Physico-Geometrical Kinetics of Solid-State Reactions in an

Article pubs.acs.org/jchemeduc

Physico-Geometrical Kinetics of Solid-State Reactions in an Undergraduate Thermal Analysis Laboratory Nobuyoshi Koga,* Yuri Goshi, Masahiro Yoshikawa, and Tomoyuki Tatsuoka Department of Science Education, Graduate School of Education, Hiroshima University, 1-1-1 Kagamiyama, Higashi-Hiroshima 739-8524, Japan S Supporting Information *

ABSTRACT: An undergraduate kinetic experiment of the thermal decomposition of solids by microscopic observation and thermal analysis was developed by investigating a suitable reaction, applicable techniques of thermal analysis and microscopic observation, and a reliable kinetic calculation method. The thermal decomposition of sodium hydrogen carbonate is selected as the suitable reaction for the student experiments, because its physico-geometrical reaction mechanism is approximately described by the twodimensional phase boundary controlled model, and its reaction kinetics are less sensitive to the measurement conditions of thermal analyses. On the basis of simple microscopic observations of this reaction, development of the physico-geometrical reaction model and derivation of the kinetic model function are imposed on students. The kinetic analysis is also performed using the kinetic rate data recorded by a single thermogravimetric run under modulated temperature conditions. This reduces the time spent in the laboratory and enables the application of a two-step kinetic calculation that provides reliable kinetic results. From the results, students can interpret the kinetics, which are closely related to the physico-geometrical characteristics of the reaction. A one-day undergraduate course in a thermal analysis laboratory is proposed from research and trials conducted in our university. KEYWORDS: Upper-Division Undergraduate, Analytical Chemistry, Laboratory Instruction, Physical Chemistry, Instrumental Methods, Kinetics, Mechanism of Reaction, Solid-State Chemistry, Thermal Analysis

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INTRODUCTION The thermal decomposition of solids is an overall kinetic process that proceeds via complex interactions of various heterogeneous phenomena, which is influenced by the morphology of solid reactant, reaction geometry, surface product layer, and solid−gas equilibrium, among others.1−3 The kinetics is usually characterized by the overall kinetics of the movement of reaction interface and its geometric characteristics,1−3 which is called here “physico-geometrical kinetics” to clearly distinguish from the homogeneous kinetics and the kinetic description for heterogeneous reactions using an empirical kinetic model function. Microscopic observations are widely used to characterize specific heterogeneous phenomena that occur during the thermal decomposition of solids,4−7 such as nucleation and growth, formation of surface product layers, removal of gaseous product by diffusion, geometric features of the reaction interface, and advancement of the reaction interface. For undergraduate chemistry courses, student laboratory experiments that utilize various microscopic techniques have been proposed that entail the study of the thermal decomposition of solids to introduce the features of solid-state reactions.8,9 By integrating microscopic observations and kinetic analysis into a laboratory activity, the conceptual and logical understanding of solid-state kinetics is expected to be enhanced.10 The development of such laboratory activities is necessary in chemistry and engineering education, because © 2013 American Chemical Society and Division of Chemical Education, Inc.

physico-geometrical kinetics of solid-state reactions is an important concept in materials science11 and is useful for students in their future research and engineering activities. Thermal analysis is a powerful tool in tracking solid-state reactions,12,13 and it is widely used in undergraduate and graduate student laboratories.14−24 The experimentally resolved shape of thermoanalytical curves recorded using thermogravimetry (TG) and differential scanning calorimetry (DSC) reveals kinetic information about reactions. For solid-state reactions, various kinetic calculation methods using the thermoanalytical measurements under linearly increasing temperatures have been proposed.25 The kinetic analysis of the thermal decomposition of solids by several simple methods has sometimes been applied in student thermal analysis laboratories.26−31 Thus, integration of kinetic analysis using thermal analysis and microscopic observations is a possible solution for designing a student experimental study to understand the heterogeneous nature of the reaction and the kinetics controlled by physico-geometry. However, the correlation of any kinetic results with the physico-geometrical characteristics of the reaction requires systematic investigation.3−7 To realize the proposed student practice, it was necessary to carefully select the solid sample and the reaction. Any thermal decomposition must possess a clear physico-geometrical Published: December 23, 2013 239

