7. Collection and Analysis of Rate Data

You can observe a lot just by watching.

—Yogi Berra, New York Yankees

7.1 The Algorithm for Data Analysis

For batch systems, the usual procedure is to collect concentration–time data, which we then use to determine the rate law. Table 7-1 gives a seven-step procedure we will emphasize in analyzing reaction engineering data.

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 pages 494–496).

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

TABLE 7-1 STEPS IN ANALYZING RATE DATA

  1. Postulate a rate law.

    1. Power-law models for homogeneous reactions

      -rA=kCAα, -rA=kCAα CBβ

    2. Langmuir-Hinshelwood models for heterogeneous reactions

      -rA=kPA1+KAPA, -rA=kPAPB(1+KAPA+PB)2

  2. Select reactor type and corresponding mole balance.

    1. If batch reactor (Section 7.2), use mole balance on Reactant A

      -rA=-dCAdt(T7-1.1)

    2. If differential PBR (Section 7.6), use mole balance on Product P (A → P)

      -rA=FPΔW=CPυ0/ΔW(T7-1.2)

  3. Process your data in terms of the measured variable (e.g., NA, CA, or PA). If necessary, rewrite your mole balance in terms of the measured variable (e.g., PA).

  4. Look for simplifications. For example, if one of the reactants is in excess, assume its concentration is constant. If the gas-phase mole fraction of reactant A is small, set. ɛ ≈ 0

  5. For a batch reactor, calculate –rA as a function of concentration CA to determine the reaction order.

    1. Differential analysis (Section 7.4)

      Combine the mole balance (T7-1.1) and power-law model (T7-1.3)

      -rA=kCAα(T7-1.3)

      dCAdt=kCAα(T7-1.4)

      and then take the natural log

      ln(-dCAdt)=ln(-rA)=lnk+αlnCA(T7-1.5)

      (1) Find -dCAdt from CA versus t data by either the

            (a) Graphical differentiation (not often used)

            (b) Finite difference method or

            (c) Polynomial fit

      (2) Either plot [ln(-dCAdt)] versus ln CA to find reaction order α, which is the slope of the line fit to the data

      or

      (3) Use nonlinear regression to find α and k simultaneously.

      (4) The method of initial rates (http://www.umich.edu/~elements/6e/07chap/prof-7-1.html). This method is used for highly reversible reactions and requires multiple experiments at different initial concentrations. Here we need to find (dCAdt)t=0 as a function of CA0

      ln(dCAdt)t=0=ln k + α ln CA0(T7-1.6)

      One can use Equation (T7-1.6) to plot the data as described in (2) above or as in (3) above use regression.

    2. Integral method (Section 7.3)

      For rA=kCAα, the combined mole balance and rate law is

      -dCAdt=kCAα(T7-1.4)

      Guess α and integrate Equation (T7-1.4). Rearrange your equation to obtain the appropriate function of CA, which when plotted as a function of time should be linear. If it is linear, then the guessed value of α is correct and the slope is the specific reaction rate, k. If it is not linear, guess again for α. If you guess α = 0, 1, and 2, and none of these orders fit the data, proceed to nonlinear regression.

    3. Nonlinear regression (Polymath; Section 7.5):

      Integrate Equation (T7-1.4) to obtain

      t=1k[CA0(1α)CA(1α)(1α)] for α1(T7-1.7)

      Whenever possible, I would encourage the use of nonlinear regression.

      Use Polymath regression to find α and k. A Polymath tutorial on regression with screen shots can be found on the CRE Web site at http://umich.edu/~elements/6e/07chap/Example_7-3_Polymath_nonlinear_regression_tutorial.pdf and at http://www.umich.edu/~elements/6e/software/nonlinear_regression_tutorial.pdf.

  6. For differential PBR, calculate rA as a function of CA or PA (Section 7.6).

    1. Calculate rA=υ0CpΔW as a function of reactant concentration, CA or partial pressure PA.

    2. Choose a model (see Chapter 10), for example,

      rA=kPA1+KAPA

    3. Use nonlinear regression to find the best model and model parameters. See example on the CRE Web site Summary Notes for Chapter 10, using data from heterogeneous catalysis.

  7. Analyze your rate law model for “goodness of fit.” Calculate a correlation coefficient.

7.2 Determining the Reaction Order for Each of Two Reactants Using the Method of Excess

Batch reactors are used primarily to determine rate-law parameters for homogeneous reactions. This determination is usually achieved by measuring concentration as a function of time and then using either the integral, differential, or nonlinear regression method of data analysis to determine the reaction order, α, and specific reaction-rate constant, k. If some reaction parameter other than concentration is monitored, such as pressure, the mole balance must be rewritten in terms of the measured variable (e.g., pressure, as shown in the example in Solved Problems on the CRE Web site at http://www.umich.edu/~elements/6e/07chap/pdf/excd5-1.pdf).

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

-rA=kACAα(7-1)

then the differential method may be used.

Assume that the rate law is of the form rA=kACAα.

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

A + B → Products

with the rate law

-rA=kACAαCBβ(7-2)

where α and β are both unknown, the reaction could first be run in an excess of B so that CB remains essentially unchanged during the course of the reaction (i.e., CBCB0) and

-rA=kCAα(7-3)

where

k=kACBβkACB0β

Method of excess

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

-rA=k"CBβ(7-4)

where k=kACAαkACA0α

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

kA=-rACAαCBβ=(dm3/mol)Sα+β1(7-5)

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

One could also carry out many experiments using the methods of initial rates (http://www.umich.edu/~elements/6e/07chap/prof-7-1.html) to find –rA0 as a function of CA0 and CB0 and then use regression to find k, α, and β as shown on the CRE Web site in Professional Reference Shelf R3 (http://www.umich.edu/~elements/6e/07chap/pdf/leastsquares.pdf).

7.3 Integral Method

The integral method is the quickest method to use to determine the rate law if the order turns out to zero, first, or second order. In the integral method, we guess the reaction order, α, in the combined batch reactor mole balance and rate law equation

dCAdt=-kCAα(7-6)

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 α we assumed is correct, then 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. We need to be familiar with the methods of obtaining these linear plots for reactions of zero, first, and second order.

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

For the reaction

A → Products

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

dCAdt=rA

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

dCAdt=k(7-7)

Integrating CA = CA0 with at t = 0, we have

CA=CA0kt(78)

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

-dCAdt=kCA

with the limits CA = CA0 at t = 0 gives

ln CA0CA=kt(79)

First order

Consequently, we see that the slope of a plot of [ln (CA0/CA)] as a function of time is indeed linear with slope k.

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

-dCAdt=kCA2

#important

We need to know these plots for zero-, first- and second-order reactions.

A graph of the zero order reaction is given.

Figure 7-1 Zero-order reaction.

A graph of the first order reaction is given.

Figure 7-2 First-order reaction.

A graph of the second order reaction is given.

Figure 7-3 Second-order reaction.

A graph of the reciprocal reactant concentration as a function of time for a reaction of unknown order.

