaction
13
Given
Then 312
CBA rrr
C3BA2
sdm
molrA
310
sdm
molrr A
35
2B
sdm
molrr AC
315
2
3
Chapter 3
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Reversible Elementary Reaction
14
e
CBAA
AA
CBAACABAAA
K
CCCk
kk
CCCkCkCCkr
32
3232
A+2B 3CkA
k-A
Chapter 3
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Reversible Elementary Reaction
15
Reaction is: First Order in A
Second Order in B
Overall third Order
sdm
molesrA 3
3dm
molesCA
smole
dm
dmmoledmmole
sdmmole
CC
rk
BA
A
2
6
233
3
2
A+2B 3CkA
k-A
Chapter 3
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16
Chapter 3
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Algorithm
17
iA Cgr Step 1: Rate Law
XhCi Step 2: Stoichiometry
XfrA Step 3: Combine to get
How to find XfrA
Chapter 3
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Arrhenius Equation
18
RTEAek
k is the specific reaction rate (constant) and is
given by the Arrhenius Equation.
where:
T k A
T 0 k 0
A 1013
k
T
Chapter 3
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Arrhenius Equation
19
where:E = Activation energy (cal/mol)
R = Gas constant (cal/mol*K)
T = Temperature (K)
A = Frequency factor (same units as rate constant k)
(units of A, and k, depend on overall reaction order)
Chapter 3
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Activation Energy Concept 1. Law of Mass Action
The rate of reaction increases with increasing concentration of reactants
owing to the corresponding increase in the number of molecular collisions.
The rate of disappearance of A, –rA, depends on temperature and
concentration. For many irreversible reactions, it can be written as the
product of a reaction rate constant, kA, and a function of the concentrations
(activities) of the various species involved in the reaction:
(3-2)
For example for the elementary reaction
E = Activation Energy, (kJ/mol)
-rA = kA T( )éë ùû fn CA,CB…( )éë ùû
AB+C ®¬ A+BC
-rAB = kABCABCC = AABe-E RTCABCC
Chapter 3
20
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Collision Theory
21
Chapter 3
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Why is there an Activation Energy?
22
We see that for the reaction to occur, the reactants
must overcome an energy barrier or activation
energy EA. The energy to overcome their barrier
comes from the transfer of the kinetic energy from
molecular collisions to internal energy (e.g.
Vibrational Energy).
1. The molecules need energy to disort or stretch
their bonds in order to break them and thus form
new bonds
2. As the reacting molecules come close together
they must overcome both stearic and electron
repulsion forces in order to react.
Chapter 3
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Activation Energy Concept 2. Potential Energy Surfaces and Energy Barriers
As two molecules, say AB and C, approach each other, the potential
energy of the system (AB, C) increases owing repulsion of the molecules.
The reaction coordinate is a measure of progress of the reaction as we go
form AB and C to A and BC as shown in Figure 3-2 below.
AB+C ®¬ A\B\C ®¬ A+BC
Chapter 3
23
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Activation Energy Concept 2. Potential Energy Surfaces and Energy Barriers
The figure below shows a 3-dimensional energy surface and barrier over
which the reaction must pass
Chapter 3
24
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25
Chapter 3
Activation Energy Concept 2. Potential Energy Surfaces and Energy Barriers
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Activation Energy Concept 3. Fraction of Molecular Collisions That Have Sufficient Energy
to React
The kinetic energy of the molecule must be transferred to potential energy to distort bonds and overcome steric forces in order for the reacting molecules to pass over the energy barrier.
The distribution of velocities of the reacting molecules relative to one another is given by the Boltzmann distribution
Where
kB = Boltzmann’s constant = 3.29 x 10–24 cal/molecule/K
m = Reduced mass, g
U = Relative velocity, m/s
T = Absolute Temperature, K
e = Energy = mU2, kcal/molecule
E = Kinetic energy, kcal/mol
f(U,T) = Fraction of relative velocities between U and U + dU
f U ,T( ) = 4pm
2pkBT
æ
è
çç
ö
ø
÷÷
3 2
exp-mU2
2kBT
é
ë
êê
ù
û
úúU2
Chapter 3
26
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Activation Energy
27
2
3 2
2 2, 42
BmU k T
B
mf U T dU e U dU
k T
Chapter 3
Concept 3. Fraction of Molecular Collisions That Have Sufficient
Energy to React
Distribution of Velocities
We will use the Maxwell-Boltzmann Distribution of Molecular Velocities.
For a species of mass m, the Maxwell distribution of velocities (relative
velocities) is:
f(U,T)dU represents the fraction of velocities between U and (U+dU).
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Activation Energy
28
Chapter 3
Concept 3. Fraction of Molecular Collisions That Have Sufficient
Energy to React
Distribution of Velocities
A plot of the distribution function, f(U,T), is shown as a function of U:
Maxwell-Boltzmann Distribution of velocities.
