7. Collection and Analysis of Rate Data

You can observe a lot just by watching.

Yogi Berra, New York Yankees

Data for homogeneous reactions is most often obtained in a batch reactor. After postulating a rate law in Step 1 and combining it with a mole balance in Step 2, we next use any or all of the methods in Step 5 to process the data and arrive at the reaction orders and specific reaction-rate constants.

Analysis of heterogeneous reactions is shown in Step 6. For gas-solid heterogeneous reactions, we need to have an understanding of the reaction and possible mechanisms in order to postulate the rate law in Step 6B. After studying Chapter 10 on heterogeneous reactions, one will be able to postulate different rate laws and then use Polymath nonlinear regression to choose the “best” rate-law and reaction-rate-law parameters (see Example 10-3 on page 452).

The procedure we should use to delineate the rate law and rate-law parameters is given in Table 7-1.

TABLE 7-1 STEPS IN ANALYZING RATE DATA

Process data in terms of the measured variable.

When a reaction is irreversible, it is possible in many cases to determine the reaction order α and the specific rate constant by either nonlinear regression or by numerically differentiating concentration versus time data. This latter method is most applicable when reaction conditions are such that the rate is essentially a function of the concentration of only one reactant; for example, if, for the decomposition reaction

A → Products

Assume that the rate law is of the form .

then the differential method may be used.

However, by utilizing the method of excess, it is also possible to determine the relationship between –r_A and the concentration of other reactants. That is, for the irreversible reaction

A + B → Products

with the rate law

where α and β are both unknown, the reaction could first be run in an excess of B so that C_B remains essentially unchanged during the course of the reaction (i.e., C_B ≈ C_B0) and

where

Method of excess

After determining α, the reaction is carried out in an excess of A, for which the rate law is approximated as

where

Once α and β are determined, k_A can be calculated from the measurement of –r_A at known concentrations of A and B

Both α and β can be determined by using the method of excess, coupled with a differential analysis of data for batch systems.

The integral method uses a trial-and-error procedure to find the reaction order.

and integrate the differential equation to obtain the concentration as a function of time. If the order we assume is correct, the appropriate plot (determined from this integration) of the concentration-time data should be linear. The integral method is used most often when the reaction order is known and it is desired to evaluate the specific reaction rate constant at different temperatures to determine the activation energy.

In the integral method of analysis of rate data, we are looking for the appropriate function of concentration corresponding to a particular rate law that is linear with time. You should be thoroughly familiar with the methods of obtaining these linear plots for reactions of zero, first, and second order.

For the reaction

A → Products

carried out in a constant-volume batch reactor, the mole balance is

It is important to know how to generate linear plots of functions of C_A versus t for zero-, first-, and second-order reactions.

For a zero-order reaction, r_A = –k, and the combined rate law and mole balance is

Integrating with C_A = C_A0 at t = 0, we have

Zero order

A plot of the concentration of A as a function of time will be linear (Figure 7-1) with slope (–k) for a zero-order reaction carried out in a constant-volume batch reactor.

If the reaction is first order (Figure 7-2), integration of the combined mole balance and the rate law

with the limit C_A = C_A0 at t = 0 gives

First order

Consequently, we see that the slope of a plot of [ln(C_A0/C_A)] as a function of time is linear with slope k.

If the reaction is second order (Figure 7-3), then

Integrating, with C_A = C_A0 initially (i.e., t = 0), yields

Second order

We see that for a second-order reaction a plot of (1/C_A) as a function of time should be linear with slope k.

In Figures 7-1, 7-2, and 7-3, we saw that when we plotted the appropriate function of concentration (i.e., C_A, ln C_A, or 1/C_A) versus time, the plots were linear, and we concluded that the reactions were zero, first, or second order, respectively. However, if the plots of concentration data versus time had turned out not to be linear, such as shown in Figure 7-4, we would say that the proposed reaction order did not fit the data. In the case of Figure 7-4, we would conclude that the reaction is not second order. After finding that the integral method for first, second, and third orders do not fit the data, one should use one of the other methods discussed in Table 7-1.

