In the Classroom
Using a Graphing Calculator To Determine a First-Order Rate Constant When the Infinity Reading Is Unknown José E. Cortés-Figueroa* Department of Chemistry, University of Puerto Rico, Mayagüez, PR 00681-9019; *
[email protected] Deborah A. Moore Department of Mathematics, University of Puerto Rico, Mayagüez, PR 00681-9018
The classroom use of Calculator-Based Laboratory technology (CBL) with Vernier sensors and graphing calculators to teach chemical and physical concepts has received attention for several years (1). This technology permits studentcentered instruction where the students actively participate in their own instruction. The technology is used in the learning process to help students construct mathematical models to explain observable facts and trends. For example, the use of CBL technology and inexpensive materials such as food colorants and bleach make the presentation of first-order reactions in the classroom possible instead of waiting to see these experiments in a laboratory session (2, 3). Since instrumental methods are usually used instead of direct chemical analysis, any physical property proportional to the concentration of the reaction species is used to monitor the progress of chemical reactions. Such physical properties include the absorbance at a fixed wavelength, the area of a NMR signal, the angle of optical rotation, and so forth. It can be shown for any chemical reaction, A→B which proceeds to completion, where all the reactants are converted to products that (4) [A]t / [A]0 = ( Pt P∞ )/( P0 P∞ )
(1)
where [A]t is the concentration of species A at time t, [A]0 is the initial concentration of species A, Pt is the value of the physical property at time t, P0 is the initial value of the physical property, and P∞ is the value of the physical property at time infinity or after ten half-lives. The equation for a first-order reaction, ln( Pt P∞ ) = ln( P0 P∞ ) kt (2) where k is the reaction rate constant, predicts linear plots of ln( Pt P∞ ) versus time. Often the P∞ value is close to zero (or is the same value as the solvent reading) and ln( Pt P∞ ) in eq 2 is substituted by ln(Pt ). Because of this common situation students (and sometimes professors and researchers) unfortunately overlook the need to consider or report the use of P∞ values in eq 2 when determining the rate constant values of first-order reactions. Nonlinear plots of ln( Pt P∞ ) versus time may result if incorrectly determined P∞ values (or no values at all) are used. The curvature of the plot may become noticeable in the vicinity of two half-lives. If the students notice this curvature they may incorrectly conclude that the reaction is not first-order. On the other hand, if they do not notice the curvature, the rate constant value would be incorrect. Besides monitoring a reaction for an insufficient number of half-lives—less than 2—wrong P∞ values are one of the main sources of error in determining the order and rate constants of chemical reactions (5). 1462
Figure 1. Calculator screen plot of αt vs time for the acid-catalyzed inversion of sucrose to about three half-lives. Abscissa is time in seconds ranging from 0 to 6500 in increments of 1000; ordinate ranges from 11 to 15 in increments of 2.
Figure 2. Calculator screen plots of the time-lag points when τ = 4200 seconds, the extrapolated regression line that best fits the time-lag points (slope = 3.887895325, vertical intercept = 22.237095325), and the graph of the 45° line. The intersection of the regression line and the 45° line is the estimated value for α∞. Both the abscissa and the ordinate range from 20 to 20 in increments of 5.
Figure 3. Calculator screen plot of ln(αt α∞) versus time is linear to five half-lives. Abscissa is time in seconds ranging from 0 to 12500 in increments of 2500; ordinate ranges from 1 to 4 in increments of 0.5.
Journal of Chemical Education • Vol. 79 No. 12 December 2002 • JChemEd.chem.wisc.edu
In the Classroom
The classical experiment of the acid-catalyzed sucrose inversion (6–8) is an excellent opportunity to introduce the concept of P∞. In this experiment the progress of the reaction is monitored by following the change of the angle of optical rotation of polarized light, α, with respect to time. The α value changes during the course of the reaction. It has positive, zero, and negative values. Since it is impossible to determine the logarithm of zero or of negative α values, students will realize the necessity of subtracting α∞ from each αt reading. The first-order behavior of this reaction is well documented. This paper will present the use of the graphing calculator to estimate the α∞value of an acid-catalyzed sucrose inversion by the Kezdy–Swinbourne method (9, 10). In addition, a closely related method to calculate the rate constant when α∞ is unknown will be presented with a comparison of the calculated rate constants. Estimating the Infinity Reading Since the acid-catalyzed inversion of sucrose is a firstorder reaction, a plot of αt versus time is exponential (Figure 1). For the purpose of illustrating the estimation of the α∞, the data presented in Table 1 are not plotted past three halflives. Time-lag methods are well known to treat first- and second-order kinetics (9, 10). For example, the Kezdy– Swinbourne method (see eq 3), predicts linear plots of αt versus αt+τ. αt = α∞[1 – exp(kτ)] + αt+τexp(k τ)
(3)
In eq 3, τ is a fixed time interval such that (t, αt ) and (t+τ, αt+τ ) are data points from the reaction. The set of all possible ordered pairs (αt+τ, αt ) can be plotted and are linear for a first-order reaction (see Table 2).1 The rate constant can be estimated from the slope of this plot (k = ln(slope)/τ). The value of τ can be chosen arbitrarily, but for best accuracy it is suggested that its value should be between two to three half-lives (11). (The discussion about the criteria to choose the correct τ value will be deferred until a later subsection.) The calculator is used to create an extrapolated regression line for the plot of αt versus αt+τ. The α∞ value can then be estimated from the intersection of the extrapolated regression line with a line at 45° (see Figure 2). At the intersection point of these two lines, αt = αt+τ. This only happens when αt = αt+τ = α∞ (see Figure 3). Once α∞ is determined, the calculator is used to find ln(αt α∞)2. Figure 3 shows the linear plot of ln(αt α∞) versus time to five half-lives as it is viewed in the calculator’s screen. The calculator is then used to determine the regression equation and the goodness of that fit that can be assessed visually and by the correlation coefficient. Once this plot and the fit are displayed on the projector screen, it is easy to remind the students that the negative of the slope of the regression line is the rate constant of the reaction. Figure 4A uses all the experimental data, showing αt versus time until the value of αt asymptotically approaches the α∞ value. Figure 4B shows both the original data and the plot ((α0 α∞ )exp[kt]) + α∞ versus time using the calculated values for k and α∞ to demonstrate the goodness of the fit.
