Pauline M. Doran
`
`ACADEMIC PRESS
`
`Harc'0m'1‘ Brace & Company, Publishers
`London SanDiego NewYo1‘k Boston
`Sydney Tokyo Toronto
`
`BEQ 1031
`Page 1
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`BEQ 1031
`Page 1
`
`

`
`ACADEMIC PRESS LIMITED
`
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`US. Edition Published by
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`Copyright © 1995 ACADEMIC PRESS LIMITED P
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`
`ISBN 0—12—220855-2
`
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`
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`
`BEQ 1031
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`BEQ 1031
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`

`
`This material may be proteded by C0pyrigh1|aw(Tfl|e 17 U_s_ Code)F_
`
`III
`
`Homogeneous Reactions
`
`The heart ofa typical bioprocess is the reactor orfermenter. Flan/eed by unit operations which carry outphysical changesfor
`medium preparation and recovery ofproducts, the reactor is where the major chemical and biochemical transformations
`occur. In many bioprocesses, characteristics ofthe reaction determine to a large extent the economicfeasibility oftheproject.
`
`Of most interest in biological systems are catalytic reactions.
`By definition, a catalyst is a substance which affects the rate of
`reaction without altering the reaction equilibrium or under-
`going permanent change itself. Enzymes, enzyme complexes,
`cell organelles and whole cells perform catalytic roles; the latter
`may be viable or non-viable, growing or non-growing.
`Biocatalysts can be of microbial, plant or animal origin. Cell
`growth is an autocatalytic reaction: this means that the catalyst.
`is a product of the reaction. The performance ofcatalytic reac-
`tions is characterised by variables such as the reaction rate and
`yield of product from substrate. These parameters must be
`taken into account when designing and operating reactors.
`In engineering analysis of catalytic reactions, a distinction
`is made between homogeneous and heterogeneous reactions. A
`reaction is homogeneous if the temperature and all concentra-
`tions in the system are uniform. Most fermentations and
`enzyme reactions carried out in mixed vessels fall into this cat-
`egory. In contrast, heterogeneous reactions take place in the
`presence of concentration or temperature gradients. Analysis
`of heterogeneous reactions requires application of mass-
`transfer principles in conjunction with reaction theory.
`Heterogeneous reactions are treated in Chapter 12.
`This chapter covers the basic aspects of reaction theory
`which allow us to quantify the extent and speed of homo-
`geneous reactions and to identify important factors affecting
`reaction rate.
`
`11.1.1 Reaction Thermodynamics
`
`Consider a reversible reaction represented by the following
`equation:
`'
`
`A+ bB x——‘ yY+zZ.
`
`(11.1)
`
`A, B, Y and Z are chemical species; b, yand zare stoichiometric
`coefficients. If the components are left in a closed system for an
`infinite period of time, the reaction proceeds until thermody-
`namic equilibrium is reached. At equilibrium there is no net
`driving force for further change; the reaction has reached the
`limit of its capacity for chemical transformation in a closed
`system. Composition of the equilibrium mixture is deter-
`mined exclusively by the thermodynamic properties of the
`reactants and products; it is independent of the way the reac-
`tion is executed. Equilibrium concentrations are related by the
`equilibrium constant, K For the reaction of Eq. (1 1.1):
`
`CYeyCZCZ
`[(__ I
`
`CAe CBeb
`
`(11.2)
`
`where CAc, CB6, CYC and Cze are equilibrium concentrations
`ofA, B, Y and Z, respectively. The value of Kdepends on tem-
`perature as follows:
`
`11.1 Basic Reaction Theory
`
`Reaction theory has two fundamental parts: reaction thermody-
`namics and reaction /einetics. Reaction thermodynamics is
`concerned with how fizr the reaction can proceed; no matter
`how fast a reaction is, it cannot continue beyond the point of
`chemical equilibrium. On the other hand, reaction kinetics is
`concerned with the rate at which reactions proceed.
`
`lnK=
`
`~AG;.
`RT
`
`(11.3).
`
`where A G ‘gm is the change in standardfree energy per mole ofA
`reacted, R is the ideal gas constant and Tis absolute tempera-
`ture. Values of R are listed in Table 2.5 (p. 20). The
`superscript ° in A G ‘Am indicates standard conditions. Usually,
`
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`

`
`258
`II Homogeneous Reactions
` _—
`
`the standard condition for a substance is its most stable form at
`
`AG=AH— TAS.