The idea is to arrange the data so that a linear relationship is obtained.

It is important to restate that, given a reaction-rate law, you should be able to quickly choose the appropriate function of concentration or conversion that yields a straight line when plotted against time or space time. The goodnessof-fit of such a line may be assessed statistically by calculating the linear correlation coefficient, r², which should be as close to 1 as possible. The value of r² is given in the output of Polymath’s nonlinear regression.

Constant-volume batch reactor

After taking the natural logarithm of both sides of Equation (5-6)

observe that the slope of a plot of [ln(–dC_A/dt)] as a function of (ln C_A) is the reaction order, α (see Figure 7-5).

Plot versus ln C_A to find α and k_A

Figure 7-5(a) shows a plot of [–(dC_A/dt)] versus [C_A] on log-log paper (or use Excel to make the plot) where the slope is equal to the reaction order α. The specific reaction rate, k_A, can be found by first choosing a concentration in the plot, say C_Ap, and then finding the corresponding value of [–(dC_A/dt)_p] on the line, as shown in Figure 7-5(b). The concentration chosen, C_Ap, to find the derivative at C_Ap, need not be a data point, it just needs to be on the line. After raising C_Ap to the power α, we divide it into [–(dC_A/dt)_p] to determine k_A

To obtain the derivative (–dC_A/dt) used in this plot, we must differentiate the concentration-time data either numerically or graphically. Three methods to determine the derivative from data giving the concentration as a function of time are

• Graphical differentiation

• Numerical differentiation formulas

• Differentiation of a polynomial fit to the data

Methods for finding from concentration-time data

We shall only discuss the graphical and numerical methods.

See Appendix A.2.

In addition to the graphical technique used to differentiate the data, two other methods are commonly used: differentiation formulas and polynomial fitting.

The three-point differentiation formulas¹ shown in Table 7-3 can be used to calculate dC_A/dt.

¹ B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (New York: Wiley, 1969), p. 129.

Equations (7-13) and (7-15) are used for the first and last data points, respectively, while Equation (7-14) is used for all intermediate data points (see Step 6A.1a in Example 7-2).

The reaction order can now be found from a plot of ln(–dC_A/dt) as a function of ln C_A, as shown in Figure 7-5(a), since

Before solving the following example problems, review the steps to determine the reaction-rate law from a set of data points (Table 7-1).

By comparing the methods of analysis of the rate data presented in Examples 7-1 and 7-2, we note that the differential method tends to accentuate the uncertainties in the data, while the integral method tends to smooth the data, thereby disguising the uncertainties in it. In most analyses, it is imperative that the engineer know the limits and uncertainties in the data. This prior knowledge is necessary to provide for a safety factor when scaling up a process from laboratory experiments to design either a pilot plant or full-scale industrial plant.

Integral method normally used to find k when order is known

We will now apply nonlinear regression to reaction-rate data to determine the rate-law parameters. Here, we make initial estimates of the parameter values (e.g., reaction order, specific rate constant) in order to calculate the concentration for each data point, C_ic, obtained by solving an integrated form of the combined mole balance and rate law. We then compare the measured concentration at that point, C_im, with the calculated value, C_ic, for the parameter values chosen. We make this comparison by calculating the sum of the squares of the differences at each point Σ(C_im – C_ic)². We then continue to choose new parameter values and search for those values of the rate law that will minimize the sum of the squared differences of the measured concentrations, C_im, and the calculated concentrations values, C_ic. That is, we want to find the rate-law parameters for which the sum of all data points Σ(C_im–C_ic)² is a minimum. If we carried out N experiments, we would want to find the parameter values (e.g., E, activation energy, reaction orders) that minimize the quantity

where

One notes that if we minimize s² in Equation (7-17), we minimize σ².

