lecture notes chemical engineering

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LECTURE NOTESin

CHEMICAL REACTION ENGINEERINGEngr Nilo T. Aldon ChE 009692

College of EngineeringColegio San Agustin-Bacolod

November 12, 2012

1 Introduction1.1 Chemical Kinetics ___ the study of the rate of reaction and mechanism by which one chemical

species is converted to another.Analyzing the influence of different reaction conditions on the reaction rate gives information about

the reaction mechanismand the transition state of a chemical reaction.In 1864, Peter Waage, a Norwegian pioneered the development of chemical kinetics by formulating

the law of mass action (the speed of a chemical reaction is proportional to the quantity of the reactingsubstances.)

Focal Points Of Chemical Kinetics

Rate of reactionKinetics deals with the experimental determination of reaction ratesfrom which a rate law and

reaction rate constant are derived. Essential rate laws exist forzero order reactions (for which reactionrates are independent of initial concentration), first order reactions, andsecond order reactions, and can

be derived for others throughcalculus. In consecutive reactions the rate-determining step often determinesthe kinetics. In consecutive first order reactions, a steady stateapproximation can simplify the rate law.The activation energy for a reaction is experimentally determined through theArrhenius equationand theEyring equation. The main factors that influence the reaction rate include: the physical state of thereactants, theconcentrations of the reactants, the temperature at which the reaction occurs, and whetheror not any catalystsare present in the reaction.

MechanismsIn chemistry, a reaction mechanism is the step by step sequence of elementary reactionsby

which overall chemicalchange occurs.Although only the net chemical change is directly observable for most chemical reactions,

experimentscan often be designed that suggest the possible sequence of steps in a reaction mechanism.An overall description of how a reaction occurs. A mechanism describes in detail exactly what

takes place at each stage of a chemical transformation. It describes the transition stateand which bondsare broken and in what order, which bonds are formed and in what order, and what the relative rates ofthe steps are. A complete mechanism must also account for all reactants used, the function of a catalyst,stereochemistry, all products formed and the amount each.

Principal Function of Reaction Kinetics from Chemical Engineers Point of View Establishing the chemical reaction mechanism Collecting experimental data Correlating rate data by mathematical equation Designing suitable reactors Specifying operating conditions, methods of control and auxiliary equipment.

1.2 Chemical reaction engineering _is the branch of engineering that is concerned with the exploitation ofchemical reactions on a commercial scale for purposes other than the production of power.

2 Fundamentals of Chemical Kinetics

2.1 Classification of Reaction:

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2.1.1 Mechanism (Elementary or non-elementary)1. Irreversible A B

2. Reversible A B

Reversible Reactions_is one which results in the formation of an equilibrium mixture. Theconcept of a reversible reaction was introduced by Berthollet(1803) after he had observed the formation ofsodium carbonate crystals at the edge of a salt lake.

2NaCl + CaCO3 Na2CO3 + CaCl2He recognized this as the reverse of the familiar reactionNa2CO3 + CaCl2 2NaCl + CaCO3

3. Simultaneous A BA C

4. Consecutive A B C5. Autocatalytic

A + B B + B6. Homogenous Catalyzed A + C R + C

2.1.2 Phases5. Homogeneous6. Heterogeneous

2.1.3 Operating Conditions

7. Isothermal @ constant volume8. Isothermal @ constant pressure9. Adiabatic10. Non-adiabatic and non-isothermal

2.1.4 Molecularity11. Unimolecular

BA

OOO:ozoneofiondecomposittheisExample

3 + 212. Bimolecular

PBA +OOHOH 2 ++

13. Trimolecular or TermolecularPCBA ++P2BA +

PA 3

222 NOONO 2 +2.1.5 Order

14. Integral (1st, 2nd, 3rd)15. Fractional or Zero

2.1.6 System16. Batch17. Flow18. Semi-batch or semi-flow

2.1.7 Equipment19. Stirred tank (single or multistage)20. Tubular (single or multiple)21. Packed bed (fixed bed, moving bed, fluidized bed-dense phase/dilute phase)

2.1.8 Catalyst22. Catalyzed23. Uncatalyzed

2.1.9 Heat evolved24. Exothermic

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25. Endothermic

2.2 Rate of reaction ___ the number of units of mass of some participating reactants whichis transformed into a product per unit time and per unit volume.

( )dt

dN

Vr A

A

1=

2.2.1 Basic Factors Affecting The Rate of Reaction :

a. Nature of the Reactants

The terms active and inactive are used in describing the nature of reactants. For instance, we saythat sodium is a very active metal which reacts violently with water. The activity of elements are

predicted by the use of the periodic table. An example of which is the activity series of metals whichranks potassium as the most active. When covalent bond formation takes place between the moleculesand when large molecules are formed, the reactions tend to be very slow.

b. Frequency and efficiency of collisions of the reactant particles.It follows that any factor (physical state, composition, temperature, pressure, area of exposure,

catalysts, etc) which affects the frequency and efficiency of collisions of the reactant particles willnecessarily alter the speed of reaction.

1. Physical StatePhysical state (solid, liquid, or gas) of a reactant is also an important factor of the rate of

change. When reactants are in the same phase, as in aqueous solution, thermal motion brings theminto contact. However, when they are in different phases, the reaction is limited to the interface

between the reactants. Reaction can only occur at their area of contact, in the case of a liquid and agas, at the surface of the liquid. Vigorous shaking and stirring may be needed to bring the reaction tocompletion. This means that the more finely divided a solid or liquid reactant, the greater its surfacearea per unit volume, and the more contact it makes with the other reactant, thus the faster thereaction. To make an analogy, for example, when you start a fire, first you use wood chips and small

branches - you don't start with big logs right away. In organic chemistry On water reactionsare theexception to the rule that homogeneous reactions take place faster than heterogeneous reactions.

2. Concentration or composition of the reactant(s)

Concentration plays an important role in reactions. According to the collision theoryof chemicalreactions, this is due to the fact that molecules must collide in order to react together. As theconcentration of the reactants increases, the frequency of the molecules colliding increases, strikingeach other faster by being in closer contact at any given point in time. Imagine two reactants being ina closed container. All the molecules contained within are colliding constantly. By increasing theamount of one or more of the reactants you cause these collisions to happen more often, increasingthe reaction rate.

Law of Mass Action _states that the rate of chemical reaction is at each instant proportional to theconcentration of the reactant with each raised to a power equal to their coefficient or the actualnumber of molecules participating in the reaction. This law can be interpreted by several complexmechanisms but it can simply be explained as follows: when two or more molecules react, it mustcome close to one another or must collide. Therefore, it is expected that the rate of reaction increasesif the molecules are crowded closely together, i.e., the concentration is high.

3. TemperatureTemperature usually has a major effect on the speed of a reaction. Molecules at a higher

temperature have more thermal energy. When reactants (reactant + reactant product) in achemical reaction are heated, the more energetic atoms or molecules have a greater probability tocollide with one another. Thus, more collisions occur at a higher temperature, making a product in achemical reaction. More importantly however, is the fact that at higher temperatures molecules havemore vibrational energy, that is, atoms are vibrating much more violently, so raising the temperaturenot only increases the number of collisions but also collisions that can result in rearrangement of


The negative sign denotes thedisappearance of reactant A .

A positive sign on the other hand

denotes formation of a product.
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atoms within the reactant molecules. For example, a refrigeratorslows down the speed of the rate ofreaction since it cools the molecules. On the other hand, an ovengives heat (energy) to the moleculeswhich in turn speeds up the rate of reaction, cooking the food faster.

Chemical kinetics can also be determined using aTemperature Jump. This involves using a sharprise in temperature and observing the relaxation rate of an equilibrium process.

The kinetic energy of particles follows the Maxwell-Boltzmann distribution. An increase intemperature not only increases the average speed of the reactant particles and the number ofcollisions, but also the fraction of particles having kinetic energy higher than the activation energy.Thus, the effective collision frequency increases.

4. PressureBy increasing the pressure, you decrease the volume between molecules and will increase the

number of collisions between reactants, increasing the rate of reaction. This is because the activity ofa gas is directly proportional to the partial pressure of the gas. This is similar to the effect ofincreasing the concentration of a solution.

5. CatalystsA catalyst is a substance that accelerates the rate of a chemical reaction but remains

chemically unchanged afterwards. The catalyst increases rate reaction by providing a differentreaction mechanism to occur with a loweractivation energy.

In autocatalysis a reaction product is itself a catalyst for that reaction leading to positivefeedback. Proteins that act as catalysts in biochemical reactions are called enzymes. Michaelis-Menten kinetics describe the rate of enzyme mediated reactions.

In certain organic molecules specific substituents can have an influence on reaction rate inneighboring group participation.

