1. 제3장
Rate Laws
and Stoichiometry
Chemical Reaction Engineering 1
반응공학 1

2. An African-American Scholar and Inventor
Booker T. Washington
“Success is measured not so much by the position one has reached in life, as by the obstacles one has overcome while trying to succeed”
An African-American Scholar and Inventor

3. Objectives After completing Chapter 3, reader will be able to:
 Write a rate law and define reaction order and activation energy.
 Set up a stoichiometric table for both batch and flow systems and express concentration as a function or conversion.
 Calculate the equilibrium conversion for both gas and liquid phase reactions.
 Write the combined mole balance and rate law in measures other than conversion.
 Set up a stoichiometric table for reactions with phase change.

4. Basic Definitions
Homogeneous reaction is one that involves only one phase
Heterogeneous reaction involves more than one phase. (reaction usually occurs at or very near the interface between the phase)
Irreversible reaction is one that proceeds in only one direction and continues in that direction until the reactants are exhausted.
Reversible reaction can proceed in either direction, depending on the concentration of reactants and products relative to the corresponding equilibrium concentration.

5. Example of Elementary Reaction
Basic Definitions
Molecularity of a reaction is the number of atoms, ions or molecules involved in a reaction step.
The term, unimolecular, bimolecular, and termolecular refer to reactions involving, respectively, one, two, or three atoms (or molecules) interacting in any one reaction step.
Example of Elementary Reaction

6. Relative Rates of Reactions
If the rate law depends on more than one species, we MUST relate the concentrations of different species to each other. A stoichiometric table presents the stoichiometric relationships between reacting molecules for a single reaction.
aA + bB cC + dD
(2-1)
In formulating our stoichiometric table, we shall take species A as our basis of calculation (i.e., limiting reactant) and then divide through by the stoichiometric coefficient of A. In order to put everything on a basis of “per mole of A.”
(2-2)
The relationship can be expressed directly from the stoichiometry of the reaction.
(3-1)

7. Relative Rates of Reactions
Example:
2NO + O NO2
rNO
rO2
rNO2 = = -1 2 -2
If rNO2= 4 mol/m3-sec
rNO= -4 mol/m3-sec
rO2= -2 mol/m3-sec

8. Reaction Order and Rate Law
The dependence of the reaction rate –rA on the concentration of the species is almost without exception determined by experimental observation.
The order of a reaction refers to the powers to which the concentrations are raised in the kinetic rate law.
(3-3)
a order with respect to reactant A
b order with respect to reactant B
n (=a+b) : the overall order of the reaction

9. Unit of Specific Reaction Rate
The unit of the specific reaction rate, kA, vary with the order of the reaction.
(Concentration)1-n k=
A  products
Time
(3-4)
(3-5)
(3-6)
(3-7)

10. Elementary and Nonelementary Reaction
Kinetic rate raw
Elementary reaction
Non-elementary reaction
2NO + O2  2NO2
CO + Cl2  COCl2
2nd order w.r.t. nitric oxide
1st order w.r.t. oxygen
overall is 3rd order reaction
1st order w.r.t. carbon monoxide
3/2 order w.r.t. chorine
overall is 5/2 order reaction
In general, first- and second-order reactions are more commonly observed.

11. Reversible Reactions aA + bB cC + dD
All rate raws for for reversible reactions must reduce to the thermodynamic relationship relating the reacting species concentrations at equilibrium. At equilibrium, the rate of reaction is identically zero for all species (i.e., -rA=0). For the general reaction
aA + bB cC + dD
The concentrations at equilibrium are related by the thermodynamic relationship

12. Arrhenius equation Activation energy, J/mol or cal/mol
Specific reaction rate (constant)
Absolute
Temperature, K
Arrhenius frequency factor or
pre-exponential factor
Gas constant
8.314 J/mol · K
1.987 cal/mol · K
8.314 kPa · dm3/mol · K
mathematical number
e= …

13. Arrhenius Equation A
T  0, kA  0
T  , kA  A kA
T (K)

14. Activation energy Activation energy E :
a minimum energy that must be possessed by reacting molecules before the reaction will occur.
The fraction of the collisions between molecules that together have this minimum energy E (the kinetic theory of gases)
Activation energy E is determined experimentally by carrying out the reaction at several different temperature.

15. Activation energy
A+BC A-B-C AB+C

16. Example 3-1: Determination of Activation energy
Decomposition of benzene diazonium chloride to give chlorobenzene and N2 Cl
N=N
Calculate the activation energy using following information for this first-order

17. Finding the activation energy
Plot (ln k) vs (1/T)
[Solution 1]
Select two data points,
(1/T1, k1) and (1/T2, k2),
and then calculate.
Therefore, E  116 kJ/mole
[Solution 2]
Find a best fit from data point
ln k=14,017/T
Slope=E/R=14,017
E=116.5 kJ/mole 1 1 2 2
( )
( ) k2 E 1 1 ln -
= - k1 R T2 T1
임의의 온도에서
속도상수를 구할 수 있다.