Table 1. Values of α for the Acid-Cataylzed Sucrose Inversion Experiment Time/s
α/deg
Time/s
α/deg
0
11.239
3600
200
10.090
3800
-1.777 2.067
400
8.932
4000
2.454
600
7.901
4200
2.809
800
6.981
4400
3.099
1000
6.013
4600
3.406
1200
5.207
4800
3.696
1400
4.433
5000
3.938
1600
3.626
5200
4.180
1800
2.901
5400
4.374
2000
2.255
5600
4.615
2200
1.691
5800
4.809
2400
1.191
6000
4.971
2600
0.594
7000
5.745
2800
10000
6.906
3000
0.030 0.422
26000
7.648
3200
0.874
7.672
3400
1.353
40000 ∞
7.679
Note: [HCl] ≈ 1 M, [sucrose] ≈ 0.3 M, T = 22.0 C Table 2. Lag-Time Table for τ 4200 seconds αt+τ/deg
αt /deg
αt+τ/deg
αt /deg
2.809
11.239
4.180
6.013
4.374
10.090
4.374
5.207
4.615
8.932
4.615
4.433
4.809
7.901
4.809
3.626
4.971
6.981
4.971
2.901
Note: estimated α = 7.797
Criteria to Choose τ Values Equation 3 predicts linear plots of αt versus αt+τ with slope equal to exp[k τ]. These plots can be constructed for various τ values and their slopes determined. This is an easy task due to the capability of the graphing calculator to construct new columns from existing ones. Once the slopes of these plots have been determined, two columns, one for the slope values and one for the corresponding τ values, are created. The pairs of τ and slope values with the corresponding k values are presented in Table 3. Since the slope of the plot of αt versus αt+τ is equal to exp[k τ], eq 4 predicts linear plots of ln(slope) versus τ. The slope of this new plot is equal to the rate constant, k. ln(slope) = k τ
(4)
The rate constant value determined from this plot (k = 3.23301 × 104 s1) is the same, within 1%, as the rate constant determined from eq 2 using the experimental α∞ value (α∞ = 7.689°, k = 3.24649 × 104 s1).
JChemEd.chem.wisc.edu • Vol. 79 No. 12 December 2002 • Journal of Chemical Education
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In the Classroom Table 3. Estimated α∞ Values, Rate Constant Values, and the Corresponding Correlation Coefficients Determined Using Various Values of τ τ/sec
Figure 4A. Calculator screen plot of αt vs time for the acid-catalyzed inversion of sucrose. Abscissa is time in seconds ranging from 4000 to 44000 in increments of 2500; ordinate ranges from 10.5 to 14.5 in increments of 2.
Slope of plot of αt versus αt+τ
α∞/ deg
k/(104 s1)
Correlation coefficient
400
1.135696456
7.8049
3.1811
.999927673
1000
1.371393183
7.8546
3.1583
.999988617
2000
1.868746382
7.9240
3.1263
.999822455
3200
2.736646105
7.8907
3.1460
.999835418
4200
3.887895325
7.7001
3.2330
.999758004
5000
5.095618188
7.6589
3.2568
.999757043
Science Foundation (grant CHE-0102167) for support of this work. Valuable comments from Keith Wayland are gratefully acknowledged. Notes Figure 4B. Calculator screen plot of original data with plot of αt using the calculated values for k and α∞ to demonstrate the goodness of the fit. Abscissa is time in seconds ranging from 4000 to 44000 in increments of 2500; ordinate ranges from 10.5 to 14.5 in increments of 2.
Based on the quality of the linear fit of the plots of αt versus αt+τ for all τ values in Table 3 (all correlation coefficients better than .999) and the close values of the rate constants determined from the slope of these plots it seems inconsequential which value of τ is chosen to estimate α∞. While choosing a small τ value permits evaluation of more data points, students and the teachers have to keep in mind that If τ