Agitating ormixing a solution will also accelerate the rate of a chemical reaction, as thisgives the particles greater kinetic energy, increasing the number of collisions between reactants andtherefore the possibility of successful collisions.

2.2.2Factors Affecting the Rate of Homogeneous Reactions :CompositionTemperaturePressure

These variables are interdependent in that the pressure is fixed, given the temperature and composition of thephase. This we may write without loss of generality.

2.2.3 Factors Affecting Heterogeneous Reactions :1. Mass transfer factors (e.g. diffusion characteristics of fluid phases)2. Contact patterns of phases ( each phase may be in one of two ideal flow patterns i.e. plug or back-mix

flow. There are a number of possible combinations of contacting patterns)3. Fluid dynamic factors (e.g. mass velocity, degree of turbulence, etc)4. Interfacial surface area5. Geometry of the reaction vessels6. Chemical kinetic factors (i.e. activation energy, concentration of reactants, etc)7. Temperature and pressure

Some of these factors are not completely independent and may interact with each other.

2.3 Mathematical Expression of Rate of Chemical Reaction

2.3.1 Stoichiometry:1. Stoichiometry may suggest whether we have a single reaction or not.2. Stoichiometry can suggest whether a single reaction is elementary or not because no elementary

reactions with molecularity greater than three have been observed.


rA=f(temperature, pressure, composition)

rA=f(temperature, composition)
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The stoichiometric equation is a chemical equation which expresses an overall chemical reactionin terms of the simplest ratio of reactant and product molecules.

Consider the irreversible elementary reaction

aA + b B rR The stoichiometric relationship of this given chemical reaction is

A B R S-r -r r r = = =a b r s

a bAA A A B

- dCr = = k C C

dt

a bBB B A B

- dCr = = k C C

dt

a bRR R A B

+ dCr = = k C C

dt

a bSS S A B

+dCr = = k C C

dt

a ba b a b a b

S A BA A B B A B R A B k C Ck C C k C C k C C= = =a b r s

From the preceding equations the values of the specific rate constants kB, kR and kS are solved as a function of kAthus,

a b a b

A A B B A Bk C C k C C=a b

; B Ab

k = ka

R A

rk = k

a

; S As

k = ka

Kinetically, the system is at equilibrium if the net rate of change of the forward and backwardelementary reactions is zero.

Consider the elementary reversible reaction

A + B R + S

The rate of formation of R by the forward reaction is:

BAR CCkr:RofFormationofRate 1=

And the rate of disappearance by the reverse reaction is

SRR CCkr-:RofnceDisappearaofRate 2= At equilibrium the net rate of formation of R is zero, then

forward backwardr =r =0

f or wa rd b ac kw ardr = r

k1CACB = k2CRCS

R S1

2 A B

C Ck=

k C C

Since for this reaction kC is defined as


k1

k2


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R S

C

A B

C Ck =

C C

at equilibrium we have

Re Se1

C

2 Ae Be

C Ckk = =

k C C

kC, which is equal to k1/k2, is a constant independent of concentration. CAe, CBe, CRe, and CSe areequilibrium concentrations with respect to A, B, R, and S, respectively. Thus, the equation for kc is very

specific only for equilibrium condition.

The rate of chemical reaction is dependent on temperature, pressure, and concentration. Inparticular, an important relationship is expressed by the Law of Mass Action. The Law of Mass Actionstates that the rate of chemical reaction is at each instant proportional to the concentration of the reactantwith each raised to a power equal to their coefficient or the actual number of molecules participating in thereaction. This law can be interpreted by several complex mechanisms but it can simply be explained asfollows: when two or more molecules react, it must come close to one another or must collide. Therefore,it is expected that the rate of reaction increases if the molecules are crowded closely together, i.e., theconcentration is high.

The statement of the Law of Mass Action is translated into its mathematical expression using thefollowing defined notations:

ri = rate of reaction of any substance ii = any substance in the reactionCi = concentration of any substance i in the reaction.

Suppose the reaction is represented bykA P

Applying the principle of the Law of Mass Action

( ) AA Cr

The preceding equation is the mathematical expression of the Law of Mass Action.Given an irreversible chemical reaction

aA + bB + cC+... dD + eE + ..k

where: A, B, C = are the reactants participating in the chemical reaction;D, E = are the products formed during chemical reaction;

a, b, c,d and e are the number of molecules of each substance involved inthe chemical reaction, then

From the given chemical reaction the rate expression of A is given as:

( ) cCb

B

a

AA CCCr

( ) cCb

B

a

AA CCkCr =Also:

( ) cCb

B

a

AB CCkCa

br =

( ) cCb

B

a

AC CCkCa

cr =

Consider the rate of change of component i involved in the chemical reaction. If the rate ofchange in the number of moles of this component is dNi/dt, the rate in the various areas of kinetics isdefined as follows



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Based on a unit volume of reacting fluid,

( ) ( )i

i

f

dN Change in No. of moles of any substance ir = =

V dt unit volume of fluid time

Based on unit volume of reactor, if different from the rate based on unit volume of fluid,

( ) ( )i

i

r

dN Chnage in the No. of moles of any substance ir = =

V dt Unit volume of reactor time

Based on unit interfacial surface in two fluid systems or based on unit surface of solid in gas-solid systems,

( ) ( )i

i

dN Change in the No. of moles of any substance ir = -

Sdt Unit surface area time

Based on unit mass of solid in fluid-solid systems,

( ) ( )i

i

s

dN Change in the No. of moles of any substance ir = =

W dt unit mass of solid time

2.3.2 Molecularity and Order of Reaction

The molecularity and order of an elementary reaction is the number of molecules involved in the ratedetermining step of a reaction. Molecularity of reactions has been found to be one, two, and occasionallythree. Molecularity refers only to an elementary reaction and can only be whole numbers.

Example:Consider the irreversible elementary reaction

aA + bB + cC +.. product

where a, b, and c are stoichiometric coefficients. We call the power to which the concentration areraised the order of reaction.

( ) rCq

B

p

AA CCkCr =

When the stoichiometric equation truly represents the mechanism of the reaction, the order and

molecularity both aren = a + b + c and individually p = a, q = b, r = c

Thus, the reaction is

ath order with respect to Abth order with respect to Bcth order with respect to Cnth order over alln = a + b + c + .

Example:

1. A P (n=1)2. A + B P (n=2)



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3. 2A + B P (n=3)4. A + B + C P (n=3)

Rates of disappearance of reactants or formation of products are related to A by the stoichiometric coefficients:

( ) ( ) ( )c

r

b

r

a

r CBA =

=

2.3.3 Rate Constant k :For Equation : aA product (s)

The rate of disappearance of reactant A:

( )n

AA

V

nkr

=

( )n

AA

V

nkr

=

( )

n

volumeunit

massunit

timeunit

1

( )

( ) ( )

1

1

=n

ntimemassunit

volumeunitk unit ofkfor an nth order chemical reaction.

( ) ( )timevolumeunitmassunit

k= for zero-order

( )timek

1= for 1st-order

( )( ) ( )timemassunit

volumeunitk= for 2nd-order

( )

( ) ( )timemassunit

volumeunitk 2

2

= for 3rd-order

The reaction rate constant could also be expressed in terms of pressure( )and mole fraction(NA):

If we letRT

N

RT

PC AAA

==

(-rA) = kc (CA)n = kp (PA)

n

( )

( )

( )

( ) nA

n

Apn

A

Ac

C

Pk

C

rk =

=

( )nP RTk=

( )

( )nA

n

AN

C

Nk=

n

N

RTk

=

Rate and Order Questions1. A reaction has the stoichiometric equation

SRA +2What is the order of reaction?Answer: The reaction is elementary, irreversible, second order

2. Given the reaction

52222

12 ONONO +



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What is the relation between the rates of formation and disappearance of three components of the reaction?Answer:

( ) [ ] [ ] 5.022

22ONOkrNO =

( ) [ ] [ ] 5.022

24

12

ONOkrO = ( )241

NOr=

( ) [ ] [ ] 5.022

22

152

ONOkr ON = ( )221

NOr=

3. A reaction with a stoichiometric equation RBA +2

1

Has the following rate expression: ( ) [ ] [ ]BArA5.0

2=What is the rate expression for this reaction if the stoichiometric equation is written as

SRBA 222 ++Answer: ( ) [ ] [ ]22 BArA =

4. For the elementary reaction: 2A+B 2S

CAo= 8 mols/liter, CBo= 6 mols/liter, CSo= 2mols/litera. What is the order of reaction with respect to A?

b. What is the over-all order of reaction?c. What is the limiting reactant?d. If 75% of the limiting reactant is reacted after 5 minutes, what will be the values of CA, CB,,and CS?e. What is (-r A)?f. What is (r B)?g. What is (r S)?

