Physico-Geometrical Kinetics of Solid-State Reactions in an Undergraduate Thermal Analysis Laboratory

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1 Supporting Information for Instructors 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, Kagamiyama, Higashi-Hiroshima, Japan. *Corresponding author, EXPERIMENTAL SECTION Materials Reagent grade sodium hydrogen carbonate (NaHCO 3: Sigma-Aldrich Japan) was sieved into different particle size fractions using stainless sieves (75 mmφ) and used without any further purification. The NaHCO 3 sample was characterized by powder X- ray diffraction, FT-IR spectroscopy, thermogravimetry-differential thermal analysis (TG- DTA), and scanning electron microscopy (SEM) as in our previous study. S1 Sometimes, several micrometer-sized small particles are observed on the surface of the columnar crystals. These are likely to be impurities, such as sodium sesquicarbonate (Na 2CO 3 NaHCO 3 2H 2O), S2 produced during preparation or storage. The samples of the sieve fractions of mesh and mesh were used for the microscopic observation and the thermal analysis in the student practice, respectively. SEM Observations (Figure 1) The NaHCO 3 sample ( mesh, 10 mg) was weighed into a platinum cell (6φ 3 mm) and isothermally heated at 383 K for 10 min in flowing N 2 (80 cm 3 min 1 ) in a TG DTA instrument (DTG50, Shimadzu Co.). The heat-treated sample was immediately cooled to room temperature. The degree of reaction of the thermally treated sample was determined by measuring the mass-loss using a microbalance. The original and thermally treated NaHCO 3 samples were coated with Pt by sputtering using a vacuum deposition instrument (JFC-1600, Jeol, 20 ma, 60s) and observed using a scanning electron microscope (JSM-6510, Jeol). TG-DTA-mass spectroscopy (TG/DTA-MS) TG/DTA MS measurements were carried out using the NaHCO 3 sample ( mesh, 4.89 mg), weighed into a platinum cell (5φ 2.5 mm), on a TG-DTA apparatus (TG8120, Rigaku Co.) connected to a quadrupole mass spectrometer (M-200QA, Anelva Co.). The sample was heated at a heating rate β of 10 K min 1 in flowing He (200 cm 3 min 1 ) for recording the TG-DTA curves, and the mass spectra of the evolved gases were continuously acquired in the range amu (EMSN: 1.0 A; SEM: 1.0 kv). The typical TG/DTA-MS records are shown in Figure S1. S1

2 Supporting Information for Instructors Figure S1. (a) TG DTA curves for the thermal decomposition of NaHCO 3 ( mesh, m 0 = 4.89 mg) at a heating rate β = 10 K min 1 in flowing He (200 cm 3 min 1 ) and (b) mass chromatograms of the evolved gases, m/z18 (H 2O + ) and m/z44 (CO 2 + ). OPTICAL MICROSCOPY OBSERVATIONS (STUDENT EXPERIMENT) Samples with larger-sized particles are preferable for morphological observation using an optical microscope. A thermally treated NaHCO 3 sample is usually provided by the instructor. The sample particles reacted approximately 20% is suitable for the observation. The partially reacted sample particles are prepared using a TG instrument by heating isothermally at K for an appropriate time. The microscope with a magnification ~ 100 can be selected from any available ones in the student laboratory, e.g., biological microscopes, stereomicroscopes, or simple digital microscopes. No further specialized equipment for the microscopy, such as a hot stage and a measuring reticle, is required as the minimum capability. Figure S2 shows the typical optical microscopic view of the original and thermally treated NaHCO 3 ( mesh sieve fraction) observed using a stereomicroscope (SZX7, Olympus). The original NaHCO 3 sample consists of columnar and transparent crystal particles with smooth surfaces (Figure S2a). Impurity, observed as small particles on the surface, can also be observed. The partially decomposed sample by thermal treatment becomes opaque because of surface product layer formation (Figure S2b). Morphology of the surface product layer can be observed with an optical microscope using ~ 100 magnification. If using a stereomicroscope, a cross section of the partially decomposed particle can be exposed by fracturing the particle using a razor blade under the microscope. The reactant crystal covered with the surface product layer and the reaction interface can be observed (Figure S2c). From this, the reaction geometry of the thermal decomposition of NaHCO 3 is easily deduced. S2

