Promoting a concept of fraction-as-measure: A study of the Learning Through Activity research program

Promoting deep understanding of fraction concepts continues to be a challenge for mathematics education. Research has demonstrated that students whose concept of fractions is limited to part-whole have difficulty with advanced fraction concepts. We conducted teaching experiments to study how students can develop a measurement concept of fractions and how task sequences can be developed to promote the necessary abstractions. Building particularly on the work of Steffe and colleagues and aspects of the Elkonin-Davydov curriculum, we focused on fostering student reinvention of a measurement concept of fractions. As a study of the Learning Through Activity research program, our goal was to promote particular activity on the part of the students through which they could abstract the necessary concepts.     

Reactivity of the Indenyl Radical (C9H7) with Acetylene (C2H2) and Vinylacetylene (C4H4)

The reactions of the indenyl radicals with acetylene (C₂H₂) and vinylacetylene (C₄H₄) is studied in a hot chemical reactor coupled to synchrotron based vacuum ultraviolet ionization mass spectrometry. These experimental results are combined with theory to reveal that the resonantly stabilized and thermodynamically most stable 1-indenyl radical (C₉H₇^.) is always formed in the pyrolysis of 1-, 2-, 6-, and 7-bromoindenes at 1500 K. The 1-indenyl radical reacts with acetylene yielding 1-ethynylindene plus atomic hydrogen, rather than adding a second acetylene molecule and leading to ring closure and formation of fluorene as observed in other reaction mechanisms such as the hydrogen abstraction acetylene addition or hydrogen abstraction vinylacetylene addition pathways. While this reaction mechanism is analogous to the bimolecular reaction between the phenyl radical (C₆H₅^.) and acetylene forming phenylacetylene (C₆H₅CCH), the 1-indenyl+acetylene→1-ethynylindene+hydrogen reaction is highly endoergic (114 kJ mol⁻¹) and slow, contrary to the exoergic (−38 kJ mol⁻¹) and faster phenyl+acetylene→phenylacetylene+hydrogen reaction. In a similar manner, no ring closure leading to fluorene formation was observed in the reaction of 1-indenyl radical with vinylacetylene. These experimental results are explained through rate constant calculations based on theoretically derived potential energy surfaces.     

Ab initio investigation on the reaction path and rate for the gas-phase reaction of HO + H2O <--> H2O + OH

This article describes an ab initio investigation on the potential surfaces for one of the simplest hydrogen atom abstraction reactions, that is, HO + H2O <--> H2O + OH. In accord with the findings in the previously reported theoretical investigations, two types of the hydrogen-bonding complexes [HOH--OH] and [H2O--HO] were located on the potential energy surface. The water molecule acts as a hydrogen donor in the [HOH--OH] complex, while the OH radical acts as a hydrogen donor in the [H2O--HO] complex. The energy evaluations at the MP2(FC) basis set limit, as well as those through the CBS-APNO procedure, have provided estimates for enthalpies of association for these complexes at 298 K as -2.1 approximately -2.3 and -4.1 approximately -4.3 kcal/mol, respectively. The IRC calculations have suggested that the [H2O--HO] complex should be located along the reaction coordinate for the hydrogen abstraction. Our best estimate for the classical barrier height for the hydrogen abstraction is 7.8 kcal/mol, which was obtained from the CBS-APNO energy evaluations. After fitting the CBS-APNO potential energy curve to a symmetrical Eckart function, the rate constants were calculated by using the transition state theory including the tunneling correction. Our estimates for the Arrhenius parameters in the temperature region from 300 to 420 K show quite reasonable agreement with the experimentally derived values.     

Theoretical Study on Gas Phase Reactions of OH Hydrogen-Abstraction from Formyl Fluoride with Di erent Catalysts

The mechanisms and kinetics of the gas phase reactions that the hydrogen atom in formyl uoride (FCHO) abstracted by OH in the presence of water, formic acid (FA), or sulfuric acid (SA) are theoretically investigated at the CCSD(T)/6-311++G(3df, 3pd)//M06-2X/6-311++G(3df, 3pd) level of theory. The calculated results show that the barriers of the transition states involving catalysts are lowered to -2.89, -6.25, and -7.76 kcal/mol from 3.64 kcal/mol with respect to the separate reactants, respectively, which re ects that those catalysts play an important role in reducing the barrier of the hydrogen abstraction reaction of FCHO with OH. Additionally, using conventional transition state theory with Eckart tun-neling correction, the kinetic data demonstrate that the entrance channel X FCHO+OH (X=H₂O, FA, or SA) is signi cantly more favorable than the pathway X OH+FCHO. More-over, the rate constants of the reactions of FCHO with OH radical with H₂O, FA, or SA introduced are computed to be smaller than that of the naked OH+FCHO reaction because the concentration of the formed X FCHO or X OH complex is quite low in the atmosphere.     

