3
ANALYSIS OF RATE DATA
Mil • in
ml >v. w ill be changing our focus. In Chapter 2, we discussed general principlei |    i|'|'lnil to a variety of reaclions. In this chapter, we will be discussing some
i.....|in . one uses to lil rate data to a rate equation. First, we will brielly discuss
........hniqucs. Then we will discuss how to analyze rate data.
II i in  i "ink was written, it was assumed that our readers had seen this material i.i    ilu-y hail measured a rate in freshman chemistry or integrated a mtC in ilieii physical chemistry or reactor design course. However, people seldom In ii in start when presented with a new problem. What needs to be measured,
i......i analyze the data? The objective of the material that follows is to
ill. i * \ ideas that one needs to understand in order to measure rale data fol .1
, .ii'in
iuiiiodUCTION
i ' 1I1   nl chemical reactions date back to Wilhelmy's (1850) measurement ol the .......inversion in the grapejuice used to make wine. They have been improved
MH ullll 1
111    huptcr we will discuss how one actually determines a rate equation. First we
II discuss some of the experimental techniques that one uses to determine a
1......... In particular, we will distinguish between direct and indirect methods, and
1   I In properties of each. Then we will discuss the data analysis. Our approach is a
.....111.in I, in that we point out that some common data analysis schemes are subjw 1
.........unity. Understanding the uncertainty is as important as understanding the
Ml ......I"t
11......hue nl the material is as follows:
VI
IIAI .h<HH >l INI I
• lii Sections i.j i i wi- will hrlclly review the experimental techniques, The r\|H'iiiiuiii.il iniini(|ius .in- evolving ill ol the timet end wc had to keep u|> with tin.' latest advances. Still, wo wanted n> give .1 pi< ture "i how the data are taken, and in particular to distinguish between direct and indirect methods.
• In Sections 3.5-3.8 wc will examine the analysis of the data. We will derive the key equations and explain how they are used.
• Then we will include solved examples. The solved examples are a key part of this chapter, and one should be sure to examine them carefully.
I have focused the discussion to make it most understandable to a senior undergraduate. Please refer to your physical chemistry or reaction engineering text for a less advanced treatment.
3.2 BACKGROUND
Studies of the kinetics of chemical reactions began in earnest in the middle part of the nineteenth century. At the time, winemaking was one of the key chemical industries, so many of the studies used wine, and winemaking equipment.
One of the key issues in a winery is controlling how sweet the wine tastes. Generally, sweeter grapes will produce a sweeter wine, so it is very important to control the sugar content of the grapes used to produce the wine.
In the 1600s, it was discovered that the refractive index of sugar solutions varied linearly with the sugar concentration. Consequently, one can measure the refractive index of a grape squeezing and use that measurement to estimate the sugar content of the grapes. If you have been to a winery, you will know that when the grapes are delivered, people use a rcfractometer to measure the sugar concentration of the grape juice.
There is a problem with that measurement, however. Sometimes when you let grape juice sit, the sucrose in the grape juice is converted to fructose and lactose via reaction (2.7). The refractive index of a fructose/lactose solution is quite different from the refractive index of a sucrose solution. Consequently, if some of the sucrose in grape juice has been converted to fructose and lactose, one will not get an accurate sugar measurement with a refractometer.
In 1850, Wilhelmy did some measurements to try to understand how quickly the sucrose was converted to lactose and fructose. He started with a sucrose solution and used a polarimeter to measure the sucrose concentration as a function of time. Wilhelmy also added various acids because it had previously been found that acids would speed the spoilage of wine or grape juice.
Figure 3.1 shows some of Wilhelmy's results. Notice that the sucrose concentration decays with time. Later in this chapter, we will show that for a first-order reaction, the sucrose concentration, Cs, should follow
Cs = C° exp(-ksT)
(3.1)
where C*<! is the initial sucrose concentration, t is time, and ks is the rate constant. Ostwald (1885) fit Wilhelmy's data to equation (3.1) and then inferred a rate equation for the conversion of sucrose to lactose and fructose.