To illustrate this technique, let’s consider the reaction

for which we want to learn the reaction order, α, and the specific reaction rate, k,

The reaction rate will be measured at a number of different concentrations. We now choose values of k and α, and calculate the rate of reaction (C_ic) at each concentration at which an experimental point was taken. We then subtract the calculated value (C_ic) from the measured value (C_im), square the result, and sum the squares for all the runs for the values of k and α that we have chosen.

This procedure is continued by further varying α and k until we find those values of k and α that minimize the sum of the squares. Many well-known searching techniques are available to obtain the minimum value .³ Figure 7-6 shows a hypothetical plot of the sum of the squares as a function of the parameters α and k:

³ (a) B. Carnahan and J. O. Wilkes, Digital Computing and Numerical Methods (New York: Wiley, 1973), p. 405. (b) D. J. Wilde and C. S. Beightler, Foundations of Optimization, 2nd ed. (Upper Saddle River, NJ: Prentice Hall, 1979). (c) D. Miller and M. Frenklach, Int. J. Chem. Kinet., 15 (1983), p. 677.

Look at the top circle. We see that there are many combinations of α and k (e.g., α = 2.2, k = 4.8 or α = 1.8, k = 5.3) that will give a value of σ² = 57. The same is true for σ² = 1.85. We need to find the combination of α and k that gives the lowest value of σ².

In searching to find the parameter values that give the minimum of the sum of squares σ², one can use a number of optimization techniques or software packages. The searching procedure begins by guessing parameter values and then calculating (C_im–C_ic) and then σ² for these values. Next, a few sets of parameters are chosen around the initial guess, and σ² is calculated for these sets as well. The search technique looks for the smallest value of σ² in the vicinity of the initial guess and then proceeds along a trajectory in the direction of decreasing σ² to choose different parameter values and determine the corresponding σ². The trajectory is continually adjusted so as to always proceed in the direction of decreasing σ² until the minimum value of σ² is reached. For example, in Figure 7-6 the search technique keeps choosing combinations of α and k until a minimum value of σ² = 0.045 (mol/dm³)² is reached. The combination that gives that minimum is α = 2 and k = 5.0 dm³/mol·min as shown in Figure 7-6. If the equations are highly nonlinear, the initial guesses of α and k are very important.

A number of software packages are available to carry out the procedure to determine the best estimates of the parameter values and the corresponding confidence limits. All we have to do is to enter the experimental values into the computer, specify the model, enter the initial guesses of the parameters, and then push the “compute” button, and the best estimates of the parameter values along with 95% confidence limits appear. If the confidence limits for a given parameter are larger than the parameter itself, the parameter is probably not significant and should be dropped from the model. After the appropriate model parameters are eliminated, the software is run again to determine the best fit with the new model equation.

Concentration-Time Data. We will now use nonlinear regression to determine the rate-law parameters from concentration-time data obtained in batch experiments. We recall that the combined rate-law stoichiometry mole balance for a constant-volume batch reactor is

We now integrate Equation (7-6) to give

Rearranging to obtain the concentration as a function of time, we obtain

Now we could use either Polymath or MATLAB to find the values of α and k that would minimize the sum of squares of the differences between the measured concentrations, C_Aim, and calculated concentrations, C_Aic. That is, for N data points,

we want the values of α and k that will make s² a minimum.

If Polymath is used, one should use the absolute value for the term in brackets in Equation (7-16), that is,

Another, and perhaps easier, way to solve for the parameter values is to use time rather than concentrations, rearranging Equation (7-20) to get

That is, we find the values of k and α that minimize

Finally, a discussion of weighted least squares as applied to a first-order reaction is provided in the Professional Reference Shelf R7.5 on the CRE Web site.

Model Discrimination. One can also determine which model or equation best fits the experimental data by comparing the sums of the squares for each model and then choosing the equation with a smaller sum of squares and/or carrying out an F-test. Alternatively, we can compare the residual plots for each model. These plots show the error associated with each data point, and one looks to see if the error is randomly distributed or if there is a trend in the error. When the error is randomly distributed, this is an additional indication that the correct rate law has been chosen. An example of model discrimination using nonlinear regression is given in Chapter 10.