`
`1 atm pressure and 25°C; however, for biochemical reactions
`occurring in solution, other standard conditions may be used
`[1]. AGE,“ is equal to the difference in stzzndardflee energy of
`fbrmzztion, G°, between products and reactants:
`
`AG;’m= yG§+zG%— G1‘,’(— /7 GE.
`~
`
`(111.4)
`
`Therefore, from Eq. (1 1.3):
`
`in K:
`
`-AH:..— +
`RT
`
`A55...
`R
`
`-
`
`(11.5)
`
`(11.6)
`
`Standard free energies offormation are available in handbooks
`such as those listed in Section 2.6.
`
`Free energy G is related to enthalpy H, entropy S and
`absolute temperature Tas follows:
`
`for exothermic reactions with negative AH;m, K
`Thus,
`decreases with increasing temperature. For endothermic reac-
`tions and positive AHgm, K increases with temperature.
`
`Example 11.1 Effect of temperature on glucose isomerisation
`
`Glucose isomerase is used extensively in the USA for production of high—fructose syrup. The reaction is:
`
`8
`
`lucose 4% fructose.
`
`AH;m for this reaction is 5.73 k] gmol”; ASfxn is 0.0176 k] gmol” K_1.
`
`(a) Calculate the equilibrium constants at 50°C and 75°C.
`(b) A company aims to develop a sweeter mixture of sugars, i.e. one with a higher concentration of fructose. Considering
`1 equilibrium only, would it be more desirable to operate the reaction at 50°C or 75°C?
`
`Solution:
`
`(a) Convert temperatures to degrees Kelvin (K) using the formula of Eq. (2.24):
`
`T: 5o°c=323.15 K
`
`T: 75°c=34s.15 K.
`
`From Table 2.5, R: 8.3144Jgmol”1 K-1: 8.3144 x 10-3 k] gmorl K“1. Using Eq. (11.6)
`__
`-1
`
`In K (50°C) :
`
`5.73 k] gmol
`(33144 x 10‘5kJgmol”1K”1)(323.15K)
`
`+
`
`K(50°C) = 0.98.
`
`Similarly for T= 75°C:
`
`0.0176 k]gmol_1K"1
`
`8.3144 X 10'5 k]gmol”1K_1
`
`In K 05°C) :
`
`-1
`_
`5.75 k] gmol
` _..___ +
`(83144 x 10"3 k]gmo1‘1K”1) (348.15K)
`
`0.0176 k] gmol_1 K”
`
`8.3144 X 1073 k] gmol_1 K71
`
`g<(75°c) =1.15.
`
`(b) As Kincreases, the fraction of fructose in the equilibrium mixture increases. Therefore, from an equilibrium point ofview,
`it is more desirable to operate the reactor at 75°C. However, other factors such as enzyme deactivation at high temperatures
`should also be considered.
`
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`

`
`II Homogeneous Reactions
`
`A limited number ofcommercially—important enzyme conver-
`sions, such as glucose isomerisation and starch hydrolysis, are
`treated as reversible reactions. In these systems, the reaction
`mixture at equilibrium‘ contains significant amounts of reac-
`tants as well as products. However, for many reactions A Gfxn is
`negative and large in magnitude. As a result, K is also very
`large, the reaction favours the products rather than the reac—
`tants, and the reaction is regarded as irreverriole. Most enzyme
`and cell reactions fall into this category. For example, the equi-
`librium constant for sucrose hydrolysis by invertase is about
`104; for fermentation of glucose to ethanol and carbon diox-
`ide, K is about 1030. The equilibrium ratio of products to
`reactants is so overwhelmingly large for these reactions that
`they are considered to proceed to completion, i.e. the reaction
`stops only when the concentration of one of the reactants falls
`to zero. Equilibrium thermodynamics has therefore only lim-
`ited application to enzyme and cell reactions. Moreover, the
`thermodynamic principles outlined in this section apply only
`to closed systems; true thermodynamic equilibrium does not
`exist in living cells which exchange matter with their sur—
`roundings. Metabolic processes in cells are in a dynamic state;
`products formed are constantly removed or broken down so
`that reactions are driven forward. Most reactions in biological
`systems proceed to completion in a finite period of time at a
`finite rate.
