9. Reaction Mechanisms, Pathways, Bioreactions and Bioreactors*


Topics

  1. Active Intermediates/Free Radicals
  2. Enzymes
  3. Bioreactors
  4. Pharmacokinetics
  5. Polymerization

Active Intermediates / Free Radicals (PSSH) top

An active intermediate is a molecule that is in a highly energetic and reactive state It is short lived as it disappears virtually as fast as it is formed. They are short lived CA 10-14s and present in very low concentrations. That is, the net rate of reaction of an active intermediate, A*, is zero.

The assumption that the net rate of reaction is zero is called the Pseudo Steady State Hypothesis (PSSH)

The active intermediates reside in the trough of the reaction coordinate as shown below for in the reaction studied by Zewoil.


Hall of Fame Reaction

The reaction

has an elementary rate law

However... Look what happens to the rate as the temperature is increased.


Why does the rate law decrease with increasing temperature?

Mechanism:

(1)
(2)
(3)


Pseudo Steady State Hypothesis (PSSH)

The PSSH assumes that the net rate of species A* (in this case, NO3*) is zero.

Solving for NO3*

 


This result shows why the rate decreases as temperature increases.

 
Pseudo Steady State Hypothesis


Enzymes top

Michaelis-Menten Kinetics    



Enzymes are protein like substances with catalytic properties. 


Enzyme unease. [From Biochemistry, 3/E by Stryer, copywrited 1988 by Lubert Stryer.  Used with permission of W.H. Freeman and Company.] 

It provides a pathway for the substrate to proceed at a faster rate.  The substrate, S, reacts to form a product P.  

A given enzyme can only catalyze only one reaction.  Urea is decomposed by the enzyme urease, as shown below.

It has been proposed that an artificial kidney to remove urea from the blood could contain encapsulated enzymes and be worn externally.

The corresponding mechanism is: 

Michaelis-Menten Equation  

Inverting yields:

Types of Enzyme Inhibition


Derive the rate law for competitive inhibition

Sketch competitive inhibition on a Lineweaver-Burk Plot


Derive the rate law for uncompetitive inhibition

Sketch uncompetitive inhibition on a Lineweaver-Burk Plot


Sketch non-competitive inhibition on a Lineweaver-Burk Plot

Uncompetitive Substrate Inhibition



E + S   E • S (Inactive)

S + E•S   S•E•S (Inactive)

E•S   P + E

The Uncompetitive Substrate Inhibition rate law is

 




The following video was made by Professor Lane's 2008 Chemical Reaction Engineering class at the University of Alabama, Tuscaloosa.

You may access the video by connecting to the internet and clicking the image below



Bifurcation analysis of substrate inhibited enzymatic reactions- Graduate Course Material, U of M
Methanol Poisoning
Polymath code for Alcohol Metabolism Living Example Problem. Example 7-7 in the 4th Edition.


Bioreactors top

Data from Laboratory of H.S. Fogler taken by P.h.D Candidate Barry Wolf. 

                   

Rate Laws

Stoichiometry

A.) Yield Coefficients

B.) Maintenance

A Word of Caution on

A.) Growth Phase

B.) Stationary Phase

Mass Balances

Cell:

Also, for most systems.

Substrate:



Polymath Setup

1.)

d(Cc)/d(t) = - D*Cc + (rg - rd)

2.)

d(Cs)/d(t) = D*(Cso - Cs) - Ysc*rg - m*Cc

3.)

d(Cp)/d(t) = - D*Cp + Ypc*rg

4.)

rg = (((1 - (Cp/Cpstar))**0.52) * mumax*(Cs/(Ks + Cs))*Cc

5.)

D = 0.2

6.)

kd = 0.01

7.)

rd = kd*CC

8.)

Cso = 250

9.)

Ypc = 5.6

10.)

m = 0.3

11.)

mumax = 0.33

12.)

Ysc = 12.5

13.)

Ks = 1.7

Polymath Screen Shots

Polymath Equations

Summary Table

Cc and Cp vs. Time

Cs vs. Time


Wash Out:

1.) Neglect Death Rate and Cell Maintenance

2.) Steady State

Washout

Maximum Production Rate

Production Rate =

Dividing by the reactor volume, V, which is constant

Substituting for CC

                    

 

How does this figure relate to drinking a lot of fluids when you have an infection or cold?

Objective Assessment of Chapter 9

 

Pharmacokinetics top

Alcohol Metabolism













Water tissue volumes, flow rates, and perfusion rates for this model

Rate-law parameters for this model

The complete description of the model and the model paramenters can be found in this journal article.       David M. Umulis, Nihat M. Gurmen, Prashant Singh, H. Scott Fogler. "A physiologically based model for ethanol and acetaldehyde metabolism in human beings," Alcohol 35 (2005).