dx.doi.org/10.1021/ed400330t | J. Chem. Educ. 2014, 91, 239−245

Journal of Chemical Education

Article

Thermal Decomposition Characteristics

reaction mechanism that can be observed by simple microscopic technique. In addition, the acquisition of thermoanalytical curve for the reaction must not be time consuming, and the kinetic calculations should provide the results with an acceptable reliability. In this study, the thermal decomposition of sodium hydrogen carbonate (NaHCO3) is proposed as a suitable reaction for students to investigate the kinetics and mechanism of the reaction. Physico-geometrical characteristics of the reaction are discussed on the basis of our previous studies,4,32,33 and the simple microscopic observation procedures needed for this student laboratory are introduced. TG under modulated temperature conditions is presented as an appropriate thermoanalytical measurement technique for this reaction. A practical method for calculating the kinetic rate data under these conditions is illustrated, demonstrating the reliability of the results. A one-day laboratory practice (8 h) for an undergraduate chemistry course is proposed on the basis of the above results. An optical microscope (∼100×) available in the student laboratory and a basic TG instrument for educational use are the minimum requirements for the laboratory practice.

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The thermal decomposition of NaHCO3 is a common example used in introducing types of chemical reactions. The endothermic reaction is characterized by a single mass-loss step that simultaneously evolves CO2 and water vapor (Figure S1 in the Supporting Information) and is given by the following reaction: 2NaHCO3(s) → Na 2CO3(s) + CO2 (g) + H 2O(g)

(1)

The reaction is initiated at the surface by nucleation and growth, and a surface product layer of Na2CO3 forms before the degree of reaction α reaches 0.1 (Figure 1B). The surface product layer is aggregates of submicrometer-sized Na2CO3 crystals (Figure 1C). The subsequent reaction proceeds by movement of the reaction interface, produced between the reactant crystal and the surface product layer, toward the center of the columnar crystals. The interstices between the Na2CO3 crystals in the surface product layer are preserved during the reaction, which serves as a possible diffusion channel for the product gases generated at the reaction interface. Kinetic Behavior

The physico-geometrical kinetics of the thermal decomposition of NaHCO3 has been investigated by many researchers as summarized by Koga et al.4 The reported kinetic parameters and kinetic model often show scattered values, 32−139 kJ mol−1, for the apparent activation energy, Ea, and a contracting geometry or a nucleation−growth model with different kinetic exponents. Recently, we reported that the reaction can be characterized by the two-dimensional phase boundary controlled model with an Ea value of ∼100 kJ mol−1.4 This is irrespective of the reactions under different temperature program modes (isothermal or linear nonisothermal) and controlled transformation rate modes,37 as long as appropriate sample and measurement conditions are used for TG data acquisition.4

THERMAL DECOMPOSITION OF SODIUM HYDROGEN CARBONATE

Sodium Hydrogen Carbonate

NaHCO3 is a readily available chemical that is widely used in our everyday lives, for example, in cooking, in medicine, and as a detergent. Reactions of NaHCO3 such as in the Solvay process34,35 are also taught in high school chemistry courses. NaHCO3 powders consisting of columnar crystals, as shown in Figure 1A, are available as a chemical reagent (CAS: 114-55-8), baking powder, or an alkaline detergent. It is often found as columnar single crystals with a monoclinic crystal structure (P21/c, a = 3.51, b = 9.71, c = 8.05, β = 111° 51′).36