⁴ B. Anderson, ed., Experimental Methods in Catalytic Research (San Diego, CA: Academic Press, 1976).

Most commonly used catalytic reactor to obtain experimental data

The differential reactor is relatively easy to construct at a low cost. Owing to the low conversion achieved in this reactor, the heat release per unit volume will be small (or can be made small by diluting the bed with inert solids) so that the reactor operates essentially in an isothermal manner. When operating this reactor, precautions must be taken so that the reactant gas or liquid does not bypass or channel through the packed catalyst, but instead flows uniformly across the catalyst. If the catalyst under investigation decays rapidly, the differential reactor is not a good choice because the reaction-rate parameters at the start of a run will be different from those at the end of the run. In some cases, sampling and analysis of the product stream may be difficult for small conversions in multicomponent systems.

Limitations of the differential reactor

For the reaction of species A going to product (P)

A → P

the volumetric flow rate through the catalyst bed is monitored, as are the entering and exiting concentrations (Figure 7-9). Therefore, if the weight of catalyst, ΔW, is known, the rate of reaction per unit mass of catalyst, , can be calculated. Since the differential reactor is assumed to be gradientless, the design equation will be similar to the CSTR design equation. A steady-state mole balance on reactant A gives

The subscript e refers to the exit of the reactor. Solving for , we have

The mole balance equation can also be written in terms of concentration

Differential reactor design equation

or in terms of conversion or product flow rate F_P

The term F_A0X gives the rate of formation of product, F_P, when the stoichiometric coefficients of A and of P are identical. Adjustments to Equation (7-28) must be made when this is not the case.

For constant volumetric flow, Equation (7-28) reduces to

Consequently, we see that the reaction rate, , can be determined by measuring the product concentration, C_P.

By using very little catalyst and large volumetric flow rates, the concentration difference, (C_A0 – C_Ae), can be made quite small. The rate of reaction determined from Equation (7-29) can be obtained as a function of the reactant concentration in the catalyst bed, C_Ab

by varying the inlet concentration. One approximation of the concentration of A within the bed, C_Ab, would be the arithmetic mean of the inlet and outlet concentrations

However, since very little reaction takes place within the bed, the bed concentration is essentially equal to the inlet concentration

C_Ab ≈ C_A0

so is a function of C_A0

As with the method of initial rates (see the CRE Web site, PRS R7.1), various numerical and graphical techniques can be used to determine the appropriate algebraic equation for the rate law. When collecting data for fluid-solid reacting systems, care must be taken that we use high flow rates through the differential reactor and small catalyst particle sizes in order to avoid mass transfer limitations. If data show the reaction to be first order with a low activation energy, say 8 kcal/mol, one should suspect the data are being collected in the mass transfer limited regime. We will expand on mass transfer limitations and how to avoid them in Chapters 10, 14, and 15.

G. C. Quarderer, Dow Chemical Co.

So far, this chapter has presented various methods of analyzing rate data. It is just as important to know in which circumstances to use each method as it is to know the mechanics of these methods. In the Expanded Material on the CRE Web site, we give a thumbnail sketch of a heuristic to plan experiments to generate the data necessary for reactor design. However, for a more thorough discussion, the reader is referred to the books and articles by Box and Hunter.⁶

⁶ G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building (New York: Wiley, 1978).

a. Guess the reaction order and integrate the mole balance equation.

b. Calculate the resulting function of concentration for the data and plot it as a function of time. If the resulting plot is linear, you have probably guessed the correct reaction order.

c. If the plot is not linear, guess another order and repeat the procedure.

2. Differential method for constant-volume systems

a. Plot –ΔC_A/Δt as a function of t.

b. Determine –dC_A/dt from this plot.

c. Take the ln of both sides of (S7-1) to get

Plot ln(–dC_A/dt) versus ln C_A. The slope will be the reaction order α. We could use finite-difference formulas or software packages to evaluate (–dC_A/dt) as a function of time and concentration.