18. Finding Activation Energy
Plot (ln k) vs (1/T)
0.01
0.001
0.0001
0.0030
0.0031
0.0032
1/T (K-1)
k (sec-1)
0.005
0.0005
ln k=14,017/T

19. Activation Energy k (sec-1) 1/T (K-1)
The larger the activation energy, the more temperature-sensitive is the rate of reaction.
Typical values of 1st order gas-phase reaction
Frequency factor ~ 1013 s-1
Activation energy ~ 300 kJ/mol
0.01
k (sec-1)
0.001
Low E
High E
0.0001
0.0030
0.0031
0.0032
1/T (K-1)

20. Specific Reaction Rate
E를 알고 나면 실험하지 않은 어느 온도에서의 k값을 구할 수 있다.
Taking the ratio

21. Present Status of Our Approach
Design Equations
Differential
form
Algebraic
form
Integral
form
Batch
CSTR
PFR
PBR

22. Stoichiometry aA + bB cC + dD (2-1) (3-1) (2-2)
The relationship can be expressed directly from the stoichiometry of the reaction.
(3-1)

23. Batch System NA = NA0 - NA0X = NA0(1-X)
NB0
NC0
ND0
NI0
At time t=0, we will open the reactor and place a number of moles of species A, B, C, D, and I (NA0, NB0, NC0, ND0, and NI0, respectively)
Species A is our basis of calculation.
t = t NA NB NC ND Ni
NA0 is the number of moles of A initially present in the reactor.
NA0X moles of A are consumed as a result of the chemical reaction.
NA0-NA0X moles of A leave in the system.
The number of moles of A remaining in the reactor after conversion X
NA = NA0 - NA0X = NA0(1-X)

24. Stoichiometric Table for a Batch System

25. Concentration of Each Species

26. Constant Volume Batch Reactor

27. Flow Systems CA =f(X) FA =f(X) Entering Leaving FA0 FA FB0 FB FC0 FC
FD0
FI0 FA FB FC FD FI
Definition of
concentration
for flow system
CA =f(X)
FA =f(X)

28. Flow Systems If v=vo CA=CAo(1-X) CB=CAo(B -(b/a)X) Batch System

29. Stoichiometric Table for a Flow System

30. Volume Change with Reaction
Gas-phase reactions that do not have an equal number
of product and reactant moles.
N2 + 3H NH3
-The combustion chamber of the internal-combustion engine
-The expanding gases within the breech and barrel

31. Batch Reactor with Variable Volume
Individual concentrations can be determined by expressing the volume V for batch system (or volumetric flow rate v for a flow system) as a function of conversion using the following equation of state.
(3-30)
T = temperature, K
P = total pressure, atm
Z = compressibility factor
R = gas constant = dm3·atm/gmol · K
This equation is valid at any point in the system at any time. At time t=0,
(3-31)
Dividing (3-30) by (3-31) and rearranging yields
(3-32)

32. The total number of moles in the system, NT
(3-33)
We divide through by NT0
(3-34)
where yA0 is mole fraction of A initially present. If all the species in the generalized reaction are in the gas phase, then
(3-23)

33. In gas-phase systems that we shall be studying, the temperature and pressure are such that the compressibility factor will not change significantly during the course of the reaction; hence Z0~Z. For a batch system the volume of the gas at any time t is
(3-38)
Eq (3-38) applies only to a variable-volume batch reactor. If the reactor is a rigid steel container of constant volume, then of course V=V0. For a constant-volume container, V=V0, and Eq. (3-38) can be used to calculate the pressure inside the reactor as a function of temperature and conversion.

34. Flow Reactor with Variable Volumetric Flow Rate
To derive the concentration of the species in terms of conversion for a variable-volume flow system, we shall use the relationships for the total concentration. The total concentration at any point in the reactor is
(3-39)
At the entrance to the reactor
(3-40)
Taking Eq (3-40)/Eq(3-39) and assuming Z~Z0,
(3-41)

35. From Table 3-4, the total molar flow rate can be written in terms of X
(3-43)
(3-44)
(3-45)

36. vo((1+X)(Po/P)(T/To))
Substituting for v using Equation (3-42) and for Fj, we have
FAo(j+vjX) Fj
Cj = = v
vo((1+X)(Po/P)(T/To))
Recalling yA0=FA0/FT0 and CA0=yA0CT0, then
(3-46)

37. Example 3-5: Determination of Cj=hj(X) for a Gas-Phase Reaction

43. Homework
P3-7B
P3-15B
P3-16B
Due Date: Next Week