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2024 | Buch

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Introduction to Statistical Thermodynamics

A Molecular Perspective

verfasst von: Marcus Elstner, Qiang Cui, Maja Gruden

Verlag: Springer International Publishing

Buchreihe : Physical Chemistry in Action

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This textbook presents the fundamentals of statistical thermodynamics and electronic structure theory and focuses on introducing the central concepts of thermodynamics and their relation to microscopic theories in a conceptually clear and simple way. The emphasis is on the description of what is going on at the microscopic level, which allows readers to understand the various facets of entropy as the fundamental driving force of all material behaviors. An atomistic perspective is introduced from the beginning, highlighting the importance of molecular structure and microscopic degrees of freedom for understanding the thermodynamic properties of materials, such as heat capacity and magnetization. Because of their importance in various research fields, classical and quantum aspects are treated equally, allowing modern research topics to be addressed with molecular simulation and electronic structure theory. It is a valuable resource for undergraduate and graduate students in chemistry,physics, and materials science, and its modular structure makes it suitable for any reader.

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Inhaltsverzeichnis

Frontmatter

Microscopic and Macroscopic Theories

Frontmatter

Chapter 1. Why Statistical Thermodynamics?

Abstract

In thermodynamics and kinetics, the thermal properties, the chemical reactivity, and the phase behaviors are described without reference to the molecular structure by a few macroscopic observables like \(p, V\), and T. Material properties such as heat capacity \(c_V\), thermal expansion coefficients \(\alpha \), or reaction constants k are treated as empirical parameters, which cannot be analyzed further in this framework. Why the materials differ in these properties cannot be questioned, since the molecular structure – on which these differences are based – is not the subject of the description.

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 2. Energies of Molecules

Abstract

The formalism of modern classical mechanics is based on that the total energyE of each mechanical system can be described by the contributions of kinetic energy T and potential energy V .

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 3. Dynamics of Molecules

Abstract

In Chap. 2 we discussed in more detail the potentialsV , which govern the way in which atoms interact with each other. Now we will do the same for the kinetic energyT. The interactions between the atoms determine (i) the structure of the molecules, as well as (ii) the forms of motion available to the molecular systems.

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 4. The Laws of Thermodynamics

Abstract

In this chapter, we will first consider very simple systems, i.e., single-component and single-phase systems capable of absorbing and releasing work and heat.

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 5. Principle of Maximum Entropy

Abstract

In this book, we reflect on two fundamental questions:

1.

We consider a system in equilibrium, e.g., a simple system as discussed in Chap. 4. For this system, we calculate the thermodynamic state variables and material constants. In statistical thermodynamics, we do this on the basis of the microscopic structure.

2.

We consider a system consisting of several components, e.g., several phases or chemical components. The latter can react with each other or mix. Here we are interested in the location of the equilibria. In this chapter we will introduce the basics for the determination of equilibria, i.e., the principle of maximum entropy.

Marcus Elstner, Qiang Cui, Maja Gruden

Fundamentals of Classical Statistical Thermodynamics

Frontmatter

Chapter 6. The Microscopic Entropy

Abstract

However, if we now want to consider processes and equilibria, as, for example, the relaxation processes in Chap. 5, we know from thermodynamics that these are not determined by the internal energy U. Rather, the central principle of maximum entropy applies. While in mechanics we can understand the system behavior exclusively from the energy, its derivatives, and the forces on the atoms, in thermodynamics the knowledge of the energy does not allow an understanding of the process. Here, the entropy and the 2nd law are required.

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 7. The Boltzmann Distribution

Abstract

In statistical thermodynamics, we determine the thermodynamic properties of substances based on their microstructure.

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 8. The Canonical Distribution

Abstract

In Chaps. 6 and 7, two central concepts were introduced by Boltzmann:

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 9. The Canonical Ensemble

Abstract

In Chap. 8, we derived the distribution.

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 10. Noninteracting and Interacting Systems

Abstract

In the last chapters, we derived methods to compute thermodynamic observables from a microscopic representation of the systems energies.