Figure 7-4 Plot of reciprocal reactant concentration as a function of time for a reaction of unknown order.

Integrating, with CA = CA0 initially (i.e., t = 0), yields

1CA1CA0=kt(7-10)

Second order

We see that for a second-order reaction a plot of (1/CA) 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., CA, ln CA, or 1/CA) 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 zero, first, and second orders also do not fit the data, one should use one of the other methods discussed in Table 7-1, such as the differential method of analysis or nonlinear regression.

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 goodness-of-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 analysis.

Example 7–1 Integral Method of CRE Data Analysis

The liquid-phase reaction

Trityl (A) + Methanol (B) → Products (C)

was carried out in a batch reactor at 25°C in a solution of benzene and pyridine in an excess of methanol (CB0=0.5mo1dm3). (We need to point out that this batch reactor was purchased at the Sunday markets in Riça, Jofostan.) Pyridine reacts with HCl, which then precipitates as pyridine hydro-chloride thereby making the reaction irreversible. The reaction is first order in methanol. The concentration of triphenyl methyl chloride (A) was measured as a function of time and is shown Table E7-1.1.

#BeCareful Watch for leaks!

TABLE E7-1.1 RAW DATA

t (min)

0

50

100

150

200

250

300

CA (mol/dm3)

0.05

0.038

0.0306

0.0256

0.0222

0.0195

0.0174

  1. Use the integral method to confirm that the reaction is second order with regard to triphenyl methyl chloride.

  2. Determine the pseudo rate constant k′ = kCB0.

  3. Assume the reaction is first order in methanol and determine the true specific reaction rate constant, k, for this over all third-order reaction.

Solution

We use the power-law model, Equation (7-2), along with information from the problem statement that the reaction is first order in methanol, (B), that is, β = 1 to obtain

-rA=kCAαCB(E7-1.1)

  1. Excess methanol: The initial concentration of methanol (B) is 10 times that of trityl (A), so even if all A were consumed, 90% of B remains. Consequently, we will take the concentration of B as a constant and combine it with k to form

    -rA=kCAαCB0=kCAα(E7-1.2)

    where k′ is the pseudo rate constant k′ = kCB0 and k is the true rate constant. Substituting α = 2 and combining with the mole balance on a batch reactor, we obtain

    -dCAdt=kCA2(E7-1.3)

    Integrating with CA = CA0 at t = 0

    t =1k[1CA-1CA0](E7-1.4)

    Rearranging

    1CA=1CA0+kt(E7-1.5)

    from Equation (E7-1.5), we see that if the reaction is indeed second order then a plot of (1/CA) versus t should be linear. Using the data in Table E7-1.1, we calculate (1/CA) to construct Table E7-1.2.

    TABLE E7-1.2 PROCESSED DATA

    t (min)

    0

    50

    100

    150

    200

    250

    300

    CA (mol/dm3)

    0.05

    0.038

    0.0306

    0.0256

    0.0222

    0.0195

    0.0174

    1/CA (dm3/mol)

    20

    26.3

    32.7

    39.1

    45

    51.3

    57.5

  2. In a graphical solution, the data in Table E7-1.2 is used to construct a plot of (1/CA) as a function of t, which will yield the pseudo-specific reaction rate k′. This plot is shown in Figure E7-1.1. Again, Excel or Polymath could be used to find k′ from the data in Table E7-1.2.

    A graph shows the reciprocal of C subscript A baseline versus t for a second order reaction.

    Figure E7-1.1 Plot of the reciprocal of CA versus t for a second-order reaction.

    We note in passing that if you used Excel or similar analysis, then the slope would most likely be part of your computer output.

    The slope of the line is the specific reaction rate k′, which using the points at t = 0 and t = 200, k is calculated to be

    k=Δ(1CA)Δt=(44-20)(dm3mol)(200-0)min=0.12dm3molmin(E7-1.6)

    We see from the Excel analysis and plot that the slope of the line is 0.12 dm3/mol · min.

  3. We now use Equation (E7-1.6), along with the initial concentration of methanol, to find the true rate constant, k.

    k=kCB0=0.120.5dm3/mol/minmol/dm3=0.24(dm3mol)2/min

    The rate law is

    rA=[0.24(dm3mol)2/min]CA2CB(E7-1.7)

    We note that the integral method tends to smooth the data.

Analysis: In this example, the reaction orders are known so that the integral method can be used to (1) verify the reaction is second order in trityl and (2) to find the specific pseudo reaction rate k′ = kCB0 for the case of excess methanol (B). Knowing k′ and CB0, we then found the true rate constant k.

7.4 Differential Method of Analysis

The differential method is an excellent technique to use when you want to look for inconsistencies in one’s experimental data. To outline the procedure used in the differential method of analysis, we consider a reaction carried out isothermally in a constant-volume batch reactor and the concentration of A, recorded as a function of time. By combining the mole balance with the rate law given by Equation (7-1), we obtain

-dCAdt=kACAα

Constant-volume batch reactor

After taking the natural logarithm of both sides of the above equation,

In(dCAdt)=In kA+α In CA(7-11)

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

Figure 7-5(a) shows a plot of [–(dCA/dt)] versus [CA] 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, kA, can be found by first choosing a concentration at a point or on the plot, say CAp, and then finding the corresponding value of [–(dCA/dt)p] on the line, as shown in Figure 7-5(b). The concentration point chosen, CAp, to find the derivative at CAp, need not be a data point; it just needs to be on the line. After raising CAp to the power α, we divide it into [–(dCA/dt)p] to determine kA

Plot ln(dCAdt) versus ln CA to find α and kA.

Two graphs of differential method to find the value for alpha and k subscript A baseline is shown.

Figure 7-5 Differential method to determine reaction order.

kA=(dCA/dt)P(CAp)α(7-12)

To obtain the derivative (–dCA/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 dCAdt from concentration–time data

We shall only discuss the graphical and numerical methods.

7.4.1 Graphical Differentiation Method

This method is very old (from slide rule days—“What’s a slide rule, Grandfather?”), when compared with the numerous software packages. So why do we use it? Because with this method, disparities in the data are easily seen. Consequently, it is advantageous to use this technique to analyze the data before planning the next set of experiments. As explained in Appendix A.2, the graphical method involves plotting (–ΔCAt) as a function of t and then using equal-area differentiation to obtain (–dCA/dt). An illustrative example is also given in Appendix A.2.

A graph representation for differentiation method is shown.

#UsedBefore Laptops

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

See Appendix

7.4.2 Numerical Method

Numerical differentiation formulas can be used when the data points in the independent variable are equally spaced, such as t1t0 = t2t1 = Δt. (See Table 7-2.)

TABLE 7-2 RAW DATA

Time (min)

t0

t1

t2

t3

t4

t5

Concentration (mol/dm3)

CA0

CA1

CA2

CA3

CA4

CA5

Raw Data

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

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

TABLE 7-3 DIFFERENTIATION FORMULAS

Initial point:

(dCAdt)t0=3CA0+4CA1CA22Δt

(7-13)

Interior points:

(dCAdt)ti=12Δt[(CA(i+1)-CA(i-1)]e.g., (dCAdt)t3=12Δt[CA4-CA2]

(7-14)

Last point:

(dCAdt)t5=12Δt[CA3-4CA4+3CA5]

(7-15)

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 5B in Example 7-2).