.........Witness (1850, measurements ct the changes in sucrose concentration in grape ,uice ,,,„
,„ iiI is lidded.
Time, hours
Figure 3.2  Concentration versus time for a simple reaction.
Most simple kinetic measurements are done using a technique that is very similai to those pioneered by Wilhelmy. One finds a way to initiate the reaction and then measure! (he concentration of some key species as a function of time. Typical data are shown in Figure 3.2. One then uses the data to infer the order of the reaction and the rate constant Over the years, the details have changed. However, one still induces a change in the system and then uses spectroscopy to measure the concentration as a function of time
In the next several sections we briefly describe the techniques, and mention how the} are used.
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Mosi kinetics books talk about the various methods one uses to measure rates ol reaction Generally, the methods discussed in most hooks are similar. One finds a way to initiate a reaction and then uses some sort of spectroscopic technique to measure the concentration of a key species as a function of time. One then tits the data to an equation to infer a rate law.
There are lots of schemes to initiate the reaction. If the reaction is slow, you can simply mix the reactants together. If the reaction is fast, you might start the reaction by mixing the reactants at low temperature and then quickly heating the mixture. People also use spark plugs and lasers to initiate reactions. For a really fast reaction, one uses a molecular beam system to measure rates.
Table 3.1 summarizes the methods discussed in most textbooks. The simplest method, called the conventional method, is to mix the reactants in a beaker, and then measure concentrations as a function of time. Generally, this is a very easy technique. However, it usually, takes several seconds to mix the reactants, and possibly several seconds to make a concentration measurement. As a result, the conventional methods are useful only for reactions that are slow enough that the reactants can be mixed and measurements made before there is signiticant conversion of reactants to products. In Table 3.1, we say that the reaction must take 10 seconds or more. That is because it usually takes several seconds to pour the reactants into a beaker and thoroughly mix them.
The next method is the stopped-flow method. In the stopped flow, one runs the reaction in a flow cell in a spectrophotometer. You start the experiment by llowing the reactants into the flow cell, and increase the velocity of the reactants until the conversion of the reactants is negligible. You wait for everything to be well mixed, and then you stop the flow and measure the concentrations of the reactants as a function of time. The stopped-flow method avoids the difficulties of the conventional method, since the reactants are already mixed at the start of the reaction. Still, it takes time to stop the flow. As a result, the stopped-flow method is limited to reactions that take 1CT3 seconds or more.
The third method is the temperature-jump method. In this method, one mixes the reactants at a low enough temperature that the reaction rate is negligible and then zaps the mixture with a C02 laser to suddenly heat the reactants to the desired reaction temperature. Again, mixing problems are avoided, but there is the difficulty that one can heat things only so fast. In practice, the temperature-jump method is usually limited to reactions that take 10~6 seconds or more, although people are developing techniques to heat liquids as fast as 10~9 seconds.
The shock tube is a way to examine fast reactions in gases. One constructs a long tube, with a thin membrane in the middle. One then puts one reactant on one side of the membrane, and a second reactant on the other side of the membrane. One adds an atmosphere or two of an inert gas, to ensure that there is a large pressure difference between the two sides of the membrane. One then ruptures the membrane, and allows the gases to suddenly mix. Generally, a shock wave forms at the interface between the high-pressure and low-pressure gas. The shock wave heats the reactants and allows them to mix. One can generally observe reactions that take between 10 3 and 10~5 seconds in this way. One cannot study faster reactions with the technique because of the finite mixing times. One cannot study slower reactions because the gas cools off after the shock wave has passed.