3. Nonlinear regression: Search for the parameters of the rate law that will minimize the sum of the squares of the difference between the measured rate of reaction and the rate of reaction calculated from the parameter values chosen. For N experimental runs and K parameters to be determined, use Polymath.

Caution: Be sure to avoid a false minimum in σ² by varying your initial guess.

4. Modeling the differential reactor:

The rate of reaction is calculated from the equation

In calculating the reaction order, α

the concentration of A is evaluated either at the entrance conditions or at a mean value between C_A0 and C_Ae. However, power-law models such as

are not the best way to describe heterogeneous reaction-rate laws. Typically, they take the form

or a similar form, with the reactant partial pressures in the numerator and denominator of the rate law.

1. Evaluation of Laboratory Reactors

2. Summary of Reactor-Ratings, Gas-liquid, Powdered Catalyst Decaying Catalyst System

3. Experimental Planning

4. Additional Homework Problems

• Learning Resources

1. Summary Notes

2. Interactive Computer Games

A. Ecology

B. Reactor Lab (www.reactorlab.net). See Reactor Lab Chapter 7 and P7-3_A.

3. Solved Problems

A. Example: Differential Method of Analysis of Pressure- Time Data

B. Example: Integral Method of Analysis of Pressure- Time Data

C. Example: Oxygenating Blood

• Living Example Problems

1. Example 7-3 Use of Regression to Find the Rate Law Parameters

• FAQ (Frequently Asked Questions)—In Updates/FAQ icon section

• Professional Reference Shelf

R7.1. Method of Initial Rates

R7.2. Method of Half Lives

R7.3. Least-Squares Analysis of the Linearized Rate Law

The CRE Web site describes how the rate law

is linearized

ln(–r_A) = ln k + α ln C_A + β ln C_B

and put in the form

Y = a₀ + αX₁ + βX₂

and used to solve for α, β, and k. The etching of a semiconductor, MnO₂, is used as an example to illustrate this technique.

R7.4. A Discussion of Weighted Least Squares

For the case when the error in measurement is not constant, we must use a weighted least-squares analysis.

R7.5. Experimental Planning

A. Why perform the experiment?

B. Are you choosing the correct parameters?

C. What is the range of your experimental variables?

D. Can you repeat the measurement? (Precision)

E. Milk your data for all it’s worth.

F. We don’t believe an experiment until it’s proven by theory.

G. Tell someone about your result.

R7.6. Evaluation or Laboratory Reactor

(a) Listen to the audios on the CRE Web site and pick one and say why it could be eliminated.

(b) Create an original problem based on Chapter 7 material.

(c) Design an experiment for the undergraduate laboratory that demonstrates the principles of chemical reaction engineering and will cost less than $500 in purchased parts to build. (From 1998 AIChE National Student Chapter Competition). Rules are provided on the CRE Web site.

(d) K-12 Experiment. Plant a number of seeds in different pots (corn works well). The plant and soil of each pot will be subjected to different conditions. Measure the height of the plant as a function of time and fertilizer concentration. Other variables might include lighting, pH, and room temperature. (Great grade school or high school science project.)

(a) Revisit Example 7-1. What is the error in assuming the concentration of species B is constant and what limits can you put on the calculated value of k? (I.e., k = 0.24 ±?)

(b) Revisit Example 7-3. Explain why the regression was carried out twice to find k′ and k.

(c) Revisit Example 7-4. Regress the data to fit the rate law

What is the difference in the correlation and sums of squares compared with those given in Example 7-4? Why was it necessary to regress the data twice, once to obtain Table E7-4.3 and once to obtain Table E7-4.4?

P7-2_A Download the Interactive Computer Game (ICG) from the CRE Web site. Play the game and then record your performance number for the module that indicates your mastery of the material. Your professor has the key to decode your performance number.

ICM Ecology Performance #_________________________.

