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TABLE 151 Rate Equations for Reactions of Simple Order
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Order Constant-volume process k(
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Differential equation ) 2(CA01/2 CA1/2) ln CA0 CA
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Zero k(
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dNA Vd )
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One-half k(
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dNA Vd ) CA0 CA 1 CZ0 1 C ) 1 CA2 1 CA02 1 C )(CC0
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kCa1/2
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First k(
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dNA Vd ) ln CACA0 CA0CB0 CACB0 CA03 1 CA C
Second k(
dNA Vd )
kCA2
dNA Vd 2k(
kCACB
CB0*
Third k(
dNA Vd ) (C
kCA2
dNA Vd
kCACBCc
C (C
CA0 CA 1 C )(CC0
1 C )(CB0
ln ln
CA CA CC0 CC0 CA0 CB0
CB0 CB0 CC0
CA0 CA0
NOTE:
C 0 and 0 are initial conditions for time and concentration, respectively kCA2 * If CA0 CB0, use expression for dNA / Vd kCA3 If CA0 CB0 CC0, use expression for dNA / Vd
CHEMICAL, ENVIRONMENTAL, PETROLEUM AND GAS ENGINEERING
require much more complicated rate equations for their description Equilibrium and Kinetics Inasmuch as all chemical reactions are limited by a chemical equilibrium, reaction kinetics really describes the rate of approach to that equilibrium rather than to a stoichiometric completeness of the reaction At equilibrium, the net rate of reaction Eq (1551) is zero, whence it follows that for a reaction whose rate law is described by the molecularity (ie, stoichiometric equation), the equilibrium constant for the reaction is related to the forward and reverse speci c rate constants Thus, C a C lBC c A C p q C PC Q Kc k k (1552)
It is clear that the larger the value of Kc , the larger is the magnitude of the forward rate constant relative to the reverse and the closer to stoichiometric completeness is the equilibrium conversion Effect of Temperature Homogeneous reactions are strongly temperaturedependent Their speci c reaction rate always increases with increasing temperature The effect of temperature is described by the semitheoretical relation of Arrhenius: k Ae
EIRT
(1553)
The coef cient A (called the frequency factor) and the exponent E (called the energy of activation) have theoretical interpretations, but they are best regarded by the process designer as empirical quantities peculiar to a particular chemical reaction and valuable from experimental rate data Thus a plot of k against 1 / T should be linear and should have a slope of E / R and an intercept of ln A, provided A and E are independent of temperature In fact, both the frequency factor and the energy of activation vary slightly with temperature, but over the temperature ranges normally encountered, they may be assigned constant average values without serious error The failure of rate data to t Eq (1553) may be accepted as evidence that 1 A reversible reaction has been treated as if it were irreversible, and the effect of temperature on the equilibrium is signi cant 2 An otherwise incorrect mechanism has been assigned to the reaction 3 The speci c rate constant has been evaluated from the experimental data incorrectly 4 The reaction is heterogeneous, and its rate is in uenced by adsorption or by some other physical process Energies of activation range from less than 1000 to greater than 100,000 cal / (g mol) For most reactions, the value will be between 10,000 and 70,000 cal / (g mol)
CHAPTER FIFTEEN
Effect of Concentration At constant temperature, the speci c rate constant is assumed to be independent of the concentration of reactants and products, so that equations like (1548) and (1550) show explicitly the effect of concentration on the progress of the reaction Homogeneous Catalyzed Reactions A catalyst is a substance that affects the rate of a chemical reaction without entering the reaction in any stoichiometric sense The catalyst may undergo net physical or chemical change in the course of the reaction, but often it does neither
22 Heterogeneous Reactions A chemical reaction is said to be heterogeneous
if more than one phase is an active participant and if transfer of materials to phase boundaries has an effect on the rate of reaction Heterogeneous reactions commonly involve uid-solid mass transfer (eg, catalysis of a uid reaction mixture on the surface of solid catalyst pellets, combustion of a solid fuel in air, acid leaching of metals from ores) or mass transfer between two uid phases (eg, absorption of gaseous sulfur dioxide by weak aqueous sodium hydroxide, nitration of toluene by nitric acid) Uncatalyzed Heterogeneous Reactions In an uncatalyzed heterogeneous reaction, chemical action occurs among components that are simultaneously being transferred physically from phase to phase The apparent rate of the reaction is in fact the rate of a more complicated process It will be in uenced not only by factors affecting chemical kinetics, but also by those affecting the rate of interphase mass transfer Catalyzed Heterogeneous Reactions Although heterogeneous reactions responsive to catalysis may involve any combination of phases, the examples most common and industrially most important are solid- uid systems in which the catalyst is the solid phase The reactants and products may be gaseous, liquid, or both The solid catalyst may be a container wall, a metal gauze or a granular mass Figure 1512 is a typical plot of effectiveness factor against a modulus
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