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MICROSCOPIC OBSERVATION AND DERIVATION OF KINETIC MODEL The morphology of the reactant crystals and the physicogeometrical reaction mechanism of the thermal decomposition of NaHCO 3 can be revealed by various microscopic techniques.8,9 In the student laboratory, microscopic observations of the reactant, partially decomposed sample particle, and its fractured surface using an optical microscope or a stereomicroscope provide useful information for constructing a physico-geometrical reaction model. Procedures for the microscopic observation and typical examples of the micrographs are described in the Supporting Information. The two-dimensional phase-boundary-controlled model1−3,38 can be deduced from the morphology of the reactant, changes in the surface texture, and the reaction geometry. By assuming an isotropy of the reaction interface shrinkage, the model can be reduced to a cylinder. A schematic illustration of the twodimensional phase boundary controlled model is shown in Figure 2. Based on the physico-geometrical model, the differential kinetic equation at constant temperature can be derived by assuming a constant linear rate for reaction interface advancement.1−3,38 dα = kf (α) dt

Figure 1. Typical SEM images of (A) an NaHCO3 sample particle (100−170 mesh) and (B, C) a particle surface of a partially decomposed NaHCO3 sample prepared by isothermal heating at 383 K for 10 min in flowing N2 (degree of reaction α ≈ 0.25).

n=2 240

with

f (α) = n(1 − α)1 − 1/ n

and (2)



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modulated temperature conditions. We thought that by utilizing this feature, a reliable kinetic analysis of the thermal decomposition of NaHCO3 would be possible within a limited laboratory time period. This is because the measurement conditions for recording the reliable kinetic data for this reaction can be selected from a relatively wide range in comparison with a range of the thermal decomposition of solids. A modern TG research instrument has various temperature program modes including a modulated temperature mode.40 Although sophisticated modulated temperature controls, such as those with a sinusoidal wave, are not available, modulated temperature control with a triangular wave can be employed using a basic TG instrument for educational use in student laboratories. This is done by utilizing a step-by-step program with repeated heating and cooling at the respective constant rates, β, on the basis of an underlying isothermal or linearly increasing temperature program. The heating and cooling rates should be sufficiently low for obtaining reliable kinetic rate data. As examples, the mass-loss data under modulated temperature conditions listed in Table 1 were recorded for the thermal

Figure 2. Physico-geometrical model of the thermal decomposition of NaHCO3 (two-dimensional phase boundary controlled model).

where f(α) and n are the differential kinetic model function expressed using the degree of reaction α and the dimension of reaction interface shrinkage, respectively. Traditional kinetic analysis is based on the rate behavior at a constant temperature, and the temperature dependence of the rate constant, k, is usually expressed by the Arrhenius equation.

⎛ E ⎞ k = A exp⎜ − a ⎟ ⎝ RT ⎠

Table 1. Temperature Programs for the Modulated Temperature TG for the Thermal Decomposition of NaHCO3

(3)

(1) isothermal + triangular wave

where A, R, and T are the preexponential factor, the gas constant, and temperature, respectively. The kinetic equation under nonisothermal conditions should include both the kinetic relationships, that is, conversion and temperature dependences of the reaction rate. The differential kinetic equation is easily derived by combining eqs 2 and 3. ⎛ E ⎞ dα = A exp⎜ − a ⎟f (α) ⎝ RT ⎠ dt

triangular wave run IM-1 IM-2 IM-3

basal T/K

period/min

amplitude/K

393 5 10 393 5 20 393 10 10 (2) linear nonisothermal + triangular wave

(4)

βa/K min−1 4 or −4 8 or −8 2 or −2

triangular wave

Equation 4 is valid for the reaction under any temperature change conditions if the reaction is an ideal single-step reaction occurring at a condition largely different from equilibrium.39 Different kinetic calculation methods have been developed on the basis of eq 4 and are classified into isoconversional methods and single-run methods.25 The development of the physicogeometrical model, on the basis of microscopic observations, and the derivation of the kinetic equations help students in their understanding of the heterogeneous characteristics of the thermal decomposition of solids. The derivation of kinetic equation and general remarks on the calculation methods based on eq 4 are described in the Supporting Information.

a

run

basal β/K min−1

period/min

amplitude/K

βa/K min−1

NM-1 NM-2 NM-3

1 1 1

5 5 10

10 20 10

5 or −3 9 or −7 3 or −1

During temperature modulation.