7.4.3 Finding the Rate-Law Parameters

Now, using either the graphical method, differentiation formulas, or the polynomial derivative, Table 7-4 can be set up.

TABLE 7-4 PROCESSED DATA

Time

t0

t1

t2

t3

Concentration

CA0

CA1

CA2

CA3

Derivative

(-dCAdt)0

(-dCAdt)1

(-dCAdt)2

(-dCAdt)3

Processed data

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

ln(-dCAdt)=ln kA+α ln CA(7-16)

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

Example 7–2 Determining the Rate Law

The reaction of triphenyl methyl chloride (trityl) (A) and methanol (B) discussed in Example 7-1 is now analyzed using the differential method.

(C6H5)3CCl+CH3OH(C6H5)3COCH3+HClA+BC     +D

The concentration–time data in Table E7-2.1 was obtained in a batch reactor.

TABLE E7-2.1 RAW DATA

Time (min)

0

50

100

150

200

250

300

Concentration of A (mol/dm3) × 103 (At t = 0, CA = 0.05 M)

50

38

30.6

25.6

22.2

19.5

17.4

The initial concentration of methanol was 0.5 mol/dm3.

Part (1) Determine the reaction order with respect to triphenyl methyl chloride.

Part (2) In a separate set of experiments, the reaction order wrt methanol was found to be first order. Determine the specific reaction-rate constant.

Solution

Part (1) Find the reaction order with respect to trityl.

Step 1 Postulate a rate law.

-rA=kCAαCBβ(E7-2.1)

Step 2 Process your data in terms of the measured variable, which in this case is CA.

Step 3 Look for simplifications. Because the concentration of methanol is 10 times the initial concentration of triphenyl methyl chloride, its concentration is essentially constant

CBCB0(E7-2.2)

Substituting for CB in Equation (E7-2.1)

rA=kCB0βkCAα
rA==kCAα(E7-2.3)

Step 4 Apply the CRE algorithm.

Mole Balance:

dNAdt=rAV(E7-2.4)

Rate Law:

-rA=kCAα(E7-2.3)

Stoichiometry: Liquid V = V0

CA=NAV0

An illustration depicts footsteps, and represents the phrase "following the algorithm".

Combine: Mole balance, rate law, and stoichiometry

-dCAdt=kCAα(E7-2.5)

Evaluate: Taking the natural log of both sides of Equation (E7-2.5)

ln[-dCAdt]=ln k+αln CA(E7-2.6)

The slope of a plot of ln[-dCAdt] versus ln CA will yield the reaction

order α with respect to triphenyl methyl chloride (A).

Step 5 Find [-dCAdt] as a function of CA from concentration–time data.

Step 5A Graphical Method. #ForOldTimesSake. To see how this problem was solved in the 1960s, go to the Chapter 7 Expanded Material on the Web site.

Step 5B Finite Difference Method. As mentioned this technique helps identify disparities in the data, especially between successive measurements. We now show how to calculate (dCA/dt) using the finite difference formulas (i.e., Equations (7-13)–(7-15)).

t=0 (dCAdt)t=0=3CA0+4CA1CA22Δt=[3(50)+4(38)30.6]×103100=2.86×104 mol/dm3mindCAdt×104=2.86 mol/dm3mint=50 (dCAdt)1=CA2CA02Δt=(30.650)×103100=1.94×104 mol/dm3mint=100 (dCAdt)2=CA3CA12Δt=(25.638)×103100=1.24×104 mol/dm3mint=150 (dCAdt)3=CA4CA22Δt=(22.230.6)×103100=0.84×104 mol/dm3mint=200 (dCAdt)4=CA5CA32Δt=(19.525.6)×103100=0.61×104 mol/dm3mint=250 (dCAdt)5=(CA6CA42Δt=(17.422.2)×103100)=0.48×104 mol/dm3mint=300 (dCAdt)6=CA44CA5+3CA62Δt=[22.24(19.5)+3(17.4)]×103100=0.36×104mol/dm3min

These are the type of calculations you would do if in the event of a power failure or malfunctioning Excel regression spreadsheet.

I know these are tedious calculations, but someone’s gotta know how to do it.

An illustration shows a young man asleep on a table.

We now enter the above values for (–dCA/dt) in Table E7-2.2 and use it to plot columns 2 and (-dCAdt×10,000) as a function of column 4 (CA × 1000) on log-log paper, as shown in Figure E7-2.1. We could also substitute the parameter values in Table E7-2.2 into Excel to find α and k′. Note that most of the points for both methods fall virtually on top of one another. This table is, in a way, redundant because it is not necessary to always find (–dCA/dt) by both techniques, graphical and finite difference.

TABLE E7-2.2 SUMMARY OF PROCESSED DATA

Graphical

Finite Difference

t (min)

-dCAdt×10000
(mol/dm3 · min)

-dCAdt×10000
(mol/dm3 · min)

CA × 1000
(mol/dm3)

0

3.0

2.86

50

50

1.86

1.94

38

100

1.20

1.24

30.6

150

0.80

0.84

25.6

200

0.68

0.61

22.2

250

0.54

0.48

19.5

300

0.42

0.36

17.4

A graph is given with a slope and plots of difference and graphical marked on it to determine alpha and k.

Figure E7-2.1 Excel plot to determine α and k.

#HowToImpress YourEmployer

From Figure E7-2.1, we found the slope to be 1.99, so that the reaction is said to be second order (α = 2.0) with respect to triphenyl methyl chloride. Recall when the cycle lengths on the x-axis and the y-axis are the same on log-log paper, then we simply measure the slope with a ruler. However, with all the software on your computer it would be more impressive to your professor or employer if you typed the data into Excel or Polymath, or some other spreadsheet software, and then printed the results.

To evaluate k′, we can evaluate the derivative in Figure E7-2.1 at CAp = 20 × 10–3 mol/dm3, which is

(-dCAdt)p = 0.5×10-4mol/dm3·min(E7-2.7)

then

k=(dCAdt)pCAp2(E7-2.8)=0.5×104mol/dm3min(20×103mol/dm3)2=0.1250dm3/molmin

As will be shown in Section 7-5, we could also use nonlinear regression on Equation (E7-1.5) to find k

k=0.122 dm3/molmin(E7-2.9)

The Excel graph shown in Figure E7-2.1 gives α = 1.99 and k′ = 0.13 dm3/mol · min. We now set α = 2 and regress again to find k′ = 0.122 dm3/mol · min.

Summary notes icon.

ODE Regression. There are techniques and software becoming available whereby an ODE solver can be combined with a regression program to solve differential equations, such as

dCAdt=kACAα(E7-2.5)

to find kA and α from concentration–time data.

