Definition of entropy

Just as the first law led to the definition of the energy, so also the second law leads to a definition of a state property of the system, the entropy. It is characteristic of a state property that the sum of the changes of that property in a cycle is zero. For example, the sum of changes in energy of a system in a cycle is given by dU = 0. We now ask whether the second law defines some new property whose changes sum to zero in a cycle. 

We begin by comparing two expressions for the efficiency of a simple reversible heat engine that operates between the two reservoirs at the thermodynamic temperatures 6 and 02 We have seen that 

The left-hand side of Eq. (8.27) is simply the sum over the cycle of the quantity Q/6. It could be written as the cyclic integral of the differential quantity dQ/Q - 

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Snch a generalization is consistent with the Second Law of Thennodynamics, since the //theorem and the generalized definition of entropy together lead to the conchision that the entropy of an isolated non-eqnilibrium system increases monotonically, as it approaches equilibrium. 

The definition of entropy and the identification of temperature made in the last subsection provides us with a coimection between the microcanonical ensemble and themiodynamics. 

By combining random flight statistics from Chap. 1 with the statistical definition of entropy from the last section, we shall be able to develop a molecular model for the stress-strain relationship in a cross-linked network. It turns out to be more convenient to work with the ratio of stretched to unstretched lengths L/Lq than with y itself. Note the relationship between these variables ... 

It must be emphasised that the heat q which appears in the definition of entropy (equation 20.137) is always that absorbed (or evolved) when the process is conducted reversibly. If the process is conducted irreversibly and the heat absorbed is q, then q will be less than q, and q/T will be less than AS the entropy change (equation 20.137). It follows that if an irreversible process takes place between the temperatures Tj and 7 , and has the same heat intake q at the higher temperature 7 2 as the corresponding reversible process, the efficiency of the former must be less than that of the latter, i.e. 

The name entropy is used here because of the similarity of Eq. (4-6) to the definition of entropy in statistical mechanics. We shall show later that H(U) is the average number of binary digits per source letter required to represent the source output. 

Planck Bed. Ber., 1908, 633) has, from the fundamental statistical definition of entropy, deduced the equation ... 

We need a quantitative definition of entropy to measure and make precise predictions about disorder. Provided that the temperature is constant, it turns out that a We generalize the definition in the change in the entropy of a system can be calculated from the following expression next section to changes in which the... 

In 1877, the Austrian physicist Ludwig Boltzmann proposed a molecular definition of entropy that enables us to calculate the absolute entropy at any temperature (Fig. 7.6). His formula provided a way of calculating the entropy when measurements could not be made and deepened our insight into the meaning of entropy at the molecular level. The Boltzmann formula for the entropy is... 

The expressions in Eq. 1 and Eq. 6 are two different definitions of entropy. The first was established by considerations of the behavior of bulk matter and the second by statistical analysis of molecular behavior. To verify that the two definitions are essentially the same we need to show that the entropy changes predicted by Eq. 6 are the same as those deduced from Eq. 1. To do so, we will show that the Boltzmann formula predicts the correct form of the volume dependence of the entropy of an ideal gas (Eq. 3a). More detailed calculations show that the two definitions are consistent with each other in every respect. In the process of developing these ideas, we shall also deepen our understanding of what we mean by disorder. ... 

We can show that the definition of entropy in Eq. 6 is quantitatively equivalent to that in Eq. 1, even though the equations look totally different. 

In physical chemistry, entropy has been introduced as a measure of disorder or lack of structure. For instance the entropy of a solid is lower than for a fluid, because the molecules are more ordered in a solid than in a fluid. In terms of probability it means also that in solids the probability distribution of finding a molecule at a given position is narrower than for fluids. This illustrates that entropy has to do with probability distributions and thus with uncertainty. One of the earliest definitions of entropy is the Shannon entropy which is equivalent to the definition of Shannon s uncertainty (see Chapter 18). By way of illustration we... 

Conventional implementations of MaxEnt method for charge density studies do not allow easy access to deformation maps a possible approach involves running a MaxEnt calculation on a set of data computed from a superposition of spherical atoms, and subtracting this map from qME [44], Recourse to a two-channel formalism, that redistributes positive- and negative-density scatterers, fitting a set of difference Fourier coefficients, has also been made [18], but there is no consensus on what the definition of entropy should be in a two-channel situation [18, 36,41] moreover, the shapes and number of positive and negative scatterers may need to differ in a way which is difficult to specify. 

