DOCSLIB.ORG

Mathematical Analysis of Chemical Reaction Systems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mathematical Analysis of Chemical Reaction Systems Mathematical Analysis of Chemical Reaction Systems Polly Y. Yu ∗ Gheorghe Craciun y May 29, 2018 Abstract. The use of mathematical methods for the analysis of chemical reaction systems has a very long history, and involves many types of models: deterministic versus stochastic, continuous versus discrete, and homogeneous versus spatially distributed. Here we focus on mathematical models based on deterministic mass-action kinetics. These models are systems of coupled nonlinear differential equations on the positive orthant. We explain how mathematical properties of the solutions of mass-action systems are strongly related to key properties of the networks of chemical reactions that generate them, such as specific versions of reversibility and feedback interactions. 1 Introduction Standard deterministic mass-action kinetics says that the rate at which a reaction occurs is directly proportional to the concentrations of the reactant species. For example, according to mass-action kinetics, the rate of the reaction X + X X is of the form kx x , where xi 1 2 ! 3 1 2 is the concentration of species Xi and k is a positive constant. If we are given a network that contains several reactions, then terms of this type can be added together to obtain a mass- action model for the whole network (see example below). The law of mass-action was first formulated by Guldberg and Waage [39] and has recently celebrated its 150th anniversary [69]. Mathematical models that use mass-action kinetics (or kinetics derived from the law of mass-action, such as Michaelis-Menten kinetics or Hill kinetics) are ubiquitous in chemistry and biology [1, 16, 28, 30, 36, 40, 45, 47, 67, 69]. The possible behaviors of mass-action systems also vary wildly; there are systems that have a single steady state for all choices of rate constants (Figure 2(a)), systems that have multiple steady states (Figure 2(b)), systems that oscillate (Figure 2(c)), and systems (e.g. a version of the Lorentz system) that admit chaotic behavior [68]. To illustrate mass-action kinetics, consider the reaction network (N1) in Figure 1. Accord- arXiv:1805.10371v1 [q-bio.MN] 25 May 2018 ing to mass-action kinetics, the network (N1) gives rise to the following system of differential 4 equations on the positive orthant R>0: ∗Department of Mathematics, University of Wisconsin-Madison; partially supported by NSERC PGS-D and NSF grant 1412643. yDepartment of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin- Madison; partially supported by NSF grant 1412643. 1 k1 X1 2X2 k2 k3 X1 + X3 2X1 (N1) k4 k6 k5 X2 + X4 Figure 1: Example network (N1). dx 1 = k x + k x2 + k x x k x2 2k x2 + k x x dt − 1 1 2 2 3 1 3 − 4 1 − 5 1 6 2 4 dx 2 = 2k x 2k x2 + k x2 k x x (1) dt 1 1 − 2 2 5 1 − 6 2 4 dx 3 = k x x + k x2 + k x x dt − 3 1 3 4 1 6 2 4 dx 4 = k x2 k x x ; dt 5 1 − 6 2 4 where xi = [Xi] is the concentration of species Xi. T At any given time, the concentration vector x(t) = (x1(t); x2(t); : : : ; xn(t)) is a point n n in R>0. Tracing the path over time gives a trajectory in the state space R>0. For example, Figure 2 shows several trajectories of three mass-action systems. For this reason, any con- T centration vector x = (x1; x2; : : : ; xn) is also called a state of the system, and we will refer to it as such. [X2] (a) [X2] (b) [X2] (c) [X1] [X1] [X1] Figure 2: Phase portraits showing possible behaviors of mass-action systems: (a) uniqueness and stability of steady state, (b) bistability, and (c) oscillation. The mass-action systems, with the rate constants labeled on the 0:7 1 1 1 3 2 2 reaction edges, are (a) ? −−! X1 −! X1 +X2 −! ?, (b) X1 +X2 −! X1 )−*− 2X1 and X1 +X2 −! X2 )−*− 2X2, 1 1 (c) a version of the Selkov model, or \Brusselator", whose network (N2) is shown in Figure 3 and rate constants given in Remark 2.5. In vector-based form, this dynamical system can also be written as 2 0x 1 0 11 0 1 1 0 1 1 1 − dBx2C B 2 C 2B 2C B 0 C B C= k1x1B C+ k2x2B− C+ k3x1x3B C dt@x3A @ 0 A @ 0 A @ 1A − x4 0 0 0 0 11 0 21 0 1 1 − − 2B 0 C 2B 1 C B 1C + k4x B C+ k5x B C+ k6x2x4B− C : (2) 1@ 1 A 1@ 0 A @ 1 A 0 1 1 − In order to write down a general mathematical formula