SOLUTION:a. Second-order

b. Third-orderc. A is the limiting reactant. 2 mols of A will react with 1 mol of B, hence, 8 mols A will react with

only 4 mols B.

d. ( )8 1 0.75A2 molCliter

= = ; ( )8 0.7562

B

3 molC

liter= = ; ( )8 0.75 2S

8 molCliter

= + =

e. ( ) ( ) ( )22 2 3AA A B

dC 12k mol r kC C k

dt literminute = = = =

f.2A

B

r 6 kmol r

literminute = =

g. ( )uteliter

molkrr As

min

12

==

5. The specific reaction rate of a first-order reaction is 2.5 x10-7/s and the initial concentration is 0.1gmol/lit.What is the initial rate expressed in terms of seconds, liters gmols.

SOLUTION:7 82.5 10 0.1 2.5 10A

A

dC x gmol x gmol kC x

dt s lit lit

= = =

6. The initial rate of a second-order reaction is 5.0x10-7 gmol/lit-s when CA is 0.2 gmol/lit. What is the specificreaction rate expressed in units of second, liter, gmol.

SOLUTION:


ANSWER:

ANSWER:

ANSWER:

ANSWER:

ANSWER:


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272

5

5.0 10 0.2( )

1.25 10

( )( )

AA A

dn x gmol gmol r kC k

Vdt lit s lit

x litk

gmol s

= = + =

=

7. The third-order gas-phase reaction 2NO+O2 2NO2 has a specific reaction rate of kC=2.65 x104 lit2/

(gmol)2(s) at 30oC. Find kP and kN.SOLUTION:

a. 33222

4

)303()08205.0()()(1065.2

)( ksgmol

litxRT

kkn

C

P== 222

)()())((

7239.1

atmsgmollit=

b.( )( )

( ) ( ) ( ) ( ) ( ) ( )

3

22

24

30 30 8 2 0 5.0

106 5.2

=

=

K

Katmli t

a tm

sgmo l

li tx

RTkk

n

CN

22 )())((7239.1

sgmollit=

8. The third-order gas-phase reaction 2NO + O2 2NO2 has a specific reaction rate at 30oC and 1 atm. Find

kpand kn.Solution;

( ) smolg

Lxk

c

..1065.2

2

2

4=

Solution:

( )n

n

p

n

ck

RTkRTk

==

( )

( )

( )satmL

molg

KKmolg

atmL

smolgLx

RT

kk

n

cp

..7239.1

303..

.08206.0

..1065.2

33

2

24

= =

==

( )3

3

.1 1.7239 1.7239

. . .

n

n p

gmol g molk k atm

L atm s L s

= = ==

9. Ex. At a given temperature the following data were taken:CH3CHO (g) CH4 (g) + CO (g)

A B C

Experiment NumberInitial Pressure, A(Torr) Initial Rate of Increase in Total

pressure (Torr)

1 300 0.61

2 200 0.27

a. Write the equationb. Find the order of reactionSOLUTION:

( )

2

200

300

log

27.0

61.0log

log

log

:;;

2

1

2

1

2

1

21

2

1

2

121

===

======

A

A

n

A

A

nA

nAn

An

An

A

P

P

r

r

n

P

P

kP

kP

r

rkPrkPrkP

dt

dr

a. r = kPA2

b. 2nd-order reaction


ANSWER:

ANSWER:

ANSWER:

Answer

Answer

Answer:


11/38

the reaction is non-elementary.

2.4 Mechanism__is the sequence of individual chemical events whose over-all result producethe observed reaction.

__the sequence of steps that takes place to complete a chemical change.

In most instances the postulated mechanism is a theory devised to explain the end results observed byexperiments. Like other theories, mechanisms are subject to change over the years as new data is uncovered or asnew concepts regarding chemical interactions are developed.

The mechanism of an overall chemical reaction consists of a series of elementary reactions, which togetherbring about the overall reaction summarized in the stoichiometric equation.

A reaction mechanism is a set of steps at the molecular level. Each step involves combinations or re-arrangements of individual molecular species. The steps in combination describe the path or route thatreactant molecules follow to reach the product molecules. The result of all steps is to produce the overall

balanced stoichiometric chemical equation for reactants producing products.

2.4.1 Steps Followed in Establishing Reaction Mechanism :1. Postulate a simple mechanism with a corresponding stoichiometry.2. If the stoichiometry appears to indicate the reaction to be one-step and elementary, proceed to acquire

kinetic data and analyze them according to the integral method.3. For non-elementary reactions, assume that the overall reaction consists of several elementary reaction

steps with formation of intermediate compounds.4. Formulate a rate expression for each of the elementary steps and sum up the individual rate

expressions to describe the overall rate.5. If the resulting rate expression agrees with experimental kinetic data, the assumed mechanism is

acceptable. Otherwise, continue to assume alternative mechanisms, and repeat step 4 until a desireddegree of agreement is obtained.6. Frequently, it may be simpler to use a purely empirical approach to correlate the data.

2.5 Elementary and Non-Elementary Reactions

2.5.1 Elementary Reactions __any such reaction in which the rate equation suggested by stoichiometricequation represents the actual mode of action and occurs in a single step.Ex. P2S2BA ++

( ) ( ) 2BAPA CKCrr ==

( ) ( ) 2BASB CKCrr 2==

2.5.2 Non-elementary Reactions __ are those where there is no direct correspondence betweenstoichiometric equation and the rate expression and occurs in two or more series of reactions.

Ex. P2S2BA ++

If ( ) ( ) 2BAPA CKCrr =

or ( ) ( ) 2BPA KCrr ==



12/38

Two Types of Non-Elementary Reaction Mechanism1. Non-chain reaction mechanism

Reactants (intermediates)*

(intermediates)* Products

Example: A P

MechanismA B*

B* P

2. Chain Reaction Mechanism

Reactants (intermediates)*

(intermediates)* + Reactant (Another intermediates)* + Product

(another intermediates)* ProductExample:

1. A + B S + P

MechanismA I*

I* + B R* + S

R* P

2. 2A + B P + 2R

MechanismA + B AB*

AB* + A S* + 2R

S* P

Example: Present mechanisms consistent with experimentally found rate equation for the following reaction:

2A2B=2AB + A2 ; ][20][

][1500

2

22

2 BAAB

BA

rA +=SOLUTION:

]][[]][[ 2[42*

3

22*

*2

2ABAkBAAkr

ABABAA

ABABA

A =+=+

+=



13/38

]][[][][

]][[][][;

][][

]][[][

0]][[]][[]][[][

24

232

242123

232

2421*

242

*

3

*

221

2

*

ABAkBAkABk

ABAkBAkBAkr

BAkABk

ABAkBAkA

ABAkBAAkABAkBAkr

A

A

+

+=

++

=

=+=

][][

]][[][

232

2

24

2

231 2

BAkABk

ABAkkBAkk

+

=

Assume k4=0

][][

][

232

2

231

2 BAkABk

BAkkrA +

= multiply by

2

2

1

1

k

k

][20][

][1500

][][

][

2

2

2

22

3

2

22

31

BAAB

BA

BAk

kAB

BAk

kk

+=

+=

Therefore, the mechanism will be:

ABABAA

ABABAk ++

+=

22

*

*

2

3

2.6. Effect of Temperature on Rate of Reaction:

2.6.1 Arrhenius Equation:

The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence ofa chemical reaction rate, more correctly, of arate coefficient, as this coefficient includes all magnitudes thataffect reaction rate except for concentration. The equation was first proposed by the Dutch chemist JacobusHendricus vant Hoffin 1884; five years later in 1889, the Swedish chemistSvante Arrhenius provided a

physical justification and interpretation for it. Nowadays it is best seen as an empirical relationship. In short,the Arrhenius equation is an expression that shows the dependence of the rate constantkofchemicalreactionson the temperatureT(in Kelvin) and activation energyEa, as shown below:

whereA is thepre-exponential factoror simply the prefactorand R is thegas constant. The units of thepre-exponential factor are identical to those of the rate constant and will vary depending on the order of thereaction. If the reaction is first order it has the units s -1, and for that reason it is often called the frequency

factororattempt frequency of the reaction. When the activation energy is given in molecular units, instead of

molar units, e.g. joules per molecule instead of joules per mol, the Boltzmann constant is used instead of thegas constant.

It can be seen that either increasing the temperature or decreasing the activation energy (for examplethrough the use ofcatalysts) will result in an increase in rate of reaction.

