J

Converting to log10, we see that Eqs. (3.48) and (3.49) become and C=C°1(r*" <3"50)

As can be seen from Eq. (3.48), a plot of ln(C/C0) versus t will yield a straight line. This is a common way of proving whether or not a reaction is first-order. The rate constant k can be determined directly from the slope of this line (k = -slope, since the rate constant cannot be negative). Similarly, a plot of log10(C/C0) versus t will yield a straight line. The rate constant k can be evaluated from this plot by multiplying the slope of the plotted line by -2.303.

For radioactive substances it is customary to express decomposition rates in terms of half-life, or the time required for the amount of substance to decrease to half its initial value. For a first-order reaction, the half-life, denoted by tm, can be found from Eq, (3.48) by inserting the requirement that at t = tm the concentration of C = |Co- This gives

There is a growing tendency to use the concept of half-life for a variety of environmental phenomena (see Sec. 5.34).

Environmental engineers also find application for the concepts of first-order reactions in areas that do not involve decomposition reactions. For example, the dissolution of gases into water and the removal of gases from water, under a given set of conditions, can be described by a first-order reaction (Sec. 2.9). In other cases, reactions that may in fact not be true first-order reactions can be approximated as such. The rate of death of microorganisms by disinfection is frequently considered to be a first-order reaction and dependent upon the concentration of live microorganisms remaining. Also, the decomposition of organic matter by bacteria in the biochemical oxygen demand (BOD) test is normally considered to be a first-order reaction dependent only upon the concentration of organic matter remaining. In actual fact, this is a very complex reaction, and the limitations of a first-order assumption should be well understood to prevent misinterpretation of BOD data. The kinetics of the BOD test are discussed in more detail in Sec. 23.2.

Strontium 90 (90Sr) is a radioactive nuclide, of public healdi sigmficance iand has a half- | EXAMPLE 3.13 life of 29 years. How long would a given amount of';oSr need to be stored:tp.obtain :a 99.9 /percent reduction in quantity? y ; ^ 'V^

From Eq. (3 51), we have k = 0.693/i1/2 = 0.693/29 yr = 2,39 X 10~2 yr"!. Time required is that for C to be reduced 99.9 percent to 0,001 fcp. Thus, from Eq. (3.4^),

-kt = ln(O.001Co/Co) = In 0.001 and t = - In 0.001/2.39 X 10"2 = 289 yr

The following are'data from an experiment to assess the disinfection .Of :a'water supply V

EXAMPLE 3.14

with a given dose of chlorine. Assuming first-order kinetics, determine the rate constant.

Time, jmiin

Percent coliform : ! V bacteria remaining

in(C/C0) '

0 ; .

•too W'.v-yH ' -

0.000 .

io ■ . ■

70

0.70 • :

>-0.357.

20 ■■•:

, 0.21. . .

:-1:561, :

30

63

0,063

-2.765 :

60

-0.6 vV

0.006

-5.116

From Eq. (3.48), a plot of ln(C/Co) versus time should yield :a -straight Une; with :a slope of —k. A plot of the data is given below.and includes the equation of the line developed from a linear regression of the data. The student should note that r2, termed the coefficient of determination, is a measure of the goodness of fit of the data with the rate expression used. The larger the r2 (with a maximum of 1.000), the better the fit. The equation of' the regression line and values for r1 can be calculated using methods described in Chap. 10, or can be determined using hand-held calculators and computer graphics software;

Gas-Liquid Mass-Transfer Kinetics. The rate of gas transfer into or from aqueous solution can be described using the first-order relationship given in Sec. 2.9:

where C = concentration in water (mass/volume)

C«!uii = concentration in water (mass/volume) that would be in equilibrium with concentration in gas phase (Henry's law) KLa - first-order, overall mass-transfer rate coefficient (time"')

Mass transfer (flux) into or from aqueous solution can be thought of as resistance in series: transfer from the bulk gas phase across a stagnant gas film to the gas-water interface, followed by transfer across a stagnant liquid film to the bulk water phase ("two-film" theory). Reaction rates can be limited by flux across the gas film or the liquid film. This is best understood using the following relationship:

a = interfacial area to volume of water, nT1 K„ = Henry's constant, L-atm/mol kg = gas-phase mass-transfer rate coefficient, m/s k„ = water-phase mass-transfer rate coefficient, m/s

For compounds with large KH, the second term will dominate the expression and water-film mass transfer will control overall rates. For compounds with small Kn, the first term dominates and gas-film mass transfer controls rates. A practical implication of such considerations is whether mixing the liquid or gas will enhance masstransfer rates. In addition to the two-film model, surface renewal and penetration models have also been used to describe gas-water mass transfer.4