`
`If we know that complete conversion will eventually take
`place, the most useful reaction parameter to know is the rate at
`which the transformation proceeds. Another important char—
`acteristic, especially for systems in which many different
`reactions take place at the same time, is the proportion of reac-
`tant that is converted to the desired products. These properties
`of reactions are discussed in the remainder of this chapter.
`
`1 1.1.2 Reaction Yield
`
`The extent to which reactants are converted to products is
`expressed as the reaction yield Generally speaking, yield is the
`amount of product formed for accumulated per amount of
`reactant provided or consumed. Unfortunately, there is no
`strict definition of yield; several different yield parameters are
`applicable in different situations. The terms used to express
`yield in this text do not necessarily have universal acceptance
`and are defined here for our convenience. Be prepared for
`other books to use different definitions.
`
`Consider the simple enzyme reaction:
`
`L—histidine —> urocanic acid + NH3
`
`catalysed by histidase. According to the reaction stoichiometry,
`1 gmol urocanic acid is produced for each gmol L—histidine
`consumed; the yield of urocanic acid from histidine is there-
`fore 1 gmol gmol‘1. However, let us assume that the histidase
`used in this reaction is contaminated with another enzyme,
`histidine decarboxylase. Histidine decarboxylase catalyses the
`following reaction:
`
`L—histidine —> histamine + CO2.
`1
`
`(1 1.8)
`
`If both enzymes are active, some L—histidine will react with
`histidase according to Eq.
`(11.7), While some will be
`decarboxylated according to Eq. (11.8). After addition of the
`enzymes to the substrate, analysis of the reaction mixture
`shows that 1 gmol urocanic acid and 1 gmol histamine are
`produced for every 2 gmol-histidine consumed. The observed
`or apparent yield of urocanic acid from L—histidine
`is
`1 8m°l/2 gmol = 0.5 gmol gmol”1. The observed yield of
`0.5 gmol gmol"1 is different from the stoic/aiometric, true or
`theoretical yield of 1 gmol gmol‘1 calculated from reaction
`stoichiometry because the reactant was channelled in two
`separate reaction pathways. An analogous situation arises if
`product rather that reactant is consumed in other reactions;
`the observed yield ofproduct would be lower than the theoret~
`ical yield. W/yen reactants orproducts are involved in additional
`reactions, the observedyield may be dzflerentfiom the t/oeoretieal
`yield.
`The above analysis leads to two useful definitions of yield for
`reaction systems:
`
`(true, stoichiometric or) __
`
`theoretical yield
`
`< total mass or moles of)
`product formed
`
`mass or moles of reactant used
`
`-
`
`to form that particular product
`
`(11.9)
`
`and
`
`< observed or
`
`_ t>.
`apparent yield _
`total mass or moles of reactant
`consumed
`
`(11.10)
`
`There is a third type of yield applicable in certain situations.
`For reactions with incomplete conversion of reactant, it may
`be of interest to specify the amount of product formed per
`amount of reactant provided to the reaction rather than actually
`consumed. For example, consider the isomerisation reaction
`catalysed by glucose isomerase:
`
`BEQ 1031
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`

`
`II Homogeneous Reactions
`
`g
`
`lucose ,——" fructose.
`
`(11.11)
`
`stoichiometry, formation of 1 gmol fructose requires 1 gmol
`glucose. The observedyieldwould also be 1 gmol gmol‘1 if the
`reaction occurs in isolation. However if the reaction is started
`
`The reaction is carried out in a closed reactor with
`
`ureP
`
`enzyme. At equilibrium the sugar mixture contains 55 mol%
`lucose and 45 mol% fructose. The theoretical
`ield of
`g
`.7
`fructose from lucose is
`1 mol mol‘1 because,
`from
`g
`g
`g
`
`with glucose present only, the equilibrium yield of fructose
`per gmol glucose added to the reactor is 0.45 gmol gm0l”1.
`This type of yield for incomplete reactions may be-denoted
`groxsyield.
`
`Example 11.2 Incomplete enzyme reaction
`
`An enzyme catalyses the reaction:
`
`A # B.
`
`At equilibrium, the reaction mixture contains 63 Wt% A.
`
`(a) What is the equilibrium constant?
`(b) If the reaction starts with A only, What is the equilibrium yield of B from A?
`
`Solution:
`
`(a) From stoichiometry the molecular weights ofA and B must be equal: therefore Wt% = mol%. From Eq. (1 1.2):
`
`K:
`
`CBC
`C
`
`CA
`
`Using a basis ofl gmol 1” 1, CAC is 0.63 gmol l”1 and CB6 is 0.37 gmol lF1. The value ofKtherefore is 057/063 = 0.59.