RT

E

eA

=k

RT

E

eA

=

k

RT

E

A

k=ln

ART

Ek lnln +=

Equation of a straight line: y = mx + bwhere: y = lnk

m = slope = -E/Rx = 1/Tb = ln A


lnk2

Lnk1

1/T22

1/T1

Slope=-E/R

=

121

211

lnTTR

E

k

k

g

Where:k = reaction rate constantE = activation energy, cal/gmolA = frequency factorT = absolute temperature, K

Ho = standard-state enthalpy change, cal/gmolR = 1.987 cal/gmol.K

ANSWER:
http://en.wikipedia.org/wiki/Rate_constanthttp://en.wikipedia.org/wiki/Rate_constanthttp://en.wikipedia.org/wiki/Svante_Arrheniushttp://en.wikipedia.org/wiki/Svante_Arrheniushttp://en.wikipedia.org/wiki/Rate_constanthttp://en.wikipedia.org/wiki/Rate_constanthttp://en.wikipedia.org/wiki/Chemical_reactionhttp://en.wikipedia.org/wiki/Chemical_reactionhttp://en.wikipedia.org/wiki/Chemical_reactionhttp://en.wikipedia.org/wiki/Absolute_temperaturehttp://en.wikipedia.org/wiki/Absolute_temperaturehttp://en.wikipedia.org/wiki/Activation_energyhttp://en.wikipedia.org/wiki/Pre-exponential_factorhttp://en.wikipedia.org/wiki/Pre-exponential_factorhttp://en.wikipedia.org/wiki/Gas_constanthttp://en.wikipedia.org/wiki/Gas_constanthttp://en.wikipedia.org/wiki/Joulehttp://en.wikipedia.org/wiki/Boltzmann_constanthttp://en.wikipedia.org/wiki/Activation_energyhttp://en.wikipedia.org/wiki/Catalysthttp://en.wikipedia.org/wiki/Catalysthttp://en.wikipedia.org/wiki/Rate_constanthttp://en.wikipedia.org/wiki/Svante_Arrheniushttp://en.wikipedia.org/wiki/Rate_constanthttp://en.wikipedia.org/wiki/Chemical_reactionhttp://en.wikipedia.org/wiki/Chemical_reactionhttp://en.wikipedia.org/wiki/Absolute_temperaturehttp://en.wikipedia.org/wiki/Activation_energyhttp://en.wikipedia.org/wiki/Pre-exponential_factorhttp://en.wikipedia.org/wiki/Gas_constanthttp://en.wikipedia.org/wiki/Joulehttp://en.wikipedia.org/wiki/Boltzmann_constanthttp://en.wikipedia.org/wiki/Activation_energyhttp://en.wikipedia.org/wiki/Catalyst


14/38

The Arrhenius EquationAE

RTok k e

= gives a good empirical description of the temperature dependence of

the rate constant of an elementary reaction. Both E A and ko are independent of temperature. This expression fitsexperimental data well over wide temperature ranges and is strongly suggested from various standpoints as being areasonable first approximation of the true temperature dependency.

1. When only two sets ofkand Tare givenFor a given k1 and T1 the equation becomes

11

A

E

RT

ok k e

=Apply the natural logarithm to both sides of the equation and simplify

1

1ln lnAE

RT

ok k e

= 1ln lnAE

RT

ok e

= + 1

ln AoE

kRT

=

Solving for lnko will give 11

ln ln AE

ko kRT

= +

Similarly for k2 and T2, Equation 1 becomes

2

2ln lnAE

RT

ok k e

= 2ln lnAE

RT

ok e

= +

22

ln ln AE

ko k

RT

= +

Equating the two equations: 1 21 2

ln lnA AE E

k kRT RT

+ = +

Simplifying further :

2 1

1 2

ln ln A AE E

k kRT RT

=

2

1 1 2

1 1ln A

k E

k R T T

=

2 2 1

1 1 2

ln Ak E T T

k R TT

=

2. When more than two sets of rate constant (k) and temperature (T) are available:AE

RTok k e

=Applying the natural logarithm to both sides of Equation and simplifying will give

ln ln AoE

k kRT

= is an equation of a straight line where ln ko and AE

R

are the intercept and the slope of

the line, respectively. The natural logarithm ofk ( )ln k is plotted against the reciprocal of temperature1

T

.

2.6.2 Collision Theory: RTE

AeTk

=2

1

For a chemical reaction to proceed, molecules must have effective collisions. The two requirements for aneffective collisions are;a.The molecules must be suitably reactive and must have enough energy,


ln k

1/T


15/38

b. The molecules must be arranged in a proper position.

Collision theory, was proposed by Max Trautz and William Lewis in 1916 that qualitatively explains howchemical reactions occur and whyreaction rates differ for different reactions. It assumes that for a reaction tooccur the reactant particles must collide, but only a certain fraction of the total collisions, the effective collisions,cause the transformation of reactant molecules into products. This is due to the fact that only a fraction of themolecules have sufficient energy and the right orientation at the moment of impact to break the existing bonds andform new bonds. The minimal amount of energy needed so that the molecule is transformed is calledactivationenergy. Collision theory is closely related tochemical kinetics.

Collision theory views the rate to be governed by the number of energetic collisions between reactants.What happens to the unstable intermediate is of no concern. It is simply assumed that this intermediate breaksdown rapidly enough into products so as not to influence the rate of the over-all process.

Tekk RTE

o

c

=Where: kis the specific rate constant, kois the collision theory constant, Tis the temperature (K),R is the

universal constant (1.987 cal/g-mol K), andECT is the collision theory activation energy (cal/g-mol).

1. When only two sets ofkand Tare available

For a given k1 and T1 equation becomes

1'1

1

CTE

RT

o

kk e

T

=

'1

11

ln ln ok E

kRTT

=

'

11

ln ln CToEk

kRTT

= +

Similarly fork2 and T2'2

22

ln ln CToEk

kRTT

=

' 2

22

ln ln CToEk

kRTT

= +

Equating2 1

2 12 1

ln lnCT CT E Ek k

RT RTT T+ = +

2 1

1 22 1

ln lnk k E E

RT RTT T =

2

2 2 1

1 1 2

1

ln

k

T E T T

k R TT

T

=

;2 1 2 1

1 2 2 1

ln CTEk T T T

k T R T T

=

The preceding equation is used to determine the activation energy (ECT) if two sets of rate constants (k) andtemperature (T) are given.

2. When more than two sets ofkand Tare given the equation can be rearranged as shown below

'CTE

RTok k T e

=

'

CTE

RTo

kk e

T

=

http://en.wikipedia.org/wiki/Chemical_reactionshttp://en.wikipedia.org/wiki/Reaction_ratehttp://en.wikipedia.org/wiki/Reaction_ratehttp://en.wikipedia.org/wiki/Activation_energyhttp://en.wikipedia.org/wiki/Activation_energyhttp://en.wikipedia.org/wiki/Activation_energyhttp://en.wikipedia.org/wiki/Activation_energyhttp://en.wikipedia.org/wiki/Chemical_kineticshttp://en.wikipedia.org/wiki/Chemical_kineticshttp://en.wikipedia.org/wiki/Chemical_reactionshttp://en.wikipedia.org/wiki/Reaction_ratehttp://en.wikipedia.org/wiki/Activation_energyhttp://en.wikipedia.org/wiki/Activation_energyhttp://en.wikipedia.org/wiki/Chemical_kinetics


16/38

Applying natural logarithm to both sides of the equation, then simplify will result into an equation of a straightline.

'ln ln CTo

Ekk

RTT=

1The plot of ln

kvs is

TT

Plot of lnk

T

as a function of (1/T) for the determination of activation energy (ECT) and constant (ko).

2.6.3 Transition-State Theory: RTE

TAek

=Activated Complex or Transition State Theory:

Henry Eyring, an American chemist, postulated an alternative to collision theory. Hehypothesized that an intermediate species called an activated complex forms during collision. This intermediatespecies exist very briefly. It dissociates to form either the product (if reaction occurs) or the original reactants(if reaction does not occur)

Most reactions proceed in many steps called elementary processes. The combined effect of all theelementary processes gives the overall reaction. The slow step determines the rate of the chemical reaction andis called the determining step. The observed rate of the overall reaction is the equivalent to the rate of the slowreaction.Transition-state theory views the reaction rate to be governed by the rate of decomposition of


1/T

lnk

T


17/38

intermediate. The rate of formation of intermediate is assumed to be so rapid that it is present in equilibriumconcentrations at all times. The governing equation is

"ACT

o

E

RTk k Te

=

where, kis the specific rate constant, ko is the activated complex theory constant, Tis the temperature (K),R

is the universal constant (1.987 cal/g-mol K), andEACTis the activation energy (cal/g-mol).