Flash photolysis is the fastest method. One zaps a mixture with a laser, and uses a variety of spectroscopic techniques to observe the resultant reactants. Generally, one
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ii,hi h Methods
1) Mix rcuclunls together in a batch reactor
2) Measure concentration versus lime
11 Sel of continuous-flow systems where
reactants are fed into the reactor, and How out again so quickly that there is negligible reaction
21 Slop the How so that the reactants can react
3) Measure conversion versus time
i) Mix reactants at such a low temperature that
the reaction rale is negligible >) Use C02 laser to suddenly heat reaction 3) Measure concentration vs. time
1) Put 10~' atm of one reactant and 10 atm at helium on one side of a diaphragm
2) Put 10"3 atm of the other reactant on the other side of the diaphragm
3) Suddenly break the diaphragm so that the gas flows from the high-pressure side to the low-pressure side
Measure the reactant concentration vs. time
1) Put the reactants into a vessel under conditions where reaction is negligible
2) Pulse a laser or flash lamp to start reaction
3) Measure the reactant concentration vs. time
1) Initiate a change with a magnetic pulse
2) Measure the decay of spins by NMR
Flow Methods 1) Continuously feed reactants into a reactor
(CSTRfc or plug flow) 1) Measure the steady-state reaction rate
1) Direct beams of reactants toward each together in a vacuum system
2) Measure the steady-state reaction rate
.ii. I. ,ii magnetic resonance. 11 ivnlinuoutly siin-ed lank reactor.
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10
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.....s0 the laser to examine the reactions of radicals or other reactive intermediates In
,, ating the intermediates with a laser and watching how they react. Generally, lasei methods are limited to reactions that take 10-9 seconds or more because laser pulses usually last perhaps 10"10 seconds. However, people are working on femtosecond lasers lo I'll around those difficulties.
I he methods described in the preceding paragraphs are called batch experiments. In ., batch experiment, one mixes a batch of reactants in a reactor and then allows the reactants to react. One can also run reactions in a flow reactor where one continuously
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lank reactors (CSTRs) and pluj; flow reactors, in eithei case, one flows reactants Into tin* reactor and measures the concentration ol the products as a function oi the residence time in the reactor x, where x is given by
(reactor volume, liters)
(volumetric flow rate of reactants, liters/hour)
(3.2)
The CSTR is the simplest reactor. The CSTR consists of a mixing tank with flow in and out, as indicated in Figure 3.3. Later in this chapter, we will show that when the general reaction A => B occurs in a CSTR, the reaction rate, rA1, in the CSTR is related to the residence time in the CSTR by
rA =
pout A
(3.3)
where C"
is the concentration of A in the inlet to the reactor, CAU| is the concentration
in the outlet of the pipe, and x is the residence time. Plug-flow reactors are basically pipes with baffels to prevent backward mixing. One feeds reactants into the pipe and takes products at the end. One can show that the residence time for a plug-flow reactor is related to the reaction rate by
fCA dCA
Plug-flow reactors (Figure 3.4) can be used to measure kinetics, although they are used less often than CSTRs.
One last experimental technique, that is especially important, is called molecular beam. Molecular beam measurements are fundamentally different than all of the other measurements in Table 3.1. Figure 3.5 shows a schematic of the molecular beam system
Stirrer
Figure 3.3  A continuously stirred tank reactor.
Reactants in
mmmmmmmmmmmm.
Figure 3.4 A plug-flow reactor.
Products *■ out
No/zol Sktinmoi
NaCI
			
■ ■ 111		| -Lrn	
			-=H
			\ mr
Differentially pumped mass
spectrometer
Down to giant pump
......n. :t.5  A molecular beam system used to measure the rate of the reaction Na + CI? -> NaCI I CI Alici
......i al (1968), and Lee et al (1968).
I  Mr1 used to examine the rate of the reaction:
Na + Cl2-> NaCI + CI
(3.3)
Sodium atoms are generated in the sodium source and flow into a vacuum system. The .minim atoms collide with the stream of chlorine, and the NaCI and CI products are detected with a mass spectrometer. A laser can also be used to measure the properties "I ill. reactant and product molecules. The system is designed so that the unreacted sodium iioins and chlorine molecules are pumped away.