Part (2) The reaction was said to be first order with respect to methanol, β = 1.

k=CB0βk=CB0k(E7-2.10)

Assuming CB0 is constant at 0.5 mol/dm3 and solving for k yields2

k0=0kCB00=00.122dm3molmin0.5moldm3k0=00.244(dm3/mol)2/min

2 M. Hoepfner and D. K. Roper, “Describing temperature increases in plasmon-resonant nanoparticle systems,” J. Therm. Anal. Calorim., 98(1), 197–202 (2009).

The rate law is

rA=[0.244(dm3/mol)2/min]CA2CB(E7-2.11)

Analysis: In this example, the differential method of data analysis was used to find the reaction order with respect to trityl (α = 1.99) and the pseudo rate constant (k′ = 0.125 (dm3/mol)/min). However, as we mentioned earlier it would probably be more impressive to your employer or professor if you typed the data into your computer and used a software package to find k. We discuss one such software package in Section 7.5. The reaction order was rounded up to α = 2 and the data was regressed again to obtain k′ = 0.122 (dm3/mol)/min, again knowing k′ and CB0, and the true rate constant is k = 0.244 (dm3/mol)2/min. We used two techniques to differentiate the concentration–time data to find (dCAdt): the graphical technique and the finite difference technique.

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

7.5 Nonlinear Regression

In nonlinear regression analysis, we search for those parameter values that minimize the sum of the squares of the differences between the measured values and the calculated values for all the data points. Not only can nonlinear regression find the best estimates of parameter values, it can also be used to discriminate between different rate-law models, such as the Langmuir-Hinshelwood models discussed in Chapter 10. Many software programs are available to find these parameter values so that all one has to do is enter the data. The Polymath software will be used to illustrate this technique. In order to carry out the search efficiently, in some cases one has to enter initial estimates of the parameter values close to the actual values. These estimates can be obtained using the linear-least-squares technique discussed on the CRE Web site Professional Reference Shelf R7.3 (http://www.umich.edu/~elements/6e/07chap/pdf/leastsquares.pdf).

Compare the calculated parameter value with the measured value.

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, Cic, obtained by solving an integrated form of the combined mole balance and rate law. We then compare the measured concentration at that point, Cim, with the calculated value, Cic, for the parameter values chosen. We make this comparison by calculating the sum of the squares of the differences at each point Σ(CimCic)2. We then continue to choose new parameter values and search for those values of the rate-law parameters that will minimize the sum of the squared differences of the measured concentrations, Cim, and the calculated concentrations values, Cic. That is, we want to find the rate-law parameters for which the sum of the squares of the difference at all data points, that is, ∑(CimCic)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

σ2 = s2N-K0=0i=1N(Cim-Cic)2N-K0000000(7-17)

where

s2 =0i=1i=N(Cim-Cic)2

N

=

number of runs

K

=

number of parameters to be determined

Cim

=

measured concentration rate for run i

Cic

=

calculated concentration rate for run i

One notes that if we minimize S2 in Equation (7-17), we minimize σ2.

To illustrate this technique, let’s consider the reaction

AProduct

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

-rA=kCAα

The reaction rate will be measured at a number of different concentrations. We now choose (guess) values of α and k, and calculate the concentration (Cic) at each point, i, that the concentration was measured experimentally. We then subtract the calculated value (Cic) from the measured value (Cim), square the result, and find the 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 σmin2.3 Figure 7-6 shows a hypothetical plot of the sum of the squares as a function of the parameters α and k:

σ2=f(k,α)00000000(7-18)

3 (a) University of Michigan colleagues 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: Pearson, 1979. (c) D. Miller and M. Frenklach, Int. J. Chem. Kinet., 15, 677 (1983).

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.

A 3-dimensional graph depicts the minimum sum of squares.

Figure 7-6 Minimum sum of squares.

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 computerized searching procedure begins by guessing parameter values and then calculating (CimCic) 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 the next set of different parameter values and determine the corresponding σ2 as shown as a contour plot in Figure 7-7. Starting at α = 1 and k = 1, the trajectory, shown by the arrows, is continually adjusted by choosing α and k. We see in Figure 7-7 that the trajectory leads us to a minimum value of σ2 = 0.045 (mol/dm3)2 at which point α = 2 and k = 5 (dm3/mol).

An illustration shows a single-story bookshelf with few books placed on it.
The trajectory to find the minimum of sigma squared is graphed.

Figure 7-7 Trajectory to find the best values of k and α.

If the equations are highly nonlinear, your initial guesses of = 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 (1) enter the experimental values into the computer, (2) specify the model, (3) enter the initial guesses of the parameters, then (4) push the “compute” button, and (5) view the best estimates of the parameter values along with 95% confidence limits. 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 as demonstrated in Chapter 10, Example 10-3. If all the model parameters are smaller than the confidence limits, that model should be eliminated and a new model equation chosen. The software is run again to analyze the new model.

7.5.1 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

dCAdt0=0-kCAα000000000(7-19)

We now integrate Equation (7-6) for the case α ≠ 1 to give

CA01α-CA1-α = (1-α)kt

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

CA=[CA01α(1α)kt]1/(1α)(7-20)

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

s20=0Σi=1N(CAim-CAic)20=0Σi=1N[CAim-[CA01α-(1-α)kti]1/(1-α)]200000(7-21)

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

If Polymath is used, one should use the absolute value, abs, for the term in brackets in Equation (7-21), that is,

s20=0Σi=1n[CAim-{abs[CA01α-(1-α)kti]}1/(1-α)]2000000(7-22)

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

tc=CA01αCA1αk(1α)(7-23)

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

s2=Σi=1N(timtic)2=Σi=1N[timCA01αCAi1αk(1α)]2(7-24)

Finally, a discussion of weighted least squares as applied to a first-order reaction is provided in the Professional Reference Shelf R7.4 on the CRE Web site.

An illustration shows a single-story bookshelf with few books placed on it.

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

See tutorial at http://www.umich.edu/~elements/6e/07chap/live.html and http://www.umich.edu/~elements/6e/07chap/Example_7-3_Polymath_nonlinear_regression_tutorial.pdf.

We shall use the reaction and data in Examples E7-1 and E7-2 to illustrate how to use regression to find α and k′.

(C6H5)3CCl+CH3OH(C6H5)3COCH3+HClA+BC         +D

The Polymath regression program is included on the CRE Web site. Recalling Equation (E7-2.5)

-dCAdt=kCAα0000000000(E7-2.3)

and integrating with the initial condition when t = 0 and CA = CA0 for α ≠ 1.0

t=1kCA0(1α) CA(1α)(1α)(E7-3.1)

Given or assuming k′ and α, Equation (E7-2.5) can be solved (cf. Equation E7-3.1) to calculate the time t to reach a concentration CA, or, alternatively we could calculate the concentration CA at time t. We can proceed two ways from this point, both of which will give the same result. We can search for the combination α and k that minimizes [σ2 = Σ(timtic)2], or we could solve Equation (E7-3.1) for CA and find α and k that minimize [σ2 = Σ(CAimCAic)2].