for mass-action systems we need to introduce more definitions and notation. The objects that are the source or the target of a reaction are called complexes. For example, the complexes in the network (N1) are X1, 2X2,X1 + X3, 2X1, and X2 + X4. Their 011 001 011 021 001 B0C B2C B0C B0C B1C complex vectors are the vectors B C, B C, B C, B C, and B C, respectively. @0A @0A @1A @0A @0A 0 0 0 0 1 Let us introduce the notation xy = xy1 xy2 xyn (3) 1 2 ··· n T n T n for any two vectors x = (x1; x2; : : : ; xn) R>0 and y = (y1; y2; : : : ; yn) R≥0. Then the 2 2 2 y monomials x1, x2, x1x3;::: , in the reaction rate functions in (2) can be represented as x , where x is the vector of species concentrations and y is the complex vector of the source of the corresponding reaction. For example, X1 +X3 is the source of the reaction X1 +X3 X4, 011 ! B0C its complex vector is B C, and the corresponding reaction rate function in (2) is k3x1x3. @1A 0 The vectors in (2) are called reaction vectors, and they are the differences between the complex vectors of the target and source of each reaction; a reaction vector records the stoichiometry of the reaction. For example, the reaction vector corresponding to X1 + X3 0 11 001 011 ! − B 0 C B0C B0C X4 is B C = B C B C. @ 1A @0A − @1A − 1 1 0 There is a naturally defined oriented graph underlying a reaction network, namely the graph where the vertices are complexes, and the edges are reactions. Therefore, a chemical reaction network can be regarded as a Euclidean embedded graph G = (V; E), where n V R is the set of vertices of the graph, and E V V is the set of oriented edges of G. ⊂ ≥0 ⊂ × For example, (N2) depicted in Figure 3 is the network for a version of the Selkov model 2 for glycolysis. Its Euclidean embedded graph G in R≥0 is shown in Figure 4. 3 Y k1 k3 k7 k2 k4 2X + Y 3X (N2) k8 k6 ∅ X k5 Figure 3: Example network (N2), a version of the Selkov model for glycolysis. It can also be regarded as a version of the \Brusselator". Y 1 • • X • 1• 2 3• Figure 4: The Euclidean embedded graph G of the network (N2) as shown in Figure 3. Given a reaction network with its Euclidean embedded graph G, and given a vector of reaction rate constants k, we can use the notation (3) to write the mass-action system generated by (G; k) as shown in (4) dx X y 0 = ky!y0 x y y : (4) dt − y!y02G The stoichiometric subspace S of a reaction network G is the vector space spanned by its reaction vectors: S = span y0 y : y y0 G : (5) Rf − ! 2 g n n The stoichiometric compatibility class of x0 R> is the set (x0+S)>0 = (x0+S) R> , 2 0 \ 0 i.e., the intersection between the affine set x0 + S and the positive orthant. Note that the solution x(t) of the mass-action system with initial condition x0 is confined to (x0 + S)>0 for all future time, i.e., each stoichiometric compatibility class is a forward invariant set [30]. We say that a network or a graph G is reversible if y0 y is a reaction whenever ! y y0 is a reaction. We say that G is weakly reversible if every reaction is part of an ! oriented cycle, i.e., each connected component of the graph G is strongly connected. The network (N1) is weakly reversible, while (N2) is reversible. When the underlying graph G is weakly reversible, we will see that the solutions of the mass-action system are known (or conjectured) to have many important properties, such as existence of positive steady states for all parameter values, persistence, permanence, and if the network satisfies some additional assumptions, also global stability [17, 18, 23, 30, 40, 42]. For example, in the next section we will see that the mass-action systems generated by network (N1) and any values of rate constants are globally stable, i.e., there exists a globally attracting steady state within each stoichiometric compatibility class. 4 2 Results Inspired by Thermodynamic Principles The idea of relating chemical kinetics and thermodynamics has a very long history, starting with Wegscheider [70], and continuing with Lewis [53], Onsager [58], Wei and Prater [71], Aris [8], Shear [63], Higgins [41], and many others. For example, the notion of \detailed- balanced systems" was studied in depth, and this notion has a strong connection to the thermodynamical properties of microscopic reversibility which goes back to Boltzmann [12, 13, 38].