Drug Drlivery

                  

See Professional Reference Shelf 7.5

 

Polymerization top

Polymers are macromolecules built up by the linking together of large numbers of much smaller molecules. The smaller molecules are called monomers and they repeat many times.

A polymer is a molecule made up of repeating structural (monomer) units.

 

Examples of Polymers

Poly (vinyl chloride)

 

Natural Polymers

Proteins
DNA/RNA
Cellulose
Fats
Starch

Synthetic Polymers

 

 

Name

 

Structural Repeating Unit (mer)

 

Uses

 

 

Poly (vinyl chloride)

 

 

Pipes

 

 

Polyethylene

 

 

High density:
Plastic cups

Low density:
Sandwich bags

 

 

Polystyrene

 

 

Coffee Cups

 

 

Poly (acrylic acid)

 

 

Superglue (Dow)

 

 

Poly (cyano acrylate)

 

 

Superglue

 

 

Poly (vinyl acetate)

 

 

Chewing gum

 

 

Poly (vinyl alcohol)

 

 

Shampoo/Thickener

 

 

Poly (ethylene glycol)

 

 

Stealth molecule

 

 

Poly (methyl methacrylite)

 

 

Plexiglas

 

 

Poly (2-hydroxyethyl methacrylate)

 

 

Contact Lenses

 

 

Poly (tetra fluoro ethylene)

 

 

Teflon

 

 

Poly (ethylene teraphthalate)

 

 

Coke bottles

Spinable fibers

 

A. Names/Nomenclature

Polymers that are synthesized from a single monomer are named by adding the prefix poly such as polyethylene. However, a parenthesis is placed after the prefix poly when the monomer has a substituted parent name or multiword name such as poly (acrylic acid) or poly (vinyl alcohol).

Homopolymers consist of a single repeating unit. All of the above are examples of homopolymers.

 

B. Polymer structure

1. Linear

Linear HDPE (70-90% crystalline)

2. Stereoregularity

Can Crystallize.

a.

Botactic = isotatic = same side

 

b.

Syndiotatic = alternating

c.

Atactic = random

Head to head (1,2 addition)

 

Head to tail (1,3 addition)

 

3. Branched Type A: Long Branches Off Backbone

 

Branched Type B: Short Branches Off the Backbone

 

Branched Type C: Branches on Branches Off the Backbone

4. Cross linked

 

C. Copolymers

More than one repeating unit.


For example, copolymers used to make records.

PVC - hard - irrigation pipes, hard to engrave
PVAc - easy to engrave
PVC + PVAc copolymer phonograph records (these are a thing of the past)

 

FIVE TYPES OF COPOLYMERS

Alternating

QSQSQS

Poly (vinyl acetate-alt-vinylchloride)

Block

QQQSSS

Poly (vinyl acetate-b-vinyl chloride)

Graph

QQQQQ
. . . . . |
. . . . . SSSS

Poly (vinyl acetate-g-vinyl chloride

Random

QSSQQQSQSSS

Poly (VAc-co-VC)

Statistical

QSSQSQQSS

 

D. What affects polymer properties

• Chemistry

• Molecular Weight () and Molecular Weight Distribution

 

Weight Average Molecular Weight

 

Molecular Weight Distribution

• Crystalinity

Amorphous Phase (Non-crystalline Phase) no order or orientation

 

 

Tg - characteristic of amorphous state

Rubbery glassy

Below glass transition temperature, Tg, there is a cessation of virtually all molecular motion (vibration , rotation).

 

Crystalline Phase gives an order to the structure.

 

 

 

Order means crystallinity

Crystalline liquid

Above the crystalline melting temperature, Tm, thre is no order. Fraction of total polymer that is in the crystalline state is the degree of crystallinity

 

• Cross linking

• Branching

• Tacticity

• Head to head attachment vs. head to tail attachment

 

E. Molecular Weight (MW)

1. Measurement

Membrane osmometry

Gel permeation chromatography

 

Viscosity

Light scattering

 

2. Calculation

 Number average molecular weight

Weight average molecular weight

 

 
Calculate the Mean Molecular Weight


Hence gives a truer picture of the average molecular weight.

3. Polydispersity   

 

TWO TYPES OF HOMOGENEOUS POLYMERIZATION: STEP AND CHAIN

Step Polymerization. Monomer must be bifunctional. Polymerization proceeds by the reaction of two different functional groups. Monomer disappears rapidly, but molecular weight builds up slowly.

