Definition of the entropy

To make progress and turn the Second Law into a qucmtitatively useful statement, we shall use the following definition of a change in entropy  

That is, the change in entropy of a system is equal to the energy transferred as heat to it reversibly divided by the temperature at which the transfer takes place. This definition can be justified thermodynamically, but we shall confine ourselves to showing that it is plausible and then show how to use it to obtain numerical values for a range of processes. 

There are three points we need to understand about the definition in eqn 2.1  

We met the concept of reversibhity in Section 1.3, where we saw that it refers to the ability of an infinitesimal chcuige in a control variable to change the direction of a process. Mechaniccd reversibhity refers to the equality of pressure acting on either side of a movable wall. Thermal reversibility, the type involved in eqn 2.1, refers to the equality of temperature on either side of a thermally conducting wall. Reversible transfer of heat is smooth, careful, restrained transfer between two bodies at the same temperature. By mcJdng the transfer reversible, we ensure that there are no hot spots generated in the object that later disperse spontaneously and hence add to the entropy. 

An organism inhabits a pond. In the course of its life, the organism transfers 100 kj of heat to the pond water at 0°C (273 K). The resulting change in entropy of the water due to this transfer is 

This is a general result. In any finite-step, cyclic expansion—compression process work is always converted to heat  

This result applies whenever the process is carried out in a nonreversible (irreversible) manner. In other words, in an irreversible cyclic process more work must be input to the system than the system produces. In all the finite gas compressions the work required is greater than I.4P1V1, which is the maximum work available from the expansion. 

Of course, all real processes are irreversible, because they cannot be carried out in an infinite number of steps without taking an infinite amount of time. In other words, all real processes are irreversible (in a thermodynamic sense). 

Another important conclusion to be drawn from the above example is that the maximum work obtainable from the gas occurs when the expansion is carried out reversibly wmax = wrev). This result is always true for PV work, as well as for any other type of work, such as electrical work performed by an electrochemical cell. We will examine this latter example in the next chapter. 

The final point that this experiment reemphasizes is that work and heat are pathway-dependent and thus are not state functions. Energy, on the other hand, is a state function. In each of these isothermal expansions and compressions between (Pls Vi) and (Pj/4, 4Vj), AE is always zero, regardless of the number of steps, since T is constant. 

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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. 

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. 

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. 

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. 

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. 

Bolometers are just thermometers weakly coupled to a thermal bath by a conductance G. Radiation is focussed on the bolometer causing its temperature to rise by AT = P/G (see Figure 9.17). The thermal time constant of a bolometer is given by t = C/G where C = dQ/dT is the heat capacity of the bolometer in ergs/K. From the definition of entropy S = k In G with Q being the state density, and dS = dQ/T, we find that... 

Thermodynamics — deals with the interrelations between -> energy and matter and the laws that govern them. The energetic changes in a system are governed by the fundamental laws of thermodynamics, which have been deduced directly from experience. The first law of thermodynamics simply states the principle of conservation of energy. The second law of thermodynamics states whether or not a process takes place in one or the other direction. For instance, heat always spontaneously flows from a higher temperature body to another one with lower temperature and never in the opposite direction. The second law of thermodynamics provides the definition of -> entropy. The third law of thermodynamics, also known as -> Nernsfs theorem, states the possibility... 

The heat absorbed by a system during some process is equal to the heat given up by the rest of the universe. Let us represent the infinitesimal heat exchange of the system by dQ For an isothermal reaction or change, dQs is simply —dQr because the heat must come from the rest of the universe. From the definition of entropy,1 dS = dQ/T, we can obtain the following relationship ... 

What we have accomplished here is to use the definition of entropy in terms of probability to derive an expression for AS that depends on volume, a macroscopic property of the gas. We can now relate the change in entropy to heat flow by noting the striking similarity between the above equation for AS and the one derived in Section 10.2 describing qrev for the isothermal expansion-compression of an ideal gas. Compare... 

This very important relationship is the macroscopic (thermodynamic) definition of AS. In our treatment we started with the definition of entropy based on probability, because that definition better emphasizes the fundamental character of entropy. However, it is also very important to know how entropy changes relate to changes in macroscopic properties, such as volume and heat, because these changes are relatively easy to measure. 

The entropy >S, like the energy C7, is a function of volume and temperature. It is not, however, possible without further assumption to predict the nature of the S-T curve in the neigh-bom hood of the absolute zero. By the definition of entropy... 

We use a statistical approach to find AS ys by applying the definition of entropy expressed by Equation 20.1. Figure 20.1 A shows a container consisting of two identical flasks connected by a stopcock, with 1 mol of neon in the left flask and an evacuated right flask. We know from experience that when we open the stopcock, the gas will expand to fill both flasks with 0.5 mol each—but whyl... 

This section summarizes the classical, equilibrium, statistical mechanics of many-particle systems, where the particles are described by their positions, q, and momenta, p. The section begins with a review of the definition of entropy and a derivation of the Boltzmann distribution and discusses the effects of fluctuations about the most probable state of a system. Some worked examples are presented to illustrate the thermodynamics of the nearly ideal gas and the Gaussian probability distribution for fluctuations. 

Recapitulating, the results of this chapter look plausible, but there is a problem while the definition of energy (1.6) may be expected and useful, using the definition of entropy as a supremum (1.31) (or by (c) in Rem. 21) will be scarcely possible. Moreover, it is not clear how to find the reference (especially nonequilibrium) state and also the existence of more possible definitions (noted in Rem. 20) complicates the situation further. 

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