First Order Kinetics Visualized by Capillary Flow and Simple Data

Communication Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX

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First Order Kinetics Visualized by Capillary Flow and Simple Data Acquisition Lea Festersen, Peter Gilch,* Anna Reiffers, and Ramona Mundt Institut für Physikalische Chemie, Heinrich-Heine-Universität Düsseldorf, Universitätstrasse 1, D-40225 Düsseldorf, Germany S Supporting Information *

ABSTRACT: First order processes are of paramount importance for chemical kinetics. In a well-established demonstration experiment, the flow of water out of a vertical glass tube through a capillary simulates a chemical first order process. Here, a digital version of this experiment for lecture hall demonstrations is presented. To this end, water flowing out of the capillary is collected in a beaker which stands on an electronic scale interfaced with a computer. The computer generates a plot of the data in real time which is projected during the lecture. Hereby, with proper explanations, a very intuitive grasp of the essence of first order processes is obtained: The rate (represented by the water flow) is proportional to the concentration of the reactant (height of the water column). With a modified setup, consecutive first order kinetics as well as the concept of a rate-limiting step are illustrated. KEYWORDS: Physical Chemistry, Kinetics, Rate Law, Demonstrations, Computer-Based Learning, Laboratory Computing/Interfacing, Second-Year Undergraduate

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INTRODUCTION Chemical kinetics is an essential part of the undergraduate education in physical chemistry.1,2 In chemical kinetics, rate laws,3 which relate concentration changes per time (rate) with present concentrations, are indispensable.4−7 Of particular importance are first order processes in which the rate is proportional to the concentration of one species.3 Many physical and chemical processes obey first order kinetics. Examples include radioactive decays,8,9 isomerizations,10 some elimination reactions,10 and many photochemical as well as photophysical processes.11 For processes of higher order, such as many bimolecular reactions, pseudo-first-order conditions are often employed, since this simplifies the experiment and the data analysis.10,12 Considering this relevance we sought an experiment which gives an intuitive grasp of a first order process and can be used in the lecture hall. Several concepts to introduce first order kinetics by analogies were reported in this Journal. Examples include the cooling of liquids,13 the transfer of water between containers,14 dice shaking,15 and capillary flow.16 The experiments as described in these papers are wellsuited for lab courses. For a lecture demonstration we do not consider them appropriate since they require repetitive measurements followed by data analysis. Here, we describe a modified capillary flow experiment suitable for the lecture hall. It is based on a paper published in this Journal more than 40 years ago.16 Relying on even earlier work,17−19 D. A. Davenport reported on simulation experiments for chemical kinetics using capillary flow.16 A glass tube is connected to a capillary by a flexible tube (cf. Figure 1a). The glass tube is mounted vertically, the capillary © XXXX American Chemical Society and Division of Chemical Education, Inc.

horizontally. This applies to all setups used here. The water column in the tube represents the concentration of the reactant. The water flow stands for the rate. In the simulations reported by Davenport, the data were recorded manually. In our “digital” version of the simulation a scale interfaced with a computer collects the data. The computer plots the kinetic trace in real time. By connecting the computer with a digital projector, the data acquisition can be shared with students in the lecture hall. Thus, students see the depletion of the glass tube, representing the concentration of a reactant (as well as the changing flow, representing the rate), simultaneously in real time. The experiments have been presented in a course on fundamental physical chemistry offered for bachelor students (according to the Bologna agreement20) of chemistry, biochemistry, and business chemistry in their second year. Within four years about 600 students attended this lecture and saw the demonstration. In the following the connection between chemical kinetics and capillary flow will be briefly summarized. A description of the setup and its application to simple first order as well as consecutive first order processes ensues (cf. Figure 1b).