  

All species are treated as polymers. Mostly used to produce polyesters and polyamides.

Chain Polymerization. Requires an initiator. Molecular weight builds up rapidly. Growing chains require 0.0001 to 1 to 10 seconds to terminate. Have high molecular weight polymers right at the start.

 

I. Step Polymerization

A. Functional Groups

1. Different functional groups on each end of monomer.


Structural Unit

 

 

Here the structural unit is the repeating unit.

2. Same functional groups on each end. Example: diamines and diols

Two structural units and

Repeating unit =

 

B. Polymerization Mechanism

Monomer dimer ----> trimer ----> tetrameter ----> Pentamer ---->

 

C. Structural Units

The number of structural units equals the number of bifunctional monomers present.

1. Monomers with different functional groups - one structural unit.

Here the repeating unit is the structural unit.

Let p = fraction of functional groups of either A or B that have reacted.

Let M = concentration of either A or B functional groups at time t.

Let M0 be the concentration of either A or B functional groups initially

Let N = total number (concentration) of polymer molecules present at time t.

Let N0 = total number of polymer molecules initially

Let MA = number of functional groups of A at time t.

Let MA0 = number of functional groups of A initially.

= number average degree of polymerization. It is the average number of structural units per chain.

 

 

therefore

the number average molecular weight.

Where is the mean molecular weight of the structural units and is the molecular weight of the end group.

 

 
Calculating Xn for Monomers with Different End Groups

D. Monomers with Same End Group

For a stoichiometric feed the number of A and B functional groups the same.

 
Calculate Xn for Monomers with the Same End Groups


E. Stoichiometry Imbalance in the Feed

1. Stoichiometry Imbalance Type 1:  Monomers with thesame end group and r not equal to  1 

The maximum number average chain length is greatly reduced if the initial feed is not exactly stoichiometric

If p = 1 then

 
Calculating Xn for Stoichiometric Inbalance

2. Stoichiometry Imbalance Type 2:  Monomers with different end groups. Monofunctional Monomer Present

 

,  [A-R-A]o =[B-R-B]o =MAo

                [B-C]o =MBo

 

3. Stoichiometry Imbalance Type 3:  Monomers with different end groups. Monofunctional Monomer Present

 

 

REACTION BETWEEN A DIOL (HOROH) AND A DIBASIC ACID (HOOCR1COOH)

Let

Then

Overall Reaction:

The Mechanism

Rate Law:

(1)

(2)

(3)

Let

-- =

~ =

The rate limiting step is Reaction (2)

Assume Reaction (1) is essentially in equilibrium

    ,   

 

Case 1: The acid itself acts as a strong acid catalyst:

[HA] º  [COOH] and Stoichiometric Feed.

As the reaction proceeds and more ester is produced, the solution becomes less polar. As a result the uncatalyzed carboxylic acid becomes the major catalyst for the reaction, and the overall reaction order at high conversion is well described by a third order reaction (Case 1). The high conversion region is of primary importance because this region is where the high molecular weight polymers are formed.

At low conversions the solution is more polar and the proton, H+ is the more effective catalyst (Case 2) than the unionized carboxylic acid. Under these conditions, the reaction is self catalyzed and the reaction is 5/2 order.

 

Case 2: Self catalyzed but acid acts as a weak acid catalyst, not completely dissociated

[HA] =  [-COOH]

 

 

Case 3: External Acid Catalyzed H+ is constant

 

F. Kinetics of Step Polymerization

(1)

k is defined wrt the reactants

Why 2k? Because there are two ways A and B can react (thus, 2k)

(2)

(3)

For all reactions of P1

 

In general for j ? 2

For j = 2

Mole balance on polymer of length j, in terms of the concentration Pj in a batch system

then

 

At t = 0, P2 = 0

If we proceed further it can be shown that

Total number of polymer molecules (i.e. functional groups of either A or B) =

Mole fraction

This is the Flory Distribution for the mole fraction of molecules with chain length j.

The weight fraction is just

W = total weight = 

 

G. Flory Distribution-Probability Approach

Rule: The probability of several events occurring successively in a particular way equals the product of the probabilities that each event happens that way.

 P = probability that an A group will has reacted

(1-P) = probability group has not reacted.

A - R - B

HO - R C OO - H

 

Probability of 1st link is P

 

Probability of 2nd link is P

 

Probability of A or B unreacted = (1 - P)

  

 

Mo = number of functional groups initially (no. of molecules)

M = number of functional groups remaining

Number distribution function.

