Interpretation of Batch Reactor Data
Conversion
• Conversion of A, XA, is the fraction of reactant A
converted to other compounds
• Suppose NAo is the initial amount of A in a reactor at time
t=0
• NA is the amount present at time t
• Then conversion of A in the constant volume system is
given by
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Integral Method of Analysis
• Integral method of analysis puts a particular rate
equation to test by integrating and comparing the
predicted C vs t curve with experimental data
• If the fit is unsatisfactory, another rate equation is
guessed and tested
• Irreversible Unimolecular-Type First-Order Reactions
• Suppose we wish to test the first-order rate equation
• Separating and integrating, we obtain
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• In terms of conversion,
• On rearranging and integrating,
• A plot of ln(1 – XA) or ln(CA/CAo) vs t gives a straight line through the
origin for this form of rate equation
• If the experimental data seems to be better fitted by a curve than by
a straight line, try another rate form because the first-order reaction
does not satisfactorily fit the data
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Irreversible Bimolecular-type Second-Order Reactions
• Consider the reaction
• The corresponding rate equation is
• The amounts of A and B that have reacted at any time t
are equal and given by CAoXA
• Letting M = CBo/CAo be the initial molar ratio of reactants
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• On separation and integration,
• After breakdown into partial fractions, integration and
rearrangement, the final result in a number of different forms is
• Two equivalent ways of obtaining a linear plot between
concentration and time for this second-order rate law
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• For second-order reaction with equal initial
concentrations of A and B or for the reaction
• The defining second-order differential equation becomes
• Which on integration yields
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• For trimolecular reactions,
• In terms of conversion, the rate of reaction becomes
• Where M = CBo/CAo. On integration,
• Similarly, for the reaction
• Integration gives
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Empirical Rate Equations of nth Order
• When the mechanism of reaction is not known, we often
attempt to fit the data with an nth-order rate equation
• Which on separation and integration yields
• The order n cannot be found explicitly, so a trial-and-
error solution must be made
• Select a value for n and calculate k, the value of n which
minimizes the variation in k is the desired value of n
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Zero Order Reactions
• A reaction is zero order when the rate of reaction is
independent of the concentration of materials
• Integrating and noting that CA can never become
negative,
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Irreversible Reactions in Parallel
• Consider A decomposing by two competing paths, both
elementary reactions
• The rates of change of the three components are
• The k values are found using all three differential rate
equations
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• Dividing the second and third equations,
• When integrated,
• The slope of a plot of CR vs CS gives the ratio k1/k2
• Knowing k1/k2 as well as k1 + k2 gives k1 and k2
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Autocatalytic Reactions
• An autocatalytic reaction is a reaction in which one of the products
of reaction acts as a catalyst
• The rate equation is
• Because the total number of moles of A and R remains unchanged
as A is consumed,
• Thus, the rate equation becomes
• Rearranging and breaking into partial fractions,
• On integration,
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• In terms of initial reactant ratio M = CRo/CAo and fractional
conversion of A,
• For an autocatalytic reaction in a batch reactor, some product R
must be present if the reaction is to proceed at all
• Starting with a very small concentration of R, the rate will rise as R is
formed
• When A is almost totally consumed, the rate must drop to zero
• Thus, the rate follows a parabola with a maximum where the
concentrations of A and R are equal
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Irreversible Reactions in Series
• Consider consecutive unimolecular-type first-order reactions
• The rate equations for the three components are
• By integration of the first equation, concentration of A is
• To find the changing concentration of R, substitute concentration of
A obtained into differential equation governing the rate of change of
R
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• The final expression for the concentration of R is
• Noting that there is no change in total number of moles,
stoichiometry relates the concentrations of reacting components by
• Concentration of S is thus
• If k2 is much larger than k1,
• The rate is determined by k1 or the first step of the two-step reaction
• If k1 is much larger than k2,
• Which is a first-order reaction governed by k2, the slower step in the
two-step reaction
• In general, for any number of reactions in series, it is the slowest
step that has the greatest influence on the overall reaction rate
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• k1 and k2 also govern the location and maximum concentration of R
• This may be found by setting dCR/dt = 0
• Time at which maximum concentration of R occurs is
• Maximum concentration of R is
• A decreases exponentially, R rises to a maximum and then falls, S
rises continuously
• The greatest rate of increase of S occurs where R is a maximum
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Tutorial Questions
• Levenspiel book: 3.5, 3.9, 3.19, 3.23
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