1. When two sets ofkand Tare givenFork1 and T1

1"

1 1

ACTE

RT

ok k T e

=

1"1

1

ACTE

RT

o

kk e

T

=

1"1

1

ln ln lnACTE

RT

o

kk e

T

= +

" 1

1 1

ln ln ACToEk

kT RT

= +

similarly fork2 and T2" 2

2 2ln ln

ACT

o

Ek

k T RT

= +

Equating2 1 2 1

1 2 2 1

ln ACTEk T T T

k T R T T

=

2. When more than 2 sets ofkand Tare available

"ACTE

RTo

kk e

T

=

''ln lnACTE

RTo

kk e

T

=

"ln ln ACToEk

kT RT

=

The1

lnk

Plot vsT T

is

Plot of lnk

T

as a function of (1/T) for the determination of activation energy (EACT) and constant (ko).

2.6.4 General Equation:RT

E

AeTkm

=

2.6.5 Vant Hoff Equation:( )

2

ln

TR

H

dT

kd

g

o=

Example:1. If a first-order reaction has an activation energy (E) of 25,000 cal/gmol and in the equation

k=Ae-E/RT, A=5x1012/s. At what temperature will the reaction has a half-life of 1-minute?SOLUTION:

RTRT

E

exAes

t

k00 0,25

12

2

110501155.02ln

6 0

12ln

1

=====


1/T

lnk

T

ANSWER:


18/38

KT 28.373=

2.7. EQUILIBRIUMThe system is in the state of equilibrium if the following characteristics are observed:1. It is a closed system ( the amount of matter in the system does not change)2. There is no change in the properties of the system as time passes.3. Two processes, which are opposite in direction, simultaneously take place at the same rate.

4. The ratio of the product of the molar concentration of the substances formed to the product of themolar concentration of the reactants is constant.

2.7.1 Factors that Affect Chemical Equilibrium:1. Temperature2. Pressure3. Composition

For Equilibrium: Pressure may be fixed given the temperature and composition of the phase, (due tothe interdependency of P and T )Therefore: r = f ( T, C ) . This assumption could also be applied to non-equilibrium systems in the

absence of better supposition (theory).

2.7.2 Le Chateliers Principle:If a stress (disturbance or change) is applied to a system in a state of equilibrium, the system will shift insuch a way to neutralize the effect of the stress.

Ex. In the decomposition of CaCO3 =CO2 + CaO, the removal of CO2 gas will trigger the system in astate of equilibrium to produce more CO2 in order to offset the reduction in pressure.

The tendency of a reaction, whether physical or chemical, is determined by a balance between these twofactors:

1. The tendency toward a state of minimum energy, or low enthalpy; and2. Tendency toward a state of maximum disorder, or high entropy.

If we represent the change in the heat content of a system as H, the change in entropy as S, andabsolute temperature as T, an equation which is helpful in predicting reaction tendency is ;

G = H - (T) ( S). (free-energy change)

In spontaneous reactions, G is negative; in systems in equilibrium , Gis zero.

2.7.3 Hesss Law : The total change in enthalpy of a system is dependent on the temperature, pressure, stateof aggregation, and state of combination at the beginning and at the end; it is independent of the numberof intermediate reactions

3 Design of Homogenous Batch Reactors

3.1 Constant-Volume Reactions:

Irreversible Reactions aA bBa. Zero Order A B

b. First- Order A B

ktC

C

Ao

A = ln ktAoA eCC=


( ) ABB kCab

dtdCr ==( ) ( ) aAAA Ckdt

dCr ==-(CA - CAo )=Kt C A = CAo Kt


19/38

c. Second OrderType 1 2A B

ktCC AoA

=11

Ao

AoA

ktC

CC

+=

1

Type 2 A + B R

( )kt

CC

CC

CC ABo

AoB

AoBo

=

ln1

d. Third OrderType I 3A B

ktCC AoA

211

22=

( ) 21

221 Ao

AoA

ktC

CC

+=

Type II 2A + B R

( ) ( ) ( )

2

2ln

22

ktCC

CC

CC

CC

CCCC AoBo

BAo

BoA

AAo

AAoAoBo =+

Type III A + 2B R

( )( )

( ) ktCCCCCC

CC

CCCCBoA

ABo

AoB

BBo

BBoBoAo

o

2

2ln

2

=+

Type IV A + B + C R

( )( ) ( ) ( ) ( )( )

ktC

C

CCCCC

C

CCCCC

C

CCCC C

Co

BoCoAoCoB

Bo

AoBoCoBoA

Ao

BoAoCoAo

=

+

+

ln1

ln1

ln1

Complex Constant-Volume Reactions

a. Side or Simultaneous Reactions

( )tkkAoA eCC

21+=

( )[ ]tkk-ABoB 1o e1kk

CkCC 2

21

1 +

++=

( )[ ]tkk-ACoC 1o e1kk

CkCC 2

21

2 +

++=

b. Consecutive Reactions CBA kk 21

ktAoA eCC

=

= tktkAoB ee

kk

CkC 21

12

1

+

= tktkAoC ekekkk

kk

CC 12 2112

12


tkAB

kk

k

AoB

C

eCk

kC

k

kCC

kk

k

k

t

oequib

Max

B

1

12

2

max

2

1

2

1

12

1

2

ln

=

=

=

CoCBoB

CoC

BoB

C

B

C

B

Ck

kC

k

kCC

CCCC

kk

dCdC

rr

2

1

2

1

2

1

=

===

BAk1

CAk2

k1

k2

BA


20/38

c. Reversible Reaction

( )( ) AoA

AoAo

CkCkk

CkCkk

kkt

212

212

12

ln1

++

+=

AeA

AeAo

CC

CC

kkt

+= ln

1

12

d. Homogeneous Catalyzed Reactions

( ) ( ) ( ) tktCkkxLnC

CLn observedcA

A

A o =+==21

1

e. Autocatalytic Reactions RRRAk ++

( )

( )( )o o

o o

oo

A o A A R

o A R

A RA o A

C C C C C Ln Ln C kt C C kt

C CC C C

= = = +

tConstanCCCCCWhere oRARAo o =+=+=:

3.2 Variable Volume Reactions:n

AA

V

nk

Vdt

dn

=

a. Zero-Order

( ) ktxC

AAA

Ao =+

1ln

b. First Order

( ) ktxV

VA

oA

==

1ln1ln

c. Second-Order

( )( )

( ) tkCxxx o

AAAAA

A =++

1ln1

1

d. Third-Order

( )

( ) ( )( ) tkCx

x

x

x

xxAoAA

A

AAA

A

AAA 22

2

2

22

1ln1

2

12

21=

+

+

Where: Ax fractional conversion=1 0

0

A A

A

x x

A

x

V Vfractional cahnge in volume

V

= =

=

= =

Sample Problems:


RAk1

CRCA k ++ 2


21/38

1. For the reaction A + B S, equal volumes of 1 molar A and 2 molar B are mixed and allowed to reactfor 1 hr which half of A had been reacted. If the reaction is 1st-order with respect to A and 1st-orderwith respect to B, how much time in hours is required for 75% of A to react?

a. 2.714 b. 2.26 c. 4.351 d. 6.732

( )( ) ( ) ( )1

2

1250

2

11==== BoAo C;.C

ixing:B after mn of A andncentratioInitial co

( )( ) ( )( )

( ) ( )( )

( ) ( ) ( )

( )( )

( )( )

( ) ( )( )

( ) ( )( )

( ) ( )( )

( )( )

( )( )h

.

..Ln

.-..CC

CCLn

CCkt

...C

...xCC

..

..Ln

.CC

CCLn

CCtk

...C

...xCC;CC

CCLn

CCkt

reactionond-orderFor

ABo

AoB

AoBo

Bo

AAoA

ABo

AoB

AoBo

Bo

AAoA

ABo

AoB

AoBo

26.212501

506250

50181090

11

6250750501

12507501501

810902501

50750

5011

11

750500501

2505015011

sec

==

=

=====

=

=

=

==

===

=

2. The reaction A +B S is of second-order and kat 0oC is 39.1 liter/(mole)(minute). An aqueoussolution is made of 0.004 molar A and 0.005 molar B. How long will it take for 90% of A to react?

a. 30 minutes b. 26 minutes c. 50 minutes d. 20minutes

( ) ( )( )( ) ( )( )

( ) ( )( )( )( ) ( )

minutes26.330.00040.005

0.0040.0014Ln

0.0040.00539.1

1

CC

CCLn

CCk

1t

0.00140.900.0040.005C0.00040.910.004x1CC

reactionorder-secondFor

ABo

AoB

AoBo

B

AAoA

=

=

=

=====

3. The reaction 2A 2R + S takes place isothermally in a constant-volume experimental reactor.

Starting with a mixture of 80% A and 20% inserts, the initial pressure of 10 atm increases by 25% in8 minutes. What conversion is attained?

a. 37.5% b. 62.5% c. 35.5% d. 64.5%

%52.68

38:358

5;5.125.010

5.00

''

''