Load more

Recommended publications
  • The Practice of Chemistry Education (Paper)
    CHEMISTRY EDUCATION: THE PRACTICE OF CHEMISTRY EDUCATION RESEARCH AND PRACTICE (PAPER) 2004, Vol. 5, No. 1, pp. 69-87 Concept teaching and learning/ History and philosophy of science (HPS) Juan QUÍLEZ IES José Ballester, Departamento de Física y Química, Valencia (Spain) A HISTORICAL APPROACH TO THE DEVELOPMENT OF CHEMICAL EQUILIBRIUM THROUGH THE EVOLUTION OF THE AFFINITY CONCEPT: SOME EDUCATIONAL SUGGESTIONS Received 20 September 2003; revised 11 February 2004; in final form/accepted 20 February 2004 ABSTRACT: Three basic ideas should be considered when teaching and learning chemical equilibrium: incomplete reaction, reversibility and dynamics. In this study, we concentrate on how these three ideas have eventually defined the chemical equilibrium concept. To this end, we analyse the contexts of scientific inquiry that have allowed the growth of chemical equilibrium from the first ideas of chemical affinity. At the beginning of the 18th century, chemists began the construction of different affinity tables, based on the concept of elective affinities. Berthollet reworked this idea, considering that the amount of the substances involved in a reaction was a key factor accounting for the chemical forces. Guldberg and Waage attempted to measure those forces, formulating the first affinity mathematical equations. Finally, the first ideas providing a molecular interpretation of the macroscopic properties of equilibrium reactions were presented. The historical approach of the first key ideas may serve as a basis for an appropriate sequencing of
    [Show full text]
  • Analyzing Binding Data UNIT 7.5 Harvey J
    Analyzing Binding Data UNIT 7.5 Harvey J. Motulsky1 and Richard R. Neubig2 1GraphPad Software, La Jolla, California 2University of Michigan, Ann Arbor, Michigan ABSTRACT Measuring the rate and extent of radioligand binding provides information on the number of binding sites, and their afÞnity and accessibility of these binding sites for various drugs. This unit explains how to design and analyze such experiments. Curr. Protoc. Neurosci. 52:7.5.1-7.5.65. C 2010 by John Wiley & Sons, Inc. Keywords: binding r radioligand r radioligand binding r Scatchard plot r r r r r r receptor binding competitive binding curve IC50 Kd Bmax nonlinear regression r curve Þtting r ßuorescence INTRODUCTION A radioligand is a radioactively labeled drug that can associate with a receptor, trans- porter, enzyme, or any protein of interest. The term ligand derives from the Latin word ligo, which means to bind or tie. Measuring the rate and extent of binding provides information on the number, afÞnity, and accessibility of these binding sites for various drugs. While physiological or biochemical measurements of tissue responses to drugs can prove the existence of receptors, only ligand binding studies (or possibly quantitative immunochemical studies) can determine the actual receptor concentration. Radioligand binding experiments are easy to perform, and provide useful data in many Þelds. For example, radioligand binding studies are used to: 1. Study receptor regulation, for example during development, in diseases, or in response to a drug treatment. 2. Discover new drugs by screening for compounds that compete with high afÞnity for radioligand binding to a particular receptor.