decomposition of NaHCO3 (initial mass m0 = 5.0 mg, 100−170 mesh) under flowing N2 (80 cm3 min−1) using a basic TG instrument (TGA-50, Shimadzu Co.). The mass-loss traces recorded under conventional isothermal and linear nonisothermal conditions are compared to those recorded under modulated temperature conditions in Figure 3. The time required for measuring the mass-loss traces under modulated temperature conditions is practically the same with those under conventional isothermal and linear nonisothermal conditions at the same T and β, respectively. It should also be noted that any possible phase shifting of the reaction rate modulation from temperature modulation46,47 is negligible in the mass-loss data measured under the conditions listed in Table 1.

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THERMOGRAVIMETRY UNDER MODULATED TEMPERATURE CONDITIONS The ability to employ various temperature control modes is a characteristic of modern thermal analysis.40 The modulated temperature TG is used as a first method to determine the apparent Ea values at different α values,41−45 thus revealing any possible changes in a reaction mechanism as a reaction advances. By applying the temperature modulation to TG, changes in the reaction rate with temperature and as reaction advances at different α values can be simultaneously recorded in a single mass-loss data.46 As the kinetic rate data, characteristics of mass-loss traces under modulated temperature conditions provide significant advantages over the conventional isothermal and linear nonisothermal measurements if the reaction under investigation behaves ideally according to eq 4 under the

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KINETIC ANALYSIS

Data Conversion

The mass-loss curves recorded by TG can be converted to the kinetic rate data of a series (reaction time t, temperature T, α, dα/dt). The α values at different t are obtained from the massloss curve by eq 5 241



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Figure 3. Mass-loss traces for the thermal decomposition of NaHCO3 (100−170 mesh, m0 = 5.0 mg) in flowing N2 (80 cm3 min−1) under (A) an isothermal condition at T = 393 K, (B) a modulated temperature condition (period = 5 min, amplitude = 10 K) based on isothermal measurement at T = 393 K, (C) a linear nonisothermal condition at β = 1 K min−1, and (D) a modulated temperature condition (period = 5 min, amplitude = 10 K) based on linear nonisothermal measurement at β = 1 K min−1.

α=

m0 − m m0 − m f

(5)

where m0, m, and mf are the initial sample mass, the sample mass at t, and the final mass after the mass-loss process is completed, respectively. Isoconversional Relationship

The kinetic rate data for the thermal decomposition of NaHCO3 under modulated temperature with a triangular wave can be analyzed using eq 4. This is because of the universal applicability of the equation to the kinetic rate data under any temperature profile.39,48 Temperature-dependent changes in the reaction rates at different α values can be extracted from the kinetic rate data under modulated temperature conditions. The data points (t, T, α, dα/dt) at the maximum and minimum temperatures, Tmax and Tmin, in each temperature modulation are extracted from the kinetic rate data. Two series of data points at Tmax and Tmin are obtained as shown in Figure 4A. Data points at different fixed α values can be obtained by mathematical interpolation within each Tmax and Tmin series of data points. This mathematical interpolation is easily carried out using scientific graph drawing software or a spreadsheet program with a macro available via the Internet.49,50 By using the two data points at a fixed α values, (1/Tmax, ln(dα/dt)max)α and (1/Tmin, ln(dα/dt)min)α, in the Tmax and Tmin series, the Ea value can be calculated using eq 6 ⎡ RTmaxTmin (dα /dt )max ⎤ Ea, α = ⎢ ln ⎥ ⎣ Tmax − Tmin (dα /dt )min ⎦α

Figure 4. (A) Data points at Tmax and Tmin from each temperature modulation that are extracted from the kinetic rate data shown in Supporting Information Figure S3A and the interpolated lines and (B) the Ea values at different α values determined using eq 6

(Table 2). The value is also comparable with that expected for the thermal decomposition of NaHCO3, ∼100 kJ mol−1. Optimization of Kinetic Parameters and Kinetic Interpretation

The direct application of eq 4 in analyzing the kinetic rate data under modulated temperature conditions permits optimization of all kinetic parameters, including the most appropriate kinetic model function, by nonlinear least-squares analysis.49 For the thermal decomposition of NaHCO3, the general form of phase boundary controlled model with the kinetic exponent n, which corresponds to the interface shrinkage dimension, can be applied as f(α); this is based on the evidence from microscopic observations and the physico-geometrical kinetic model developed by the students.