Second-Order Reactions

A second-order reaction is one in which the rate of the reaction is proportional to the square of the concentration of one of the reactants or to the product-of the concentrations of two different reactants. Thus, if the overall second-order reaction were of the form

1 ^ RT KLa Ktlkga kwa

"W. J. Weber, Jr., and F, A. DiGiano, "Process Dynamics in Environmental Systems," John Wiley & Sons, New York, 1996.

then the rate law for this situation might be dC, dt = KCl (3.54)

where Ca and Cb are the concentrations of A and B, respectively. The decrease in B could be formulated in a similar manner:

The integrated forms of Eqs. (3.54) and (3,56) are

Thus, if a plot of 1/A or 1/B versus f gives a straight line, a second-order reaction is implicated. The interested student can find the integrated forms of Eqs. (3.55) and (3.57) in most textbooks on physical chemistry.

An important use of a second-order reaction is in describing cometabolic biotransformation of some halogenated organic compounds or biological transformations in general where the concentration of the compound transformed is very low (< 1 mg/L). In this case, the rate expression is

where C is the concentration of the organic compound transformed and X is the concentration of bacteria. This reaction is of the type described by Eq. (3.55). During such biotransformations, X might also be considered constant and the rate expression becomes

where k' is equal to kX, and the reaction in this form is typically termed a pseudofirst-order reaction with k' being the pseudo-first-order rate constant. Of course, the reaction is first-order with respect to C regardless of whether X is constant. This type of rate expression is used to describe other reactions of importance to environmental engineers and scientists.

Consecutive Reactions

Consecutive reactions are complex reactions of great environmental importance, and so equations describing the kinetics of such reactions are of real interest. In consecutive reactions, the products of one reaction become the reactants of a following reaction:

Here reactant A is converted to product B at a rate determined by rate constant Jfc,. Product B in turn becomes the reactant for the second step and is converted to product C as determined by rate constant k2. If the rates of each of the consecutive reactions are considered to be first-order, then the differential equations which describe the rates of decomposition and formation of the reactants and products are as follows:

If at t = 0 we have Ca = C° Cb = ¿t and Cc = C?, then a solution for the concentration of each constituent at some time / is as follows:

Equation (3.67) is widely used to describe the oxygen deficit in a stream caused by organic pollution. In this case Cb can be considered the oxygen deficit being created in the first step by the biological oxidation of organic matter with concentration Ca. At the same time, the oxygen deficit is being decreased by atmospheric reaeration [see Eq. (3.52)] to give the second step in the consecutive reactions. Equation (3.67) when used for this case is the well-known Streeter-Phelps equation.

A consecutive reaction can also be used to describe the bacterial nitrification of ammonia. Here ammonia is oxidized by Nitrosomonas bacteria to nitrite, which is then oxidized in the second step by Nitrobacter bacteria to nitrate as indicated by the following sequence:

J HIiraoHumu Nutobaaer

The buildup and decay of the various forms of nitrogen in this consecutive reaction are sometimes assumed for simplicity to follow first-order kinetics. The changes in nitrogen forms which would occur with this assumption are illustrated in Fig. 3.10. The concentrations of NOJ and NO3 were set equal to zero when t = 0, and k( was assumed to equal 2In actual fact, the kinetics for nitrification are much more complex, so one should consider the limitations of this assumption before applying these equations in practice. The changes in nitrogen forms indicated in Fig. 3.10 are typical of those frequently noted in trickling filters where ammonia is oxidized or in rivers downstream from an ammonia discharge.

Figure 3.10

Nitrogen changes during nitrification, assuming consecutive first-order reactions.

Figure 3.10

Nitrogen changes during nitrification, assuming consecutive first-order reactions.

Consecutive-type kinetics are frequently used to describe the growth and decay of microorganisms in biological treatment processes. They can also describe the consecutive steps in the decomposition of organic matter as it occurs in anaerobic waste treatment. Thus, consecutive-type kinetics are widely applicable in environmental engineering and science practice.

Enzyme Reactions

Another complex kinetics expression to describe the rate of biological waste treatment was first used by Michaelis and Menten5 to describe enzyme reactions. Since bacterial decomposition involves a series of enzyme-catalyzed steps, the MichaelisMenten expression can be empirically extended to describe the kinetics of bacterial growth and waste decomposition. The resulting equation is termed the Monod equation. Figure 3.11 indicates the normally observed relationship between substrate or waste concentration, designated as S, and speed of waste utilization per unit mass of enzyme or bacteria, designated as VIE.

The Michaelis-Menten relationship for enzyme reactions in a simplified form assumes the following reaction, where is free enzyme, S is substrate, and ECS is enzyme-substrate complex:

EJ5 is formed at rate when free enzyme and substrate combine. The complex is unstable and decomposes either back to the original free enzyme and substrate at rate k_u or into free enzyme and reaction products at rate k. The total enzyme

'Michaelis and Menten, Biochem. Zeit., 49: 333 (1913).