The advantage of a molecular beam system is that the sodium atoms are in close proximity to the CI2 molecules for perhaps only 10"" seconds. Consequently, very fas) pun esses can be studied via molecular beam techniques. Further, one can catch the produi 1 NaCI molecules before the products collide with the walls or other molecules. One can directly measure the properties of the molecules that have reacted, and simultaneously determine all of the properties of the product molecules that form. Therefore, one can ill more information about what happens during the reaction than with conventional techniques.
In practice, the beam systems have allowed people to examine much faster processes than could be examined with conventional techniques. Also, many of the details of a reaction can be probed only by using molecular beam techniques. Consequently, molecular beam techniques have been popular in the literature.
HUH i i ANI i INI Hl II il Ml IIIIHUI
I InfoiInnately. thCTC is iiisiiIIu h-mI room l<» Ul tO discuss nioleculai beam techniques here. One should refer i<> IK-r^lih;uih (I*>(><>| I .evine ami Uemsteln (1987) or Scoles (1988) lor a discussion ol nioleculai lieani techniques and how thej contribute lo kinetics. We also do not have room to discuss all of the experimental techniques in Table 3. I in detail. Several older books, including the books by Moore and Pearson (1981) and Laidler (1987), discuss the experimental techniques.
Still, there is one other important point to recognize in the table — one needs lo use a different experimental method for a fast reaction than for a slow reaction. If a reaction takes 10-2 seconds, one needs to initiate the reaction within 10~3 seconds. A fast measurement system is needed. A much slower measurement technique can be used if the reaction takes several minutes. Table 3.1 indicates which methods can be used to measure rates for fast reactions and which ones are limited to slow reactions. Consequently, the reader should memorize the information in Table 3.1 before proceeding.
3.4   DIRECT AND INDIRECT METHODS
The listing in Table 3.1 separates the experimental methods according to the timescale of the measurement. However, there is another key distinction: whether the technique produces a direct or indirect measurement of the rate equation. I am assuming that the reader probably is not familiar with the terms direct and indirect methods. Therefore, I thought that I would define them before proceeding.
Recall that, by definition, the rate equation is an expression for the rate as a function of the concentration of the reactants. One can measure the rate as a function of concentration directly or indirectly. The experimental methods in Section 3.2 were mainly indirect measurements. If one loads a reactant into a reactor and measures the reactant concentration as a function of time, one is not measuring the rate as a function of concentration. Instead, one is measuring the concentration as a function of time, and fitting that data to infer a rate law. This is an indirect measurement of the rate law, so I will refer to it as an indirect method.
On the other hand, it is possible to directly measure the rate as a function of concentration. In that case, one can fit the rate equation directly. I will call such a measurement a direct rate measurement. More precisely, I will define a direct method as any experimental method where one actually measures the rate of reaction as a function of the concentrations in the reactor. I will define an indirect method as a method where one does not actually measure the rate. Instead, one measures some other property, for example, a concentration as a function of time, and fits that data to infer a rate law.
In the literature, it has become common to refer to a direct method as a differential method, while an indirect method is referred to as an integral method.
It is useful to consider an example, the decomposition of arsine (AsHj), to illustrate the difference between direct and indirect measurements. The decomposition of arsine on silicon or gallium arsenic is quite important to semiconductor device manufacture. During the reaction, the arsine decomposes to yield an arsenic film and liberate hydrogen:
2AsH,
2As + 3H2
(3.6)
Reaction (3.6) is used to deposit arsenic as a dopant for silicon in the manufacture of integrated circuits. Reaction (3.6) is also used as an arsenic source in gallium arsenide production for light-emitting diodes (LEDs), flat-panel displays, and compact-disk (CD) players.
PlgUK 1 6 shows ,i typical rcitcloi used lot ilns reaction I he reaclor consists ol a i|imM/ lube ill a lube liiinacc You load silicon wafers uilo I he reactor, eviieunle, turn on il>. ..viii. and lei the wafers heal lo 1(1(10 < You (hen feed arsine onlo Ihe hoi wafers 11.. mine ileconi|)oses, depositing arsenic onto Ihe wafer.