P7-3_A Go to Professor Herz’s Reactor Lab on the Web at www.reactorlab.net. Do (a) one quiz, or (b) two quizzes from Division 1. When you first enter a lab, you see all input values and can vary them. In a lab, click on the Quiz button in the navigation bar to enter the quiz for that lab. In a quiz, you cannot see some of the input values: you need to find those with “???” hiding the values. In the quiz, perform experiments and analyze your data in order to determine the unknown values. See the bottom of the Example Quiz page at www.reactorlab.net for equations that relate E and k. Click on the “???” next to an input and supply your value. Your answer will be accepted if it is within ±20% of the correct value. Scoring is done with imaginary dollars to emphasize that you should design your experimental study rather than do random experiments. Each time you enter a quiz, new unknown values are assigned. To reenter an unfinished quiz at the same stage you left, click the [i] info button in the Directory for instructions. Turn in copies of your data, your analysis work, and the Budget Report.

P7-4_A When arterial blood enters a tissue capillary, it exchanges oxygen and carbon dioxide with its environment, as shown in this diagram.

The kinetics of this deoxygenation of hemoglobin in blood was studied with the aid of a tubular reactor by Nakamura and Staub (J. Physiol., 173, 161).

Although this is a reversible reaction, measurements were made in the initial phases of the decomposition so that the reverse reaction could be neglected. Consider a system similar to the one used by Nakamura and Staub: the solution enters a tubular reactor (0.158 cm in diameter) that has oxygen electrodes placed at 5-cm intervals down the tube. The solution flow rate into the reactor is 19.6 cm³/s with C_A0 = 2.33 × 10^–6mol/cm³.

(a) Using the method of differential analysis of rate data, determine the reaction order and the forward specific reaction-rate constant k₁ for the deoxygenation of hemoglobin.

(b) Repeat using regression.

P7-5_B The liquid-phase irreversible reaction

A → B + C

is carried out in a CSTR. To learn the rate law, the volumetric flow rate, υ₀, (hence τ = V/υ₀) is varied and the effluent concentrations of species A are recorded as a function of the space time τ. Pure A enters the reactor at a concentration of 2 mol/dm³. Steady-state conditions exist when the measurements are recorded.

(a) Determine the reaction order and specific reaction-rate constant.

(b) If you were to repeat this experiment to determine the kinetics, what would you do differently? Would you run at a higher, lower, or the same temperature? If you were to take more data, where would you place the measurements (e.g., τ)?

(c) It is believed that the technician may have made a dilution factor-of-10 error in one of the concentration measurements. What do you think? How do your answers compare using regression (Polymath or other software) with those obtained by graphical methods?

Note: All measurements were taken at steady-state conditions.

P7-6_A The reaction

A → B + C

was carried out in a constant-volume batch reactor where the following concentration measurements were recorded as a function of time.

(a) Use nonlinear least squares (i.e., regression) and one other method to determine the reaction order, α, and the specific reaction rate, k.

(b) Nicolas Bellini wants to know, if you were to take more data, where would you place the points? Why?

(c) Prof. Dr. Sven Köttlov from Jofostan University always asks his students, if you were to repeat this experiment to determine the kinetics, what would you do differently? Would you run at a higher, lower, or the same temperature? Take different data points? Explain.

(d) It is believed that the technician made a dilution error in the concentration measured at 60 min. What do you think? How do your answers compare using regression (Polymath or other software) with those obtained by graphical methods?

P7-7_A The following data were reported [from C. N. Hinshelwood and P. J. Ackey, Proc. R. Soc. (Lond)., A115, 215] for a gas-phase constant-volume decomposition of dimethyl ether at 504°C in a batch reactor. Initially, only (CH₃)₂O was present.

(a) Why do you think the total pressure measurement at t = 0 is missing? Can you estimate it?

(b) Assuming that the reaction

(CH₃)₂O → CH₄ + H₂ + CO

is irreversible and goes virtually to completion, determine the reaction order and specific reaction rate k. (Ans.: k = 0.00048 min^–1)

(c) What experimental conditions would you suggest if you were to obtain more data?

