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Cover Foundations of Science Mathematics

Data analysis  

This chapter focuses on data analysis. One of the things that sets science apart from other systems of trying to make sense of the world is the use of empirical evidence. A scientific theory must therefore be able to make predictions that can be compared with observations. Since the amount of data obtained in an experiment is limited, and subject to noise in the measurement process and uncertainty in the parameters required for their analysis, the conclusions reached have to be couched in conditional terms with an appropriate statement about their reliability. The chapter then looks at the basic rules of drawing inferences, common probability distributions, parameter estimation, propagation of errors, and model comparison.


Cover Physical Chemistry: Quanta, Matter, and Change

Peter Atkins, Julio de Paula, and Ronald Friedman

Physical Chemistry starts off by looking at the foundations of the subject and provides the reader with some mathematical background. It then looks at quantum mechanics, taking into consideration the quantum mechanics of motion, molecular structure, and molecular symmetry. It then considers Fourier transforms, molecular spectroscopy, magnetic resonance, statistical thermodynamics, probability theory, the first law of thermodynamics, and multivariate calculus. The final part of the book examines physical equilibria, chemical equilibria, molecular motion, chemical kinetics, reaction dynamics, and processes in fluid systems and solid states.


Cover Introductory Statistical Mechanics

Maxwell distribution of molecular speeds  

This chapter starts with ideas from James Clerk Maxwell's publication the distribution of the speeds of particles in a gas, which he obtained by analysing collision processes between the particles. It recounts how Maxwell showed that there could be a distribution of speeds and calculated the distribution, contradicting people's speculation that the particles of the gas moved with the same speed. It also mentions Ludwig Boltzmann, who treated the kinetic theory of gases from a microscopic point of view using the notion of probability. The chapter details how Boltzmann derived Maxwell's distribution of speeds and related it to the temperature of the particles. It examines particles using quantum mechanics and derives the Maxwell distribution of speeds.


Cover The Chemistry Maths Book

Probability and statistics  

This chapter introduces probability theory—the mathematical theory of statistics—that gives the theoretical models and analytical tools for the organization, interpretation, and analysis of statistical data. Probability theory is applied in chemistry for the description of the collective behaviour of very large numbers of particles in statistical mechanics, the quantum mechanical descriptions of changes of state and of rate processes, the physical interpretation of wave functions, and the enumeration of the ways of assembling basic chemical units to form large molecules. The chapter starts with a brief discussion on descriptive statistics. Then it tackles frequency and probability, as well as the combinations of probabilities. Furthermore, the discussion covers the binomial distribution. It describes permutations and combinations, as well as continuous distributions, then explains the Gaussian distribution. Lastly, it discusses the method of least squares.


Cover Reaction Dynamics

Microcanonical rate coefficients  

This chapter focuses on microcanonical rate coefficients. It begins by looking at the cumulative reaction probability. The chapter then considers transition state theory. The transition state may be thought of as a bottleneck somewhere on the potential energy surface through which reactants must pass if they are to form products. More precisely, it is a dividing surface, which separates reactants from products. The key dynamical assumption of transition state theory is that of 'direct dynamics', which is often referred to as the no-recrossing rule. Trajectories which cross the transition state dividing surface from the reactant side are assumed to go on to form products and cannot be reflected back into the reactant valley. Finally, the chapter studies the measurement of microcanonical rate coefficients.


Cover Reaction Dynamics


This introductory chapter provides an overview of elementary reactions in the gas phase. Most elementary reactions can be categorized as either unimolecular or bimolecular. Another class of reactions is association reactions. Association reactions and chemical activation can be modelled using the same theories as those developed to rationalize unimolecular reactions. The chapter then looks at reaction kinetics and dynamics, considering thermal rate coefficients and simple collision theory. It also offers some insight into the significance of the reaction cross-sections. Finally, the chapter highlights the role of the reaction probability on collision in determining the magnitude of the reaction cross-section and, hence, the thermal rate coefficient.