Now, lei's consider Hying lo measure Ihe rale of reaction (3.6). One can measure ili< inie ol reaction (3.6) using ihe apparatus in figure 3.7. The apparatus consists ol . i. ,i. lot containing a very sensitive balance called a 'mlcrobalance.' During a kinetic l iircmcnt, one loads a microchip onlo Ihe microbalance, runs the reaction, and iheu ■. nle. the chip as a function of lime. The change in weight of Ihe chip is equal lo the •. nlil ol ihe arsenic lhal is deposited. Therefore, one can use Ihe change in weight ol il.. . hip lo determine how much arsenic is deposited.
I li. i< are Iwo ways to run Ihe reactor. First, one could continuously Iced arsine into the .. i..i insure a constant concentration of the arsine in Ihe reactor and then determine the
.....lit ul ihe chip as a function of lime. In that case, one would gel a steady-slate reaclion
i.in Second, one could load a fixed amount of arsine into the reactor and determine how ......li at sine is deposited as a function of time. In that case, one would determine I
n nr.lent reaction rate.
In the lusi case, arsenic would be continuously deposited onto the chip. IIOne measured ih. w eighl of the chip as a function of time, one could calculate Ro, the arsenic deposition
Pressure gauge
Door (loadlock)
3-Zone oven
Silicon wafers
Figure
Feed      Holder (boat)
3.6  A typical arsine decomposition reactor.
Microbalance
Wafer
Heat lamp
Feed
Exhaust
Figure 3
7  A possible apparatus to examine the decomposition of arsine (AsH3) on silicon.
fill AI i/M i'.ľ.i H MAM llAIA
i llllll i ani i ini iiiii l i Ml 11 H H ľ . OH
rate in mol/t lit nu . in I. I.....i
I I dWeight^ (Au ) MWAs dt
(3.7)
where Aw is the area of the chip, MWAS is the molecular weight of arsenic, and Weight, is the weight of the chip. In this way, one could measure the rate of arsenic deposition at any arsine concentration directly.
If one wanted to use this method to determine a rate law, one would repeat the measurements dozens of times, varying the arsine and hydrogen concentration within the reactor. Eventually, one could get a plot of the rate of reaction as a function of the arsine and hydrogen concentration within the reactor. Notice that during such an experiment, one is directly measuring a rate of reaction as a function of the concentration in the reactor. Therefore, I like to call such a measurement a direct measurement of the rate equation. In the literature, people also call it a differential method because one generally has to differentiate data to calculate a rate.
An alternate experiment is to load a fixed amount of arsine into the reactor and measure the arsine pressure as a function of time while the arsine is deposited. Figure 3.8 shows typical data. In this case, the rate varies with time because the arsine is being used up. One could still differentiate the data to get the rate as a function of time. However, an alternative is to derive a theoretical equation for the pressure versus time, and compare the theoretical equation to the data.
Later in this chapter, we will integrate the rate equation to show that, for a first-order reaction, the weight of the chip will vary as follows:
x e
(3.8)
where PAsH, is the measured pressure of arsine, which varies with time; PAsH is the initial arsine pressure; ki is the rate constant for arsine decomposition; and t is time. One could measure the weight as a function of time and fit the data to equation (3.8) to calculate
100
CD
10      15      20      25      30 35 Time, hours
Figure 3.8 Typical batch data for reaction (3.7). [Data of Tamaru et al. (1955).]
ti. nil iiMisluiil In ílu- Iili'iiililiv, people i,ill mu It an analysis an Integral method mih ľ .nu ill iiwil equation ( IK) by uitegialliig llie rale equation.
I like to instead call it an indirect iiiellmil because one is not measuring the tale as i linn lion ol concentration directly. Instead, one is determining how the rate vanes Willi .........Irution indirectly by lining the measurements to a theoretical expression.