=

=

===

==+

++

Ao

AAoAA

AA

AA

P

PPxP

xxf

xxA

x'-

0028i

S2RI2A

4. If the same feed in the preceding problem is introduced in an isothermal variable-volume batch reactor,

what is the time required for the same conversion?a. 9.3 minutes b. 8.5 minutes c. 6.5 minutes d. 10.5 minutes

@ constant-volume

02604.08

1

3

1

8

1111=

=

=

oAACCt

k

Variable Volume:



22/38

( )

( )

( )( )

( ) ( )

( ) ( )

( )( ) ( ) min32.9625.014.0

625.01

625.04.01

802604.0

11

1

11

;4.08.02

23

0

01

=

+

+

==

+

+

=

=

=

==

==

LntxLnx

x

kPt

V

VV

AA

A

AA

A

x

xx

A

o

A

AA

3.3. REACTOR DESIGN

3.3.1 Material Balance:

Types of Reactors

1. Single Batch Reactors (Continuously-Stirred Tanks)a. Feed is introduced before the start of reaction and product is drawn out after reaction iscompleted or terminated.

b. No feed is introduced and no product is drawn out during reaction timec. Concentration of reactants and products varies as a function of timed. Concentration of reactants and products is uniform throughout the reactor volume

2. Back-Mix Reactors (Continuously-Stirred Tanks, steady-state)a. Feed is introduced simultaneously as product is drawn out

b. The volumetric feed rate is equal to the product draw-out rate


ReactantEnters

ReactantLeaves

ReactantDisappears

by reaction

ReactantAccumulates

oC

Aodt

fC

Adt

(-rA)Vdt

d (VCA)

REACTOR


23/38

c. Reactor volume and reactant concentration remain constant and do not vary as function oftime.

d. Concentration of reactants and products is uniform throughout the reactor volume.

3. Plug-Flow or Tubular Reactorsa. Feed is introduced simultaneously as product is drawn out

b. Concentration of the reactants and products varies as a function of location or length of thereactor

c. The concentration of the reactants and products at the specific length of the reactor does notvary as function of timed. The volumetric feed rate may not necessarily be equal to the volumetric product draw-outrate.

REACTOR COMPARISON

1. When fluid density is constant plugflowplugflowbatchbatch ttt kvk === ==

2. For all conditions plugflowbatchtt

k ==

3. For Batch stt+=

4. For Back-Mix ( )AAxt += 1

Where:

= space time or cycle time; the time required to process one reactor volumet=holding time or reaction time (batch and plug-low)

t= holding time, mean residence time (for back-mix)ts=shutdown time; time required for loading and unloading of reactants and products

respectively, cleaning etc. Applicable for batch reactorss= space velocity; volume of entering feed at a specified conditions per unit time per void

volume of reactor.

=1/

3.3.2 Single Batch Reactors (Continuously-Stirred Tanks)In a batch reactor, neither the reactants nor the products flow into or leave the system when the

reaction is carried out. They are either the constant-volume or constant-pressure reactors. The operation of abatch reactor is unsteady-state. Although the composition throughout the reactor is uniform at a giveninstant, it changes with time.

Material Balance

Page 23 /38 nilotaldon2/7/2013BATCHREACTOR

Reactant

Accumulatesd (VCA)

Reactantdisappears

by reaction(-rA)V dt


24/38

0 = (-rA)V dt + d (VCA)

( )

( )

( )A

A

A

A

r

dC

rV

VCd

dt

=

=where: (-rA) = kCA

n

Holding or Reaction time, t_ retention time of reactants inside the reactor volumeConstant Volume

a. Zero-Order ( )

k

xC

k

CCt

AAAA oo

=

=1

b. First-order

( )A

A

AxLn

kC

CLn

kt o

== 1

11

c. Second-Order

=

=

A

A

AAAx

x

kCCCkt

oo1

1111

d. Third-Order

( )

=

=

2

2

2221

2

2

111

2

1

A

AA

AAAx

xx

kCCCkt

oo

Variable -Volume Reactors

a. Zero-Order

( )AAA

AxLn

k

Ct o

+= 1

b. First Order

( )

==

oA

AV

VLn

kxLn

kt

11

11

c. Second-Order

( )

( )( )

+

+

=AA

A

AA

A

xLnx

x

kCt

o

11

11

d. Third-Order



25/38

( )

+

= oAAACC

r

( ) ( )( )

( )( )

( )

+

+=

AA

A

AAA

A

AAA

A

xLnx

x

x

xx

kCt

o

11

2

12

211 22

2

22

2

Note: For Batch, Space Time or cycle Time,

st t = + where: ts= shutdown time

3.3.3 Back-Mix Reactors (Continuously-Stirred Tanks, steady-state)

Material Balance

( ) dtVrdtCdtC AAfAoo +=

@ steady-state: o =f ; = CA, V are constant

o (CAo-CA)dt = (-rA)Vdt

( ) ( )A

AA o

A

AAo

o r

xC

r

CCV

=

==

For single reactor

_ equation of a straight line


BACK-MIXREACTOR

ReactantEnters ReactantLeaves

ReactantDisappears

by Reaction

oC

Aodt

fC

Aodt

(-rA)Vdt

(-rA)

CAn

inter-yCAo

=


26/38

For reactors in series:

Back-Mix Constant Volume: (Steady-State)

1. Zero-Order

( ) krA

=

For Reactors in Series:

2. First-Order

( )AA

kCr =

For Uniform Volume Reactors in series:

3. Second-Order

( ) 2AA

kCr =

For Reactors in series:

4. Third-Order

( ) 3AA

kCr =


( )

1+=

nAnAA

CCr

kCCnn AA

=1

( ) nAo

Ank

CC

+=

1

( )22 1 AA

A

A

AA

xkC

x

kC

CC

o

o

=

=

+=

k

CkC

oA

A2

411

n

A

k

CkC

n

An

2

4111

+

=

( )AA

A

AA

xk

x

kC

CCo

=

=

1

( )323 1 AAo

A

A

AA

xkC

x

kC

CCo

=

=

kC

C

oA

A

+=

1

1

kCCAoA

=

k

xC

k

CCAAAA oo =

=


27/38

Back-Mix Variable Volume: (Steady-State)

General Equation:

10. Zero-Order

11. First-Order

12. Second-Order

13. Third-Order

Note: a. When density is constant or A = 0, t=

b. When density varies with conversion,

10.5.4 For Back-Mix Reactors, (Unsteady State)

Where: V = Vo + (o-f)t

10.6. Plug-Flow or Tubular Reactors


( )

( )

( ) nA

n

AAA

nAoA

AA

x

xx

kCr

CCo

+

=

= 1

11

1

k

xC

k

CC AAAA oo =

=

( ) ( )AA

Ao

A

AA

Ao

A

xk

xC

xk

CCt

+=

+

=

11

( )( )A

AAA

x

xx

k +

=1

11

( )A

A

x

x

kt

=

1

1

( )

( )2

2

1

11

A

AAA

A x

xx

kC

o+

=

( )( ) 2111

A

AAA

oAx

xx

kCt

+

=

Space Time, Mean Holding Time,

t

( )

( )3

3

21

11

A

AAA

Ao x

xx

kC

+=

( )

( ) 3

2

1

11

2

A

AAA

x

xx

kCt

oA+

=

( )AA xt += 1

oCAo dt =fCAdt + (-rA)Vdt + VdCA + CAdV


28/38

10.6.1 Material Balance

Material Balance for Incremental Volume:

( )dVrdFFF AAAA ++=

( )dVrdF AA =

( )AAoA

AAoA

dxFdF

xFF

=

=and

1:but

Substituting values:

( )dVrdxF AAAo =

= )(1

A

A

Ar

dxdV

Fo

also :

oo AoACF =

o

V

=

= )( AA

A r

dx

Co

General equation:

=

)(A

A

Ar

dXC

o

An

A

n

AA

n

A

dx

x

x

kC o )1(

)1(11

+=

10.6.2 Space Time and Holding Time


Space Time, Holding Time, t

xA

FA

xA +

dxA

xA

FA

+dF

A

oC

Ao

FAo

oC

Af

FAf

dV

Reactant

Enters

(-rA)dV

ReactantDisappears

by Reaction

Reactant

Leaves


29/38

1. Zero-Order

k

xC

k

CCAAAA oo =

= ( )AA

A

AxLn

k

Ct o += 1

2. First-Order

( )( )

+=

AA

A

Ax

xk

1

1ln1

1

3. Second-Order

( ) ( )( )

( )

+

+++=A

AA

AAAAA

Ax

xxx

kCo

1

11ln12

12

2

( )

( )( )

+

+

=AA

A

AA

A

xLnx

x

kCt

o

11

11

4. Third-Order( )( )

( )( )

( )( )

+

++=

A

AAAAAAA

A

AAAAA

Ao x

xxxln

x

xx

kC 1

3613

12

2631212

32

2

322

2

( ) ( )

( )

( )( )

( )

+

+= AA

A

AAA

A

AAA

Ao

xLnx

x

x

xx

kCt 1

1

2

12

211 22

2

22

2

Sample Problems:

1. The reaction 2A 2R + S takes place isothermally in a constant-volume experimental reactor. Starting witha mixture of 80% A and 20% inert, the initial pressure of 10 atm increases by 25% in 8 minutes. Whatconversion is attained?