    [Show full text]
  • Fugacity Nov 2 2011.Pdf
    Fugacity - Wikipedia, the free encyclopedia 頁 1 / 5 Fugacity From Wikipedia, the free encyclopedia In chemical thermodynamics, the fugacity (f) of a real gas is an effective pressure which replaces the true mechanical pressure in accurate chemical equilibrium calculations. It is equal to the pressure of an ideal gas which has the same chemical potential as the real gas. For example, [1] nitrogen gas (N2) at 0°C and a pressure of 100 atm has a fugacity of 97.03 atm. This means that the chemical potential of real nitrogen at a pressure of 100 atm has the value which ideal nitrogen would have at a pressure of 97.03 atm. Fugacities are determined experimentally or estimated for various models such as a Van der Waals gas that are closer to reality than an ideal gas. The ideal gas pressure and fugacity are related through the dimensionless fugacity coefficient .[2] For nitrogen at 100 atm, the fugacity coefficient is 97.03 atm / 100 atm = 0.9703. For an ideal gas, fugacity and pressure are equal so is 1. The fugacity is closely related to the thermodynamic activity. For a gas, the activity is simply the fugacity divided by a reference pressure to give a dimensionless quantity. This reference pressure is called the standard state and normally chosen as 1 atmosphere or 1 bar, Again using nitrogen at 100 atm as an example, since the fugacity is 97.03 atm, the activity is just 97.03 with no units. Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure.
    [Show full text]
  • Chemical Kinetics I: Basics
    2 CHEMICAL KINETICS I: BASICS c1 In the previous chapter, we c2discussed some the fundamental biological pro- cesses c3undertaken by cells such as transcription and translation. The purpose of this chapter is to introduce the basic concepts needed to model the dynamics of such processes. c4These next two chapters explain how to describe chemical re- actions mathematically, both at a deterministic level where stochastic effects are ignored, as well as probabilistically where stochasticity is explicitly incorporated. 2.1 Law of mass action Consider a reaction where two kinds of molecules, c5 A and B, irreversibly re- act to produce a third kind c6of molecule, C. Schematically, such a reaction is represented as k A+B C. (2.1) ! The parameter k is the rate of the reaction. c7In general, kinetic parameters such as k depend on the environment through thermodynamic quantities such as the pressure and temperature. However, since cells often operate in environments where these quantities do not vary much, for simplicity, we will neglect these dependencies in what follows. According to the law of mass action,the rate of increase of the concentration of the product is given by d[C] = k[A][B] (2.2) dt where we follow the standard convention in chemistry texts: the concentration of the chemical X is represented by [X]. Note that the accompanied decrease of the concentrations of A and B is given by the same expression: k[A][B]. Namely, d[A] d[B] = = k[A][B] (2.3) dt dt − c1Pankaj: This is a test c2Pankaj: have introduced some of the basic c3Pankaj: Text added.
    [Show full text]
  • 1301 Dynamic Equilibrium, Keq, and the Mass Action Expression
    1301 Dynamic Equilibrium, Keq, and the Mass Action Expression The Equilibrium Process Dr. Fred Omega Garces Chemistry 111 Miramar College 1 Equilibrium 05.2015 Concept of Equilibrium & Mass Action Expression Extent of a Chemical Reaction Many reactions do not convert 100% of reactants to products. There is often a point in a reaction when the products will back react to form reactants. The extent of the reaction i.e., 20% or 80%, can be determined by measuring the concentration of each component in solution. In general the extent of the reaction is a function of: Temperature, Concentration and degree of organization which is monitored by some constant value called the - The equilibrium constant (Keq). 2 Equilibrium 05.2015 (Dynamic) Equilibrium Chemical Equilibrium is a dynamic state in which the rates of the forward and the reverse reaction are equal. A teeter-totter is an analogy of a system in balance. The left is balanced with the right. In a chemical reaction this means that the Left (reactant) is changing at the same rate as the right (products). Reactant Product 3 Equilibrium 05.2015 How Equilibrium is achieved Consider the reaction A → B Forward A → B rate = kf[A] Reverse B → A rate = kr[B] Overall A D B rate forward (kf[A]) = rate reverse (kr[B]) Example: N2O4 + E D 2 NO2 Effect of temperature on the N2O4/NO2 equilibrium. The tubes contain a mixture of NO2 and N2O4 . As predicted by Le Chatelier’s principle, the equilibrium favors colorless N2O4 at lower temperatures, but shifts to the darker brown NO2 at higher temperature.