3 Supporting Information for Instructors Figure S2. Optical microscopy pictures of the original and thermally treated NaHCO 3 samples. (a) Original NaHCO 3 crystal ( mesh sieve fraction), (b) sample heated to 393 K at 5 K min 1 in flowing N 2 (80 cm 3 min 1 ), and (c) fractured surface of the thermally treated NaHCO 3. DERIVATION OF KINETIC MODEL FUNCTION (STUDENT PRACTICE) From the above microscopic observations, a geometrical model of the thermal decomposition of NaHCO 3 can be developed by simplifying it to a cylinder, as shown in Figure 2 in the main article. Based on this model, the kinetic equation for the twodimensional phase boundary controlled model can be derived. S3-S6 For this cylinder with an original radius r 0 and length, L, the degree of reaction α is expressed by Eq. S ( πr0 πr ) L α = (S1) 2 πr0 L where r is the radius of residual reactant crystal at time t. Assuming that reaction interface advancement is regulated by the chemical reaction, a constant rate law is applied to the linear advancement rate of reaction interface. r r vt = 0 where v is the rate constant for the linear advancement. Substituting Eq. S2 in Eq. S1, the integral kinetic equation at a constant temperature is derived. (S2) S3

4 Supporting Information for Instructors 1/ 2 v g ( α ) = 1 ( 1 α ) = t kt (S3) r = 0 where g(α) and k are the kinetic model function in integral form and the apparent rate constant, respectively. The first derivation of Eq. S3 gives the differential kinetic equation. dα 1/ 2 = k 2( 1 α ) = kf ( α ) (S4) dt Different types of physico-geometrical kinetic model functions listed in Table S1 can be introduced after the students derive the kinetic model function for the phase boundary controlled reaction, R n. Table S1. Typical physico-geometrical kinetic model functions for solid-state reactions S3-S7 Model Phase Boundary Controlled Reaction Nucleation & Growth (Avrami-Erofeev) Symbol R n (n=1, 2, and 3) A m (m=0.5, 1, 1.5, 2, 2.5, 3 and 4) ( ) g( α ) f α α = 0 dα f 1 1/ n n( 1 α ) 1 ( 1 α ) 1/ n m ( α ) 1 1/ m ( 1 α )[ ln( 1 α )] [ ln( 1 α )] 1/ m 1D- Diffusion D 1 2D- Diffusion D 2 3D-Diffusion (Jander) 3D-Diffusion (Ginstring- Brounshtein) D 3 D α 1 ln 1 ( α ) 2 / ( α ) ( 1 α ) 3 3 [ ] 1/ / 3 ( 1 α ) 1 α ( 1 α ) ln( α ) α [ 1 ( 1 α ) ] 1/ 2 2α 1 α 3 ( 1 ) 2 / 3 GENERAL REMARKS ON KINETIC ANALYSIS OF THERMOANALYTICAL DATA Different kinetic calculation methods have been developed on the basis of Eq. 4 and are classified into isoconversional methods and single-run methods. S8 The former is based on multiple thermoanalytical measurements at different T or β, and enables the determination of E a at different α values. S9 Taking the logarithm of Eq. 4, the following equation is obtained. dα ln = ln dt [ Af ( α )] Ea RT (S5) Eq. S5 indicates that, at a fixed α value, a plot of ln(dα/dt) versus reciprocal temperature among the different series of kinetic rate data has a slope of E a/r and an intercept of ln[af(α)]. S10,S11 From the plots at different α values, any variation in the E a value during the reaction can be determined. The observation of constant E a during the S4