`(b) From stoichiometry, the true yield of B from A is 1 gmol gmol_1. However the gross yield is 037/10 = 0.37 gmol gmol” 1.
`
`11.1.3 Reaction Rate
`
`Consider the general irreversible reaction:
`
`aA+ B —> yY+zZ.
`
`system, M0 is mass flow rate out of the system, RG is mass rate
`of generation by reaction and RC is mass rate of consumption
`by reaction. Let us apply Eq. (6.5) to compound A, assuming
`that the reaction of Eq. (11.12) is the only reaction taking
`place that involves A. Rate of consumption RC is equal to R ,
`and RC = 0. The mass—balance equation becomes:
`
`(11.12)
`
`The rate ofthis reaction can be represented by the rate ofconver-
`sion of compound A; let us use the symbol RA to denote the rate
`ofreaction with respect toA. RA has units of, for example kg s”1.
`How do we measure reaction rates? For a general reaction
`system, rate of reaction is related to rate of change of mass in
`the system by the unsteady—state mass—balance equation
`derived in Chapter 6:
`
`dM A
`TA=MAi —MA0 “ RA‘
`
`(11.13)
`
`Therefore, rate ofreaction RA can be determined ifwe measure
`the rate of change in mass ofAA, dMA/d1, and the rates of flow of
`A in and out Aof the systern,MAi and MA0. In a dorm’ system
`where MN = MAO = 0, Eq. (11.13) becomes:
`
`In'Eq (6.5), Mis mass, tis time,
`
`is mass flow rate into the
`
`(6-5)
`
`—dM
`dtA
`
`RA 2
`
`BEQ 1031
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`
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`
`

`
`II Homogeneous Reactions
`
`and reaction rate is measured simply by monitoring the change
`in mass ofA in the system. Most measurements of reaction rate
`are carried out in closed systems so that the data can be analysed
`according to Eq. (11.14). dMA/dt is negative when A is con-
`sumed by reaction; therefore the minus sign in Eq. (11.14) is
`necessary to make RA a positive quantity. Rate of reaction is
`sometimes called reaction velocity. Reaction velocity can also be
`measured in terms ofcomponents B, Y or Z. In a closed system:
`
`—dM
`RB="“B
`
`dM
`’*Y="d7Y
`
`dM
`RZ=TZ
`
`(11.15)
`
`where M , MY and M2 are masses of B, Y and Z, respectively.
`When reporting reaction rate, the reactant being monitored
`should be specified. Because RY and R2 are based on product
`accumulation, these reaction rates are called production rate: or
`productivity.
`Eqs (11.14) and (11.15) define the rate of reaction in a
`closed system. However, reaction rate can be expressed using
`different measurement bases. In bioprocess engineering there
`are three distinct ways of expressing reaction rate which can be
`applied in different situations.
`
`(1)
`
`Total rote. Total reaction rate is defined in Eqs (11.14)
`and (11.15) and is expressed as either mass or moles per
`unit time. Total rate is useful for specifying the output of a
`particular reactor or manufacturing plant. Production
`rates for factories are often expressed as total rates; for
`example: ‘The production rate is 100 000 tonnes per year’.
`If additional reactors are built so that the reaction volume
`
`in the plant is increased, then clearly the total reaction rate
`would increase. Similarly, if the amount of cells or enzyme
`used in each reactor were also increased, then the total pro-
`duction rate would be improved even further.
`Volumetric rote. Because the total mass of reactant con-
`
`verted in a reaction mixture depends on the size of the
`system, it is often convenient to specify reaction rate as
`the rate per unit volume. Units ofvolumetric rate are, e.g.
`kg m_5 s_1. Rate of reaction expressed on a volumetric
`basis is used to account for differences in volume between
`
`reaction systems. Therefore, if the reaction mixture in a
`closed system has Volume I/:
`
`RA _ —1dMA
`Tf
`dt
`V
`
`VA:
`
`(11.16)
`
`(11.17)
`
`where CA is the concentration of A in units of, e.g.
`kg m"3. Volumetric rates are particularly useful for com-
`paring the performance of reactors of different size. A
`common objective in optimising reaction processes is to
`maximise volumetric productivity so that the desired
`total production rate can be ‘achieved with reactors of
`minimum size and therefore minimum cost.
