1. 반응공학1 Youn-Woo Lee School of Chemical and Biological Engineering
Seoul National University
, 1 Gwanangro, Gwanak-gu, Seoul, Korea  

2. Seoul National University
第3章
Rate Laws(속도법칙)
Reaction Engineering I
反應工學I
Seoul National University

3. 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

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개요.
제2장에서 전화율의 함수로서 반응속도를 알고 있다면, 흐름계에서는 특정한 전화율을 달성하는 데 필요한 반응기 부피를 계산할 수 있고, 회분식계에서는 주어진 전화율을 달성하는 데 필요한 시간을 구할 수 있다는 것을 보였다.
불행하게도, 실험자료로 부터 -rA = f(X)를 바로 구하기는 매우 드물다. 그러나 이를 두려워 말라. 이 장에서 전화율의 함수로서 반응속도를 구하는 방법을 배우게 될 것이다. 반응속도와 전화율 사이의 관계는 아래 두 단계로 얻어진다.
제3장에서 다루는 1단계에서는, 반응속도를 반응물 (reacting species)의 농도와 온도에 관련 짓는 속도법칙 (rate law)을 정의한다. 제4장에서 다루는 2단계에서는, 흐름계와 회분식계에 대한 농도를 정의하고 화학양론표 (stoichiometric table)를 작성하여, 농도를 전화율의 함수로 표현할 수 있도록 한다. 1단계와 2단계를 결합하면, 반응속도를 전화율의 함수로 표현할 수 있으며, 제2장에서 전개된 방법들을 사용하여 반응계를 설계할 수 있다는 것을 알게 된다.
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Objectives
여러분이 이 장을 마친 뒤에는
 어떤 반응이 일어날 때, 각 성분의 반응 속도를 서로 연관지울 수 있고…
 농도의 함수로 속도법칙을 구할 수 있으며…
 온도의 함수로서의 속도 상수를 구하기 위해 아레니우스 식을 사용할 수 있을 것이다.
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Reactor Size
Design Equations
Levenspiel plot
Batch
CSTR
Graphical method
PFR
For vs. X, the volume of a CSTR and the volume of a PFR can be represented as the shaded areas in the Levenspiel plots.
PBR
If we know the molar flow rate to the reactor and the reaction rate as a function of conversion, then we can calculate the reactor volume necessary to achieve a specific conversion.
PFR
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7. Design Isothermal Reactor
CHAPTER 2
-rA=f(X)
CA = f(X)
NA = f(X)
V= f(X)
stoichiometry
CHAPTER 4
Rate Laws
CHAPTER 3
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8. 3.1.1 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)
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9. 3.1.1 Relative Rates of Reactions
2 NO + O NO2
a A + b B c C + d D
If rNO2= 4 mol/m3/s, (formation of NO2)
Then
rNO= -4 mol/m3/s (disappearance of NO)
rO2= -2 mol/m3/s (disappearance of O2)
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10. 3.2 Reaction Order and Rate Law
Let’s take A as the basis of calculation
a species A is one of the reactants that is disappearing as a result of the reaction. The limiting reactant is usually chosen as our basis for calculation.
The rate of disappearance of A, -rA, depends on temperature and concentration and it can be written as the product of the reaction constant k and
Rate raw (Kinetic expression) : the algebraic equation that relates –rA to the species concentration
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11. 3.2.1 Power Law Models and Elementary Rate Laws
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
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12. 농도 대 활동도 (concentration vs. activity)
엄밀히 말하면,
반응속도는 활동도 ai (ai=iCi)의 항으로 나타내야 한다.
그러나 많은 반응계의 경우에 활동도계수 i는 반응이 진행되는 동안 현저하게 변화하지 않으므로 반응속도 kA에 흡수된다. 즉,
(3-3)
kline and Fogler, JCIS, 82, 93 (1981), JCIS, 82, 103 (1981)
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13. 농도 대 활동도 (concentration vs. activity)
활동도는 상대방에 따라서 달라진다.
초등학교 동창회 vs. 사은회
학생A – 학생B, 학생A-교수
상견례 vs. 노래방
학생A-상대부모, 학생A-학생C
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14. Unit of Specific Reaction Rate
The unit of the specific reaction rate, kA, vary with the order of the reaction.
A  products
k =
(Concentration)1-n
Time

15. Unit of Specific Reaction Rate
The unit of the specific reaction rate, kA, vary with the order of the reaction.
A  products 0차
(3-4) 1차
(3-5)
(3-6) 2차 3차
(3-7)
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16. Elementary and Non-elementary Reaction
Kinetic rate raw
“Elementary reaction”
“Non-elementary reaction”
기초반응
비기초반응
O + CH3OH  CH3O + OH
CO + Cl2  COCl2
-rO = k CO CCH3OH
1st order w.r.t. atomic oxygen
1st order w.r.t. methanol
overall is 2nd 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.
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17. Elementary Reaction -r O = k CO C
An elementary reaction is one that evolves a single step such as the bimolecular reaction between oxygen and methanol
O + CH3OH  CH3O + OH
The stoichiometric coefficients in this reaction are identical to the powers in the rate law.
-r O = k CO C
The reaction is 1st order in oxygen free radical and 1st order in methanol; therefore, we say both the reaction and rate law are elementary.