Weight distribution function

 

       

On a number average basis there will always be more monomer than polymer.


 

II. Chain Polymerization

A. Free Radical

Example: Polyethylene

Linear addition

Back biting

 

            

          

Branched Polyethylene

resulting low density (0.92)

 

B. Cationic Polymerization

 

C. Anionic Polymerization

 

D. Ziegler-Natta Polymerization

Ziegler-Natta Catalyst

Steps in Polymer Chain Growth

(4) Desorption from active site

      +   

to produce linear polymer: Eq. High Density Polyethylene (0.98) (HDPE)

Chain polymerizations require an initiator.

FREE RADICAL POLYMERIZATION

1. The Reaction

INITIATION

This reaction produces the formation of the Primary Radical

PROPAGATION

TERMINATION

Transfer

To solvent

To monomer

To chain transfer agent

To initiator

Addition

Disproportionation

MORE ABOUT INITIATION

Types of initiators, homiletic, photo


Typical Initiators for Homolytic Dissociation

 

Temperature Range

Initiator

(1)

50-70

Azobisisobutyranitride (ABIN)

 

(2)

70-90

Acetyl Peroxide

 

(3)

80-95

Benzyl Peroxide


Kinetics


Initiator Efficiency "f"

f = fraction of radicals produced in the homolysis reaction that initiate polymer chains. It is a measure of waste of initiator.

 
Reasons for "f" Less than One

How to determine f experimentally

  1. In AIBN measure N2 evolution compare number of radicals produced with number of polymer molecules obtained.
  2. Tag initiator 14C or 35S
  3. Use scavengers - to stop growth.

 

Lab-on-a-Chip

2. Free Radical Polymerization Kinetics

INITIATION

PSSH applied to initiator

  (1)
  (2)
     
     

PROPAGATION

For all radicals the total rate of propagation

= total concentration of radicals (R1 + R2 + R3 . . . Rn)

Monomer balance

Long Chain Approximation (LCA)

The rate of disappearance of monomer, -rm,

TERMINATION

1. Chain transfer

A.

To monomer

 
 

Total rate of transfer for all radicals

   

B.

To solvent

 
   

C.

Transfer to a chain transfer agent

 
   

D.

Transfer to initiator

 
   

2. Dispropriation Termination

   
   

Disproportionation

Define kd wrt reactants

 

i.e.

   
 

Net rate of termination of all radicals by dispropriation. For every dead polymer molecule that is formed, one live polymer radical is lost.


3. Addition Termination

where ka is defined wrt to the reactant.

The net rate of termination of j radicals will all the Rk radical (k = 1, 2, ?)

The net rate of termination of all radicals is

 

PSSH Applied to All Free Radicals

Recall

Example  Termination by the Initiator Primary Radicals, I

Independent of Initiator Concentration

                     

Dead ended polymerization occurs when the initiator concentration decreases to such a low value, the half life of the polymer chains approximates half life of initiator

Polymerization of Isoprene initiated by azobisisolulyronitride.

Kinetic Chain Length

The kinetic chain length v is the average number of monomer molecules consumed (polymerized) per radial that initiates a polymer chain.

Chain Transfer

In Principles of Polymerization 3/e by George Odian, the definition of Rt, the factor of 2 is incorrect but does not matter because it cancels out since Rt is dived by 2 in the denominator in later equations. Also note Eqn. (3-118a) in Odians is altogether incorrect.

Let the transfer coefficient be defined by

(1)      (2)     (3)      (4)

Term (1)

Canceling terms

Term (4)

Solving for I2

The Mayo-Walling Equation

Let?s neglect the term

For styrene

In benzyl peroxide

CM = 0.00006

CI = 0.055

In benzene

CS = 2.3 ´ 10?6

In Butyl mercaptan

CS = 21.1

Increasing the initiator concentration decreases

Increasing the monomer concentration increases

Further determination of Rate Constant

1. Dilatometry

Volume charge

2. Spectroscopically measure I2(t) and M(t)

               

B. Batch - Method of Initial Rates

Many experiments

Steady State Measurement

CSTR

No Chain Transfer

Determining Chain transfer Constants

A. Only transfer to chain transfer agent S

Hold M and I constant, vary S

B. Transfer to monomer, initiation and a chain transfer agent

Hold I and S constant, vary M

Change I and repeat, the intercept will be the same but slope will be different, S2. Change S and repeat S3. Three equations and 3 unknowns. Could also use regression.