5 Supporting Information for Instructors reaction gives evidence for a single-step reaction and supports the use of an ideal kinetic equation, such as Eq. 4. S9 On the basis of these results, the kinetic analysis is further extended to determination of the reaction type illustrated by a kinetic model function, f(α), and the preexponential factor, A, in the Arrhenius equation. S7,S12 As compared to the rather laborious kinetic analysis by the isoconversional method, the single-run method provides a set of kinetic parameters, E a, A, and f(α), from a single thermoanalytical run under nonisothermal conditions. However, a mutually dependent change in the determined kinetic parameters and the possible determination of superficial values have to be considered when it is applied. S13,S14 Several advanced calculation techniques have also been proposed to overcome the intrinsic problems in the single-run method. S15,S16 GENERAL REMARKS ON TG MEASUREMENT To obtain reliable kinetic results from the TG data, the NaHCO 3 crystals have to be sieved before use. This means that assumption of having homogeneously sized reactant particles is valid in the kinetic calculation. If the particle size distribution is wide, the experimentally resolved shape of the TG and derivative TG (DTG) curves is distorted because of the distribution in α value among different reactant particles during the reaction. S17 The thermal decomposition of NaHCO 3 can be accelerated and decelerated by the effect of water vapor and CO 2, respectively. S18 To minimize these effects, the sample mass for the measurements is maintained below 10.0 mg, and the measurements are carried out at moderate reaction rates in flowing inert gas at a sufficiently high flow rate ( cm 3 min 1 ). When the mass-loss measurements are performed under linearly increasing temperatures, the heating rate β should be less than 10 K min 1. MEASUREMENTS OF MODULATED TEMPERATURE TG (STUDENT EXPERIMENT, FIGURE 3) A hanging-type TG instrument (TGA-50, Shimadzu Co.) was used for tracking the thermal decomposition of NaHCO 3 under modulated temperature conditions. The sample ( mesh, 5.0 mg) was weighed in a platinum cell (6 mmφ 2.5 mm) and heated according to the conditions listed in Table 1 in the main article in flowing N 2 (80 cm 3 min 1 ). When baseline fluctuation in recorded TG and DTG curves caused by the temperature modulation is significant, these data are corrected using blank data recorded for an empty platinum cell under the same measurement conditions. DATA CONVERSION The mass-loss trace under modulated temperature conditions is converted to the kinetic rate data using Eq. 5; this is the same as for data acquired under isothermal and linear nonisothermal conditions. Figure S3 shows the typical kinetic rate data under modulated temperature conditions converted from the mass-loss curves shown in Figure 3. S5

6 Supporting Information for Instructors Figure S3. Kinetic rate data converted from the mass-loss traces under modulated temperature conditions. (a) Modulated temperature condition (period = 5 min, amplitude = 10 K) based on isothermal measurement at T = 393 K, from Figure 3b, and (b) modulated temperature condition (period = 5 min, amplitude = 10 K) based on linear nonisothermal measurement at β = 1 K min 1, from Figure 3d. INFORMATION FOR KINETIC CALCULATIONS (STUDENT PRACTICE) An MS-Excel file for kinetic calculation of the kinetic rate data under modulated temperature conditions is available in the Supporting Information. The file consists of four spreadsheets with sample data. Sheet 1: Sample data of the modulated temperature TG shown in Figure 3b of the main article. Sheet 2: Kinetic rate data converted from the data in Sheet 1 and the kinetic rate data at T max and T min. Sheet 3: Interpolated data at different α values from the data at T max and T min and the calculation of E a values by Eq. 6 in the main article. Sheet 4: Optimization of kinetic parameters using Excel Solver. OUTLINE OF STUDENT LABORATORY (1) Introduction (30 min, pp. IS1 2): Several examples of physico-geometrically controlled reactions are introduced with simple demonstrations using phenomena familiar to students. Students are asked to provide other examples and the characteristics of those phenomena. The thermal decomposition of NaHCO 3 is introduced as reaction to be studied. Students discuss the nature of the reaction based on their knowledge and the physico-geometry viewpoints already introduced. The discussion is directed into a discussion of the reaction kinetics and possible experimental methods for determining the kinetic behavior of the reaction. As a result of the discussion, approaches using thermal analysis and microscopic observation are proposed. (2) Thermal Analysis (60 min: pp. IS7 9): The instructor provides guidance for the measurement of kinetic rate data using TG under modulated temperature conditions (30 min). Students prepare for the TG measurement and start it with the S6