We shall choose the former. So, substituting for the initial concentration CA0 = 0.05 mol/dm3 into Equation (E7-3.1)

t=1k(0.05)(1α)CA(1α)(1α)(E7-3.2)

Tutorial with screen shots.

A brief tutorial on how to input data in Polymath on the CRE Web site (http://umich.edu/~elements/6e/07chap/Example_7-3_Polymath_nonlinear_regression_tutorial.pdf) shows screen shots of how to enter the raw data in Table E7-2.1 and how to carry out a nonlinear regression on Equation (E7-3.2).

For CA0 = 0.05 mol/dm3, that is, Equation (E7-3.1) becomes

tc=1k(0.05)(1-α)-CA(1-α)(1-α)(E7-3.3)

We want to minimize s2 to give α and k′.

s2=Σi=1N(tim-tic)2=Σi=1N[tim-0.05(1-α)-CAic(1-α)k(1-α)]2(7-25)

We shall carry out a first regression to find approximate values of α and k. Next, we use a common procedure, which is to round α off to an integer and then carry out a second regression to find the best value of k after α has been set equal to an integer.

TABLE E7-3.1 RESULTS OF 1ST REGRESSION

Images

TABLE E7-3.2 RESULTS OF 2ND REGRESSION WITH α = 2

Images

The results shown are

The results of alpha and k prime are shown.

The first regression gives α = 2.04, as shown in Table E7-3.1. We shall round off α to make the reaction second order, (i.e., α = 2.00). Now having fixed α at 2.0, we must do another regression (cf. Table E7-3.2) on k′ because the k′ given in Table E.7-3.1 is for α = 2.04. We now regress the equation

t=1k[1CA-1CA0]

The second regression gives k′ = 0.125 dm3/mol · min. We now calculate k

k=kCA0=0.125(dm3/mol)0.5(dm3/mol)min=0.25(dm3mol)2/min

Analysis: In this example, we showed how to use nonlinear regression to find k′ and α. The first regression gave α = 2.04, which we rounded to 2.00 and then regressed again for the best value of k′ for α = 2.0, which was k′ = 0.125 (dm3/mol)/min giving a value of the true specific reaction rate of k = 0.25 (dm3/mol)2/min. We note that the reaction order is the same as that in Examples 7-1 and 7-2; however, the value of k is about 8% larger. The r2 and other statistics can be found in Polymath’s output.

7.5.2 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 whether the error is randomly distributed or whether 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, Example 10-3.

7.6 Reaction-Rate Data from Differential Reactors

Data acquisition using the method of initial rates and a differential reactor is similar in that the rate of reaction is determined for a specified number of predetermined initial or entering reactant concentrations. A differential reactor (PBR) is normally used to determine the rate of reaction as a function of either concentration or partial pressure. It consists of a tube containing a very small amount of catalyst, usually arranged in the form of a thin wafer or disk. A typical arrangement is shown schematically in Figure 7-8. The criterion for a reactor being differential is that the conversion of the reactants in the bed is extremely small, as is the change in temperature and reactant concentration through the bed. As a result, the reactant concentration through the reactor is essentially constant and approximately equal to the inlet concentration. That is, the reactor is considered to be gradientless,4 and the reaction rate is considered spatially uniform within the bed.

Image of a differential reactor.

Figure 7-8 Differential reactor.

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

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, as discussed in Chapter 10, 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

The differential reactor is operated at steady state.

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, rA, 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

In the differential catalyst bed, the entering volumetric flow rate and concentration are marked as F subscript A0 and C subscript A0, whereas the exiting volumetric flow rate and concentration are marked as F subscript Ae, F subscript p, and C subscript p. The length of the catalyst bed at the top is marked as delta L, and the length at the bottom is marked as delta w.

Figure 7-9 Differential catalyst bed.

[Flowratein][Flowrateout]+[Rate ofgeneration]=[Rate ofaccumulation][FA0][FAe]+[(Rate of reactionMass of catalyst)(Mass of catalyst)]=0FA0FAe+(rA)(ΔW)=0

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

rA=FA0-FAeΔW(7-26)

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

rAυ0CA0υCAeΔW(727)

Differential reactor design equation

or in terms of conversion or product flow rate FP

rA=FA0XΔW=FpΔW(7-28)

The term FA0X gives the rate of formation of product, Fp, when the stoichio-metric 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

-rA=v0(CA0-CAe)ΔW=v0CPΔW(7-29)

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

By using very little catalyst and large volumetric flow rates, the concentration difference, (CA0CAe), 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, CAb

-rA=-rA(CAb)(7-30)

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

CAb=CA0+CAe2(7-31)

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

CAbCA0

so -rA is a function of CA0

-rA=-rA(CA0)(7-32)

Caution.

As with the method of initial rates (see the CRE Web site, http://www.umich.edu/~elements/6e/software/nonlinear_regression_tutorial.pdf), we find –rA0 as a function of CA0 and then use various numerical and graphical techniques 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 Chapter 10, and also in Chapters 14 and 15.

Example 7–4 Using a Differential Reactor to Obtain Catalytic Rate Data

The formation of methane from carbon monoxide and hydrogen using a nickel catalyst was studied by Pursley.5 The reaction

3H2 + CO → CH4 + H2O

5 J. A. Pursley, “An investigation of the reaction between carbon monoxide and hydrogen on a nickel catalyst above one atmosphere,” Ph.D. thesis, University of Michigan.

was carried out at 500°F in a differential reactor where the effluent concentration of methane was measured. The raw data is shown in Table E7-4.1.

TABLE E7-4.1 RAW DATA

Run

PCO (atm)

PH2 (atm)

CCH4 (mol/dm3)

1

1

1.0

1.73 × 10–4

2

1.8

1.0

4.40 × 10–4

3

4.08

1.0

10.0 × 10–4

4

1.0

0.1

1.65 × 10–4

5

1.0

0.5

2.47 × 10–4

6

1.0

4.0

1.75 × 10–4

PH2 is constant in Runs 1, 2, 3.

PCO is constant in Runs 4, 5, 6.

The exit volumetric flow rate from a differential packed bed containing 10 g of catalyst was maintained at S0 = 300 dm3/min for each run. The partial pressures of H2 and CO were determined at the entrance to the reactor, and the methane concentration was measured at the reactor exit.

Determine the rate law and rate-law parameters. To better understand how we will determine the rate law and its parameters, we will break the solution into three parts, (a), (b), and (c), as shown below.

  1. Relate the rate of reaction to the exit methane concentration. The reaction-rate law is assumed to be the product of a function of the partial pressure of CO, f(CO), and a function of the partial pressure of H2, g(H2), that is,

    rCH4=f(CO)g(H2)(E7-4.1)

  2. Determine the rate-law dependence on carbon monoxide, using the data generated in part (a). Assume that the functional dependence of rCH4 on PCO is of the form

    rCH4PCOα(E7-4.2)

  3. Generate a table of the reaction rate as a function of partial pressures of carbon monoxide and hydrogen, and then determine the rate-law dependence on H2.