`
`Specific rote. Biological reactions involve enzyme and cell
`catalysts. Because the total rate of conversion depends on
`the amount of catalyst present, it is sometimes useful to
`specify reaction rate as the rate per quantity of enzyme or
`cells involved in the reaction. In a closed system, specific
`reaction rate can be measured as follows:
`
`(11.18)
`
`where rA is the specific rate ofreaction with respect to A, X
`is the quantity of cells, E is the quantity of enzyme and
`‘IMA/dtis the rate of change of mass ofA in the system. As
`quantity of cells is usually expressed as mass, units of spe-
`cific rate for a cell—catalysed reaction would be, e.g. kg
`(kg cells)_1 sil or simply ST1. On the other hand, the
`mass of a particular enzyme added to a reaction is rarely
`known; most commercial enzyme preparations contain
`several components in unknown and variable proportions
`depending on the batch obtained from the manufacturer.
`To overcome these difficulties, enzyme quantity is often
`expressed as units of activity measured under specified
`conditions. One unit of enzyme is usually taken as the
`amount which catalyses conversion of 1 pmol substrate
`per minute at the optimal temperature, pH and substrate
`concentration. Therefore, if E in Eq. (11.18) is expressed
`as units of enzyme activity, the specific rate of reaction
`under process conditions could be reported as, e.g. kg
`(unit enzyme)—1 sil.
`In a closed system where the
`volume of reaction mixture remains constant, an alterna-
`
`tive expression for specific reaction rate is:
`
`1
`
`“A: "<7 7)?!‘
`
`1
`
`CIC
`
`where rA is the volumetric rate of reaction with respect to
`A. When Vis constant, Eq. (11.16) can be written:
`
`where x is cell concentration and e is enzyme concentration.
`
`(11.19)
`
`BEQ 1031
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`

`
`II Homogeneous Reactions
`
`Volumetric and total rates are not a direct reflection of cata-
`
`this is represented by the specific rate.
`lyst performance;
`Specific rates are employed when comparing different cells or
`enzymes. Specific rate is the rate achieved per unit catalyst and,
`under usual circumstances, is not dependent on the size of the
`system or the amount ofcatalyst present. Some care is necessary
`when interpreting results for reaction rate. For example, if two
`_ fermentations are carried out with different cell lines and the
`volumetric rate of reaction is greater in the first fermentation
`than in the second, you should notjump to the conclusion that
`the cell line in the first experiment is ‘better’, or capable of
`greater metabolic activity. It could be that the faster Volumetric
`rate is due to the first fermenter being operated at a higher cell
`density than the second, leading to measurement of a more
`rapid rate per unit volume. Different strains of organism
`should be compared in terms ofspecific reaction rates.
`Total, volumetric and specific productivities are inter-
`related concepts in process design. For example, high total
`productivity could be achieved with a catalyst of low specific
`activity if the reactor is loaded with a high catalyst concentra-
`tion. If this is not possible, the volumetric productivity will be
`relatively low and a larger reactor is required to achieve the
`desired total productivity. In this book, the symbol RA will be
`used to denote total reaction rate with respect to component
`A; rA represents either volumetric or specific rate.
`
`influence reaction rate, such as temperature. When the kinetic
`equation has the form of Eq. (11.20), the reaction is said to be
`of oroierawith respect to component A and order [7 with respect
`to B. The order of the overall reaction is (a+ 12). It is not usually
`possible to predict the order of reactions from stoichiometry.
`The mechanism of single reactions and the functional form of
`the kinetic expression must be determined by experiment. The
`dimensions and units of /edepend on the order of the reaction.
`
`11.1.5 Effect of Temperature on Reaction
`Rate
`
`Temperature has a significant kinetic effect on reactions.
`Variation of the rate constant kwith temperature is described
`by the Arrhenius equation:
`
`k=Ae—E/RT
`
`(11.21)
`
`where k is the rate constant, A is the Arr/aenins constant or fre-
`qnencyfizctor, Eis the activation energy for the reaction, R is the
`ideal gas constant, and Tis absolute temperature. Values of R
`are listed in Table 2.5 (p. 20). According to the Arrhenius
`equation, as Tincreases, kincreases exponentially. Taking the
`natural logarithm of both sides of Eq. (1 1.21):
`
`11.1.4 Reaction Kinetics
`
`lnk=ln/1 —
`
`RT
`
`As reactions proceed, the concentrations of reactants decrease.
