In the Classroom edited by
JCE DigiDemos: Tested Demonstrations
Ed Vitz Kutztown University Kutztown, PA 19530
Using a CBL Unit, a Temperature Sensor, and a Graphing Calculator To Model the Kinetics of Consecutive First-Order Reactions as Safe In-Class Demonstrations submitted by:
W
Deborah A. Moore-Russo Department of Learning and Instruction, State University of New York at Buffalo, Buffalo, NY 14260-1000 José E. Cortés-Figueroa* Organometallic Chemistry Research Laboratory, University of Puerto Rico, Mayagüez, PR 00681-9019; *
[email protected] checked by:
Michael J. Schuman Department of Chemistry, East Stroudsburg University, East Stroudsburg, PA 18301-2999
Calculator-Based Laboratory (CBL) technology and the graphing calculator have been used to teach chemical and physical concepts for several years (1–5). This technology has been used in the laboratory (1, 2), classroom (3), and group discussion sessions (2, 3). Its use permits student-centered instruction and exploration of concepts and natural phenomena that otherwise might be too time consuming or difficult. The graphing capabilities of the calculator help students develop skills to construct mathematical models and to discover trends in data. Three recent articles from our group have reported the use of CBL technology and the graphing calculator (i) to determine a first-order rate constant when the infinity reading is unknown (2), (ii) to teach the kinetics of consecutive first-order reactions (1), and (iii) to link the concept of temperature with kinetics concepts of a first-order reaction (3). The first article addressed the estimation of the infinity reading of the acid-catalyzed sucrose inversion. Because of the use of a polarimeter and because of safety considerations, the actual acid-catalyzed sucrose inversion experiment must be done in a laboratory. The second article addressed the solution for a set of first-order consecutive ligand exchange reactions in a metal carbonyl complex. This teaching experience was designed also for a laboratory setting. The third article presented the use of CBL technology, a graphing calculator, and a cooling piece of metal in air as a classroom demonstration to teach key concepts of a simple first-order chemical reaction. The mathematical function to describe spontaneous cooling is
(Tt (T0
− T∞ ) = exp ( −ksc t ) − T∞ )
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k1
k2
B I P where B is the reactant, I is the intermediate, and P is the product. The reaction is safe to use as an in-class demonstration. The mathematical function that gives the concentration of the nonsteady state intermediate species, I, as function of time, t , for a biphasic consecutive set of first-order chemical reactions, as shown above, is
[I]
k1 [B ]0
exp(−k1 t ) − exp (−k2 t ) (2) k2 − k1 where [B]0 is the initial concentration of reactant B (t = 0) (6). It can be shown that if a physical property such as absorbance, A, is used to monitor the progress of the chemical reactions, then =
At = αabs exp (−k1t ) ± βabs exp (−k2 t) + A ∞ (3a) where At is the absorbance at time t, A∞ is the absorbance at time infinity (6),
(ε − ε P ) k2 αabs = ( ε I − ε B ) k1 + Β [B] (3b) k2 − k1 0 εi is the absorptivity of species i, and
(1)
where Tt is the temperature at time t, T0 is the initial temperature (t = 0), T∞ is the temperature at time infinity (ambient temperature), and ksc is a constant for the cooling process in air. Here we present the use of CBL technology, the graphing calculator, and the cooling and heating (or heating and cooling) of water to model the behavior of consecutive (and 64
biphasic) first-order reactions,
− ε I ) k1 [B]0 (3c) k 2 − k1 This activity can be used in a variety of educational contexts outside of chemistry such as in pharmacokinetics in the description of the mechanisms of drug absorption, distribution, and elimination and in population ecology in the predator– prey population kinetics. For example, the plasma drug con-
Vol. 83 No. 1 January 2006
βabs =
•
(ε P
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In the Classroom
centration, Cdrug,t , in mg L᎑1, t seconds after drug injection is Cdrug,t =
f Ddrug
ka exp( −ke t) − exp( −ka t ) Vdrug ka − ke
(4) where Ddrug is the dose (mg), f is the fractional absorption, Vdrug is the ratio of (drug in body)(drug in blood), ka is the absorption rate constant (s᎑1), and ke is the elimination rate constant (s᎑1) (7). In general the pharmacokinetics model that best describes the plasma concentration profile after absorption is biexponential (8, 9). A recent article reported a compartment model for phenolsulfonpthalein (PSP) absorption from liver surface to be biexponential where the plasma concentration of PSP, CPSP,t, is given by
C PSP,t =
DPSP (α PSP − k21) Vc (α PSP − βPSP ) +
exp (−α PSP t )
DPSP ( k21 − βPSP) Vc (α PSP − βPSP)
exp(− βPSP t )
(5)
where DPSP is the administration dose of PSP, Vc is the volume of the central compartment, and αPSP and βPSP are related to first-order rate constants (k12 and k21 for transfer compartment and kel for elimination from the central compartment). Notice that while eq 4 is mathematically equivalent to eq 2, eq 5 is mathematically equivalent to eq 3a. Comparison of eqs 3a and 5 relates
αabs =
βabs =
cal property, P∞ is the value of the physical property at time infinity or after 10 half-lives, and kr is the reaction rate constant. Not only can this relatively inexpensive and safe activity be performed in a variety of teaching settings, it also allows all students to participate, often prompting vivid discussions, since students can easily grasp the qualitative idea of temperature (12, 13). The calculator is used for the activity since students can easily transfer data from the instructor’s calculator to their own to personally engage in the analysis and determination of the mathematical model that best fits the situation. Furthermore, students cannot blindly use the statistical regression capabilities of the calculator to determine the equation of best fit. They must first consider the important factors contributing to the model including ambient temperature, initial temperature, final temperature, the relative conductivity of the water container’s wall, and the biphasic first-order behavior that describes the cooling or heating of water.
