7. Reaction Mechanisms, Pathways, Bioreactions and Bioreactors^*

Topics

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

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 C_A 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.

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)

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

Michaelis-Menten Kinetics (Section 7.4.2)   

Enzymes are protein like substances with catalytic properties. 

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 movie was made by the students of Professor Alan Lane's chemical reaction engineering class at the University of Alabama Tuscaloosa

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.

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.) 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?

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

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

Crystalline Phase gives an order to the structure.

• 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 X_n 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 X_n 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 X_n for Stoichiometric Inbalance

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

,  [A-R-A]_o =[B-R-B]_o =M_Ao

                [B-C]_o =M_Bo

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 P_j 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

M_o = 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.

 

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 . . . R_n)

Monomer balance

Long Chain Approximation (LCA)

The rate of disappearance of monomer, -r_m,

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 k_d 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 k_a is defined wrt to the reactant.

The net rate of termination of j radicals will all the R_k 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 R_t, the factor of 2 is incorrect but does not matter because it cancels out since R_t 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

C_M = 0.00006

C_I = 0.055

In benzene

C_S = 2.3 ´ 10^?6

In Butyl mercaptan

C_S = 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 < C_M < .0015

C_M 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

C_I is a function of both the initiator and the reaction

0.0008 < C_I < 0.3

Peroxides are usually the strongest chain transfer agents.

C. To Chain Transfer Agent

For styrene

0.000002 < C_S < 21

(Benzene) < C_S < (n Butyl mercaptan)

Chain Transfer to Polymer, C_P

Chain transfer to polymer produces branched polymers.

It is not important in determining C_I, C_M and C_S because they are determined at low conversion.

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

C_P is the order of 10^?4

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

For C_P = 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 k_P

In either solvents is 10-100 times larger than k_P

		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 I_o 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 = I_o

2. No termination

          

Case 1     k_o >> k_p 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 = k_P M dt

t = 0, q = 0, R₁ = I_o

Convert back to real time from scaled time

     

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

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

Distribution of molecular weights of living polymers

Next consider a different set of initiation conditions

Case 2     k_o = k_p

        

Anionic Polymerization in a CSTR

Monomer Balance

Balance on R₁

Balance on R_j

 

Psuedosteady State Hypothesis (PSSH)

Case 1     k_o is essentially (i.e., k_o >> k_p) infinite. I_o is reacted immediately upon mixing with monomer to form R₁₀

There is no initator, I, in the reactor

where

        

Substituting for

 

Case 2     k_o is finite

j = 1

Objective Assessment of Chapter 7

^* All chapter references are for the 4th Edition of the text Elements of Chemical Reaction Engineering .

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