`In general, rate of reaction depends on reactant concentration
`so that the specific rate of conversion decreases simultaneous:
`ly. Reaction rate also varies with temperature; most reactions
`speed up considerably as the temperature rises. Reaction
`/einetics refers to the relationship between rate of reaction and
`conditions which affect reaction velocity, such as reactant
`concentration and temperature. These relationships are con-
`veniently described using /einetic ‘expressions or kinetic
`eqnationx.
`irreversible reaction of Eq.
`Consider again the general
`(1 1.12). Often but not always, the volumetric rate of this reac-
`tion can be expressed as a function of reactant concentrations
`using the following mathematical form:
`
`rA= k C; C;
`
`(11.20)
`
`where /eis the rate constant or rate coefiicient for the reaction. By
`definition, the rate constant is independent of the concentra-
`tion of reacting species but is dependent on other variables that
`
`(11.22)
`
`Thus, a plot of ln /e versus 1/Tgives a straight line with slope
`-5/R. For many reactions the value of E is positive and large,
`indicating a rapid increase in reaction rate with temperature.
`
`1 1.2 Calculation of Reaction Rates From
`Experimental Data
`
`As outlined in Section 11.1.3, the volumetric rate of reaction
`
`in a closed system can be found by measuring the rate of
`change in the mass of reactant present, provided the reactant is
`involved in only one reaction. Most kinetic studies of biolog-
`ical reactions are carried out in closed systems with a constant
`volume of reaction mixture; therefore, Eq. (1 1.17) can be used
`to evaluate the volumetric reaction rate. The concentration of
`
`a particular reactant or product is measured as a function of
`time. For a reactant such as A in Eq. (11.12), the results will be
`similar to those shown in Figure 11.1(a); the concentration
`will decrease with time. The vo_lumetric rate of reaction is
`equal to dc/\/dt, which can be evaluated as the slope of a
`
`BEQ 1031
`Page 8
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`

`
`II Homogeneous Reactions
`
`(a) Change in reactant concentration with time during reaction. (b) Graphical differentiation ofconcentration
`Figure 11.1
`data by drawing a tangent.
`
`Slope of tangent:
`reaction rate at I,
`
`smooth curve drawn through the data points. The slope of the
`curve in Figure 11.1(a) changes with time; the reaction' rate is
`greater at the beginning of the experiment than at the end.
`One obvious way to determine reaction rate is to draw tan-
`gents to the curve ofFigure 1 1.1(a) at Various times and evaiuate
`the slopes of the tangents; this is shown in Figure 1 1.1 Ifyou
`have ever attempted this you will know that it can be extremely
`difficult, even though correct in principle. Drawing tangents to
`curves is a highly subjective procedure prone to great inaccur— .
`acy, even with special drawing devices designed for the purpose.
`The results depend strongly on‘ the way the data are smoothed
`and the appearance of the curve at the points chosen. More reli-
`able techniques are available for grap/aieai cliflerentiation of rate
`data. Graphical differentiation is valid only if the data can be
`presumed to differentiate smoothly.
`
`Table 11.1 Graphical differentiation using the average
`rate—equal area construction
`
`Time
`
`(1: min)
`
`Oxygen
`concentration
`
`ACA
`
`At
`
`ACA/At
`
`dCA/dt
`
`(CA, ppm)
`8.00
`7.55
`7.22
`6.96
`6.76
`6.61
`6.49
`6.33
`6.25
`
`0.0
`1.0
`2.0
`3.0
`4.0
`5.0
`6.0
`8.0
`10.0
`
`-0.45
`-0.33
`-0.26
`-0.20
`-0.15
`-0.12
`--0.16
`-0.08
`
`1.0
`1.0
`1.0
`1.0
`1.0
`1.0
`2.0
`2.0
`
`-0.45
`-0.33
`-0.26
`-0.20
`-0.15
`-0.12
`-0.08
`-0.04
`
`-0.59
`-0.38
`-0.29
`-0.23
`-0.18
`-0.14
`-0.11
`-0.06
`-0.02
`
`11.2.1 Average Rate—Equal Area Method
`
`This technique for determining rates is based on the average
`rate—equal area construction, and will be illustrated using data
`for oxygen uptake by immobilised cells. Results from measure-
`ment of oxygen concentration in a closed system as a function
`of time are listed in the first two columns ofTable 11.1.
