VOLUME 2 Post-Treatment, Reuse, and Disposal
2.3 Reaction Rates and Order of Reaction
4. Calculate the quantity of CaO and CO2produced.
CaO=9072 mole CaO×56 g/mole CaO× 1
453.6 g/lb=1120 lb CaO CO2=9072 mole CO2×44 g/mole CO2× 1
453.6 g/lb=880 lb CO2
5. Demonstrate that the mass is conserved.
−2000 lb CaCO3+1120 lb CaO+880 lb CO2=0
EXAMPLE 2.3: OXYGEN CONSUMPTION
Calculate the theoretical amount of oxygen required to completely burn 50 lb of sugar (C6H12O6).
Solution
1. Write the stoichiometric reaction.
C6H12O6+6 O26 CO2+6 H2O 1 mole of sugar reacts with 6 moles of O2. 2. Calculate the mole of sugar incinerated.
nsugar=50 lb sugar×453.6 g/lb× 1
180 g/mole sugar=126 mole sugar 3. Calculate the theoretical amount of oxygen (O2) required.
nO2=126 mole sugar× 6 mole O2
1 mole sugar=756 mole O2
Theoretical amount of oxygen, O2=756 mole O2×32 g/mole O2× 1
453.6 g/lb=53.3 lb O2
where
A =reactant A that is converted to some unknown product
r =overall rate of reaction, mole/L·t. The time (t) may be in an unit of s, min, h, or d.
[A]=molar concentration of A at any timet, mole/L k =reaction rate constant, (mole/L)1−n·t−1
n =an exponent that is typically determined through an experiment and used to define the order of the reaction with respect to the concentration of reactant(s), dimensionless
The order of reaction in chemical kinetics is the power to which its concentration term in the rate equa- tion is raised. Depending upon the condition, the reaction rate may be:
n=0 for the zero-order reactions, n=1 for thefirst-order reactions, and n=2 for the second-order reactions.
2.3.1 Reaction Rates
The definition and basic information on irreversible and reversible reactions are presented in Section 2.2.1.
The reaction rates of different order reactions are presented in Examples 2.4 through 2.9.
EXAMPLE 2.4: RATE OF SINGLE REACTION
A generalized stoichiometric single irreversible reaction is given by the expression:aA+bB→cC+dD (Equation 2.1). Express (a) overall rate of reaction, and define all related terms, (b) relationships between the rates of overall reaction and each individual reaction, and (c) relationships between the reaction rates of reactants and products.
Solution
1. The overall reaction rate is defined by Equation 2.6.
r=k[A]α[B]β (2.6)
where
r =overall rate of reaction, mole/L·t
k =reaction rate constant, (mole/L)1−(α+β)·t−1 [A] and [B]=molar concentrations of reactants A and B, mole/L
αandβ =empirical exponents that are used to define the order of reaction with respect to reac- tants, dimensionless. The exponentsαandβare usually 0, 1 or 2.
2. Express the relationships between the rates of overall reaction and each individual reaction.
r= −rA
a = −rB
b =rC
c =rD
d or r=|rA| a =|rB|
b =rC
c =rD
d
3. Express the relationships between the reaction rates of reactants and products.
Using the reactant A and product C as an example, the ratio of the rates must equal to the ratio of the stoichiometric coefficient of reactant A to that of product C.
−rA
rC=a
c or |rA| rC =a
c
Similarly, the ratios of reaction rates between any other reactants and products can also be established.
EXAMPLE 2.5: REACTION ORDER AND UNITS OF REACTION RATE CONSTANT The overall rate of reaction of a single reversible reaction is given by the equation:r=k[A][B]2. Deter- mine (a) the order of reaction, and (b) units of reaction rate constant.
Solution
1. Determine the order of reaction.
The overall reactionr=k[A][B]2.
The reaction is thefirst order with respect to reactant A.
The reaction is the second order with respect to reactant B.
The reaction is the third order with respect to overall reaction.