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CONNECTION BETWEEN RATE LAWS AND CAPILLARY FLOW Consider a simple isomerization in which a molecule A transforms into B (see, e.g., ref 4). Received: July 24, 2017 Revised: January 19, 2018

A

DOI: 10.1021/acs.jchemed.7b00556 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

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Figure 1. Photographs and schemes of the setups used for the lecture demonstrations. (a) Setup for simple first order kinetics. (b) Setup for consecutive first order kinetics. In this photograph and scheme, the second capillary is longer than the first one, so that k1 > k2. For a third demonstration (not shown) the capillary lengths are selected such that k1 ≪ k2. k1

A→B

water mass in the beaker with the product concentration [B]. The flow rate dV/dt, which is defined as a positive quantity, is governed by the Hagen−Poiseuille equation16−19

(1)

Assuming first order kinetics, the rate of consumption of A is related to the concentration [A] by d[A] = −k1[A] dt

dV πr 4 = (p − pE ) dt 8ηl F

(2)

with k1 being the first order rate constant. The integrated rate law describes the concentration decrease as a function of time [A] = [A]0 e−k1t

Here, r is the inner radius and l the length of the capillary (cf., Figure 1). η denotes the viscosity of water, and pF and pE are the pressures at the front and the end of the capillary, respectively. The pressure drop along the capillary (pF − pE > 0) due to friction can be related to the hydrostatic pressure of the water column in the glass tube given by ρgh

(3)

Here, [A]0 represents the initial concentration of the reactant A. The initial concentration of product [B]0 is assumed to be zero. On the basis of mass balance the product concentration [B] has to rise according to [B] = [A]0 (1 − e−k1t )

(5)

pF − pE = ρgh

(6)

ρ is the density of water, g the gravitational acceleration, and h the height of the water column. The flow rate dV/dt can be also expressed as the change of that height multiplied by the inner cross section of the glass tube A,16,18

(4)

As will be shown now, the height of the water column in the capillary setup (Figure 1a) behaves according to first order kinetics. A vertical glass tube filled with water is connected to a horizontally mounted capillary by a flexible tube. The horizontal orientation avoids variations in the hydrostatic pressure along the capillary. Water flowing out of the capillary is collected in a beaker. The beaker stands on an electronic scale interfaced to a computer. The water level in the glass tube can be associated with the reactant concentration [A], and the

dV dh = −A dt dt

(7)

The negative sign accounts for the reduction of the height h during the flow. Combining eqs 5−7) yields a differential equation B


Journal of Chemical Education −πr 4ρg dh = h = −kch dt 8ηlA

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mass in the tube can also be generated (red ●, concentration [A]). This mass is related to the height of the water column. The relation between this height and the flow rate given by the slopes (red lines) is also illustrated in Figure 2. Initially, the water level is of course maximal. This results in the highest slope and thereby flow rate. For the water level decreased to half of its initial value, the slope is also reduced by a factor of 2. By relating further water heights at a given time with the respective slopes of the curve (see Figure 2), the proportionality between the two quantities can be established. This proportionality is the essence of a first order process. Its time dependence is given by a decaying exponential (see above). Overlaying the experimental data with such an exponential (cf. eq 4) confirms this. Here, a rate constant kc of 0.0225 s−1 was used. The value slightly deviates from the one predicted above. The value of 0.0225 s−1 corresponds to an inner radius r of 0.6125 mm (nominal value 0.6 mm). The deviation is well within the production tolerance of ∼10%. Considering this, there is good agreement between data and prediction which underscores the quality of this “simulator” introduced by D. A. Davenport. With two capillaries and glass tubes, consecutive first order kinetics can also be demonstrated (Figure 1b). These and all other capillaries used in the following were cut from the same rod, so that one can rely on the same inner radius r (0.6125 mm). The flow out of the first capillary is guided into the second glass tube by a glass funnel. The kinetics scheme now involves two rate constants

(8)

identical in form to the first order rate law (eq 2). The rate constant kc may be computed from the parameters of the experiment. We recommend discussing this mathematical background with the students prior to the demonstrations. With this background students may grasp that first order equations can be derived from physical principles.