    [Show full text]
  • Math for Biology - an Introduction
    Math for Biology - An Introduction Terri A. Grosso Outline Math for Biology - An Introduction Differential Equations - An Overview The Law of Terri A. Grosso Mass Action Enzyme Kinetics CMACS Workshop 2012 January 6, 2011 Terri A. Grosso Math for Biology - An Introduction Math for Biology - An Introduction Terri A. Grosso Outline 1 Differential Equations - An Overview Differential Equations - An Overview The Law of 2 The Law of Mass Action Mass Action Enzyme Kinetics 3 Enzyme Kinetics Terri A. Grosso Math for Biology - An Introduction Differential Equations - Our Goal Math for Biology - An Introduction Terri A. Grosso Outline We will NOT be solving differential equations Differential Equations - An Overview The tools - Rule Bender and BioNetGen - will do that for The Law of us Mass Action This lecture is designed to give some background about Enzyme Kinetics what the programs are doing Terri A. Grosso Math for Biology - An Introduction Differential Equations - An Overview Math for Biology - An Introduction Terri A. Grosso Differential Equations contain the derivatives of (possibly) unknown functions. Outline Differential Represent how a function is changing. Equations - An Overview We work with first-order differential equations - only The Law of Mass Action include first derivatives Enzyme Generally real-world differential equations are not directly Kinetics solvable. Often we use numerical approximations to get an idea of the unknown function's shape. Terri A. Grosso Math for Biology - An Introduction A few solutions. Figure: Some solutions to f 0(x) = 2 Differential Equations - Starting from the solution Math for 0 Biology - An A differential equation: f (x) = C Introduction Terri A.
    [Show full text]
  • Chapter 14. CHEMICAL EQUILIBRIUM
    Chapter 14. CHEMICAL EQUILIBRIUM 14.1 THE CONCEPT OF EQUILIBRIUM AND THE EQUILIBRIUM CONSTANT Many chemical reactions do not go to completion but instead attain a state of chemical equilibrium. Chemical equilibrium: A state in which the rates of the forward and reverse reactions are equal and the concentrations of the reactants and products remain constant. ⇒ Equilibrium is a dynamic process – the conversions of reactants to products and products to reactants are still going on, although there is no net change in the number of reactant and product molecules. For the reaction: N2O4(g) 2NO2(g) N 2O4 Forward rate concentration NO Rate 2 Reverse rate time time The Equilibrium Constant For a reaction: aA + bB cC + dD [C]c [D]d equilibrium constant: Kc = [A]a [B]b The equilibrium constant, Kc, is the ratio of the equilibrium concentrations of products over the equilibrium concentrations of reactants each raised to the power of their stoichiometric coefficients. Example. Write the equilibrium constant, Kc, for N2O4(g) 2NO2(g) Law of mass action - The value of the equilibrium constant expression, Kc, is constant for a given reaction at equilibrium and at a constant temperature. ⇒ The equilibrium concentrations of reactants and products may vary, but the value for Kc remains the same. Other Characteristics of Kc 1) Equilibrium can be approached from either direction. 2) Kc does not depend on the initial concentrations of reactants and products. 3) Kc does depend on temperature. Magnitude of Kc ⇒ If the Kc value is large (Kc >> 1), the equilibrium lies to the right and the reaction mixture contains mostly products.