7 Supporting Information for Instructors assistance and under the supervision of the instructor (30 min). During automatic TG measurement, the students proceed to steps (3) and (4), while also observing the timely progress of the TG measurement. (3) Microscopic Observation and Construction of the Physico-Geometrical Model (90 min: pp. IS2 6): Students are provided with unreacted and partially decomposed NaHCO 3 samples. They observe the surface morphologies of the samples and the reaction interface by crushing the partially decomposed sample and viewing under an optical microscope (40 min). Surface morphology and reaction geometry characteristics are discussed in a group with the instructor (20 min), keeping the digital micrographs taken during the observations as reference. Based on this discussion, students are asked to develop a schematic model of the reaction process. With guidance from the instructor, if necessary, the kinetic model function for the phase boundary controlled reaction is obtained by each student (30 min). (4) Preparation for Kinetic Calculation (150 min: pp. IS10 13): After deriving the fundamental kinetic equation, Eq. 4, the kinetic calculation method using the massloss data under modulated temperature conditions is presented (60 min). The relationship to the conventional kinetic calculation method for homogeneous reactions, which is known to students, is emphasized. Then, the students work on developing the four spreadsheets for the kinetic calculation. This consists of conversion of the mass-loss data to the kinetic rate data using Eq. 5, extraction of data points at T max and T min and the interpolation at different α, determination of E a values at different α values using Eq. 6, and optimization of kinetic parameters by the nonlinear least squares analysis based on Eqs. 4, 7, and 8. The spreadsheets are checked using sample mass-loss data, obtained under modulated temperature condition, provided to the students (90 min). (5) Kinetic Calculation and Interpretation (30 min: pp. IS10 13): The spreadsheets can now be used to analyze the data collected in step (2) (15 min). The determined kinetic parameters are compared among the students and possible interpretation of the kinetic parameters is discussed with reference to the kinetic model derived by students themselves (15 min). (6) Concluding Discussion (30 min: pp. IS14 15): The students verbally describe the results of their kinetic analysis and their interpretation to the instructor. Their understanding is further enhanced through answering questions posed by the instructor. Students are requested to submit a laboratory report within one week. PREPARATION FOR MICROSCOPIC OBSERVATION (FOR ONE GROUP) Chemical Reagent Reagent Remark NaHCO 3(s) (ca. 10 mg) CAS The sample particles sieved to a particular particle size fraction are used ( mesh or larger particle size fraction). Partially decomposed NaHCO 3(s) (ca. 10 mg) The sample is prepared by heating NaHCO 3(s) isothermally at 383 K in TG for an appropriate time. The sample decomposed to a degree of reaction of α 0.2 is preferred. S7

8 Supporting Information for Instructors Apparatus Apparatus Optical microscope (1) Digital Camera (1) Glass slide (1) Tweezers (1) Razor blade (1) Remark or Stereomicroscope A digital camera is useful to record the microscopic views of the samples. The partially decomposed sample particle is fractured using a razor blade under a microscope to observe the cross-sectional surface. PREPARATION FOR TG MEASUREMENT Chemical Reagent Reagent Remark NaHCO 3(s) (ca. 10 mg) CAS Sample particles sieved to a particular particle size fraction are used ( mesh or so). Nitrogen gas An inert gas is used in the TG measurement for purging the evolved gases during the thermal decomposition. Apparatus Apparatus Electronic micro-balance (1) Sample cell for TG measurement (1) Spatula (1) Tweezers (1) TG instrument with data acquisition system Remark Minimum reading ~0.1 mg A sample pan with an appropriate size for TG instrument (material: aluminum, stainless, vitreous silica, or platinum) An instrument capable of multi-step temperature program (at least 50 steps) is required. REFERENCES FOR SUPPORTING INFORMATION S1. Koga, N.; Maruta, S.; Kimura, T.; Yamada, S. Phenomenological Kinetics of the Thermal Decomposition of Sodium Hydrogencarbonate. J. Phys. Chem. A 2011, 115, S2. Kimura, T.; Koga, N. Thermal Dehydration of Monohydrocalcite: Overall Kinetics and Physico-Geometrical Mechanisms. J. Phys. Chem. A 2011, 115, S3. Šesták, J. Thernophysical Properties of Solids: Their Measurements and Theoretical Thermal Analysis; Elsevier: Amsterdam, S4. Galwey, A. K. Structure and Order in Thermal Dehydrations of Crystalline Solids. Thermochim. Acta 2000, 355, S5. Koga, N.; Tanaka, H. A Physico-Geometric Approach to the Kinetics of Solid-State Reactions as Exemplified by the Thermal Dehydration and Decomposition of Inorganic Solids. Thermochim. Acta 2002, 388, S8

