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### Chapter ## Ab initio methods

Contents Configuration interaction 251 Brief illustration 29.1: Configuration interaction 252 Example 29.1: Finding the energy lowering due to CI 252 Many-body perturbation theory 253 Example 29.2: Setting up Møller–Plesset perturbation theory 254 Checklist of concepts 254 Checklist of equations 255 Why do...

### Chapter ## Absolute entropy

This chapter studies the molecular interpretation of entropy and how it relates to the third law of thermodynamics. The third law of thermodynamics states that the entropies of all perfectly crystalline substances are the same at T = 0. By convention (and as justified statistically), S(0) = 0 for all perfectly ordered crystalline materials. The standard molar entropy is the molar entropy of a substance in its standard state (pure, at 1 bar) at the temperature of interest. Meanwhile, the statistical entropy is the entropy calculated from the Boltzmann formula, as the logarithm of the weight of a configuration. The chapter then looks at the residual entropy of a solid, which is the contribution to the entropy at T = 0 from positional disorder that is frozen in.

### Chapter ## Acid–base equilibria of salts in water

This chapter studies the acid–base equilibria of salt solutions. The ions that a dissolved salt provide are themselves either acids or bases, sometimes both. Acidity constants can be used to predict the pH of solutions, and that information in turn can be used to account for the variation of pH during the course of a titration. That information is also helpful as a guide to the selection of solutes that stabilize the pH of solutions. The chapter looks at acid–base titrations, explaining how the pH of a mixed solution of a weak acid and its conjugate base is given by the Henderson–Hasselbalch equation. It then considers the buffer action, examining the buffer solution and differentiating between an acid buffer and a base buffer.

### Chapter ## Activities

This chapter describes how the extension of the concept of chemical potential to real solutions involves introducing an effective concentration called an ‘activity’. In certain cases, the activity may be interpreted in terms of intermolecular interactions; an important example is a solution containing ions. Such solutions often deviate considerably from ideal behaviour on account of the strong, long-range interactions between the charged species. The chapter shows how a model can be used to estimate the deviations from ideal behaviour when the solution is very dilute, and how to extend the resulting expressions to more concentrated solutions. It looks at the Margules equations, the Debye–Hückel theory, and the Debye–Hückel limiting law.

### Chapter This chapter describes adiabatic processes, which occur without the transfer of energy as heat. It focuses on reversible adiabatic changes involving perfect gases. The temperature of a gas falls when it expands adiabatically in a thermally insulated container. Work is done, but as no heat enters the system, the internal energy falls, and therefore the temperature of the gas also falls. In molecular terms, the kinetic energy of the molecules falls as work is done, so their average speed decreases, and hence the temperature falls too. The chapter then looks at how an adiabat is a curve showing how pressure varies with volume in an adiabatic process.

### Chapter Contents Adsorption isotherms 922 The Langmuir isotherm 922 Using the Langmuir isotherm 923 The isosteric enthalpy of adsorption 924 Measuring the isosteric enthalpy of adsorption 925 The BET isotherm 925 Using the BET isotherm 927...

### Chapter This chapter looks at the extent to which molecules attach themselves to a surface, which is crucial to understanding the way in which a surface influences chemical processes. It discusses the extent of adsorption that can be explored with the aid of some simple models that allow quantitative predictions to be made about how the extent of surface coverage varies with both pressure and temperature. It also demonstrates how surfaces can affect the rates of chemical reactions by assessing the extent of surface coverage and the factors that determine the rates at which molecules attach to and detach from solid surfaces. The chapter outlines the extent of surface coverage that can be expressed in terms of isotherms derived on the basis of dynamic equilibria between adsorbed and free molecules.

### Chapter This chapter explains the basic theory of continuous wave (CW) electron paramagnetic resonance (EPR), illustrating the power of the technique to study a wide range of paramagnetic systems. It cites several experiments based on pulsed techniques similar to those routinely employed in nuclear magnetic resonance (NMR) spectroscopy. It also talks about how Pulse EPR can offer significant advantages over CW methods, such as direct detection of relaxation times and access to longer distances between paramagnetic centres. The chapter talks about the independent control of the electron and nuclear spins via the application of short microwave (MW) and radiofrequency (RF) pulses. It presents the vector model and product operator formalism used in pulse techniques.

### Chapter ## The analysis of molecular shape

Contents Symmetry operations and symmetry elements 275 Brief illustration 31.1: Symmetry elements 276 The symmetry classification of molecules 276 Brief illustration 31.2: Symmetry classification 277 The groups C 1, C i, and C s 278 Brief illustration 31.3: C...

### Chapter ## Anisotropic EPR spectra in the solid state

This chapter explores the origins of the anisotropies in g and A for a spin. It explains how symmetry derived anisotropies in the solid state are manifested through g and how the interpretation of this tensor provides valuable information on the symmetry of the paramagnetic centre. It also concentrates on the lineshapes for powder spectra and the origins of the hyperfine A tensor. The chapter considers the electron paramagnetic resonance (EPR) spectra of a paramagnetic vanadyl and presents the theory explaining the origins of anisotropies. It describes a tensor as a mathematical object that illustrates a physical property and outlines the rank of the tensor that depends on the number of directions needed to describe that property.