The definition of entropy in the grand canonical ensemble follows from the definition of — S/k according to (50), combined with (46) and (49), as... 

The second and third laws of thermodynamics The second law and the definition of entropy... 

As was the case with energy, the definition of entropy permits only a calculation of differences, not an absolute value. Integration of Equation (6.48) provides an expression for the finite difference in entropy between two states ... 

The definition of entropy requires that information about a reversible path be available to calculate an entropy change. To obtain the change of entropy in an irreversible process, it is necessary to discover a reversible path between the same initial and final states. As S is a state function, AS is the same for the irreversible as for the reversible process. 

The first satisfactory definition of entropy, which is quite recent, is that of Kittel (1989) entropy is the natural logarithm of the quantum states accessible to a system. As we will see, this definition is easily understood in light of Boltzmann s relation between configurational entropy and permutability. The definition is clearly nonoperative (because the number of quantum states accessible to a system cannot be calculated). Nevertheless, the entropy of a phase may be experimentally measured with good precision (with a calorimeter, for instance), and we do not need any operative definition. Kittel s definition has the merit to having put an end to all sorts of nebulous definitions that confused causes with effects. The fundamental P-V-T relation between state functions in a closed system is represented by the exact differential (cf appendix 2)... 

Traditional thermodynamics gives a clear definition of entropy but unfortunately does not tell us what it is. An idea of the physical nature of entropy can be gained from statistical thermodynamics. Kelvin and Boltzmann recognised diat there was a relationship between entropy and probability (cf., disorder) of a system with the entropy given by... 

Adiabatic expansion or compression of an air mass maintains a constant potential temperature. From the definition of entropy, S, as dS = dqKV/T, these processes are also constant-entropy processes since no heat is exchanged between the system (i.e., the air parcel) and its surroundings. Hence the term isen-tropic is used to describe processes that occur without a change in potential temperature. 

The term entropy, which literally means a change within, was first used in 1851 by Rudolf Clausius, one of the formulators of the second law of thermodynamics. A rigorous quantitative definition of entropy involves statistical and probability considerations. However, its nature can be illustrated qualitatively by three simple examples, each demonstrating one aspect of entropy. The key descriptors of entropy are randomness and disorder, manifested in different ways. 

The material covered in this chapter is self-contained, and is derived from well-known relationships such as Newton s second law and the ideal gas law. Some quantum mechanical results and the statistical thermodynamics definition of entropy are given without rigorous derivation. The end result will be a number of practical formulas that can be used to calculate thermodynamic properties of interest. 

The next important thermodynamic function that we must obtain is the entropy S. The statistical thermodynamic definition of entropy is... 

The thermodynamic definition of entropy says that the change in entropy dS in a process carried out reversibly is the heat absorbed in the process d Qrev divided by the temperature... 

C. E. Shannon (1916-2001) developed an information-theoretic definition of entropy that (although not equivalent to the physical quantity) carries similar associations with microstates and probability theory. Shannon recognized that Boolean bit patterns (sequences of l s and 0 s) can be considered the basis of all methods for encoding information. ... 

A more intuitive definition of entropy is in terms of probability a more random system has higher probability and therefore higher entropy. 

In words when a system undergoes a change, the increase in entropy of the system is equal to or greater than the heat absorbed in the process divided by the temperature. On the other hand, the equality, which provides a definition of entropy increment, applies to any reversible process, whereas the inequality refers to a spontaneous (or irreversible) process, defined as one which proceeds without intervention from the outside. Example 1 illustrates the reversible and irreversible reactions. 

Chapter 5 gives a microscopic-world explanation of the second law, and uses Boltzmann s definition of entropy to derive some elementary statistical mechanics relationships. These are used to develop the kinetic theory of gases and derive formulas for thermodynamic functions based on microscopic partition functions. These formulas are apphed to ideal gases, simple polymer mechanics, and the classical approximation to rotations and vibrations of molecules. 

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