9 Supporting Information for Instructors S6. Khawam, A.; Flanagan, D. R. Solid-State Kinetic Models: Basics and Mathematical Fundamentals. J. Phys. Chem. B 2006, 110, S7. Málek, J. The Kinetic Analysis of Non-Isothermal Data. Thermochim. Acta 1992, 200, S8. Khawam, A.; Flanagan, D. R. Basics and Applications of Solid-State Kinetics: A Pharmaceutical Perspective. J. Pharm. Sci. 2006, 95, S9. Koga, N. Ozawa's Kinetic Method for Analyzing Thermoanalytical Curves: History and Theoretical Fundamentals. J. Therm. Anal. Calorim Doi: /s S10. Friedman, H. L. Kinetics of Thermal Degradation of Cha-Forming Plastics from Thermogravimetry, Application to a Phenolic Plastic. J. Polym. Sci. C 1964, 6, S11. Tatsuoka, T.; Koga, N. Energy Diagram for the Catalytic Decomposition of Hydrogen Peroxide. J. Chem. Educ. 2013, 90, S12. Gotor, F. J.; Criado, J. M.; Málek, J.; Koga, N. Kinetic Analysis of Solid-State Reactions: The Universality of Master Plots for Analyzing Isothermal and Nonisothermal Experiments. J. Phys. Chem. A 2000, 104, S13. Koga, N.; Šesták, J.; Málek, J. Distortion of the Arrhenius Parameters by the Inappropriate Kinetic Model Function. Thermochim. Acta 1991, 188, S14. Koga, N. A Review of the Mutual Dependence of Arrhenius Parameters Evaluated by the Thermoanalytical Study of Solid-State Reactions: The Kinetic Compensation Effect. Thermochim. Acta 1994, 244, S15. Serra, R.; Nomen, R.; Sempere, J. The Non-Parametric Kinetics: A New Method for the Study of Thermoanalytical Data. J. Therm. Anal. Calorim. 1998, 52, S16. Perez-Maqueda, L. A.; Criado, J. M.; Sanchez-Jimenez, P. E. Combined Kinetic Analysis of Solid-State Reactions: A Powerful Tool for the Simultaneous Determination of Kinetic Parameters and the Kinetic Model without Previous Assumptions on the Reaction Mechanism. J. Phys. Chem. A 2006, 110, S17. Koga, N.; Criado, J. M. Kinetic Analyses of Solid-State Reactions with a Particle-Size Distribution. J. Am. Ceram. Soc. 1998, 81, S18. Yamada, S.; Koga, N. Kinetics of the Thermal Decomposition of Sodium Hydrogen Carbonate Evaluated by Controlled Rate Evolved Gas Analysis Coupled With Thermogravimetry. Thermochim. Acta 2005, 431, S9

10 Thermal Analysis Laboratory Kinetics of Thermal Decomposition of Sodium Hydrogen Carbonate 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, Kagamiyama, Higashi-Hiroshima, Japan. INTRODUCTION Sodium hydrogen carbonate (NaHCO 3) is widely used in our everyday lives in cooking, in medicine, and as a detergent, to name just a few of its use. NaHCO 3 powders are often determined as columnar single crystals with a monoclinic crystal structure. On heating, NaHCO 3 decomposes to sodium carbonate (Na 2CO 3) evolving CO 2 and H 2O. The reaction is given by the following chemical equation: 2NaHCO 3(s) Na 2CO 3(s) + CO 2(g) + H 2O(g) (1) Figure 1 shows thermogravimetry (TG) differential thermal analysis (DTA) curves for the thermal decomposition of NaHCO 3, together with mass-chromatograms of evolved H 2O and CO 2. Thermal decomposition proceeds in a single mass loss step at a temperature higher than 373 K, indicating 36.9% mass loss, as expected from Eq.(1). The reaction is endothermic and simultaneously evolves H 2O and CO 2 during the mass loss process. Figure 1. (a) TG DTA curves for the thermal decomposition of NaHCO 3 ( mesh, m 0 = 4.89 mg) at β = 10 K min 1 in flowing He (200 cm 3 min 1 ) and (b) mass chromatograms of the evolved gases, m/z18 (H 2 O + ) and m/z44 (CO 2 + ). IS1