2. Determine the units of reaction rate constants.
The reaction rates of and units for individual reactants can be written as follows:
d[A]
dt =rA=kA[A](first-order reaction) Unit forkA= rA
[A]=mole/L·t mole/L =t−1 d[B]
dt =rB=kB[B]2(second-order reaction) Unit forkB= rB
[B]2= mole/L·t (mole/L)2= L
mole·t r=k[A][B]2(third-order reaction) Unit fork= r
[A][B]2= mole/L·t
(mole/L)(mole/L)2= L2 mole2·t
EXAMPLE 2.6: REACTION RATE FOR SINGLE REACTION
A single irreversiblefirst-order reaction has only one reaction step. A generalized stoichiometric single reaction is given below. Determine the rates of reaction for the reactant and each product as a function of reaction rate constantkand molar concentration of A.
3 A2 B+C Solution
1. Express the overall rate of reaction.
r=k[A]
2. Determine the rates of reaction of individual reactants.
r= −rA
a =rB
b =rC
c =k[A]
rA= −ak[A] = −3k[A] rB=bk[A] =2k[A]
rC=ck[A] =k[A]
Note:TherAhas negative sign because the concentration of [A] is reduced.
EXAMPLE 2.7: REACTION RATE OF CONSECUTIVE REACTION
In a consecutivefirst-order reaction, the product of one step becomes the reactant of the subsequent reaction steps. A consecutive irreversible reaction is given below. Determine the stoichiometric reaction relationships.
aA−−−−k1 bB−−−−k2 cC Solution
1. Express the overall rates of reactions, assuming thefirst-order reaction.
r1= −rA
a =rB1
b =k1[A] r2= −rB2
b =rC
c =k2[B]
Note: The rB1 represents generation of B; and the rate rB2 represents reduction in B and has negative sign.
2. Express the rates of reaction of individual reactants with respect to overall rate.
rA= −ar1
rB=rB1+rB2 =br1−br2
rC= −cr2
3. Express the rates of reaction of individual reactants with respect to concentration remaining.
rA= −ak1[A]
rB=bk1[A]−bk2[B]
rC= −ck2[B]
Proper signs should be used in the above reaction rates.
EXAMPLE 2.8: NITRIFICATION OF AMMONIA, A CONSECUTIVE REACTION A classic example of consecutive reaction in environmental engineering is nitrification of ammonia expressed by the following expression. Develop the stoichiometric reaction relationships.
NH3−−−−−−−−−−−−−−−O2&Nitrosomonas, k1
NO−2 −−−−−−−−−−−−−−−O2&Nitrobacter, k2
NO−3 Solution
1. Express the overall rates of reactions, assumingfirst-order reaction.
r1= −rNH3=r(NO−2)1=k1[NH3] r2= −r(NO−2)2=rNO−3 =k2NO−2
2. Express the rates of reaction of NH3, NO−2, and NO−3. rNH3= −k1[NH3]
rNO−2 =r(NO−2)1+r(NO−2)2
rNO−2 =k1[NH3] −k2NO−2 rNO−3 =k2NO−2
EXAMPLE 2.9: SINGLE REVERSIBLE REACTION
A single reversiblefirst-order reaction is given below. Express the reaction rates of reactants.
aAk1
k2
bB Solution
1. Express the overall reaction rates.
r1= −rA1
a =rB1
b =k1[A], r2= −rB2
b =rA2
a =k2[B], where
rA1=expresses decrease in concentration of A due to forward reaction rA2=expresses increase in concentration of A due to reverse reaction rB1=expresses increase in concentration of B due to forward reaction rB2=expresses decrease in concentration of B due to reverse reaction 2. Express the individual rates of reaction in terms of overall reaction rate
rA=rA1+rA2= −ar1+ar2
rB=rB1+rB2=br1−br2
3. Express the individual rates of reactions in terms of concentration.
rA= −ak1[A] +ak2[B] rB=bk1[A] −bk2[B]
2.3.2 Saturation-Type or Enzymatic Reactions
The saturation-type reactions reach a maximum rate. After reaching the maximum rate, the reaction becomes independent of the concentration of the reactants. The reaction rate constant of a simplified saturation-type reactionaA→bB is given by Equation 2.7a.
r= k[A]
Ks+ [A] (2.7a)
where
r =overall reaction rate of saturation reaction, mole/L·t k =maximum reaction rate, mole/L·t
Ks=half-saturation constant or substrate concentration at one-half the maximum reaction rate, mole/L
The relationship between the constantskandKsare indicated inFigure 2.1.