Nm\ inn- might ask. "Mow are the direct method and the indirect method fundamental!)
ililli n nl '" Notice that when one uses the direct method, one is actually incasutin;' the
i a. i.iu directly. In other words, when one differentiates the weigh) \cimis time iI.m.i. one . . i i direct measurement of the rate for each arsine concentration. One docs not have lo nil. .my assumptions about the form of the rate equation lo calculate a rale; one just has i- hi data Alternatively, with the indirect method, the concentration is varying, so one h i in m.ike .m assumption about the form of the rate equation to dense equation i 1 ki iii.ii In less accurate. However, the advantage is that one does not need to differential!
.......gel useful information.
In the literature, it lias become common to refer to any method where one gets .i dim t Hli . 111 < n u-m ol the rate at each concentration a direct method or differential method
ii il niie does not have to actually differentiate data to calculate a rale. Alternatively,
■.....leis to any method where one gets an indirect measurement of the rale as an indheel
method m integral method, even though one would not necessarily need to integrate > i .i. . quation to analyze the data.
3.4.1   Advantages of Direct and Indirect Methods
\\ hen one plans an experiment to determine the rate equation for a given reaction, one
i.....i eds to do is to decide whether to use a direct or indirect method.
I able 3.2 compares the advantages and disadvantages of direct and indirect methods
......Lilly, indirect measurements require easier experiments than do direct measurements,
bul the resulting rate equations are less accurate. With an indirect method, one shnph loads icaclants into a reactor and measures concentration versus time. Those are usuall) relatively easy measurements. In contrast, direct measurements generally require -i llnu system and a very sensitive measurement device to directly measure the tale i . msequently, the actual experiments are harder with a direct method than with an Indirect method. The experiments are impossible with very fast or very slow reactions. Also, one i.....• a direct measurement gets only one point on the rate versus concentration curve
I able 3.2 Comparison of the advantages and disadvantages of direct and indirect methods
I hi n i Method
Indirect Mctluul
Advantages
(let rate equation directly
l-.asy to tit data to a rate law
High confidence on final rate equation
I >is;idvanlages
I liliicult experiment Need many runs
Not suitable for very fast or very slow reactions
Disadvantages
Must infer rate equation
Hard to analyze rate data
Low confidence on final rate equation Advantages
Easier experiment
Can do a few runs and get important
information Suitable for all reactions including ver)
fast or very slow ones
i im iw < limn ii i i mi'imii .Al   MAU   i AW.    .iui im   Ml Al   1 AN 1 h M
al .1 time, no h taken man) metiNuremcnlN i" ill ii rn.....     n mli i quution (I.e., the rati i
a function til concenlralion) One can )'cl iiselul il.il.i hum .1 suiflc 11111 Willi nil indued
measurement.
Still, the advantage of a direct method is that one measures the rale equation (i.e., the rate as a function of the reactant concentration) directly. One does not have to make any assumptions about the form of the rate equation to get an answer.
Generally direct measurements are much easier to fit to a rate equation than indirect measurements because, in a direct measurement, one determines the rate equation directly, while in an indirect measurement, one needs to infer the rate equation by fitting a curve to the data. The latter process can introduce some degree of error.
In my experience, when you are trying to determine kinetics for a new system, it is usually better to start with an indirect method. The indirect method gives you an approximate rate equation with a quick and easy experiment. That is often good enough. A direct method is required only when you need a precise rate equation. If you are designing a process where a 10% change in rate matters, you need to do direct rate measurements. If you can accept a 10% error and adjust the process accordingly, an indirect method will suffice. In my experience, direct measurements take 10-100 times longer than indirect measurements, so direct methods are useful only when a high degree of accuracy is needed.
3.5   EXAMPLES OF DIRECT AND INDIRECT METHODS
Indirect methods are the most common kinetic measurements in the older literature. You measure a concentration as a function of time and then fit the data to a rate equation. Direct measurements are harder. One has to find a way to measure the rate of reaction directly.