Solution

(a) Calculate the Rates of Reaction. In this example the product composition, rather than the reactant concentration, is being monitored. The term (rCO) can be written in terms of the flow rate of methane from the reaction

-rCO=rCH4=FCH4ΔW

Substituting for FCH4 in terms of the volumetric flow rate and the concentration of methane gives

rCH4=υ0CCH4ΔW(E7-4.3)

Since υ0, CCH4, and ΔW are known for each run, we can calculate the rate of reaction.

For run 1

rCH4=(30dm3min)1.73×10-410gcatmol/dm3=5.2×10-3mol CH4gcat×min

The rate for runs 2 through 6 can be calculated in a similar manner (Table E7-4.2).

TABLE E7-4.2 RAW AND CALCULATED DATA

Run

PCO (atm)

PH2 (atm)

CCH4 (mol/dm3)

rCH4(mol CH4gcat-min)

1

1.0

1.0

1.73 ×10–4

5.2 × 10–3

2

1.8

1.0

4.40 × 10–4

13.2 × 10–3

3

4.08

1.0

10.0 ×10–4

30.0 × 10–3

4

1.0

0.1

1.65 × 10–4

4.95 × 10–3

5

1.0

0.5

2.47 × 10–4

7.42 × 10–3

6

1.0

4.0

1.75 × 10–4

5.25 × 10–3

(b) Determining the Rate-Law Dependence in CO. For constant hydrogen concentration (runs 1, 2, and 3), the rate law

rCH4=kg(PH2)kpCOα

can be written as

rCH4=kPCOα(E7-4.4)

Taking the natural log of Equation (E7-4.4) gives us

ln(rCH4)=lnk+αlnPCO

We now plot ln (rCH4) versus ln PCO using runs 1, 2, and 3, for which H2 the concentration is constant, in Figure E7-4.1. We see from the Excel plot that α = 1.22.

The plots of r prime subscript CH4 versus P subscript CO are depicted on a graph.

Figure E7-4.1 Reaction rate as a function of concentration.

Had we not been so pressed for time and included more points, we would have found that the reaction is essentially first order with α = 1, that is

-rCO=kPCO(E7-4.5)

If we had used more data (not given here) we would have found α = 1.

From the first three data points in Table E7-4.2 where the partial pressure of H2 is constant, we see the rate is linear in partial pressure of CO

rCH4=kPCOg(H2)

Now let’s look at the hydrogen dependence.

(c) Determining the Rate-Law Dependence on H2. It will not be intuitively obvious how to do this analysis until you have studied Chapter 10 where we have rate laws with concentrations or partial pressures in both the numerator and denominator of the rate law. From Table E7-4.2 it appears that the dependence of rCH4 on PH2 cannot be represented by a power law. Comparing run 4 with run 5 and then run 5 with run 6, we see that the reaction rate first increases with increasing partial pressure of hydrogen, and subsequently decreases with increasing PH2. That is, there appears to be a concentration of hydrogen at which the rate is maximum. One set of rate laws that is consistent with these observations is:

  1. At low H2 concentrations where rCH4 increases as PH2 increases, the rate law may be of the form

    rCH4PH2β1(E7-4.6)

  2. At high H2 concentrations where rCH4 decreases as PH2 increases, the rate law may be of the form

    rCH41PH2β2(E7-4.7)

Sketch of rCH4 as a function of PH2

A graph of r prime subscript CH4 versus P subscript upper H2 shows a concave down curve. Initially, the curve increases as the value of P subscript H2 increases, but in the end, it decreases as P subscript H2 increases.

We would like to find one rate law that is consistent with reaction-rate data at both high and low hydrogen concentrations. After we have studied heterogeneous reactions in Chapter 10, we would recognize that Equations (E7-4.6) and (E7-4.7) can be combined into the form

rCH4PH2β11+bPH2β2(E7-4.8)

We will see in Chapter 10 that this combination and similar rate laws that have reactant concentrations (or partial pressures) in the numerator and denominator are common in heterogeneous catalysis.

Let’s see if the resulting rate law (E7-4.8) is qualitatively consistent with the rate observed.

  1. For condition 1: At low PH2, [b(PH2)β2 ≪ 1] and Equation (E7-4.8) reduces to

    rCH4PH2β1(E7-4.9)

    Equation (E7-4.9) is consistent with the trend in comparing runs 4 and 5.

  2. For condition 2: At high PH2, [b(PH2)β2 ≫ 1] and Equation (E7-4.8) reduces to

    rCH4(PH2)β1(PH2)β21(PH2)β2-β1(E7-4.10)

    where β2 > β1. Equation (E7-4.10) is consistent with the trends in comparing runs 5 and 6.

Combining Equations (E7-4.8) and (E7-4.5)

rCH4=aPCOPH2β11+bPH2β2(E7-4.11)

Typical form of the rate law for heterogeneous catalysis

We now use the Polymath regression program to find the parameter values a, b, β1, and β2. The results are shown in Table E7-4.3.

The corresponding rate law is

rCH4=0.025PCOPH20.611+2.49PH2(E7-4.12)

TABLE E7-4.3 FIRST REGRESSION

Images

Summary notes icon.

Polymath and Excel regression tutorials are given in the Chapter 7 Summary Notes on the CRE Web site.

We could use the rate law given by Equation (E7-4.12) as is, but there are only six data points, and we should be concerned about extrapolating the rate law over a wider range of partial pressures. We could take more data, and/or we could carry out a theoretical analysis of the type discussed in Chapter 10 for heterogeneous reactions. If we assume hydrogen undergoes dissociative adsorption on the catalyst surface (cf. Figure 10-11), we would expect a dependence on the partial pressure of hydrogen to be to the ½ power. Because 0.61 is close to 0.5, we are going to regress the data again, setting β1 = ½ and β2 = 1.0. The results are shown in Table E7-4.4.

TABLE E7-4.4 SECOND REGRESSION

Images

After studying the heterogeneous rate laws in Chapter 10, we will become more adept at postulating and analyzing this type of data for heterogeneous reactions.

Note: A complete tutorial of how to enter this example into Polymath can be found on the Web site at http://www.umich.edu/~elements/6e/software/nonlinear_regression_tutorial.pdf.

The rate law is now

rCH4=0.018PCOPH21/21+1.49 PH2

where rCH4 is in (mol/g-cat · s) and the partial pressures are in (atm).

We could also have set β1 = ½ and β2 = 1.0 and rearranged Equation (E7-4.11) in the form

PCOPH21/2rCH4=1a+baPH2(E7-4.13)

Linearizing the rate law to determine the rate-law parameters

A plot of PCOPH21/2/rCH4 as a function of PH2 should be a straight line with an intercept of 1/a and a slope of b/a. From the plot in Figure E7-4.2, we see that the rate law is indeed consistent with the rate-law data.

A graph depicts the linearized plot of two variables.

Figure E7-4.2 Linearized plot of data.