### Chapter ## Appendix

Solutions to problems 1 Classical mechanics 1.1 (a) v 1 = –3/4u v 2 = 1/4u (b) Fraction of original KE lost = 1/8 1.2 Equatorial gravitational field strength = –9.841m s–2 Polar gravitational field strength = –9.906 m s–2...

### Chapter ## Applications of symmetry

Contents Vanishing integrals 291 Integrals over the product of two functions 292 Example 33.1: Deciding if an integral must be zero 1 292 Decomposition of a direct product 293 Brief illustration 33.1: Decomposition of a direct product 293 Integrals over products...

### Chapter ## Applications of symmetry

This chapter explores the applications of symmetry while referring to group theory as a significant tool for constructing molecular orbitals and formulating spectroscopic selection rules. It explains how group theory provides simple criteria for deciding whether certain integrals necessarily vanish. Additionally, character tables can be used to determine whether an integral is necessarily zero. The integrand must include a transforming component as the totally symmetric irreducible represent to acknowledge an integral to be non-zero. The chapter also expounds on the concept of symmetry-adapted linear combination (SALC) as a linear combination of atomic orbitals constructed from equivalent atoms and having a specified symmetry.

### Chapter ## The approach to equilibrium

This chapter addresses how all forward reactions are accompanied by their reverse reactions. Close to the start of a reaction, when little or no product is present, the rate of the reverse reaction is negligible. However, as the concentration of products increases, the rate at which they decompose into reactants becomes greater. At equilibrium, the reverse rate matches the forward rate and the reactants and products are present in abundances given by the equilibrium constant for the reaction. The chapter then considers the term relaxation, which denotes the return of a system to equilibrium. It is used in chemical kinetics to indicate that an externally applied influence has shifted the equilibrium position of a reaction, normally abruptly, and that the reaction is adjusting to the equilibrium composition characteristic of the new conditions. Relaxation methods include the temperature jump technique.

### Chapter ## The Arrhenius equation

Contents The temperature dependence of reaction rates 816 Example 85.1: Determining the Arrhenius parameters 817 Brief illustration 85.1: The Arrhenius equation 817 The interpretation of the Arrhenius parameters 818 A first look at the energy requirements of reactions 818 Brief illustration 85.2:...

### Chapter ## The Arrhenius equation

This chapter discusses how rate constants of most reactions increase with increasing temperature. It introduces the ‘Arrhenius equation’, which captures this temperature dependence by using two parameters that can be determined experimentally. It also reviews the exploration of the dependence of reaction rates on temperature that leads to the formulation of theories that reveal the details of the processes that occur when reactant molecules meet and undergo reaction. The chapter looks at the temperature dependence of the rate of a reaction that depends on the activation energy and the minimum energy needed for reaction to occur in an encounter between reactants. It emphasizes how chemical reactions usually go faster as the temperature is raised, which is almost always due to the increase of the rate constant with temperature.

### Book ## Atkins’ Physical Chemistry (Twelfth Edition)

Physical Chemistry provides a comprehensive overview of this topic. It starts off with looking into the properties of gases. It then covers the First, Second, and Third Laws. Next it looks into physical transformations of pure substances, simple mixtures, and chemical equilibrium. The text also considers quantum theory, atomic structure and spectra, molecular structure, molecular symmetry, and molecular spectroscopy. There follows a chapter about magnetic resonance. The text then looks at statistical thermodynamics. The last quarter of the book considers molecular interactions, solids, molecules in motion, chemical kinetic, and reaction dynamics. The last chapter covers processes at solid surfaces.

### Chapter ## Atomic orbitals

This chapter takes a look at the atomic orbital, which is a one-electron wavefunction describing the spatial distribution of an electron in an atom. Atomic orbitals are used throughout chemistry in discussions of the electronic structure of atoms in general and in discussions of molecular electronic structure. This chapter extends the discussion of atomic structure to include the effect of nuclear charge by considering one-electron ions with higher atomic numbers. It shows how hydrogenic atoms are important because the Schrödinger equation can be solved for them. Furthermore, the concepts learned from a study of hydrogenic atoms can be used to describe the structures of many-electron atoms and of molecules too. To that end the chapter takes a look at the energy levels of hydrogenic atoms as well as the wavefunctions of hydrogenic atoms.

### Chapter ## Atomic spectra

This chapter explains the general idea behind atomic spectroscopy. It details how lines in the spectrum (in either emission or absorption) occur when the electron distribution in an atom undergoes a transition wherein its energy changes by ΔE. The spectra of many-electron atoms are more complicated than that of hydrogenic atoms. Even though similar principles apply, Coulombic and magnetic interactions between the electrons give rise to a variety of energy differences, which are summarized by constructing term symbols. The chapter notes how total angular momentum in light atoms is obtained based on Russell–Saunders coupling, while jj-coupling is used for heavy atoms.

### Chapter ## Atomic spectroscopy

This chapter considers atomic spectroscopy as an important way of determining the energies of electrons in atoms and reviews the spectra of hydrogenic atoms and many-electron atoms. It highlights the concept of selection rules which is used to predict which spectroscopic transitions can be observed. It also analyzes the spectra of many-electron atoms which are more complicated than those of hydrogen as they are influenced by the Coulombic and magnetic interactions of electrons. The chapter describes the term symbols and shows how these are based on the various contributions to the total angular momentum of the electrons. It details how spectroscopic measurements confirm the theoretical prediction that the energy levels of atoms correlate with the contributions to the total angular momentum of their electrons.