In a saturation-type reaction, the reaction raterreaches the maximum rate of reactionkand the half- saturation constantKsis equal to the substrate concentration at which the reaction rateris one-half of maximum (r=(1/2)k).
Equation 2.7a may be modified to develop a linear relationship to obtain the coefficients k and Ks. The procedure involves inversing Equation 2.7a to develop a linear relationship that is expressed by Equation 2.7b.
1 r =Ks
k 1 [A]+1
k (2.7b)
A plot of 1/rversus 1/[A] gives a linear relationship as shown inFigure 2.1b. The slope of the line is Ks/kand the intercept is 1/k. From these relationships, the coefficientskand Kscan be obtained.
More complex saturation reactions are used to express the specific substrate utilization rate, and specific biomass growth rate with respect to substrate concentration remaining and kinetic coefficients. This topic is covered in detail inChapter 10, Section 10.3.2, Examples 10.28 through 10.30.
EXAMPLE 2.10: TWO LIMITING CASES OF SATURATION-TYPE REACTION Two limiting cases of enzymatic reaction are commonly developed from Equation 2.7a. Express these limiting cases.
Solution
Two limiting cases of enzymatic reactions are dependent upon the substrate concentration. These are described below:
1. Expression for the substrate in excess.
When the substrate concentration is much greater [A]..KsandKs+[A]≈[A], the reaction rate is approaching the maximum rate and is independent of the concentration of [A]. It is derived from Equation 2.7a and is expressed by the equation:
r= k[A]
Ks+ [A]≈k[A]
[A] , or r=k (This is a pseudo zero-order reaction.) 2. Expression for the limited substrate.
When the substrate concentration is small [A],,KsandKs+[A]≈Ks(or a constant), the rate of enzymatic reaction is derived from Equation 2.7a and is expressed by the equation:
r= k[A]
Ks+ [A]≈k[A]
Ks
, or r=k′[A] and k′= k Ks
(This is a pseudo first-order reaction.) Many enzymatic reactions are function of the product. Examples are the number of microorganisms that increases in proportion to the number present. These reactions can befirst order, second order, or saturation type.
(a) (b)
1/r
1/[A]
Ks/k 1/k
k/2 k
ks Concentration [A], mole/L
Reaction rate (r), mole/L·d
FIGURE 2.1 The relationship between substrate concentration and reaction rate in saturation-type reaction:
(a) reaction raterreaching to a maximum rate ofkand (b) linear representation of the data.
EXAMPLE 2.11: REACTION RATES OF AUTOCATALYTIC REACTIONS
A reaction is autocatalytic if the rate is a function of the product concentration. A generalized single reac- tion is given by: aA→bB. Express the stoichiometric reaction relationships of these autocatalytic reactions.
Solution
The autocatalytic reactions can befirst order, second order or saturation type. Thefirst- and second-order reactions are as follows:
1. First-order autocatalytic reaction.
a. Express the overall reaction rate.
r=k[B]
b. Express the individual reactant and product reaction rates.
rA= −ar= −ak[B]
rB=br=bk[B]
2. Second-order autocatalytic reaction or partially autocatalytic reaction.
a. Express the overall reaction rate.
r=k[A][B]
b. Express the individual reactant and product reaction rates.
rA= −ar=ak[A][B] rB=br=bk[A][B]
Other examples of pseudofirst-order reaction are when the concentration of one component remains constant during the reaction. Example of such reactions are (1) the initial con- centration of one reactant is much higher than that of the other, (2) concentration of one reac- tant is buffered such as alkalinity remaining unchanged by dissolution of CaCO3, and (3) one reactant is supplied continuously. In these situations, a pseudofirst-order reaction is used.
EXAMPLE 2.12: PSEUDO FIRST-ORDER REACTION RATE
Consider an irreversible elementary reaction. Assume [A]≈[A0] and [A0]≫[B0], where A0and B0are initial concentrations. Express the modified pseudofirst-order overall reaction rate.
Solution
1. Express the irreversible elementary reaction.
aA+bBcC+dD
2. Express the rate law for the reaction.
r=k[A]a[B]b
3. If concentration of A does not change significantly during the reaction, the concentration of [A]
remains essentially constant. Assume [A]a≈K, the overall reaction rate is expressed below.
r=kK[ ]Bb=k′[B]bandk′=k[A]