`
`(i) Tabulate values of ACA and At for each time interval as
`shown in Table 1 1.1. A CA values are negative because CA
`decreases over each interval.
`
`(ii) Calculate average oxygen uptake rates, AC!‘/A, for each
`time interval.
`
`(iii)
`Plot ACA/At on linear graph paper. Over each time inter-
`V val a horizontal line is drawn to represent ACA/Atfor that.
`interval; this is shown in Figure 1 1.2.
`V
`Draw a smooth curve to cut the horizontal lines in such a
`manner that the shaded areas above and below the curve
`
`(iv)
`
`are equal for each time interval. The curve thus developed
`gives values of ‘ICA/dt for all points in time. Results for
`dc!‘/dt at the times of sampling can be read from the curve
`and are tabulated in Table 1 1.1.
`
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`

`
`11. Homogeneous Reactions
`
`Figure 11.2 Graphical differentiation using the average
`rate—equal area construction.
`
`Figure 11.3 Average rate—equal area method for data with
`experimental error.
`
`:7 II
`0-0:
`012345678910
`
`I
`I
`012345678910
`
`Time (min)
`
`Time (min)
`
`A disadvantage of the average rate—equal area method is that it
`is not easily applied if the data show scatter. if the concentra—
`tion measurements are not very accurate, the horizontal lines
`representing ACA/Atmay be located as shown in Figure 11.3.
`Ifwe were to draw a curve equalising areas at each ACA/Arline,
`the rate curve would show complex behaviour oscillating up
`and down as indicated by the dashed line in Figure 11.3.
`Experience suggests that this is not a realistic representation of
`reaction rate. Because of the inaccuracies in measured data, we
`
`need several concentration measurements to define a change
`in rate. The data of Figure 11.3 are better represented using a
`smooth curve to_equalise as far as possible the areas above and
`below adjacent groups of horizontal lines. For data showing
`even greater scatter, it may be necessary to average consecutive
`pairs ofACA/Atvalues to simplify graphical analysis.
`
`A second graphical differentiation technique for evaluating
`dCA/dr is described below.
`
`11.2.2 Mid—Point Slope Method
`
`In this method, the raw data are smoothed and values tabulat—
`
`ed at intervals. The mid—point slope method is illustrated
`using the data of Table 1 1.1.
`
`(i)
`
`Plot the raw data and smooth by hand. This is shown in
`Figure 11.4.
`(ii) Mark off the smoothed curve at time intervals of 8. 8
`should be chosen so that the number of inteivals is less
`
`the less
`than the number of data points measured;
`accurate the data the fewer should be the intervals. In this
`example, sis taken as 1.0 min until t: 6 min; thereafter 8
`
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`

`
`II Homogeneous Reactions
`
`Figure 11.4 Graphical differentiation using the mid—point
`slope method.
`
`smoothed curve. When t= 6 min, 8= 1.0; concentrations
`for the difference calculation are read from the curve at
`t - E = 5 min and t+ 8= 7 min. For the last rate deter-
`mination at 1f: 8 min, S= 2.0 and the concentrations are
`read from the curve at t -* 8= 6 min and t+ £= 10 min.
`
`(iv) The slope or rate is determined using the central—differ—
`ence formula:
`
`[(CA)t+e“ (CA)t—e1
`
`RCA) t+8 T (CA) t—s]
`2 8
`
`'
`
`Time (min)
`
`Table 11.2 Graphical differentiation using the mid—point
`slope method
`
`Time
`
`(t, min)
`
`Oxygen
`concentration
`
`[(CA)t+8_ A)r~e]
`
`dCA/dt
`
`0.0
`1.0
`2.0
`3.0
`4.0
`5.0
`6.0
`8.0
`10.0
`
`(Cm PPH1)
`8.00
`7.55
`7.22
`6.96
`6.76
`6.61
`6.49
`6.33
`6.25
`
`1.0
`1.0
`1.0
`1.0
`1.0
`1.0
`1.0
`2.0
`2.0
`
`= 2.0 min. The intervals are marked in Figure 11.4 as
`dashed lines. Values of 8 are entered in Table 1 1.2.
`
`In the mid—point slope method, rates are calculated mid-
`way between two adjacent intervals of size 6. Therefore,
`the first rate determination is made for
`t = 1 min.