Transfer Constants

A. To Monomer

0.00005 < CM < .0015

CM is generally small, however chain transfer to monomer for vinyl chloride is sufficiently high to limit the molecular weight so that the maximum molecular weight of PVC is 50,000 to 500,000.

B. To Initiator

CI is a function of both the initiator and the reaction

0.0008 < CI < 0.3

Peroxides are usually the strongest chain transfer agents.

C. To Chain Transfer Agent

For styrene

0.000002 < CS < 21

(Benzene) < CS < (n Butyl mercaptan)

Chain Transfer to Polymer, CP

Chain transfer to polymer produces branched polymers.

It is not important in determining CI, CM and CS because they are determined at low conversion.

CP involves the determination of the number of branches produced relative to the number of polymer molecules polymerized.

CP is the order of 10?4

The branching density, r , is the number of branches per monomer molecule polymerized.

For CP = 10?4 and 80% conversion, there will be 1.0 branches per 104 monomer units polymerized. There is one branch for every 4,000 to 10,000 monomer units and for a polymer molecular weight of 105 ? 106 this corresponds to 1 polymer chain in 10 containing a branch.

Polyethylene

1. Short branches (less than 7 carbon atoms)

Formed by backbiting. Short branches out number the long branches by a fact of 20-50. They affect crystallinity giving maximum crystallinity of 60-70%.

2. Long branches ? formed by normal chain transfer to polymer.

Energetics (Free Radical)

Therefore chain transfer becomes more significant as temperature increases!!!

Chain Polymerization

Ionic Polymerization

Cationic

Used for monomers with electron releasing substituents

(a) (b) (c) (d)

e.g. alkoxy, 1,1?dealky

(a) covalent species, (b) tight ion pair, (c) loose ion pair, (d) free and highly solvated ion

Anionic

Used with monomers possessing electron withdrawing groups, e.g. nitride, carboxyl.

Anionic

High molecular weight. No chain-chain termination.

Initiation

Alkyllithium used because soluble in hydrocarbon solvents.

Potassium amide

Propagation

No effective termination - complete consumption of monomer to form living polymers.

Termination by:

a. Impurities

Moisture

b.  Deliberate addition of chain transfer agent

c.  Spontaneous

Hydride elimination, i.e.

Comparison with free radical polymerization

Free Radical:  

Concentration of radicals is 10?9 to 10?7 mol/dm3

Anionic:  

Concentration of propagation anions is 10?4 to 10?2 mol/dm3

In hydrocarbon solvents is 10-100 times smaller than kP

In either solvents is 10-100 times larger than kP

   

Benzene

Tetrahydrofuran

1,2-Dimethoxyethane

 

2

550

3,800

Other values are given in Odian p.412

Why do the rates of polymerization vary by several orders of magnitude in different solvents?

Kinetics of Ion/Ion Pair Initiation/Polymerization

Initiation

Summing over all radicals

where is the concentration of and all radicals initiated with and is the concentration of and radicals initiated with

We assume the ion and the ion pair are in equilibrium with the "salt."

Let be the total concentration of all types of anionic living propagating centers.

where Io is the total amount of initiator added.

For small degrees of dissociation

If K ~ 10?8 and

then  

If K ~ 10?6 then

Polymerization of Styrene

 

160

80

22

2.2

1.5

0.02

6.5

6.5

6.5

Data of Bhattacharya et al., J. Phys Chem. 69, p.612 (1965)

So we see that different solvents bring about different degrees of dissociation of the initiator resulting in different specific reaction rates.

Anionic Polymerization

1. Determining the living polymer concentration as a function of time

For complete dissociation of the iniator

Assumptions

Initiation is instantaneous, R10 = Io

2. No termination

          

Case 1     ko >> kp Immediate rate formulation of primary radical

                        

     

 

Propagation with No termination

For the live polymer with the largest chain length n

Summing all these equations

Constant live polymer concentration

Let dq = kP M dt

t = 0, q = 0, R1 = Io

Convert back to real time from scaled time

     

Very small t (i.e., small IokPt)

Very large t (i.e., large IokPt)

Distribution of molecular weights of living polymers

Next consider a different set of initiation conditions

Case 2     ko = kp

        

Anionic Polymerization in a CSTR

Monomer Balance

Balance on R1

Balance on Rj

 

Psuedosteady State Hypothesis (PSSH)

Case 1     ko is essentially (i.e., ko >> kp) infinite. Io is reacted immediately upon mixing with monomer to form R10

There is no initator, I, in the reactor

where

        

Substituting for

 

Case 2     ko is finite

j = 1

 

* All chapter references are for the 1st Edition of the text Essentials of Chemical Reaction Engineering .

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