Most direct methods involve differentiating a rate equation. However, that is not a necessity. For example, in the reactor in Figure 3.8, one could measure how much arsine flows into the reactor and how much flows out. If one knows how many moles per hour of arsine flow into the reactor and how many moles per hour flow out of the reactor, one can calculate the rate from a mass balance:
Ras = -—(Fi„ — Foui)
Aw
(3.9)
where RAs is the rate of arsenic deposition per unit area, Aw is the area of the wafer, Fin is the flow rate of arsine into the reactor in mol/hour, and, FOUi is the flow rate of arsine out of the reactor in mole/hour.
Notice that one is still directly measuring the rate. Therefore, one would call the measurement a direct method or a differential method, even though you are not differentiating anything to get a direct measurement of the rate.
Note, however, that equation (3.9) applies only if the rate is constant across the wafer and there is no reaction anywhere else in the reactor. If there were a reaction somewhere else in the reactor, one would have to do analysis to eliminate those effects. One would call the measurement integral methods or indirect methods since one needs to do an analysis to determine the rate. Generally, one would only call a measurement a direct determination of the rate equation when one can directly measure the rate as a function of concentration. If one has to do some analysis, it will be an indirect method.
Mi, 11 .in iii.inv variation* Ol thil IdM Today, most direct kinetic meusiiiemeuls mi'
......la m .1 continuously stirred tank reacioi kstki Figure 1.4 shows .1 diagram ol 1
1 m i< Basically, you continuously feed reactants into the reactor. Some ol the 1 caelum
,11,1i11 iiles read, while other reactant molecules jusl llow through Ihe reaelor. You ilien ,11. hsiiii- Ihe reactanl concentration in the Intel and outlet of the reactor Ihe average n ,11 inm rale rA in the reactor is given by
A
( l III)
......In 11 urns the reactor so that Ihe mixture stays well mixed. In thai case, the rate ll
- ill 1.mi throughout the reactor, so one can calculate the rale directly from equation < 1 10) \iiniliii important direct method is Ihe method of initial rales In the unn.il rata IIH 111.»1. one runs Ihe reaction in a batch reactor, as in an indirect measurement. Ilowevei
......ni.ilv/es the rate differently. Consider the data in Figure 3.2. Figure 1.2 shown
....., , 1 ii 1.11 it>ii time data. Notice that according to equation (2.4), the rale at any liinc
.  iii.   lope of Ihe target to the line. Therefore, one can use the slope to gel a rata In the initial rate method, one fits a (dashed) line to the initial pari of the concentration
1 .us ihe tlata. The rate is the slope of the line. One then changes the initial concentre!.....
111,1 generates ihe rate versus concentration data. The advantage of this approach is thai ..... , in operate direct data quickly, by running several small reactors at once.
.1(1   EXAMPLES OF INDIRECT MEASUREMENTS
Indirect measurements are generally made in a batch system. Most of the rate measure
minis you did in your chemistry lab were indirect measurements. For example, yoi.....ghl
have loaded some species into a beaker, measured the concentration versus time, anil hi Ihe dala to a first-order or second-order rate law. That is an indirect measurement. The Hush photolysis experiments that students sometimes do are also indirect measurements ol iii, rate law. Direct measurements require that you actually measure the rate. The measure mi nis are harder. They are seldom done in undergraduate labs. In Table 3.1, we briell> mentioned several techniques that one uses to measure rates of reaction. As an exen Isi 1I1, under should go back and decide whether each method is a direct or indirect one
3.7   FITTING DATA TO EMPIRICAL RATE LAWS: SINGLE REACTANTS
\i this point, we will be changing topics. We will assume that you have used eiihei .1 ilueci or an indirect method to measure the rate data for a given reaction. We will now discuss how one fits the rate data to a rate equation. The general scheme will be to
1. Determine the order of the reaction. (In a complicated case, one also lias in determine the form of the rate equation.)
2. Fit the constants.
The hard part is to determine the order of the reaction.