`
`from
`[(CA) H8 — (CA) t_ 8]
`Calculate the differences
`Figure 11.4, where (CA) “S denotes the concentration of
`A at time t+ 8, and (CA) t“ 8 denotes the concentration at
`time t— 8. A difference calculation is illustrated in Figure
`11.4 for t== 3 min. Note that the concentrations are not
`
`taken from the list of original data but are read from the
`
`(11.23)
`
`These results are listed in Table 11.2.
`
`Values of dCA/dt calculated using the two differentiation
`methods compare favourably. Application of both methods
`allows checking of the results.
`
`1 1.3 General Reaction Kinetics For
`Biological Systems
`
`The kinetics ofmany biological reactions are either zero—order,
`first—order or a combination of these called Michaelis—Menten
`
`kinetics. Kinetic expressions for biological systems are exam-
`ined in this section.
`
`11.3.1 Zero—Order Kinetics
`
`If a reaction obeys zero—order kinetics, the reaction rate is inde-
`pendent of reactant concentration. The kinetic expression is:
`
`“A: /30
`
`(11.24)
`
`where VA is the volumetric rate ofreaction with respect to A and
`k0 is the zero—order rate constant. ko as defined in Eq. (1 1.24) is a
`volumetric rate constant with units of, e.g. kgmol m_3 s_1.
`Because the volumetric rate of a catalytic reaction depends on
`the amount ofcatalyst present, when Eq. (1 1.24) is used to rep-
`resent the rate of a cell or enzyme reaction, the value of /co
`includes the effect of catalyst concentration as well as the
`specific rate of reaction. We could write:
`
`/e0=k6e or k0=k6'x
`
`>
`
`(11.25)
`
`where 1% is the specific zero—order rate constant for enzyme
`reaction and e is
`the concentration of enzyme. Corres-
`pondingly, for cell reaction, k8 is the specific zero—order rate
`constant and xis cell concentration.
`
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`Page 11
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`

`
`II Homogeneous Reactions
`
`Let us assume we have collected concentration data for a
`
`particular reaction, and wish to determine the appropriate
`kinetic constant. If the reaction takes place in a closed, con—
`stant—volume system, rate of reaction can be evaluated directly
`as the rate of change in reactant concentration using the meth-
`ods for graphical differentiation described in Section 11.2.
`From Eq. (11.24), if the reaction is zero—order the rate will be
`constant and equal to ko at all times during the reaction.
`Because the kinetic expression for zero—order reactions is rela-
`tively simple, rather than differentiate the concentration data
`it is easier to integrate Eq. (11.24) with rA = ”dCA/dt to obtain
`
`Example 11.3 Kinetics of oxygen uptake
`
`an equation for CA as a function of time. The experimental
`data can then be checked directly against the integrated equa-
`tion. Integrating Eq. (11.24) with initial condition CA = CA0
`at t“: 0 gives:
`
`CA= i—rAdt = CA0 — /eot.
`
`(11.26)
`
`Therefore, when‘ the reaction is zero order, a plot of CA versus
`time gives a straight line with slope — k0. Application of Eq.
`(1 1.26) is illustrated in Example 1 1.3.
`
`Serrntizz marcescens is cultured in minimal medium in a small stirred fermenter. Oxygen consumption is measured at a cell con—
`centration of22.7 g 1-1 dry weight.
`
`Time
`
`(min)
`
`0
`2
`5
`8
`10
`12
`15
`
`Oxygen concentration
`(mmol 1- 1)
`
`0.25
`0.23
`0.21
`0.20
`0.18
`0.16
`0.15
`
`(a) Determine the rate constant for oxygen uptake.
`(b) Ifthe cell concentration is reduced to 12 g 1“ 1, what is the value of the rate constant?
`
`Solution:
`
`(a) As indicated in Section 9.5.1, microbial oxygen consumption is a zero—order reaction over a wide range of oxygen concen-
`trations above Ccfit. To test if the measured data can be fitted using the zero—order model of Eq. (11.26), plot oxygen
`concentration as a function of time as shown in Figure 11133.1.
`
`Figure 11133.1 Kinetic analysis of oxygen uptake.
`
`
`
`
`
`Oxygenconcentration(mol1'‘)
`
`.9NJ
`
`.0
`
`10
`
`Time (min)
`
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`

`
`II Homogeneous Reactions
`
`(b) For cells of the