Analysis: The reaction-rate data in this example were obtained at steady state, and as a result, neither the integral method nor differential method of analysis can be used. One of the purposes of this example is to show how to reason out the form of the rate law and to then use regression to determine the rate-law parameters. Once the parameters were obtained, we showed how to linearize the rate-law (e.g., Equation (E7-4.13)) to generate a single plot of all the data, Figure (E7-4.2).

7.7 Experimental Planning

Four to six weeks in the lab can save you an hour in the library.

—G. C. Quarderer, Retired from Dow Chemical Co.

An illustration shows a single-story bookshelf with few books placed on it.

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 and in the Professional Reference Shelf (http://www.umich.edu/~elements/6e/07chap/prof-7-5.html) 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.

7.8 And Now… A Word from Our Sponsor–Safety 7 (AWFOS–S7 Laboratory Safety)

This chapter focused on obtaining and analyzing experimental data that is collected safely in both academia and industrial laboratories. To help promote safer laboratory practices, the Web site (http://umich.edu/~safeche/index.html), Process Safety Across the Chemical Engineering Curriculum, provides a number of videos on safety. These videos are classified into two types:

  1. Video Snippets that are 1–5 minutes instructional videos

  2. Case histories of academic accidents that resulted in fatalities or serious injuries and how they may have been prevented.

To access the Snippets Videos, go to the Safety Web site home page and click on Laboratory and Personal Safety (http://umich.edu/~safeche/lab_safety.html), which is under Course Specific Safety Modules. Upon clicking on Laboratory and Personal Safety, you will pull up the menu as shown here:

Safety Snippets Videos

Food in the Lab

Photograph of a person with a sketch of an eye painted on his forehead.

The Sloppy Lab

Photograph of a person who has full of bruises on his face.

Safety Gear

Photograph of a person who has full of burn injuries on his face.

Evacuate

Photograph of a skeleton in a hoodie holding an ax.

Videos of Case Histories

After the Rainbow

Sketch of a person lighting up some lamps.

Experimenting with Danger

An image shows flames engulfing the body of a woman as she tries to move away from the danger.

Guidelines

Active Attacker (https://www.dpss.umich.edu/content/prevention-education/safety-tips/active-attacker/)

UM Undergraduate Lab Safety Guidelines (PDF) (http://umich.edu/~safeche/assets/pdf/LabsSafetyGuidelines.pdf)

Introduction to Laboratory Safety for Graduate Students (PDF) (https://pubs.acs.org/doi/pdfplus/10.1021/acs.jchemed.8b00774)

ACS Safety Guidelines (https://www.acs.org/content/dam/acsorg/about/governance/committees/chemicalsafety/publications/acs-safety-guidelines-academic.pdf?logActivity=true)

To enhance your laboratory safety prowess, view each of the snippet videos and then write a one-sentence takeaway lesson for each video.

Summary

  1. Integral method:

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

    2. 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.

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

  2. Differential method for constant-volume systems:

    -dCAdt=kCAα(S7-1)

    1. Plot –ΔCAt as a function of t.

    2. Determine –dCA/dt from this plot.

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

      ln(-dCAdt)=lnk+αln CA(S7-2)

    Plot ln(–dCA/dt) versus lnCA. The slope will be the reaction order α. We could use finite-difference formulas or software packages to evaluate (–dCA/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.

    σ2=i=1N-Pi(measured)-Pi(calculated)]2N-K(S7-3)

    s2=i=1N(tim-tic)2=i=1N[tim-CA01α-CAi1-αk(1-α)]2(S7-4)

    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

    -rA=FA0XΔW=FPΔW=υ0(CA0-CAe)ΔW=CPυ0ΔW(S7-5)

    In calculating the reaction order, α

    -rA=kCAα

    the concentration of A is evaluated either at the entrance conditions or at a mean value between CA0 and CAe. However, power-law models such as

    -rA=kCAαCBβ(S7-6)

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

    -rA=kPAPB1+KAPA+KBPB

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

CRE WEB SITE MATERIALS

(http://umich.edu/~elements/6e/07chap/obj.html#/)

A figure represents the useful links and evaluation of the CRE website.

Interactive Computer Games (http://umich.edu/~elements/6e/icm/index.html)

The window of the Interactive Computer Games (ICGs) shows the problems related to Wetland Analysis.

Ecology (http://umich.edu/~elements/6e/icm/ecology.html)

QUESTIONS, SIMULATIONS, AND PROBLEMS

The subscript to each of the problem numbers indicates the level of difficulty: A, least difficult; D, most difficult.

A = • B = ▪ C = ♦ D = ♦♦

An illustration shows a pile of books.

Questions

Q7-1A QBR (Question Before Reading). If you need to determine the rate law, what methods would you use to collect the data and how would you analyze it?

Q7-2A i>clicker. Go to the Web site (http://www.umich.edu/~elements/6e/07chap/iclicker_ch7_q1.html) and view at least five i>clicker questions. Choose one that could be used as is, or a variation thereof, to be included on the next exam. You also could consider the opposite case: explaining why the question should not be on the next exam. In either case, explain your reasoning.

Q7-3A

  1. Listen to the audios Headphone icon. on the CRE Web site. Pick one and say why it could be eliminated.

  2. Create an original problem based on Chapter 7 material.

  3. 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.

  4. 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.)

Creative thinking icon.

Q7-4B 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 ±?)

Q7-5A Example 7-3. Explain why the regression was carried out twice to find k′ and k.

Q7-6A AWFOS–S. Which of the four Safety Snippets Videos most impressed you with getting across the safety aspect that the video was trying to highlight? Which of the four 1- to 2-minute Snippets Videos was the most humorous while at the same time getting its point across?

Q7-7A Go to the LearnChemE screencasts link for Chapter 7

(http://www.umich.edu/~elements/6e/07chap/learn-cheme-videos.html).

  1. View the two screencasts on nonlinear regression and describe or list any differences between the two videos. Would you recommend these screencasts be assigned to next year’s class?

  2. In the nonlinear regression screencast, what type of data do you need to regress for the reaction order?

Computer Simulations and Experiments

P7-1A

  1. LEP Example Problem 7-3A. The reaction A + B → C occurs in the liquid phase. The proposed empirical rate law is rA=A exp(E/RT)CAαCBβ. Look at the values of s2 as a function of α. At what value of α is s2 minimized?

    1. Vary β keeping A fixed and describe what you find.

    2. Vary A keeping β fixed and describe what you find.

    3. Suggest a rate law that minimizes s2.

  2. Example 7-4: Nonlinear Regression

    Follow the Polymath tutorial to regress the data given in this example to fit the rate law

    rCH4=kPCOαPH2β

    What is the difference in the correlation and sums of squares compared with those given in Example 7-4?

Interactive Computer Games

P7-2A

  1. Download the Interactive Computer Game (ICG) from the CRE Web site (http://www.umich.edu/~elements/6e/icm/ecology.html). 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 # ________________.

  2. 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.

Problems

P7-3A Skiing Hand Warmers. Once a year, Professor Dr. Sven Köttlov loves to go skiing with his students in the mountains of Jofostan. Before going, he retires to his basement in his small but adequate university housing to make hand warmers for everyone.