Marcus Elstner, Qiang Cui, Maja Gruden

Application of Classical Statistical Thermodynamics

Frontmatter

Chapter 11. State Sum (Partition Function) for Noninteracting Systems

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 12. Application to Ideal Gases of Polyatomic Molecules

Abstract

In this chapter, we will apply the formalism developed so far to the ideal gas in detail, i.e., we will calculate the state sum.

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 13. Chemical Equilibria and Kinetics

Abstract

Most quantum mechanical applications (Chaps. 12, 14 and 15), that is, applications that involve the quantum mechanical representation of energy, consider homogeneous one-component systems where thermodynamic properties such as U and \(c_V\) are calculated. The system is already in a state of equilibrium at a temperature T and a pressure p. The energy is calculated as a function of the structure and composition of the molecules.

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 14. Application to Solid Bodies

Abstract

We have described molecules in the gas phase using the discrete quantum mechanical representation. We also use this approach for solids. Again, we use models that allow an additive representation of energy; this leads to a product of state sums, which in turn allows the thermodynamic properties to be calculated in a relatively simple way.

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 15. Application to 2-Level Systems: Spin Systems

Abstract

At relatively low temperatures, systems with only two quantum states exhibit characteristic behavior in terms of energy and heat capacity. This behavior is the subject of our study in this chapter.

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 16. Spatial Distributions

Abstract

In Chap. 2, we discussed the fundamental molecular interactions, which can be represented approximately as classical potentials V . These potentials appear in a classical treatment (Hamiltonian formalism) of the Hamiltonian function \(H = T +V\), that represents the total energy of a system. Its counterpart in quantum mechanics is the Hamiltonian operator. Operating on the wavefunction with the Hamiltonian produces the Schrödinger equation. In the time independent Schrödinger equation, the operation produces specific values for the energy called energy eigenvalues \(E_n\). In the calculation of the state sumQ, the exponent is either H or the eigenenergies \(E_n\): In both cases the potentials must be very simple to get compact solutions.

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 17. Thermodynamics of Real Gases, Liquids, and Polymers

Abstract

In this chapter, we continue with the description of interacting systems using classical mechanics.

Marcus Elstner, Qiang Cui, Maja Gruden

Quantum statistics and the electronic structure of molecules

Frontmatter

Chapter 18. Quantum Statistics

Abstract

Why are we now talking about quantum statistics? In what sense have we done classical statistics so far? We have treated the gases and the solid body quantum mechanically, what is missing here? One must distinguish two aspects:

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 19. Electron Densities and Electron Correlations

Abstract

This refers to the orbital structure, i.e., the occupation of these orbitals with electrons of opposite spins, according to the Aufbau principle.

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 20. Wave-Function-Based Methods

Abstract

This chapter follows directly from Chap. 19. We have explained there that the inclusion of the electron correlation is the central requirement of an accurate calculation of the electronic structure, and that the most used methods of quantum chemistry are based on two different approaches.

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 21. Density Functional Theory (DFT)

Abstract

The goal of density functional theory is to express energy as a functional of the electron density. The Hohenberg–Kohn theorems show that this is possible in principle. Practically, however, it is a bit more complicated: The theorems do not tell us what these functionals look like, i.e., they leave us pretty much in the dark when looking for them.

Marcus Elstner, Qiang Cui, Maja Gruden

Chapter 22. Non-covalent Interactions

Abstract

Non-covalent interactions are crucial for many materials, especially for the class called soft matter. The typical interaction energies are much lower in magnitude than for covalent bonds, leading to quite different material properties.

Marcus Elstner, Qiang Cui, Maja Gruden

Backmatter

Titel: Introduction to Statistical Thermodynamics
verfasst von: Marcus Elstner
Qiang Cui
Maja Gruden
Copyright-Jahr: 2024
Verlag: Springer International Publishing
Electronic ISBN: 978-3-031-54994-6
Print ISBN: 978-3-031-54993-9
DOI: https://doi.org/10.1007/978-3-031-54994-6

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