(6)

Equation 6 is derived from eq 4 and equivalent to the equation used for the rate-jump and temperature-jump methods in kinetic calculations. The Ea values calculated according to eq 6 at different α values slightly fluctuate during the reaction (Figure 4B). However, the Ea values averaged over different α values are relatively consistent among the different thermoanalytical runs 242



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Table 2. Average Ea Values Determined by the Isoconversional Relationship According to Eq 6 (0.1 ≤ α ≤ 0.9) and the Kinetic Parameters Optimized by Nonlinear Least-Squares Analysis According to Eqs 4, 7, and 8 isoconversional run

Ea/kJ mol−1

IM-1 IM-2 IM-3 NM-1 NM-2 NM-3

95 ± 7 103 ± 6 101 ± 10 100 ± 6 95 ± 3 97 ± 13

f (α) = n(1 − α)1 − 1/ n

optimization by nonlinear least-squares analysis Ea/kJ mol−1 98 100 101 104 101 102

± ± ± ± ± ±

A / 109 s−1

1 1 1 1 1 1

1.00 1.95 1.99 5.91 2.72 3.46

± ± ± ± ± ±

0.03 0.06 0.01 0.10 0.14 0.03

R2

n 2.22 2.21 2.29 2.20 2.37 2.25

± ± ± ± ± ±

0.02 0.03 0.03 0.01 0.01 0.01

0.9680 0.9558 0.9700 0.9822 0.9854 0.9838

(7)

The Ea, A, and n values in eqs 4 and 7 are optimized by minimizing the square sum of the residue F when fitting the calculated curve (dα/dt)cal versus time to the experimental curve (dα/dt)exp versus time 2 ⎡⎛ ⎞ ⎛ dα ⎞ ⎤ d α F = ∑ ⎢⎜ ⎟ −⎜ ⎟ ⎥ ⎢⎝ dt ⎠exp,i ⎝ dt ⎠cal, i ⎥⎦ i=1 ⎣ N

(8)

where N is the number of data points in a modulated temperature TG measurement. Parameter optimization can be easily carried out using a general-purpose spreadsheet program with a Solver function such as Microsoft Excel in the student laboratory.51−53 The standard deviation of the optimized parameters can also be calculated using a macro available from the Internet.49,50 It is good practice for students to prepare the spreadsheet used for the kinetic calculations. An example of the spreadsheet is available in the Supporting Information. In such calculations, appropriate initial parameters are always necessary to avoid superficial optimizations due to the local minimum F value.49 In this kinetic calculation, the average Ea value obtained from the isoconversional relationship according to eq 6 can be used as the default Ea value. By this initial setting of the Ea value, the isoconversional relationship is reflected in the subsequent optimization. The n value is set to 1, 2, or 3. The order of the default A value is determined by comparing the experimental and calculated curves on a graph in the spreadsheet. After setting all of the default values, the Solver function is run to minimize the F value. Figure 5 graphically shows typical results of the optimization calculation. The optimized kinetic parameters for the thermal decomposition of NaHCO3 under modulated temperature conditions are also listed in Table 2. The fitting of the calculated curve to the experimental curve is satisfactory, with an R2 value greater than 0.95. The optimized Ea values tend to be closer to the expected Ea value for the thermal decomposition of NaHCO3, ∼100 kJ mol−1, and the variation in the Ea values from the different measurements are reduced in comparison to that determined from the simplified isoconversional method according to eq 6. The kinetic exponent n in the kinetic model function of the phase boundary controlled reaction model is slightly greater than 2, indicative of ideal twodimensional shrinkage of the reaction interface, and agrees with our systematic previous study. The nonintegral n value is interpreted by anisotropy in the linear advancement of the reaction interface, the fractal dimension of interface shrinkage, and the contribution of the reaction from the end surfaces of the crystals.54,55 For the thermal decomposition of NaHCO3, the contribution of the reaction that proceeds from both ends