Substrate concentration, S—»-

Figure 3.11

The relationship between substrate concentration and reaction rate for enzyme-type reactions.

Substrate concentration, S—»-

Figure 3.11

The relationship between substrate concentration and reaction rate for enzyme-type reactions.

concentration in the system, E, remains constant and is equal to [£,] + I&S]. On the basis of the Eq. (3.70), the rate of formation of enzyme-substrate complex is d{EcS]ldt = k{ lEf]S ™ (&_{ + k)[EcS]

The rate of complex formation is generally much faster than the overall reaction rate, so for the purpose of determining the overall reaction rate, the complex can be considered as being at pseudo-steady-state concentration, that is, d[EcS\!dt - 0. Therefore, k,S(E - M) = (*_, + k)[EcS]

Rearranging, we obtain

The rate of product formation is equal to the overall velocity (rate) of the reaction and is given by V = and thus from the relationship given by Eq. (3.73), the overall rate as a function of E and S becomes kES

The significance of the constants k and Ks is indicated in Fig. 3.11. The constant k gives the maximum rate of the reaction, and Ks is equal to the substrate concentration at which the reaction rate is one-half of maximum. K, is commonly called the "half-velocity" constant. Two limiting cases for Eq. (3.74) are apparent:

Equation (3.75) indicates that when the substrate concentration is low compared to Ks, the rate of the enzyme reaction is directly proportional to S. Therefore, the reaction can be described as first-order with respect to substrate. However, when S is much greater than K„ the reaction rate is a maximum and independent of the concentration S. The reaction is then said to be zero-order with respect to substrate.

Both the continuous Eq. (3.74) and the discontinuous set of Eqs. (3.75) and (3.76) are frequently used to describe biological reaction rates. All are somewhat empirical when used to describe complex biological processes, but they give a sufficiently adequate description of the overall process to yield practical results.

A study;was made to evaluate the.constants.so that the Michaelis-Menten relationship| EXAMPLE 3.15 could be used to describe waste utilization by bacteria. It was found that 1 g of bacteria could decompose the1 waste at a maximum rate of 20 g/day when the waste concentration was highi Also, it was found that this same quantity of bacteria would decompose waste at a rate of 10 g/day when the Waste concentration surrounding the bacteria was 15 mg/L. .. What would be the rate of waste decompositiion by 2 g of bacteria if the waste concentration were maintained at 5: mg/L?•: V; . ■ : • •• ..

. The constant k gives the maximum rate of waste utilization, and so for this case it is equal to 20 g/day-g. The constant.is equal to the substrate concentration at which the :; rate is § of maximum or 10 g/day-g. Therefore, for this example, = 15 mg/^, and from Eq (3 74), and assuming £ = weight of bacteria t= 2 g, and S = 5 mg/L, , .

While Example 3.15 is interesting, the student should be aware that it is not good practice to attempt to determine reaction rate constants from only one or two measurements.

Temperature Dependence of Reaction Rates

In general, the rates of most chemical and biological reactions increase with temperature. An approximate rule is that the rate of a reaction will about double for each 10°C rise in temperature. In biological reactions, this rule will hold more or

less true up to a certain optimum temperature. Above this, the rate decreases, probably owing to destruction of enzymes at the higher temperatures.

The change in rate constant with temperature can be expressed mathematically by the Arrhenius equation, d In ktdT = EJRT1 (3.77)

where d In ktdT = change in natural log of rate constant with temperature R = universal gas constant

Eu = constant for reaction termed activation energy Integrating between limits gives ltl h = ^P* ~ Ti). (3,78)

m k, RT2Tt where and kx are the rate constants at temperatures T2 and respectively. Temperature is expressed in kelvins.

Most processes of concern to environmental engineers and scientists operate over a small temperature range near ambient temperatures. For this case the product T,r, changes very little, and for practical purposes it can be considered constant. Thus, EJRT-iTi can be considered equal to a constant A, so that an approximate formula for temperature dependence of reaction rates can be used:

Another common fonn of the equation describing temperature dependence is

Of course, the values for A are different in Eqs. (3.80) and (3.81). The value of A in Eq. (3.81) is equal to eE-KT^. Both Eqs. (3.80) and (3.81) are commonly used to express the effect of temperature on reaction rates. Although A is supposed to be a constant, it sometimes varies significantly even over a limited temperature range. For example, for the BOD reaction rate (see Chap. 23), A [Eq. (3.80)] has been indicated6 to vary from 0.135 in the temperature range from 4 to 20°C, down to 0.056 in the temperature range from 20 to 30°C. Thus, caution must be exercised in using an A value beyond the temperature range for which it was evaluated.

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