Laser Flash Photolysis Mechanism of Anthraquinone-2-Sodium Sulfonate in Pyridine Ionic Liquid/Water Mixed System

The photochemical reaction process of anthraquinone-2-sodium sulfonate (AQS) in the mixture of water (H₂O) and N-butylpyridinium tetrafluoroborate ([BPy][BF₄]) was studied using the laser flash photolysis technique. Experimental results show that the excited triplet of AQS (³AQS^*) could react rapidly with H₂O and the transient absorption spectra greatly changed by increasing the volume fraction of the ionic liquid (V_IL) in [BPy][BF₄]/H₂O mixtures. The absorbance at 510 nm increased gradually with increasing V_IL when 0< V_IL< 0.1. By contrast, the absorbance decreased gradually when V_IL>0.1. Otherwise, the absorbance of the band near 380 nm steadily increased. The apparent kinetic parameters of transient species B and ³AQS^* are obtained approximately. ³AQS^* abstracting hydrogen from [BPy]⁺ was also explored. It was deduced that the 350-420 nm band was the superposition of the peaks of ³AQS^* and AQSH^·. The two reactions of ³AQS^* with [BPy][BF₄] and H₂O are a pair of competitive reactions. We also concluded that the entire reaction processes slow down in the case of high [BPy][BF₄] concentrations.     

Direct dynamic study on the hydrogen abstraction reaction CH3CN + OH-->CH2CN + H2O

The reaction of acetonitrile with hydroxyl has been studied using the direct ab initio dynamics methods. The geometries, vibrational frequencies of the stationary points, as well as the minimum energy paths were computed at the BHandHLYP and MP2 levels of theory with the 6-311G(d, p) basis set. The energies were further refined at the PMP4/6-311+G(2df, 2pd) and QCISD(T)/6-311+G(2df, 2pd) levels of theory based on the structures optimized at BHandHLYP/6-311G(d, p) and MP2/6-311G(d, p) levels of theory. The Polyrate 8.2 program was employed to predict the thermal rate constants using the canonical variational transition state theory incorporating a small-curvature tunneling correction. The computed rate constants are in good agreement with the available experimental data.     

DFT and ab initio direct dynamics studies on the hydrogen abstraction reactions of chlorine atoms with CH(4-n)F(n) (n = 1-3)

A direct dynamics study is carried out for the hydrogen abstraction reactions Cl + CH(4-n)F(n) (n = 1-3) in the temperature range of 200-1,000 K. The minimum energy paths (MEPs) of these reactions are calculated at the BH&H-LYP/6-311G(d,p) level, and the energies along the MEPs are further refined at the QCISD(T)/6-311+G(2df,2p) and QCISD(T)/6-311+G(d,p) (single-point) level. The rate constants obtained by using the improved canonical variational transition state theory incorporating small-curvature tunneling correction (ICVT/SCT) are in good agreement with the available experimental results. It is shown that the vibrational adiabatic potential energy curves for these reactions have two barriers, a situation similar to the analogous reactions CH(3)X+Cl (X=Cl, Br). The theoretical results show that for the title reactions the variational effect should not be neglected over the whole considered temperature range, while the small-curvature tunneling effect is only important in the lower temperature range. The effects of fluorine substitution on the rate of this kind of reactions are also examined.     

烷基过氧化氢中氢提取反应类大分子体系的反应能垒与速率常数的精确计算

自由基与分子反应是一类具有负活化能的非基元反应,通常认为是通过反应复合物的两步过程,在大气化学和碳氢燃料燃烧机理中广泛存在,且在理论计算和实验上广泛研究.以碳氢燃料燃烧机理中重要反应类羟基自由基提取烷基过氧化氢α位氢的反应为研究对象,通过量化计算揭示其反应规律,计算得到其精确动力学参数.在所研究反应类中,定义第一步反应复合物的生成反应的标准摩尔吉布斯自由能变化等于零时所对应的温度为其转折温度T_c,并表明了当T >> T_c时可采用稳态近似法处理该类反应体系,得到总包反应速率常数.所有反应涉及的物种几何结构优化和频率分析均在BHandHLYP/6-311G（d,p）水平下得到,并在所研究反应类中选取了5个代表反应,通过CCSD（T）/CBS单点能计算,得到其最高转折温度为195.17 K,远远低于碳氢燃料燃烧模拟通常关注温度范围的最低温度650 K,表明用稳态近似法处理该类负活化能反应体系是合理的.计算还表明,该类反应的过渡态反应中心几何结构守恒,因此可将等键反应方法引入类反应,通过对低水平从头算得到的反应能垒进行校正,以得到高精度的结果.为了验证等键反应方法的可靠性,选取5个反应作为研究对象,将低水平BHandHLYP/6-311G（d,p）的校正结果和高水平CCSD（T）/CBS直接计算的结果进行比较,反应能垒最大绝对偏差由校正前的19.99 kJ·mol^－1降到校正后的1.47 kJ·mol^－1,表明用等键反应方法,只需在低水平从头算水平下就可以得到高水平的计算结果,从而可解决大分子体系精确动力学参数缺乏的问题.利用等键反应方法计算了20个反应的反应能垒,并结合过渡态理论计算得到了总包反应的速率常数,并揭示了该类反应只在低温段呈现负活化能关系.