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LECTURE DEMONSTRATIONS By selecting the parameters of the capillary and the glass tube, the rate constant kc and thereby the time constant τ = 1/kc can be defined. For the first demonstration described here, a capillary with a nominal inner radius r of 0.6 mm and a length l of 30 cm was selected. Such viscometer capillaries can, for instance, be purchased from Schott AG, Germany. The cross section A of the glass tube amounted to π (5 mm)2. Inserting the viscosity η (1.019 mPa s21) and density ρ (0.998 g/cm321) of water for the temperature during the experiments (19.3 °C) yields a predicted rate constant kc of 0.021 s−1, i.e., τ = 48.2 s. In the flexible tube, which should form a loop, a glassware valve blocks the flow. The setup is filled with colored water. The scale EJ200 of A&D selected for the demonstration comes with a RS232-interface and the free software WinCT-RsWeight 5.12 which plots the mass of water in the beaker as a function of time. During the lecture this plot and the changing fluid level in the glass tube, recorded by a digital camera, are shown with the aid of a digital projector. Synchronized with the opening of the valve, plotting of the data is started. The plot and the video (for videos of simple and consecutive kinetics see Supporting Information) show that with decreasing water height the flow rate diminishes. To establish a quantitative relation between the flow rate and the filling of the glass tube, the recorded curve is inspected in detail. The curve gives the increasing water mass, representing the product concentration [B], as a function of time in Figure 2 (black ●). On the basis of this input and the total water mass (∼24 g here), a plot of the decreasing water

k1

k2

(9)

A→I→B

The concentration of the intermediate I is represented by the water column in the second glass tube. The reading of the scale is now proportional to3 ⎞ ⎛ k1 k2 [B] = [A]0 ⎜ e−k 2t − e−k1t + 1⎟ k 2 − k1 ⎠ ⎝ k 2 − k1

(10)

With capillary lengths l of 10 and 30 cm, one expects rate constants of k1 = 0.0676 s−1 and k2 = 0.0225 s−1. The recorded behavior is in good agreement with the prediction based on eq 10 (see Figure 3). Relying on these two rate constants, plots for the filling of the first and second glass tube may be generated. During the demonstration, it is easy to grasp why the rate for the formation of the product B (water mass in the beaker) is initially zero, reaches a maximum, and then drops again (see plotted slopes in Figure 3). The behavior is perfectly reflected in the filling of the second glass tube. Initially, the second glass tube is empty and the formation rate of B is zero. As this tube fills the rate increases until the maximal level is reached. The water level in the second glass tube then drops, and the rate concomitantly decreases. With a slightly modified setup the concept of the ratedetermining step may be visualized. To this end, a capillary with a length of l1 = 100 cm and one with l2 = 10 cm are inserted. For this selection the rate constants are k1 = 0.00676 s−1 and k2 = 0.0676 s−1, i.e., k1 ≪ k2. The rise of the mass in the beaker for this situation is plotted in Figure 4. During this experiment it becomes obvious that the filling of the second glass tube representing the intermediate I remains low (∼0.1 of the maximum filling of the first tube). The rise of water in the beaker is again well-described by the prediction. On the basis of the prediction, plots for the filling of the two tubes may also be generated. The actual behavior strongly resembles a simple

Figure 2. Water mass in the beaker (product concentration [B]) and in the glass tube (reagent concentration [A]) as a function of time as obtained with the setup depicted in Figure 1a. Data points are overlaid with the expected first order behavior. Tangent segments (red lines) represent the maximal rate at t = 0 (filled tube), the rate for a half-filled tube, and an empty one. C



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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.7b00556. Student handouts (PDF) Video showing experimental setup and demonstration for the simple first order process (AVI) Video showing simulations for consecutive kinetics and the condition k1 > k2 (AVI)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Figure 3. Water mass in the beaker (product concentration [B]), the first glass tube (reagent concentration [A]), and the second glass tube (intermediate concentration [I]) as a function of time as obtained with the setup depicted in Figure 1b, for the condition k1 > k2. The data points are overlaid with the expected consecutive first order behavior. Tangent segments (green lines) represent the rate at t = 0 (empty second tube), the rate for maximal filling of the second tube, and for long times.

Peter Gilch: 0000-0002-6602-9397 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank our former PhD student Lars Czerwinski who was instrumental in getting this project started as well as Klaus Kelbert for technical support.