( )AxLnk

t = 11


30/38

( ) ( )

62.5%x100%8

5

P

PPn%conversio

atm358P

PP5x'

101.250.5xx2x8final

0.5xxx

0028initial

S2RI2A

Ao

AAo

A

AAoA

AAA

AAA

==

=

====

==+++

++

2. If the same feed in the preceding problem is introduced in an isothermal variable-volume batch reactor, whatis the time required for the same conversion?

( )

:

A Ao

Solving for k

2nd order constant volume reactor

1 1 1 1kt = k 8

C C 3 8

k 0.026

= =

=

for 2nd order variable volume reactor

( )( )

( ) tkCxlnxx

AoAAA

A

A =+

+1

1

1

Solving for fractional change in volume :A

A A

A

A

x =1 x =0

x =0

V -V 3-2 = = 0.8=0.4

V 2

Substituting values:

( )

( )( )A A A A Ao

A

1+x + ln 1-x =kC t

1-x

( )

( )( ) ( ) ( )

1 0.40.625 0.4ln 1 0.625 0.026 8 t

1 0.625

t 9.3 minutes

++ =

=

3. For a gas reaction 2A R+2S taking isothermally in a constant volume reactor. Starting with 3 atm A and 1

atm inerts, the pressure rises to 4.5 atm in 60 minutes. What space time, space velocity and holding time isrequired to effect this conversion in a) plug flow b) back-mix reactor.

I + 2A R + 2Sinitial 1 3 -xA xA 2xAFinal = =1+3 -xA+ xA+ 2xA=4.5

xA= 1.0PA = PAo - xA

= 3-1= 2Fractional conversion

= %33.333

1

P

PP

aA

AoA

==

For 2nd-order constant volume reactor:

360

1

3

1

2

1

60

1

11

0

=

=

=

k

k tPP AA


Answer

Answer

Answer


31/38

( ) ( )( )

( )

++++=

A

AA

AAAAA

Ax

xxx

kCo

1

11ln12

12

2

a. For 2nd-order variable volume plug-flow reactor:

37504

3

2

23.A =

=

( ) ( ) ( ) ( ) ( ) ( )( )

esminut68.770.3331

0.33320.37510.33320.3750.3331ln0.37510.3752

3

360 =

++++=

( )

( )( )

+

+

=AA

A

AA

A

xLnx

x

kCt

o

11

11

( )

( )( ) tesLnt minu64.15333.01375.0

333.01

333.0375.01

3

360=

+

+

=

b. for back-mix

( )

( )

( ) ( )

( )

2

2

2

2

1 0.333 1 (0.375) (0.333)1 360113.66min

3 1 0.3331o

A A A

AA

x xutes

kC x

+ += = =

( )

( )

( ) ( )

( )min

A A A

2 2

A Ao

x 1 x 0.333 1 (0.375)(0.333)1 360t 101.66 utes

kC 31 x 1 0.333

+ += == =

or:

( ) ( )minutes101.66

33)(0.375)(0.1

113.66

x1

t

AA

=+

=+

=

4. The isothermal irreversible aqueous phase A+B E @ 100oF obeys rE=KCACB; k=15ft3/lbmol.h. Using a

1000ft3 stirred tank reactor with aqueous feed of 2000 ft3/h, solve for the active concentration of E if theinlet concentration of A and B are both 0.25 lbmol/ft3.Solution:

h.h/ft

ftV50

2000

1000

3

3

===

For 2nd-order Back-mix reactor: 2A

AAo

kC

CC =

( )( )( )

( )( )

3/1233.01277.025.0

1277.05.0152

25.05.015411

2

411

ftlbmolCCC

k

CkC

AAoE

AoA

===

=+

=+

=

5. A gas decomposes @ 900oC according to the reaction: 2A(g) 2R(g) + S(g) with a rate constant of 1000cm3/gmol.s. Solve for the time required in minutes for 80% of reactant A in a batch reactor @ 900oC and 1atm.Solution:

For 2nd-order variable-volume batch reactor:

( )( )

( ) tkCxlnxx

AoAAA

A

A =+

+1

1

1

Solving for fractional change in volume :A


Answer

Answer

Answer

Answer

Answer

Answer


32/38

sgmol

L

cm

L

sgmol

cmk

x

V

VV

A

x

xx

A

AA

A

.

1

1000.

1000

80.0

5.02

23

3

3

0

01

=

=

=

=

=

=

=

==

Ao 1Ao

1 1

n TP 1 273 1 gmolC = = = =0.01039

V T PV 273+900 1 22.4 L

Substituting known values in the equation:( )

( )( )A A A A Ao

A

1+x + ln 1-x =kC t

1-x

( ) ( )

( )

( )( ) ( ) ( )

1+0.51t= 0.8 + 0.5 ln 1-0.8 =500 s

1 0.01039 1-0.8

minutes= 500 =8.33 minutes

60 s

11. GlossaryActivation energy_the critical amount of energy required for a reaction to take place. The amount of energy in

excess of the average energy level which the reactants must have in order for the reaction to proceed.Akinetic energy greater than a certain minimum,

Antimatter_ matter composed of elementary particles that are, in a special sense, mirror images of the particlesthat make up ordinary matter as it is known on earth. Antiparticles have the same mass as theircorresponding particles but have opposite electric charges or other properties related to electromagnetism.For example, the antimatter electron, or positron, has opposite electric charge and magnetic moment (a

property that determines how it behaves in a magnetic field), but is identical in all other respects to theelectron. The antimatter equivalent of the chargeless neutron, on the other hand, differs in having amagnetic moment of opposite sign (magnetic moment is another electromagnetic property). In all of theother parameters involved in the dynamical properties of elementary particles, such as mass, spin, and

partial decay, antiparticles are identical with their corresponding particles.The existence of antiparticles


Answer


33/38

was first proposed by the British physicist Paul Adrien Maurice Dirac, arising from his attempt to apply thetechniques of relativistic mechanics.

Catalyst_ acts by lowering the activation energy requirement of a particular reaction or shortening of the reactionpathway. It either hastens or retards the rate of a chemical reaction without itself undergoing a permanentchange.a. A substance that changes the rate of chemical reaction

b. A substance which changes the rate of reaction without itself undergoing a permanent chemicalchange.

c. A substance which lowers the activation energy requirement of a reaction.Chemical Kinetics ___ the study of the rate of reaction and mechanism by which one chemical species isconverted to another. chemical kinetics or reaction kinetics is the study ofreaction rates in a chemicalreaction. Analyzing the influence of different reaction conditions on the reaction rate gives informationabout the reaction mechanism and the transition state of a chemical reaction. In 1864, Peter Waage

pioneered the development of chemical kinetics by formulating the law of mass action, which states thatthe speed of a chemical reaction is proportional to the quantity of the reacting substances.

Chemical equation_ is a symbolic representation of a chemical reaction. The coefficients next to the symbols andformulae of entities are the absolute values of the stoichiometric numbers. The first-ever chemicalequation was diagrammed by Jean Beguin in 1615.

Chemical reaction_ is a process that results in the interconversion of chemical substances.The substance orsubstances initially involved in a chemical reaction are called reactants. Chemical reactions are usuallycharacterized by a chemical change, and they yield one or moreproducts which are, in general, differentfrom the reactants.

Chemical reaction engineering _is the branch of engineering that is concerned with the exploitation of chemicalreactions on a commercial scale for purposes other than the production of power.

Collision theory of chemical reaction_ rate is directly proportional to the number of collisions per seconda. Only certain collisions between particles results in the formation of productsb. The molecules must have proper orientation

c. A more concentrated solution contains a greater number of particlesCollision Theory of a Chemical reaction:

For a chemical reaction to proceed, molecules must have effective collisions. The tworequirements for an effective collisions are; a.The molecules must be suitably reactive and must haveenough energy, b. The molecules must be arranged in a proper position.

a. The molecules must be suitably reactive and must have enough energy. The energy that themolecules must possess is known as activation energy. The critical amount of energy required for areaction to take place. The amount of energy in excess of the average energy level which the reactants

must have in order for the reaction to proceed.Collision theory is a theory, proposed by Max Trautz and William Lewis in 1916 that qualitativelyexplains how chemical reactions occur and why reaction rates differ for different reactions. Itassumes that for a reaction to occur the reactant particles must collide, but only a certain fraction ofthe total collisions, the effective collisions, cause the transformation of reactant molecules into

products. This is due to the fact that only a fraction of the molecules have sufficient energy and theright orientation at the moment of impact to break the existing bonds and form new bonds.