18. Apparent Reaction Order
Sometimes reactions have complex rate expressions that cannot be separated into solely temperature-dependent and concentration-dependent portions.
2N2O N O2 Pt
Kinetic rate raw
Limiting conditions: depending on oxygen concentration
(1)
“apparent” 1st-order w.r.t. N2O
1st order overall
“apparent” reaction-order
-1 w.r.t. O2 , +1 w.r.t. N2O,
overall apparent zero order
(2)

19. 3.2.2 Non-elementary Rate Laws and Reactions
H2 + Br HBr
CH3CHO CH4 + CO
Non-elementary reaction (free radical)
Non-elementary reaction (free radical)
gas-phase decomposition of acetaldehyde
@500oC
(3-8)

20. 3.2.2 Nonelementary Rate Laws and Reactions
In many gas-solid catalyzed reactions, it is sometimes preferable to write the rate law in terms of partial pressures rather than concentrations.
C6H5CH(CH3) C6H6 + C3H6
cumene benzene propylene
C B + P
Langmuir-
Hinshelwood
kinetics
KP = the pressure equilibrium constant [atm]
KB = the adsorption constants [atm-1]
k = the specific reaction rate [mol/kg cat·s ·atm]

21. 3.2.2 Nonelementary Rate Laws and Reactions
At equilibrium, -r’C =0; the rate law for the reversible reaction is indeed thermodynamically consistent:
Solving for Kp yields
which is identical to the expression obtained from thermodynamics.
To express the rate of decomposition of cumene, -r’C as a function of conversion, replace the partial pressure with concentration, using the ideal gas law:
and then express concentration in terms of conversion.

22. Determination of Reaction Rate Law
It is important to remember that the rate laws are determined by experimental observation! They cannot be deduced from reaction stoichiometry.
They are function of the reaction chemistry and not the type of reactor in which the reactions occur.
Even though a number of reactions follow elementary rate laws, at least as many reactions do not. One must determine the reaction order from the experiments or from literature.
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23. Where do you find rate laws?
The activation energy, frequency factor, and reaction order
 Floppy disks and CDROMs by National Institute of Standards and Technology (NIST)
 Standard Reference Data 221/A320 Gaithersburg, MD 20899
 Tables of Chemical Kinetics: Homogeneous Reaction,
National Bureau of Standards Circular 510 (Sept. 1951)
Suppl. 1 (Nov. 14, 1956), Suppl. 2 (Aug. 5, 1960), Suppl. 3 (Sept. 15, 1961)
Washington, D.C., U.S. Government Printing Office
 Chemical Kinetics and Photochemical Data for Use in Stratospheric Modeling,
Evaluate No. 10, JPL Publication 92-20, Aug. 15, 1992, Jet Propulsion
Laboratories, Pasadena, CA, USA
 International Journal of Chemical Kinetics, Journal of Physical Chemistry
 Journal of Catalysis, Journal of Applied Catalysis
 AIChE Journal, Chemical Engineering Science, Korean Journal of Chemical Engineering
 Chemical Engineering Communications
 Industrial and Engineering Chemistry Research
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Example of Rate Law
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25. Seoul National University
Example of Rate Law
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26. B. Second Order Reaction
Example of Rate Law
A. First Order Reaction
B. Second Order Reaction
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27. 3.2.3 Reversible Reactions aA + bB cC + dD
All rate raws 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

28. 3.2.3 Reversible Reactions 2C6H6 C12H10 +H2 2B D +H2 C r H - = +
The rate of disappearance of benzene The rate of formation of benzene 2 , 10 12 6 B
forward k C r H - = +
The net rate of formation of benzene

29. The rate law for the rate of disappearance of benzene
Reversible Reactions
The rate law for the rate of disappearance of benzene B -B

30. thermodynamically consistent with the equilibrium constant
2B D +H2
We need to check
whether the above rate law is thermodynamically consistent at equilibrium.
At equilibrium, -rB=0,
Rearranging, we obtain
identical

31. Thermodynamically Consistent with Equilibrium Constant
2B D +H2
The rate of formation of diphenyl is
(3-15)
Using the relationship
We can obtain the relationship between the various specific reaction rates,
kB, kD:
(3-16)

32. Temperature Dependence of Concentration Equilibrium Constant
Appendix C, Eq (C-9) T KC
Exothermic
reaction T KC
Endothermic
reaction

33. 3.3 kA: The specific Reaction Rate (the Rate Constant)
The specific reaction rate is always referred to that species in the reactions and normally should be subscripted w.r.t. that species. The reaction rate constant k is not truly a constant, but is merely independent of the concentrations of the species involved in the reaction.
The quantity k is also referred to as the specific reaction rate (constant). It is almost always strongly dependent on temperature. In gas-phase reactions, it depends on the catalyst and total pressure. In liquid systems, it depends on the total pressure, ionic strength and choice of solvent.
These other variables normally exhibit much less effect on the specific reaction rate than temperature does with the exception of supercritical solvents, such as supercritical water.
In this text, it will be assumed that kA depends only on temperature.