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REFERENCES

(1) Miller, S. R. Rethinking undergraduate physical chemistry curricula. J. Chem. Educ. 2016, 93 (9), 1536−1542. (2) ACS Committee on Professional Training. ACS Guidelines for Bachelor’s Degree Programs: Physical Chemistry Supplement (2015). https://www.acs.org/content/dam/acsorg/about/governance/ committees/training/acsapproved/degreeprogram/physical-chemistrysupplement.pdf (accessed Jan 2018). (3) Amdur, I.; Hammes, G. G. Chemical Kinetics: Principles and Selected Topics; McGraw-Hill: New York, 1966. (4) Atkins, P. W.; De Paula, J. Atkins’ Physical Chemistry, ninth ed.; Oxford University Press: Oxford; New York, 2010. (5) Engel, T.; Reid, P. J. Physical Chemistry, 3rd ed.; Pearson: London, 2013. (6) Houston, P. L. Chemical Kinetics and Reaction Dynamics; Dover Publications: Mineola, 2006. (7) Upadhyay, S. K. Chemical Kinetics and Reaction Dynamics; Springer Netherlands: Dordrecht, 2006. (8) Myers, R. L. The Basics of Physics; Greenwood Press: Westport, CT, 2006. (9) Giancoli, D. C. Physics: Principles with Applications; Addison Wesley: Boston, 2009. (10) Maskill, H. The Physical Basis of Organic Chemistry; Oxford University Press: Oxford, 1985. (11) Klán, P.; Wirz, J. Photochemistry of Organic Compounds. From Concepts to Practice; Wiley: Chichester, 2009. (12) Connors, K. A. Chemical Kinetics: The Study of Reaction Rates in Solution; VCH: New York, 1990. (13) Birk, J. P. Coffee cup kinetics - general chemistry experiment. J. Chem. Educ. 1976, 53 (3), 195−196. (14) Birk, J. P.; Gunter, S. K. Water dipping kinetics - physical analog for chemical kinetics. J. Chem. Educ. 1977, 54 (9), 557−559. (15) Schultz, E. Dice-shaking as an analogy for radioactive decay and first-order kinetics. J. Chem. Educ. 1997, 74 (5), 505−507. (16) Davenport, D. A. Capillary flow - versatile analog for chemical kinetics. J. Chem. Educ. 1975, 52 (6), 379−381. (17) Coffin, C. C. Experiments simulating first and second order gas reactions. J. Chem. Educ. 1948, 25 (3), 167. (18) Lemlich, R. A kinetic analogy. J. Chem. Educ. 1954, 31 (8), 431.

Figure 4. Water mass in the beaker (product concentration [B]), the first glass tube (reagent concentration [A]), and the second glass tube (intermediate concentration [I]) as a function of time as obtained with the setup depicted in Figure 1b, for the condition k1≪ k2. Data points are overlaid with the simple as well as the consecutive first order behavior.

exponential rise with a rate constant k1. This is what one expects for the first process being rate-determining.

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CONCLUSIONS The described setups are well-suited for qualitative and quantitative simulations of first order chemical kinetics in the lecture hall. Thanks to the simple and fast data acquisition with an electronic scale, quantitative comparisons with integrated rate laws are straightforward. The simulation can also be used as an opportunity to familiarize students with data acquisition and processing using scientific instruments and computers. Students may be given access to this data via the Internet and do the data analysis themselves. Part of this analysis could be fits of the data sets, thereby augmenting the numerical simulations presented in this paper. D



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(19) Lago, R. M.; Wei, J.; Prater, C. D. Demonstrating the dynamic behavior of coupled systems of chemical reactions. J. Chem. Educ. 1963, 40 (8), 395. (20) Pinto, G. The Bologna Process and its impact on university-level chemical education in Europe. J. Chem. Educ. 2010, 87 (11), 1176− 1182. (21) Release on the IAPWS Formulation 2008 for the Viscosity of Ordinary Water Substance Berlin, Germany, http://twt.mpei.ac.ru/ mcs/worksheets/iapws/wspDVPT.xmcd (accessed Jan 2018).

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First Order Kinetics Visualized by Capillary Flow and Simple Data

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