Activated Complex or Transition Theory:

Henry Eyring, an American chemist, postulated an alternative to collision theory. Hehypothesized that an intermediate species called an activated complex forms during collision. Thisintermediate species exist very briefly. It dissociates to form either the product (if reaction occurs) orthe original reactants (if reaction does not occur)

Most reactions proceed in many steps called elementary processes. The combined effect of all the

elementary processes gives the overall reaction. The slow step determines the rate of the chemicalreaction and is called the determining step. The observed rate of the overall reaction is theequivalent to the rate of the slow reaction

b. The molecules must be arranged in a proper position. This affects the chance for the reaction tooccur.

http://en.wikipedia.org/wiki/Reaction_ratehttp://en.wikipedia.org/wiki/Reaction_ratehttp://en.wikipedia.org/wiki/Chemical_reactionhttp://en.wikipedia.org/wiki/Chemical_reactionhttp://en.wikipedia.org/wiki/Reaction_mechanismhttp://en.wikipedia.org/wiki/Transition_statehttp://en.wikipedia.org/wiki/Transition_statehttp://en.wikipedia.org/wiki/Chemical_reactionhttp://en.wikipedia.org/wiki/Chemical_reactionhttp://en.wikipedia.org/wiki/Chemical_reactionhttp://en.wikipedia.org/wiki/Peter_Waagehttp://en.wikipedia.org/wiki/Peter_Waagehttp://en.wikipedia.org/wiki/Law_of_mass_actionhttp://en.wikipedia.org/wiki/Chemical_reactionhttp://en.wikipedia.org/wiki/Stoichiometric_coefficienthttp://en.wikipedia.org/wiki/Jean_Beguinhttp://en.wikipedia.org/wiki/Chemical_substancehttp://en.wikipedia.org/wiki/Reactantshttp://en.wikipedia.org/wiki/Chemical_changehttp://en.wikipedia.org/wiki/Product_(chemistry)http://en.wikipedia.org/wiki/Chemical_reactionshttp://en.wikipedia.org/wiki/Chemical_reactionshttp://en.wikipedia.org/wiki/Chemical_reactionshttp://en.wikipedia.org/wiki/Reaction_ratehttp://en.wikipedia.org/wiki/Reaction_ratehttp://en.wikipedia.org/wiki/Reaction_ratehttp://en.wikipedia.org/wiki/Reaction_ratehttp://en.wikipedia.org/wiki/Chemical_reactionhttp://en.wikipedia.org/wiki/Chemical_reactionhttp://en.wikipedia.org/wiki/Reaction_mechanismhttp://en.wikipedia.org/wiki/Transition_statehttp://en.wikipedia.org/wiki/Chemical_reactionhttp://en.wikipedia.org/wiki/Peter_Waagehttp://en.wikipedia.org/wiki/Law_of_mass_actionhttp://en.wikipedia.org/wiki/Chemical_reactionhttp://en.wikipedia.org/wiki/Stoichiometric_coefficienthttp://en.wikipedia.org/wiki/Jean_Beguinhttp://en.wikipedia.org/wiki/Chemical_substancehttp://en.wikipedia.org/wiki/Reactantshttp://en.wikipedia.org/wiki/Chemical_changehttp://en.wikipedia.org/wiki/Product_(chemistry)http://en.wikipedia.org/wiki/Chemical_reactionshttp://en.wikipedia.org/wiki/Reaction_rate


34/38

Jacobus vant Hoff

Complex chemical reactions_ reactions where actual mechanisms are not represented by simple stoichiometricequations. These are usually non-elementary reactions and involve more than one step to accomplish.

Elementary Particles, in physics, particles that cannot be broken down into any other particles. The termelementary particles also is used more loosely to include some subatomic particles that are composed ofother particles. Particles that cannot be broken further are sometimes called fundamental particles to avoidconfusion. These fundamental particles provide the basic units that make up all matter and energy in theuniverse.

Structure of MatterModern physics has revealed successively deeper layers of structure in ordinary matter. Matter is composed, on a

tiny scale, of particles called atoms. Atoms are in turn made up of minuscule nuclei surrounded by a cloudof particles called electrons. Nuclei are composed of particles called protons and neutrons, which arethemselves made up of even smaller particles called quarks. Quarks are believed to be fundamental,meaning that they cannot be broken up into smaller particles.

Elementary Reactions __ any such reaction in which the rate equation suggested by stoichiometric equationrepresents the actual mode of action and occurs in a single step.

Equilibrium_ the system is in the state of equilibrium if the following characteristics are observed:1. It is a closed system ( the amount of matter in the system does not change)2. There is no change in the properties of the system as time passes.3. Two processes, which are opposite in direction, simultaneously take place at the same rate.

For chemical equilibrium: The ratio of the product of the molar concentration of the substances formedto the product of the molar concentration of the reactants is constant.

Equilibrium constant _ the ratio of the forward and reverse rate constant for a reversible reaction.Fermion_ one of the two main classes of fundamental particles that make up matter and energy. Fermions play an

important role in the structure of matter. The particles that make up atoms (electrons, protons, andneutrons) are all fermions. Fermions are named for Italian-born American physicist Enrico Fermi. In the1920s Fermi calculated a set of rules that define the behavior of fermions.

Fractional conversion (xA )_ the amount of the reactant converted to the desired product divided by the originalamount of reactants.

Fractional change in volume (A)_ the resulting change in volume when all of the reactants are converted dividedby the original volume occupied by the reactants before the start of the reaction .

A =0

01

=

==

A

AA

x

xx

V

VV

General Theory of Relativity_ The principle of equivalence holds that forces produced by gravity are in every wayequivalent to forces produced by acceleration, so that it is theoretically impossible to distinguish between


Dutch chemistJacobus vant Hoffwon the 1901 Nobel Prize in

chemistry. vant Hoff investigated the structure of organic

compounds and came to be known as the father of chemical

kinetics.


35/38

gravitational and accelerational forces by experiment. In 1915 Einstein developed the general theory ofrelativity in which he considered objects accelerated with respect to one another. He developed this theoryto explain apparent conflicts between the laws of relativity and the law of gravity. To resolve theseconflicts he developed an entirely new approach to the concept of gravity, based on the principle ofequivalence.

Holding time or reaction time (t) = (batch and plug-low); t = holding time, mean residence time (for back-mix)_the average time required by the reactants to stay inside the rector in order to attain the desiredconversion to the desired product

Hadron, family of elementary particles. All known particles of matter and energy are classified as hadrons,

leptons, or force carriers. Scientists believe leptons and force carriers are fundamental particles, meaningthey cannot be divided into smaller particles. Hadrons, on the other hand, have an underlying structure ofsmaller particles called quarks. The most common hadrons are protons and neutrons, the building blocks ofthe nucleus in an atom.

Hesss Law _ The total change in enthalpy of a system is dependent on the temperature, pressure, state ofaggregation, and state of combination at the beginning and at the end; it is independent of the number ofintermediate reactions.

Irreversible reactions_ reactions that proceed only in one directionLaw of chemical equilibrium_ the ratio of the products of the molar concentration of the substances formed to the

product of the molar concentration of the reactants is constant.Law of Mass Action _states that the rate of chemical reaction is at each instant proportional to the concentration of

the reactant with each raised to a power equal to their coefficient or the actual number of moleculesparticipating in the reaction. This law can be interpreted by several complex mechanisms but it cansimply be explained as follows: when two or more molecules react, it must come close to one another ormust collide. Therefore, it is expected that the rate of reaction increases if the molecules are crowdedclosely together, i.e., the concentration is high.

Le Chateliers Principle_ If a stress (disturbance or change) is applied to a system in a state of equilibrium, thesystem will shift in such a way to neutralize the effect of the stress .

Lepton_ type of elementary particle, the most basic building block of matter, that does not experience the strongforce. The strong force holds particles in the nucleus of an atom together. Physicists have discovered thefollowing leptons: electrons, muons, tau particles, and three corresponding types of neutrino (electronneutrinos, muon neutrinos, and tau neutrinos).

Limiting reactant_ the reactant that determines the rate of the chemical reaction. The chemical that determineshow far the reaction will go before the chemical in question gets used up, causing the reaction to stop. Thechemical of which there are fewer mols than the proportion requires is the limiting reagent.

Mechanism __ is the sequence of individual chemical events whose over-all result produce the observedreaction.

_the sequence of steps that takes place to complete a chemical change.Metabolic pathway_ is a series ofchemical reactions occurring within a cell. In each pathway a principal chemicalis modified by chemical reactions. These reactions are accelerated, more accu

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