    [Show full text]
  • Chapter 5 Conservation of Mass: Chemistry, Biology, and Thermodynamics
    Chapter 5 Conservation of Mass: Chemistry, Biology, and Thermodynamics 5.1 An Environmental Pollutant Consider a region in the natural environment where a water borne pollutant enters and leaves by stream flow, rainfall, and evaporation. A plant species absorbs this pollutant and returns a portion of it to the environmentafter death. A herbivore species absorbs the pollutant by eating the plants and drinking the water, and it return a portion of the pollutant to the environ- ment in excrement and after death. We are given (by ecologists) the initial concentrations of the pollutant in the ambient environment, the plants, and the animals together with the rates at which the pollutant is transported by the water flow, the plants and the animals. A basic problem is to determine the concentrations of the pollutant in the water, plants, and animals as a function of time and the parameters in the system. Figure 5.1 depicts the transport pathways for our pollutant, wherethe state variable x1 denotes the pollutant concentration in the environment, x2 its concentration in the plants, and x3 its concentration in the herbivores. These three state variables all have dimensions of mass per volume (in some consistent units of measurement). The pollutant enters the environment at the rate A (with dimensions of mass per volume per time) and leaves at the rate ex1 (where e has dimensions of inverse time). Likewise the remaining 37 38 Conservation of Mass A b Plants Animals Environment c x1 x2 f x3 a e d Figure 5.1: Schematic representation of the transport of an environmental pollutant.
    [Show full text]
  • Chemical Reaction Kinetics: Mathematical Underpinnings
    Molecular Life Sciences DOI 10.1007/978-1-4614-6436-5_564-1 # Springer Science+Business Media New York 2014 Chemical Reaction Kinetics: Mathematical Underpinnings John W. Cain* Department of Mathematics and Computer Science, University of Richmond, Richmond, VA, USA Synopsis Mathematical modeling and simulation of biochemical reaction networks facilitates our understand- ing of metabolic and signaling processes. For closed, well-mixed reaction systems, it is straightfor- ward to derive kinetic equations that govern the concentrations of the reactants and products. The usual way of deriving kinetic equations involves application of the principle of conservation of mass in conjunction with the law of mass action. Here, examples of kinetic models for several basic processes are discussed. Introduction A century has passed since Michaelis and Menten described a mechanism for enzyme- mediated conversion of a substrate into a product. The kinetics of such biochemical reaction processes can be analyzed mathematically and simulated on computers, usually by appealing to the law of mass action. Kinetic models are typically presented as systems of differential equations (DEs) or contin- uous time Markov chains (“▶ Mathematical Models in the Sciences”). There are advantages to both types of models, but in what follows, the former are developed because the equations are (i) easy to write down by applying the law of mass action and (ii) deterministic in the sense that the kinetic parameters and initial state of the reaction system completely determine the future states. Conservation of Mass At the most basic level, models of chemical reaction kinetics boil down to invoking the principle of conservation of mass.
    [Show full text]
  • (LAW of MASS ACTION) to BIOLOGICAL PROBLEMS the Concept of Balanced Chemical Reac
    APPLICATION OF THE LAW OF CHEMICAL EQUILIBRIUM (LAW OF MASS ACTION) TO BIOLOGICAL PROBLEMS FRANKLIN C. MCLEAN Department of Physiology, University of Chicago, Chicago, Illinois The concept of balanced chemical reactions, introduced by Wenzel in 1777l and made more exact by Berthollet in 1801, was put into the quantitatively useful form of the law of mass action by Guldberg and Waage in 1867. In 1877 van’t Hoff showed how this law could be derived from the principles of thermodynamics, without introducing the vague idea of chemical force, and in 1887 Arrhenius applied the same law to the dissociation of electrolytes in solution. The earliest applications of the law of chemical equilibrium, which was originally derived from the law of mass action, to physiological problems were that of Hiifner (1890) to the dissociation of oxyhemoglobin, and that of L. J. Henderson (1908), characterized by Peters and Van Slyke (1931) as “the first unified explanation of the physiological and physico- chemical mechanisms by which the body maintains its normal acid-base balance.” This review will attempt to set forth the concepts essential to a work- ing knowledge of the law of chemical equilibrium, and to illustrate the adapt ability of this law to specific problems, rather than to assemblethe many important though unrelated contributions which have made use of it. The equally important applications of the law of mass action to the kinetics of chemical reactions will be considered only in so far as they have been applied to physiological problems studied also by the law of chemical equilibrium. THE LAW OF CHEMICAL EQUILIBRIUM2.