(d) How would the data and your answers change if the reaction were run at a higher temperature? A lower temperature?

P7-8_A In order to study the photochemical decay of aqueous bromine in bright sunlight, a small quantity of liquid bromine was dissolved in water contained in a glass battery jar and placed in direct sunlight. The following data were obtained at 25°C:

(a) Determine whether the reaction rate is zero, first, or second order in bromine, and calculate the reaction-rate constant in units of your choice.

(b) Assuming identical exposure conditions, calculate the required hourly rate of injection of bromine (in pounds per hour) into a sunlit body of water, 25,000 gal in volume, in order to maintain a sterilizing level of bromine of 1.0 ppm. [Ans.: 0.43 lb/h]

(c) Apply one or more of the six ideas in Preface Table P-4, page xxviii, to this problem.

(Note: ppm = parts of bromine per million parts of brominated water by weight. In dilute aqueous solutions, 1 ppm ≡ 1 milligram per liter.) (From California Professional Engineers’ Exam.)

P7-9_C The reactions of ozone were studied in the presence of alkenes [from R. Atkinson et al., Int. J. Chem. Kinet., 15(8), 721 (1983) ]. The data in Table P7-9_C are for one of the alkenes studied, cis-2-butene. The reaction was carried out isothermally at 297 K. Determine the rate law and the values of the rate-law parameters.

P7-10_A Tests were run on a small experimental reactor used for decomposing nitrogen oxides in an automobile exhaust stream. In one series of tests, a nitrogen stream containing various concentrations of NO₂ was fed to a reactor, and the kinetic data obtained are shown in Figure P7-10_A. Each point represents one complete run. The reactor operates essentially as an isothermal backmix reactor (CSTR). What can you deduce about the apparent order of the reaction over the temperature range studied?

The plot gives the fractional decomposition of NO₂ fed versus the ratio of reactor volume V (in cm³) to the NO₂ feed rate, F_NO2,0 (g mol/h), at different feed concentrations of NO₂ (in parts per million by weight). Determine as many rate law parameters as you can.

P7-11_A The thermal decomposition of isopropyl isocyanate was studied in a differential packed-bed reactor. From the data in Table P7-11_A, determine the reaction-rate-law parameters.

• Additional Homework Problems are on the CRE Web site

New Problems on the Web

CDP7-New From time to time, new problems relating Chapter 7 material to everyday interests or emerging technologies will be placed on the Web. Solutions to these problems can be obtained by emailing the author. Also, one can go to the Web site, nebula.rowan.edu:82/home.asp, and work the home problem specific to this chapter.

Green Engineering

BURGESS, THORNTON W., Mr. Toad and Danny the Meadow Mouse Take a Walk. New York: Dover Publications, Inc., 1915.

FOGLER, H. SCOTT and STEVEN E. LEBLANC, Strategies for Creative Problem Solving. Englewood Cliffs, NJ: Prentice Hall, 1995.

KARRASS, CHESTER L., In Business As in Life, You Don’t Get What You Deserve, You Get What You Negotiate. Hill, CA: Stanford Street Press, 1996.

ROBINSON, J. W., Undergraduate Instrumental Analysis, 5th ed. New York: Marcel Dekker, 1995.

SKOOG, DOUGLAS A., F. JAMES HOLLER, and TIMOTHY A. NIEMAN, Principles of Instrumental Analysis, 5th ed. Philadelphia: Saunders College Publishers, Harcourt Brace College Publishers, 1998.

2. The design of laboratory catalytic reactors for obtaining rate data is presented in

RASE, H. F., Chemical Reactor Design for Process Plants, Vol. 1. New York: Wiley, 1983, Chap. 5.

3. The sequential design of experiments and parameter estimation is covered in

BOX, G. E. P., W. G. HUNTER, and J. S. HUNTER, Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building. New York: Wiley, 1978.