T2>T1T2
T1
U
𝑓 𝑈, 𝑇
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Activation Energy
29
Chapter 3
Concept 3. Fraction of Molecular Collisions That Have Sufficient
Energy to React
Distribution of Velocities
Given
Let
f(E,T)dE represents the fraction of collisions that have energy between E and (E+dE)
2
3 2
2 2, 42
BmU k T
B
mf U T dU e U dU
k T
2
2
1mUE
1 2
3 2
2,
2
B
E
k T
B
f E T dE E e dEk T
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Activation Energy Concept 3. Fraction of Molecular Collisions That Have Sufficient Energy
to React
In terms of energy per mole, E, instead of energy per molecule, e, we have
(3-20)
where E is in (cal/mol) or (J/mole), R is in (cal/mol/K), and f(E,T) is in mol/cal.
The distribution function f(E,T) is most easily interpreted by recognizing that
[f(E,T) dE] is the fraction of collisions with energies between E and E + dE.
(3-21)
This distribution is shown as a function of E for two temperatures in the next
slide.
f E ,T( ) =2p1
pRT
æ
èç
ö
ø÷
3 2
E1 2exp-E
RT
é
ëê
ù
ûú
f E ,T( )dE =2p1
pkBT
æ
è
çç
ö
ø
÷÷
3 2
E1 2exp-E
kBT
é
ëêê
ù
ûúúdE
Chapter 3
30
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Activation Energy Concept 3. Fraction of Molecular Collisions That Have Sufficient Energy
to React
This fraction is shown by the shaded area in Figure 3-4(a) and is approximated
by the average value of f(E,T) at E = 0.3 kcal/mole is 0.81 mol/kcal.
We see 8.1% of the collisions have energies between 0.25 and 0.35 kcal/mol
Fractionwithenergiesbetween(0.25)and0.35)kcal
mol
æ
èç
ö
ø÷= f
0.25
0.35ò E ,T( )dE
f E ,T( )dE = f 0.3,300K( )DE =0.81mol
kcal0.35
kcal
mol-0.25
kcal
mol
æ
èç
ö
ø÷=0.081
Chapter 3
31
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Activation Energy Concept 3. Fraction of Molecular Collisions That Have Sufficient Energy
to React
Fraction of Molecules with Energy Greater than EA
We can also determine the fraction of collision that have energies greater than
a certain value, EA
(3-22)
For EA > 3RT, we can obtain an analytical approximation for the fraction of
molecules of collision with energies greater than EA by combining Equations (3-
21) and (3-22) and integrating to get
(3-23)
Equation (3-23) is plotted in Figure 4-4(b) as a function of activation energy,
EA, as shown in the next slide.
FractionofMoleculeswithE >EA( ) = F E >E
A,T( ) = f
EA
¥
ò E ,T( )dE
FractionofcollisionwithenergiesgreaterthanE
A
æ
èç
ö
ø÷= F E >E
A,T( ) @
2
p
EA
RT
æ
è
çç
ö
ø
÷÷
1 2
exp -EA
RT
æ
è
çç
ö
ø
÷÷
Chapter 3
32
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Activation Energy Concept 3. Fraction of Molecular Collisions That Have Sufficient
Energy to React
Fraction of Molecules with Energy Greater than EA
One observes for an activation energy EA of 20 kcal/mol and a temperature
of 300 K the fraction of collisions with energies greater than 20 kcal/mol is
1.76 x 10–14 while at 600 K, the fraction increases to 2.39 x 10–7, which is
a 7 orders of magnitude difference.
Chapter 3
33
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End of Lecture 3
34
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Supplementary Material
35
BC AB
kJ/
Molecule
rBC
kJ/
Molecule
r0
Potentials (Morse or Lennard-Jones)
VABVBC
rAB
r0
Chapter 3 Professional
Reference Shelf
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Supplementary Material
36
For a fixed AC distance as B moves away from C the distance of
separation of B from C, rBC increases as N moves closer to A. As
rBC increases rAB decreases and the AB energy first decreases
then increases as the AB molecules become close. Likewise as B
moves away from A and towards C similar energy relationships
are found. E.g., as B moves towards C from A, the energy first
decreases due to attraction then increases due to repulsion of the
AB molecules as they come closer together. We now
superimpose the potentials for AB and BC to form the following
figure:
One can also view the reaction coordinate as variation of the BC
distance for a fixed AC distance:
l
BCA
l
CAB BCAB rr
l
CBA
Chapter 3 Professional
Reference Shelf
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Reaction Coordinate
37
The activation energy can be thought of as a barrier
to the reaction. One way to view the barrier to a
reaction is through the reaction coordinates. These
coordinates denote the energy of the system as a
function of progress along the reaction path. For the
reaction:CABCBABCA ::::::
The reaction coordinate is:
Chapter 3 Professional
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Supplementary Material
38
Energy
r
BC
ABS2
S1
E*
E1P
E2P∆HRx=E2P-E1P
Ea
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