    [Show full text]
  • 2020-2021 AP Chemistry Curriculum Map Units of Study and Timeline Aug 26
    2020-2021 AP Chemistry Curriculum Map Units of Study and Timeline Aug 26 - Sept 23 Unit 1 - Chemistry Foundations Sept 23 - Oct 20 Unit 2 - Reactions in Aqueous Solutions Oct 20 - Oct 30 Unit 3 - Atoms, Electrons, and Trends Oct 30 - Nov 23 Unit 4 - Bonding and Molecular Geometry Nov 23 - Dec 18 Unit 5 - Gases Jan 4 - Jan 12 Unit 6 - Thermodynamics (Part 1) Jan 12 - Feb 2 Unit 7 - IMF’s and Solutions Feb 2 - Feb 17 Unit 8 - Kinetics Feb 17 - Mar 4 Unit 9 - Equilibrium Mar 4 - March 24 Unit 10 - Acids and Bases Mar 24 - Apr 14 Unit 11 - Electrochem Apr 14 - Apr 27 Unit 12 - Thermodynamics (Part 2) Apr 27 - May 7 Review and Mock Exams May 7 AP Chemistry Exam (Morning) AP Chemistry Learning Targets BIG IDEA 1 LO 1.1 The student can justify the observation that the ratio of the masses of the constituent elements in any pure sample of that compound is always identical on the basis of the atomic molecular theory. LO 1.2 The student is able to select and apply mathematical routines to mass data to identify or infer the composition of pure substances and/or mixtures. LO 1.3 The student is able to select and apply mathematical relationships to mass data in order to justify a claim regarding the identity and/or estimated purity of a substance. LO 1.4 The student is able to connect the number of particles, moles, mass, and volume of substances to one another, both qualitatively and quantitatively. LO 1.5 The student is able to explain the distribution of electrons in an atom or ion based upon data.
    [Show full text]
  • We Put the Reactants and Products in Quotes Since, Depending on the Initial Distribution of Chemical Species, the Reaction Can Really Go in Either Direction
    Chemical Reaction Equilibrium (CRE) Chemical Reaction Equilibrium • Explain the relationship between energy and entropy in reacting systems (i.e., show why the Gibss Free Energy is still the proper state function for equilibirium) • Write balance chemical reaction expressions with associate reaction stoichiometry • Relate extent of reaction expressions to the equilibrium constant(s) • Use thermochemical data to calculate the equilibrium constant and its dependence on temperature • Determine the equilibrium composition for a single-phase, single-reaction system (i.e., calculate the extent of reaction) ◦ in vapor phase reactions ◦ in liquid phase reactions • Determine the equilibrium composition for a multiphase, single-reaction system • (not covered) Determine the equilibrium composition for a multiple-reaction system CRE: Stoichiometry and Extent of Reaction Stoichiometry and Extent of Reaction (Reaction Coordinate) We know (from years of Chemistry classes) how to balance an equation: We will write this type of reaction in the following form, replacing each chemical species with a generic identifier: where we are using to denote the stoichiometric coefficients where we would consider , , and so that, in general, "reactants" would have and "products" would have . NOTE: We put the reactants and products in quotes since, depending on the initial distribution of chemical species, the reaction can really go in either direction. Here, we will consider "products" to mean chemical species on the right-hand-side. Since there is one degree of freedom when determining the values (that is, you can arbitrarily multiply all of them by any value you like as long as they maintain the same ratios), it is useful to consider changes in the number of moles of each species as ratios, so: or that the ratio of the change in moles of any two species is equal to the ratio of their stoichiometric coefficients.
    [Show full text]