Finite entropy change

By integrating both sides of Eq. 4.5.3 between the initial and final states of the irreversible process, we obtain a relation for the finite entropy change corresponding to many infinitesimal cycles of the Carnot engine ... 

How is equation (13.2) used to calculate a finite entropy change, AS ... 

Third Law of Thermodynamics. Also referred to as the Nernst heat theorem, this law states that it is impossible to reduce the temperature of any system, via a finite set of operations, to absolute zero. For any changes involving perfectly crystalline solids at absolute zero, the change in total entropy is zero (thus, A5qk = 0). A corollary to this statement is that every substance, at T > 0 K, must have a positive and finite entropy value. The entropy of that substance is zero only at absolute zero when that substance is in pure, perfect crystalline form. See Entropy... 

It is easy to recognize from general thermodynamic principles that both CP and T are intrinsically non-negative. The integral on the right-hand side of (5.75) is therefore a positive number for any finite T. It is also easy to see that the entropy change A5rxn(T) for any chemical reaction is... 

The choice of magnetic field variation as the process that distinguishes the two curves in Figs, la and lb is of practical, but not of theoretical, significance. If we could find any reversible isothermal process whose entropy change remained finite as 0 K was approached, it would be described by a diagram similar to Fig. la and theoretically permit the attainment of 0 K in a finite number of steps. The third law therefore requires the following ... 

In the special case of a mechanically reversible process (Sec. 2.9), the entropy change of the system is correctly evaluated from J dQj T applied to the actual process, even though the heat transfer between system and surroundings is irreversible. The reason is that it is immaterial, as far as the system is concerned, whether the temperature difference causing the heat transfer is differential (making the process reversible) or finite. The entropy change of a system caused by the... 

One additional feature of the Second Law must be taken up at this point, which relates to the entropy change incurred in a finite, possibly irreversible process in an adiabatically isolated system. Consider such a system in an initial state characterized by the set of deformation variables xx and entropy Si and its transition to a final state 2 associated with x2... 

In a reversible process, the maximum external work is dw = —dG. Irreversible change in the system decreases the maximum external work of the system by TdiS, the energy associated with creation of entropy in the system. For a finite state change, the available external work is... 

It is generally stated that the Nemst formulation of the Third Law of Thermodynamics, according to which all entropy changes vanish at 0 K, and the unattainability formulation thereof, according to which 0 K is unattainable in a finite number of finite operations, are equivalent. But we should note that there are dissensions to this viewpoint [108-113].9... 

These two laws of thermodynamics were developed in the early to middle 19th century. The third law of thermodynamics was postulated in the first decade of the 20th century by the German chemist Walther Hermann Nernst (1864-1941). It treats substances at very low temperature (approaching absolute zero, 0 K, -273°C or -459°F). It predicts that absolute zero cannot be reached in a finite number of steps and that, close to absolute zero, the entropy change between two stable states also approaches zero. This allows chemists to calculate absolute entropies for substances at any given temperature. Nernst received the 1920 Nobel Prize in chemistry for this work. 

What is of very wide interest here is that the entropy change which in a phase transition would have occurred at a fixed temperature has in the order-disorder transition been spread out over a range of temperature. The increase in potential energy which in the former would have manifested itself as a latent heat absorbed at the constant transformation temperature is in the latter absorbed over a finite range and thus manifests itself as an anomalously great specific heat. In the region of temperature where such phenomena are in process of evolution the specific heat generally follows a curve of the form shown in Fig. 28. 

If one includes the entropy change of the system, 5(t) = —In p x x),x), the equation above holds even for finite times in the steady state. 

We know that during a reversible process of a closed system, each infinitesimal entropy change dS is equal to dq/ and the finite change AS is equal to the integral /(dq/ 7b)— but what can we say about d5 and AS for an irreversible process ... 

The net saving in entropy is most apparent in a graphic comparison of the entropy change produced in a traditional Adiabatic column and an optimized Diabatic column, that is, one with heat exchangers along the column. The reversible limit is still clearly lower than the finite-time system, but the separation part of that entropy is very similar for the optimized realistic and reversible columns the difference is almost entirely in the heat exchange. This is shown in Fig. 14.5. 

The function describing the change in entropy, as a function of temperature, involves the use of a prescription that contains a formula specific to a particular phase. At each phase transition temperature the function suffers a finite jump in value because of the sudden change in thermodynamic properties. For example, at the boiling point 7b the sudden change in entropy is due to the latent heat of evaporation (see Figure 2.8). 

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