11 Thermal decomposition of solids involves several stages, including the chemical act of breaking bonds, followed by the destruction of the reactant crystal lattice, formation of nuclei of the solid product and growth of these nuclei, adsorption desorption of gaseous products, diffusion of gases, and heat transfer. Usually, the reactant surface is most reactive and the reaction initiates on the surfaces by the nucleation and growth of solid product. The subsequent reaction proceeds by the advancement of the reaction interface, characterized as a zone of locally enhanced reactivity located at the reactant/product contact. Thus, the kinetics of the thermal decomposition of solids is controlled by the geometry of the reaction interface advancement and the physico-chemical events at the reaction interface. In this laboratory, the kinetics of the thermal decomposition of NaHCO 3 is investigated in view of the reaction geometry and the kinetics of reaction interface advancement. A geometric reaction model and kinetic equation for the thermal decomposition of NaHCO 3 are derived through microscopic observations of the changes in morphology of a sample particle and the texture of the surface. A TG measurement under a modulated temperature condition is used to record kinetic rate data. By analyzing the kinetic rate data using the derived kinetic equation, the physico-geometric features of the thermal decomposition of NaHCO 3 are determined. MICRSCOPIC OBSERVATION Through microscopic observations of the sample morphology, the surface texture of the partially decomposed sample, the texture of cross-sectional surface of the partially decomposed sample, and the geometric features of the thermal decomposition of NaHCO 3 are revealed. <Samples> Sample Remark NaHCO 3 (s) CAS Sample particles sieved to a particular particle size fraction are used ( mesh or larger particle size fraction). Partially decomposed NaHCO 3 (s) The sample was prepared by isothermally heating NaHCO 3 (s) at 383 K in TG for an appropriate time for decomposition to occur to the degree of reaction of α 0.2. <Apparatus> Apparatus Optical microscope (or Stereomicroscope) (1) Digital Camera (1) Glass slide (1) Tweezers (1) Razor blade (1) Remark Operate according to the manual. Microscopic views of the samples are recorded using a digital camera. The partially decomposed sample particle is fractured using a razor blade under a microscope to observe the cross-sectional surface. IS2

12 <Procedure> (1) Observe the morphology and surface texture of a NaHCO 3 sample particle under a microscope and either record the digital image or sketch the features. (2) The same is performed for the partially decomposed NaHCO 3 sample particle. Carefully observe the changes in the surface texture and structure owing to the thermal decomposition. (3) Fracture the partially decomposed NaHCO 3 sample particle using a razor blade under the microscope. Observe the cross-sectional surface with special attention to the surface product layer and the reaction interface. <Results> (1) Morphology and surface texture of the sample particle Digital image or sketch: Description: (2) Surface texture of the partially decomposed sample Digital image or sketch: Description: IS3

13 (3) Texture of the cross-sectional surface of the partially decomposed sample Digital image or sketch: Description: <Discussion> Discuss the geometric features of the thermal decomposition of NaHCO 3 on the basis of the microscopic observations. IS4

14 DERIVATIONS OF A KINETIC MODEL AND KINETIC EQUATION On the basis of the microscopic observations, the kinetic model and kinetic equation for the thermal decomposition of NaHCO 3 are derived. <Derivation> (1) Assume the morphology of the NaHCO 3 particle to be cylindrical (Figure 2). Then, the degree of reaction, α, is expressed by the volume ratio of the product to the initial volume of the reactant using the initial radius, r 0, of the reactant solids; the radius, r, of the reactant at time, t; and the length, L, of the cylinder. α = (2) Figure 2. Physico-geometrical model of the thermal decomposition of NaHCO 3 (two-dimensional phase boundary-controlled model). (2) Assume that the reaction interface advancement is regulated by the chemical reaction. Then, the constant rate low is applied to the linear advancement rate of reaction interface. The r value is expressed as a function of time using the rate constant, v, for linear advancement. r = (3) (3) Substituting Eq. (3) in Eq.(2), the integral kinetic equation for the two-dimensional phase boundary-controlled reaction at a constant temperature is derived. g ( α ) = kt (4) IS5

15 with g ( α ) = (5) and k = (6) where g(α) and k are the kinetic model function in integral form and the rate constant, respectively. (4) The corresponding differential kinetic equation is expressed by: dα = kf dt ( α ) (7) with ( α ) ( α ) 1 dg f = = (8) dα where f(α) is the kinetic model function in differential form. For a given interface shrinkage dimension, n, Eq. (8) is expressed by a general form. f ( α ) = (9) (5) The temperature dependence of k is usually expressed by the Arrhenius relationship. E = a k A RT exp (10) where A, E a, and R are the preexponential factor, the apparent activation energy, and the gas constant, respectively. Combining Eq.(7) with Eq.(10), we obtain the fundamental kinetic equation applicable to the kinetic rate data under any temperature profile. dα = dt (11) IS6