34. Specific reaction rate (constant) pre-exponential factor
Arrhenius equation
Activation energy, J/mol or cal/mol
Specific reaction rate (constant)
Absolute
Temperature, K
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= …

35. Arrhenius equation kA kA T  0, kA  0 T  , kA  A A T (K) 1/T (K-1)
high E kA
Low E
Slope= -E/R
1/T (K-1)
T  0, kA  0
T  , kA  A

36. SVANTE AUGUST ARRHENIUS
1884 (25) : Ph.D. “ionic theory”
1887 (27) : work with Boltzmann
1888 (28): work with Ostwald and Van't Hoff
1889 (30) : introduced the concept of activation
energy as the critical energy that
chemicals need to react
1903(44) : Nobel Prize in chemistry for ionic theory
prove the influence of the electrolytic dissociation on the osmotic pressure, the lowering of the freezing point and increase of the boiling point of solutions containing electrolytes. He also pointed out the existence of a "greenhouse effect" in which small changes in the concentration of carbon dioxide in the atmosphere could considerably alter the average temperature of a planet.

37. SVANTE AUGUST ARRHENIUS
Svante August Arrhenius was born in Vik, Sweden in At age 25 he turned in his PhD thesis at the University of Uppsala, Sweden. His PhD examining committee did not think very highly of his thesis and rated it 4th class. His oral thesis defense did not fair much better as they rated it as only 3rd class. Arrhenius left Sweden for five years to work with Oswald, Boltzmann and van't Hoff. In 1889 his interpretation of temperature-dependent equation by van't Hoff led to the universal accepted Arrhenius equation for kinetic rate laws in chemistry. He received the Nobel Prize in From 1905 until his death in 1927 he was director of Physical Chemistry at the Nobel Institute

38. 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.

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

40. Energy distribution of reacting molecules
f(E,T) EA
Energy distribution function
for kinetic energies
of the reacting molecules
Fraction of collisions that
have energy EA or great

41. Energy distribution of reacting molecules
Fraction of collisions at T2
that have energy EA or greater
Fraction of collisions at T1
that have energy EA or greater

42. Activation energy

43. (4-hydroxyl phenol) azobenzene
Azo Dye
0~5oC
HNO3
H2SO4
NO2
3H2
Pd/C*
NH2
HNO3
HCl
N=N-Cl
Benzene
diazonium
Aniline
bp=184oC
0~5oC OH
0~5oC OH
-N=N- OH
-N=N-
-OH
Yellow
(4-hydroxyl phenol) azobenzene
Bright red
mp=132oC

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

45. Finding the activation energy
Plot (log k) vs (1/T)
1/T (K-1)
0.01
0.001
0.0001
0.0030
0.0031
0.0032
k (sec-1)
Slope = -E/R

46. Finding the activation energy
Plot (log k) vs (1/T)
When k1=0.005 : 1/T1=
When k2= : 1/T2=
Therefore,
To use the decade method,
choose 1/T1 and 1/T2 so that k2=0.1 k1.
Then, log(k1/k2)=1

47. Finding the 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

48. 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)

49. Specific reaction rate
어느 다른 온도에서의
k 값을 구할 수 있다.
Taking the ratio

50. 3.4 Present Status of Our Approach to Reactor Sizing and Design
Design Equations
Differential
form
Algebraic
form
Integral
form
Batch
CSTR
PFR
PBR

51. Seoul National University
Reactor Size
Design Equations
Levenspiel plot
Batch
CSTR
Graphical method
PFR
For vs. X, the volume of a CSTR and the volume of a PFR can be represented as the shaded areas in the Levenspiel plots.
PBR
If we know the molar flow rate to the reactor and the reaction rate as a function of conversion, then we can calculate the reactor volume necessary to achieve a specific conversion.
PFR
Seoul National University

52. Design Isothermal Reactor
CHAPTER 2
-rA=f(X)
CA = f(X)
NA = f(X)
V= f(X)
stoichiometry
CHAPTER 4
Rate Laws
CHAPTER 3
Seoul National University

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마무리
이 장을 마친 후에, 여러분은 속도식을 농도와 아레니우스 온도 의존성의 항으로 나타낼 수 있다. 이제 등온 화학반응과 등온반응기를 연구하기 위한 알고리즘 블록 쌓기의 첫 번째 2개의 기본적인 블록을 완성하였다.
제4장에서는 세 번째 블록인 화학양론에 초점을 맞출 것이며, 화학양론표를 사용해 농도를 전환율의 항으로 나타내고 마지막으로는 반응 속도와 전환율의 관계를 도출할 것이다.
몰 수지
속도법칙
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