DPSP (α PSP − k21)
(6a)
Vc (α PSP − βPSP ) DPSP ( k21 − βPSP)
(6b)
Vc (α PSP − βPSP)
The Activity The actual activity demonstrates the spontaneous and consecutive heating then cooling (or cooling then heating) of water. The equipment for this demonstration is shown in Figure 1. For best results, the temperature probe and the vial should be placed as near as possible to each other, avoiding physical contact between them. However, satisfactory results are obtained even if the temperature sensor is far away from the vial and close to the beaker’s wall. Although the top of the vial does not have to be completely submerged in the beaker filled with water, the water inside the vial should be completely under the beaker’s water line. (The vial is partially filled.) Stirring the water in the beaker does not give good results, probably due to perturbation of the temperature probe by bubbles or the movement of the circulating water. The CBL with its sensors and the graphing calculator are user-friendly, and explicit directions on the use of this technology are readily available in the literature (1, 2, 10, 11). By measuring the temperature of water in the beaker after a sealed vial partially filled with water at temperature
The description of the CBL unit and its use with the graphing calculator is presented in detail elsewhere (10, 11). Since a detailed description of the analysis of consecutive firstorder reactions is published (1), this article will focus on the actual modeling of various types of first-order consecutive reactions. The activity described below uses the temperature of water to create the mathematical model. The behavior of the cooling or heating of water (eq 1) is mathematically equivalent to eq 7 that describes the behavior of a first-order reaction when a physical property (Pt ) proportional to the concentration of the chemical species is used to monitor the reaction,
[ A ]t [ A ]0
=
Pt − P∞ = exp(−kr t ) P0 − P∞
(7)
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 physiwww.JCE.DivCHED.org
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Figure 1. Picture of equipment used to gather temperature versus time experimental data.
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In the Classroom
Figure 2. Plot of temperature versus time for the consecutive cooling then heating of water. The continuous trace is for temperature values obtained from the equation that best describes the relation between temperature and time (see eq 8 and values in the text).
Figure 3. Plot of temperature versus time for the consecutive heating then cooling of water. The continuous trace is for temperature values obtained from the equation that best describes the relation between temperature and time (see eq 10 and values in the text).
Tv is submerged in it (Tv ≠ Tb, Tv ≠ ambient temperature), plots of temperature versus time that model the kinetics behavior of first-order biphasic reactions and other phenomena of nature can be produced. In this demonstration, the temperature measured is the temperature of the water inside the beaker. It is analogous to the concentration of the nonsteady state intermediate species in consecutive first-order chemical reactions (14). For example, the temperature of the water in the beaker as a function of time for one of many possible demonstrations is presented in the Supplemental Material.W The corresponding graph of temperature versus time for those particular data, from the consecutive cooling then heating of water, is given in Figure 2 with the corresponding plot of temperature values that were obtained from the equation that best describes the relation between temperature and time,
The reader should bear in mind that these are mere mathematical analogies that can be used to explain biphasic processes such as in basic chemical kinetics or in pharmacokinetics systems using the concept of temperature. The reader should search the Internet to find other cases where pharmacokinetics models or chemical reactions follow a biexponential “biphasic” behavior. As in chemical reactions, where the observation of a biphasic behavior depends on the relative values of k1 and k2 and on the relative values of αabs and βabs, whether the plot of Tb versus time is biphasic should depend on the relative values of kc and kh and on the relative values of αT and βT. The relative values of kc and kh should be related to the thermal conductivity of the vial and the beaker walls. Since the values of αabs and βabs are related to k1 and k2, by analogy αT and βT should be related to kc and kh as well. This expectation was tested by altering the thermal conductivity of the beaker (by covering the entire surface of the beaker with styrofoam). The observed plot for the expected consecutive cooling then heating of water using the insulated beaker was not biphasic (i.e., a single exponential of temperature decrease) suggesting that kc >> kh. The reader can perform experiments altering the thermal conductivity of the vial. For example, if kc