16 THERMOGRAVIMETRY UNDER A MODULATED TEMPERATURE CONDITION For the kinetic analysis of the thermal decomposition of NaHCO 3, a TG measurement under a modulated temperature condition is performed to record the kinetic rate data. <Sample> Sample Remark NaHCO 3 (s) (~10 mg) CAS Sample particles sieved to a particular particle size fraction are used ( mesh or so). <Apparatus> Apparatus Electronic micro-balance (1) Sample cell for TG measurement (1) Spatula (1) Tweezers (1) TG instrument with data acquisition system Remark Minimum reading ~0.1 mg Platinum pan (6φ 2.5 mm) Operate according to the manual. <Procedure> (1) Weigh and record the precise mass of NaHCO 3 (~5.0 mg) in a sample cell for a TG measurement using an electric microbalance. (2) Place the sample in a TG instrument and start the flow of purge gas (nitrogen) at a constant rate suggested by the instructor. (3) Select one of the temperature programs for TG measurement under modulated temperature condition from Table 1 and input the temperature program in a stepby-step manner. (4) After confirming the stability of the sample mass signal in the TG instrument, start the thermoanalytical run. Observe the timely progress of the measurement. (5) After the end of the thermoanalytical run, wait until the furnace is cooled to lower than 323 K. Remove the sample from the TG instrument and stop the purge gas flow. (6) Observe the morphology and surface texture of the sample after performing the TG measurement as in the microscopic observations above. IS7

17 Table 1. Typical temperature programs of temperature modulated TG for the thermal decomposition of NaHCO 3 (1) Isothermal + Triangular wave Run Basal T / K Triangular wave Period / min Amplitude / K β a / K min 1 IM or 4 IM or 8 IM or 2 (2) Linear Nonisothermal + Triangular wave Run Basal β / K min 1 Triangular wave Period / min Amplitude / K β a / K min 1 NM or 3 NM or 7 NM or 1 a during temperature modulation. <Results> Measurement Condition Date Experimenter Instrument Sample Pan Temp. Program Sample Name Sample Mass / mg Sample Remark File Name Data Interval / s Meas.Time / min Atmosphere IS8

18 (1) Plot changes in the measured temperature, sample mass, and derivative mass change against time. Insert Figure. (2) Surface texture of the partially decomposed sample Digital image or sketch: Description: IS9

19 KINETIC CALCULATION By analyzing the experimentally resolved TG and the derivative TG (DTG) curves, the kinetic features of the thermal decomposition of NaHCO 3 are revealed. The kinetic calculation is performed using thermal analysis software attached to the TG instrument and MS-Excel. Excel spreadsheets for the kinetic calculation have to be prepared using the sample spreadsheets provided as a reference. If scientific graph software is available, this software can be used instead of MS-Excel. [1] Data Treatment (thermal analysis software of the TG instrument) (1) If necessary, base-line correction for the TG curve is performed by subtracting the blank data recorded under the same measurement conditions. (2) If necessary, the TG curve is smoothed mathematically. [Caution: Heavy smoothing deforms the kinetic information.] (3) The thermoanalytical data are saved in an ASCII format readable when using MS- Excel. [Sample Data: Sheet1] [2] Conversion to Kinetic Rate Data (MS-Excel or scientific graph drawing software) (1) Open the thermoanalytical data file using MS-Excel. (2) Draw a measured mass vs. time plot as shown in Figure 3. Figure 3. Determination of m 0 and m f. (3) Read the initial time, t 0, and final time, t f, of the mass change. (4) Delete the data points after t f. (5) Delete the data points before t 0 and set the start time to zero by subtracting t 0 from the data. IS10

20 (6) Read the initial mass, m 0, and final mass, m f. (7) Calculate the degree of reaction, α, at different data points using Eq. (12). m0 m = m α, (12) 0 m f where m is the sample mass at a data point. (8) Draw an α vs. time plot and numerically differentiate to obtain (dα/dt) at different data points. (9) Finally, obtain the series of kinetic rate data, (time, temperature, α, dα/dt).[sample Data: Sheet2] [3] Extraction of Data Points (MS-Excel or scientific graph drawing software) [Sample spreadsheet: Sheet2] (1) Draw a (dα/dt) vs. time plot as shown in Figure 4. Figure 4. Extraction of data points at T max and T min. (2) Extract a series of data points at the maximum temperature, T max, in each temperature modulation. (3) Similarly, extract a series of data points at the minimum temperature, T min, in each temperature modulation. (4) As a result, two different series of kinetic rate data, T max and T min series, are obtained. IS11