B. Opegdra, A.M.R.P. Journal of Chemical Elevation on Time, D01:10.1021/acs.jchemed.8600012. J. Gordon and K. Chancy, J. Chem. Educ., 82(7), 1065 (July 2005).

Here he mixes iron (steel wool), sodium chloride, and a few other proprietary ingredients, then seals them in an airtight Ziploc® compartment. When the warmers are needed, the bag is broken and the iron is exposed to air generating heat by the exothermic reaction

4Fe(s)+3O2(g)  2Fe2O3

To determine the reaction kinetics, a steel wool sample was weighed, cleaned, and put in a sealed container where the percent of oxygen was measured as a function of time and the following data was recorded:

t(min)

0

3

5

8

15

20

25

[% O2]

21

15

12

10

5

3

2

For a specified steel wool weight and surface area,the proposed relationship for the percent oxygen in the bag as a function of time is

[% O2]=3 exp(~kt)(P7-3.1)

  1. Assuming the surface area of steel wool is approximately constant and the bag is sealed, use the CRE algorithm to derive Equation (P7-3.1).

  2. Compare the rate law and data. By inspection, without carrying out any data analysis, can you determine immediately that the rate law is correct or not? Explain.

  3. Use the data to determine the rate law and rate-law parameters.

  4. The data shows the reaction is virtually complete after 25 minutes, while the students will be skiing for 5–7 hours. Professor Köttlov had finger nail polish removed from his nails with acetic acid immediately before performing the experiment. It is known that acid accelerates the reaction. Can you take this information into account? If so, how would you do it?

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

An illustration depicts the deoxygenation of hemoglobin in the blood. The hemoglobin molecules are shown with oxygen and carbon dioxide, where the exchange of oxygen and carbon dioxide takes palaces within the environment.

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).

HbO2k1k1Hb+O2

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 CA0 = 2.33 × 10–6 mol/cm3.

Electrode position

1

2

3

4

5

6

7

Percent decomposition of HbO2

0.00

1.93

3.82

5.68

7.48

9.25

11.00

  1. 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.

  2. Repeat using regression.

P7-5A The irreversible isomerization

AB

was carried out in a constant-volume batch reactor and the following concentration–time data were obtained:

t (min)

0

5

8

10

12

15

17.5

20

CA (mol/dm3)

4.0

2.25

1.45

1.0

0.65

0.25

0.06

0.008

Determine the reaction order = and the specific reaction rate k in appropriate units.

P7-6B OEQ (Old Exam Question). 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.

Run

1

2

3

4

5

τ (min)

15

38

100

300

1200

CA (mol/dm3)

1.5

1.25

1.0

0.75

0.5

  1. Determine the reaction order and specific reaction-rate constant.

  2. What experiments would you do to determine the activation energy?

  3. If you were to take more data, where would you place the measurements (e.g., τ)?

  4. 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?

Note: All measurements were taken at steady-state conditions. Be careful; this problem is sneaky.

P7-7A 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.

t (min)

0

5

9

15

22

30

40

60

CA (mol/dm3)

2

1.6

1.35

1.1

0.87

0.70

0.53

0.35

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

  2. Nicolas Bellini wants to know, if you were to take more data, where would you place the points? Why?

  3. 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 what you think he is expecting them to say and why.

  4. 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?

P7-8A OEQ (Old Exam Question). 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.

Time (s)

390

777

1195

3155

Total Pressure (mmHg)

408

488

562

799

931

  1. Why do you think the total pressure measurement at t = 0 is missing? Can you estimate it?

  2. 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)

  3. What experimental conditions would you suggest if you were to obtain more data?

  4. How would the data and your answers change if the reaction were run at a higher temperature? A lower temperature?

P7-9A OEQ (Old Exam Question). 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:

Time (min)

10

20

30

40

50

60

ppm Br2

2.45

1.74

1.23

0.88

0.62

0.44

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

  2. Assuming identical exposure conditions, calculate the required hourly rate of injection of bromine (in pounds per hour) into a sunlit body of water, 25000 gal in volume, in order to maintain a sterilizing level of bromine of 1.0 ppm. (Ans: 0.43 lb/h)

    Member batch with an outline map of California.
  3. Apply to this problem one or more of the six ideas discussed in Table P-4 in the Complete Preface-Introduction on the Web site (http://www.umich.edu/~elements/6e/toc/Preface-Complete.pdf).

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.)

P7-10C The reactions of ozone were studied in the presence of alkenes (from R. Atkinson et al., Int. J. Chem. Kinet., 15(8), 721). The data in Table P7-10C 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.

TABLE P7-10C RATE AS A FUNCTION OF OZONE AND BUTENE CONCENTRATIONS

Run

Ozone Rate
(mol/s · dm3 × 107)

Ozone Concentration
(mol/dm3)

Butene Concentration
(mol/dm3)

1

1.5

0.01

10–12

2

3.2

0.02

10–11

3

3.5

0.015

10–10

4

5.0

0.005

10–9

5

8.8

0.001

10–8

6

4.7*

0.018

10–9

*Hint: Ozone also decomposes by collision with the wall.

P7-11A OEQ (Old Exam Question). 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 P7-11A. 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?

A graph compares the decomposition of Nitrogen dioxide and the ratio of reactor volume to the Nitrogen dioxide feed rate at different feed concentrations.

Figure P7-11A Auto exhaust data.

The plot gives the fractional decomposition of NO2 fed versus the ratio of reactor volume V (in cm3) to the NO2 feed rate, FNO2.0 (g mol/h), at different feed concentrations of NO2 (in parts per million by weight). Determine as many rate-law parameters as you can.

P7-12A The thermal decomposition of isopropyl isocyanate was studied in a differential packed-bed reactor. From the data in Table P7-12A, determine the reaction-rate-law parameters.

TABLE P7-12A RAW DATA

Run

Rate
(mol/s · dm3)

Concentration
(mol/dm3)

Temperature
(K)

1

4.9 × 10–4

0.2

700

2

1.1 × 10–4

0.02

750

3

2.4 × 10–3

0.05

800

4

2.2 × 10–2

0.08

850

5

1.18 × 10–1

0.1

900

6

1.82 × 10–2

0.06

950

Jofostan Journal of Chemical Engineering, Vol. 15, page 743 (1995).

Supplementary Reading

  1. A wide variety of techniques for measuring the concentrations of the reacting species may or may not be found in

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

    H. SCOTT FOGLER AND STEVEN E. LEBLANC, with BENJAMIN RIZZO, Strategies for Creative Problem Solving, 3rd ed. Upper Saddle River, NJ: Pearson, 2014.

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

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

    DOUGLAS A. SKOOG, 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

    H. F. RASE, 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

    G. E. P. BOX, J. S. HUNTER, and W. G. HUNTER, Statistics for Experimenters: Design, Innovation, and Discovery, 2nd ed. Hoboken, NJ: Wiley, 2005.

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