Figure 5. Kinetic optimization results for the kinetic rate data under (A) modulated temperature condition (period = 5 min, amplitude = 10 K) based on isothermal measurement at T = 393 K and (B) modulated temperature condition (period = 5 min, amplitude = 10 K) based on linear nonisothermal measurement at β = 1 K min−1.

of the columnar crystal to the center of columnar crystal should be considered in addition to the two-dimensional shrinkage of the reaction interface from the edge surfaces. The optimized A values have the same order of magnitude irrespective of the measurement conditions. The kinetic results obtained by these kinetic calculations can be directly correlated to the physico-geometrical reaction model, which is developed through microscopic observations of the students. The determined kinetic exponent of ∼2 in the kinetic model function of the phase boundary controlled model indicates the two-dimensional shrinkage of the reaction interface. On the basis of the reaction geometry, students interpret the physical meaning of the Ea value of ∼100 kJ mol−1 as the apparent value for the linear advancement of the reaction interface, which is controlled by a chemical reaction. For interpreting the apparent meaning of the A value, comparison of the values for samples with different particle sizes is useful because the value changes depending on the particle size; this 243



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ACKNOWLEDGMENTS The present work was supported partially by a grant-in-aid for scientific research (A)(25242015), (C)(25350202, 25350203) and challenging exploratory research (23650511) from the Japan Society for the Promotion of Science.

change can be explained by the different radius of columnar crystals.

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OUTLINE OF STUDENT LABORATORY This thermal analysis laboratory is suitable for students who have studied chemical kinetics in a physical chemistry course and have conducted a fundamental thermal analysis experiment in a laboratory course. From trials in our undergraduate thermal analysis laboratory, we found that a one day course (8 h) is required to facilitate the understanding of the physicogeometrical nature of the reaction kinetics in solid-state reactions. The student practice can be divided into two 4 h sessions, in which the first session is for the kinetic modeling based on the microscopic observation and the second session is for the kinetic analysis of the thermoanalytical curve. Training in a group (four or five members), consisting of guidance, experimentation, practice, and discussion, is effective as a project-based inquiry. The details of proposed student practices are described in the Supporting Information. The student practice uses no hazardous material, and microscopic observation and thermoanalytical measurement are safely carried out as long as the instruments are used according to the manuals.

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CONCLUSION An experimental approach for investing the kinetics and mechanism of the thermal decomposition of NaHCO3 is a suitable subject in a thermal analysis laboratory for an undergraduate chemistry course. The physico-geometrical reaction mechanism is clearly described by the two-dimensional phase boundary controlled reaction and is easily observed using an optical microscope or a stereomicroscope. This enables students to develop the physico-geometrical reaction model and derive the kinetic model function. The reliable kinetic rate data for the reaction can be recorded using TG under relatively wide range of measurement conditions; therefore, the mass-loss measurements under modulated temperature conditions are applicable for recording the kinetic rate data. The kinetic rate data converted from a single TG curve under modulated temperature conditions can be analyzed by a simplified isoconversional analysis and then the subsequent optimization of all kinetic parameters by the nonlinear least-squares analysis. The kinetic results obtained by this two-step kinetic calculation are reproducible, and the expected kinetic parameters and kinetic exponent in the phase boundary controlled model are obtained. On the basis of the microscopic observation and kinetic analysis using TG under modulated temperature conditions, students can physico-geometrically interpret the kinetics of the thermal decomposition of NaHCO3. ASSOCIATED CONTENT

S Supporting Information *

Notes for instructors; instruction for the students; Microsoft Excel spreadsheets for the kinetic calculation with sample data. This material is available via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*N. Koga. E-mail: [email protected]. Notes

The authors declare no competing financial interest. 244



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Physico-Geometrical Kinetics of Solid-State Reactions in an

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