21 [4] Examination of the Isoconversional Relationship (MS-Excel with a macro or scientific graph drawing Software) [Sample spreadsheet: Sheet3] (1) Using an MS-Excel macro available via the Internet, obtain two series of kinetic rate data at different fixed α values, for example, 0.10 α 0.90 in steps of 0.01, by mathematical interpolation from each series of kinetic rate data at T max and T min generated in steps [3]. (2) Use the data points at fixed α, α fix, in the two series of kinetic rate data, (α fix, (dα/dt) max, T max) and (α fix, (dα/dt) min, T min), to calculate the E a value. On the basis of Eq. (11), the E a value is calculated using Eq.(13). T = T T T ( dα / dt) max ( dα / dt) min α max min E a, α ln (13) max min Calculate the E a values at different fixed α values using Eq.(13). (3) Average the E a values for different fixed α values and calculate the standard error. <Result> E = ± (kj mol 1 ) a [5] Optimization of Kinetic Parameters (MS-Excel with a macro or scientific graph drawing software) [Sample spreadsheet: Sheet4] (1) Use the original kinetic rate data generated in step [2] as the experimental data (time, temperature, α, dα/dt) exp. (2) Set the parameters, E a, A, and n, in Eqs. (9) and (11) as variables to be optimized, and input arbitrary values. (3) Calculate the values of α and (dα/dt) using Eqs. (9) and (11) with the parameters input in step (2) to obtain the series of calculated data (time, temperature, α, dα/dt) cal. 1) Set α = at t 0. 2) Calculate the (dα/dt) value at t 0 using Eqs. (9) and (11) with the parameters input in step (2) and the α value set in step (3)-1). 3) Calculate the subsequent α values of i-th data according to: αi = αi-1 + (dα/dt)i 1 (t i ti 1). 4) Calculate the (dα/dt) values of i-th data using Eqs. (9) and (11) with the parameters input in step (2) and the α ι value. IS12

22 (4) Calculate the F value in Eq. (14). F = N dα dt dα dt = 1 exp, cal, i i i 2 (14) (5) Set the default values of the parameters to be optimized as follows. 1) E a: Use the average E a value determined in step [4]. 2) n: Set n = 2 owing to the two-dimensional shrinkage of the reaction interface observed microscopically. 3) A: Determine the appropriate order of A by graphically comparing the curves of (dα/dt) exp vs. time and (dα/dt) cal vs. time. (6) Optimize the parameters, E a, A, and n, by running Excel Solver to minimize the F value. (7) Calculate the standard error of the optimized kinetic parameters using an Excel macro available via the Internet. <Results> E = ± (kj mol 1 ) a A = ± (s 1 ) n = ± IS13

23 DISCUSSION (1) Summarize the geometric features of the thermal decomposition of NaHCO 3 by considering the initial reaction site, surface reaction, surface product layer, reaction interface, and reaction geometry. (2) Discuss the physico-chemical meaning of the experimentally determined Arrhenius parameters in relation to the physico-geometrical reaction model derived for the thermal decomposition of NaHCO 3. IS14

24 (3) Discuss the possible dependence of experimentally determined A value on the radius of cylindrical sample. FOR READING Fundamental Chemical Kinetics D.A. McQuarrie, J.D. Simon, Physical Chemistry: A Molecular Approach, Univ. Science Books, California, 1997, pp P. Atkins, J. de Paula, Physical Chemistry, 7 th ed., Oxford Univ. Press, Oxford, 2002, pp Solid-State Reactions S.E. Dann, Reactions and Characterization of Solids, RS-C, Cambridge, Microscopic Observation of Solid-Sate Reactions H. Tanaka, N. Koga, and A.K. Galwey, Thermal Dehydration of Crystalline Hydrates: Microscopic Studies and Introductory Experiments to the Kinetics of Solid-State Reactions. J. Chem. Educ. 1995, 72, Thermal Analysis M.E. Brown, Introduction to Thermal Analysis, Kluwer, Dordrecht, IS15