An Account Of Thermodynamic Entropy

Alberto Gianinetti

• Publisher: Bentham Science Publishers
• Year: 2017
• Print ISBN: 9781681083940
• eBook ISBN: 9781681083933
• Topic: Physical Sciences
• Subtopic: Chemistry
• Index: Chemistry

The second law of thermodynamics is an example of the fundamental laws that govern our universe and is relevant to every branch of science exploring the physical world. This reference summarizes knowledge and concepts about the second law of thermodynamics and entropy. A verbatim explanation of chemical thermodynamics is presented by the author, making this text easy to understand for chemistry students, researchers, non-experts, and educators.

Boltzmann Entropy & Equilibrium in Non-Isolated Systems

Alberto Gianinetti

Council for agricultural research and economics, Genomics research centre, via San Protaso 302, 29017 Fiorenzuola d'Arda, Italy

Abstract

The microscopic approach of statistical mechanics has developed a series of formal expressions that, depending on the different features of the system and/or process involved, allow for calculating the value of entropy from the microscopic state of the system. This value is maximal when the particles attain the most probable distribution through space and the most equilibrated sharing of energy between them. At the macroscopic level, this means that the system is at equilibrium, a stable condition wherein no net statistical force emerges from the overall behaviour of the particles. If no force is available then no work can be done and the system is inert. This provides the bridge between the probabilistic equilibration that occurs at the microscopic level and the classical observation that, at a macroscopic level, a system is at equilibrium when no work can be done by it.

Keywords: Approximate equiprobability, Approximate isoenergeticity, Boltzmann entropy, Boltzmann factor, Canonical ensemble, Canonical partition function, Dominating configuration, Energetic (im)probability, Equal probabilities, Equilibrium fluctuations, Fundamental thermodynamic potential, Gibbs free energy, Grand canonical ensemble, Helmholtz free energy, Maximization of entropy, Microcanonical partition function, Microcanonical system, Minimization of energy, Temperature, Thermostatic bath.

Relationship between the microcanonical (isoenergetic) and canonical (isothermal) ensembles

The classical Ω is calculated, for a system at equilibrium defined by given N, E, and V (i.e. a microcanonical system), by counting, among all the possible micro-

states (i.e. every possible combination of positions, orientations, and velocities for all of the particles), only the number of microstates whose energy equals E, that is, those that belong to the microcanonical ensemble [9]. In fact, in the microcanonical ensemble, there are absolutely no fluctuations in energy, since it assumes that N, E, and V are constant, so that, by hypothesis, no fluctuations of E can occur over microstates. These microstates are assumed to be equiprobable, and Ω(E, V, N) represents the microcanonical partition function associated with such a condition [9]. In most real systems, however, the assumption that no fluctuations of E can occur over microstates cannot be guaranteed, because they are not isolated. This is why, in practice, the canonical partition/ensemble, which assumes that N, T, and V, are constant for a system, appears more realistic, since it is quite easy to guarantee the stability of the system temperature by, for example, a large thermostatic bath. In the canonical ensemble the probability of each microstate m is proportional to exp(−βE_m), where E_m is the energy of that microstate, that is, in the canonical ensemble the system visits each microstate with a frequency proportional to its Boltzmann factor [9]. The energy fluctuates in the canonical ensemble, even if only by very small amounts, while it is the temperature that is kept constant by coupling with an infinite heat bath [9]. Note, in fact, that, whereas the internal energy of a system, E, is an exact quantity (whatever the size of the system is) that can be theoretically kept perfectly constant if the system is perfectly insulated, the system’s temperature, T, is an overall property that emerges from the statistical average behaviours of all the particles. So, temperature becomes increasingly reliable as a macroscopic property as the number of particles that contribute to determine it gets larger and relative deviations from this average value get smaller. Ultimately, temperature can be considered an exact property only in the limit of an infinite number of particles, that is, in the presence of an infinite thermal reservoir (the heat bath) with which the system (even a small one) is thermally equilibrated.

The fact that, in actuality, the microstates of every non-isolated system are not necessarily all equiprobable would suggest that the original Boltzmann equation, which assumes Ω as the number of microstates the system can access at equilibrium, provides a biased estimation of the maximum entropy outside microcanonical conditions. Specifically, it would seem to underestimate its value since there are many possible (although very improbable) microstates that are excluded [9] because they have higher energies than the one theoretically assumed for the microcanonical ensemble of the system. Indeed, the Boltzmann equation requires that entropy is calculated as S(E, V, N) for an isolated system at equilibrium, that is, the energy, the volume, and the number of particles are exactly defined. However, E can be truly considered constant only if the insulation of the system is really perfect, which is a very difficult task to accomplish. On the other hand, real systems can easily be kept at a given temperature by means of a thermostatic bath, that is, it is easy to keep them in canonical conditions, wherein T, V, and N are fixed, but it is not really possible to guarantee that every microstate of a system has a given, exact energy level, as the microcanonical assumptions would require, since microscopic fluctuations in the system (resulting from microscopic transfers of thermal energy between the system and the bath, or the surroundings, which are in dynamic thermal equilibrium) can alter, even if passingly, this condition. In other words, strictly speaking, microcanonical conditions would seem quite imaginary for most ordinary systems. Nonetheless, it is important from a theoretical point of view to consider entropy under microcanonical assumptions, as S(E, V, N) is a fundamental thermodynamic potential (i.e., all the thermodynamic properties of the system can be derived from it, specifically in terms of its derivative, given the values of the fixed state functions), whereas S(T, V, N) is not [9]. This means that the microcanonical entropy is a much more meaningful parameter than the canonical one, and, for this reason, it should be preferentially adopted for thermodynamic interpretations when possible.

An excellent compromise to these contrasting exigencies (have E theoretically fixed, so that entropy has a deep thermodynamic value, and to fixing the temperature as an actua...

Table of contents

• Welcome
• Table of Contents
• Title
• BENTHAM SCIENCE PUBLISHERS LTD.
• Preface
• Introduction
• Entropy in Classical Thermodynamics: The Importance of Reversibility
• Heat vs. Work
• More on Reversibility
• More a Matter of Lost Work Than of Waste Heat
• Entropy in Statistical Mechanics
• The Effect of Temperature
• From Nature’s Tendency To Statistical Mechanics
• Distributions of Particles and Energy
• Boltzmann Entropy & Equilibrium in Non-Isolated Systems
• Entropy Increase as Tendency: Drive and Effector
• The Probabilistic Driver and the Role of Restraints
• The Motional Effector
• Spreading & Sharing is a Common Outcome of a Physical Function Levelling Down Available Energy
• Changes of Entropy: The Fundamental Equation and the Chemical Potential
• Instances of Entropy Change
• The Limits of the Spreading and Sharing Metaphor
• Some Special Instances of Entropy Change
• Quantization
• The Role of Probability in Defining Entropy
• Outlines of a Verbal Account of the Thermodynamic Entropy for a Pedagogical Approach
• Notes
• References