Reversible process adiabatic

In the neighborhood of every equilibrium state of a thermodynamic system, there exist states unattainable from it by any adiabatic process (reversible or irreversible). 

We have proved Caratheodory s principle. Caratheodory s principle states From an arbitrary point (state), there is a finite region (set of states of finite measure) which cannot be reached by an adiabatic process, reversible or irreversible. This region may be taken arbitrarily near to the initial point. 

In Section 3.2, we showed that two reversible adiabats cannot cross. Since a reversible adiabat corresponds to constant entropy, the curve representing T = 0 is a reversible adiabat as well as an isotherm (curve of constant temperature). This is depicted in Figure 3.12, in which the variable X represents an independent variable specifying the state of the system, such as the volume or the magnetization. A reversible adiabat gives the temperature as a function of X. Since two reversible adiabats cannot intersect, no other reversible adiabat can cross or meet the T = 0 isotherm. Therefore, no reversible adiabatic process can reduce the temperature of the system to zero temperature. Furthermore, since we found in Section 3.2 that irreversible adiabatic processes lead to higher temperatures than a reversible adiabat, no adiabatic process, reversible or irreversible, can lead to zero temperature. 

Flere the subscripts and/refer to the initial and final states of the system and the work is defined as the work perfomied on the system (the opposite sign convention—with as work done by the system on the surroundings—is also in connnon use). Note that a cyclic process (one in which the system is returned to its initial state) is not introduced as will be seen later, a cyclic adiabatic process is possible only if every step is reversible. Equation (A2.1.9), i.e. the mtroduction of t/ as a state fiinction, is an expression of the law of conservation of energy. 

Obviously die first law is not all there is to the structure of themiodynamics, since some adiabatic changes occur spontaneously while the reverse process never occurs. An aspect of the second law is that a state fimction, the entropy S, is found that increases in a spontaneous adiabatic process and remains unchanged in a reversible adiabatic process it caimot decrease in any adiabatic process. 

Now the system is thennally insulated and the magnetic field is decreased to zero in this adiabatic, essentially reversible (isentropic) process, the temperature necessarily decreases since... 

These derivatives are of importance for reversible, adiabatic processes (such as in an ideal turbine or compressor), since then the entropy is constant. An example is the Joule-Tnomson coefficient. 

Total Pressure is the pressure that would occur if the fluid were brought to rest in a reversible adiabatic process. Many texts and engineers use the words total and stagnation to describe the flow characteristics interchangeably. To be accurate, the stagnation pressure is the pressure that would occur if the fluid were brought to rest adia-baticaUy or diabatically. 

Adiabatic A process for which there is no heat transfer between a system and its surroundings. An adiabatic process that is reversible is isentropic. 

Isentropic A reversible adiabatic process, in which there is no change in the entropy of the system. 

State is adiabatic and reversible. Such an adiabatic reversible process is called an isentropic state change one in which the entropy remains constant. 

The second part of the second law states that where system undergoes an adiabatic process (system surrounded by insulating walls), i—and the process is reversible, the entropy is not changed, while when the adiabatic process is not reversible the entropy must increase ... 

A reversible adiabatic process is known as isentropic. Thus, the two conditions are directly related. In actual practice compressors generate friction heat, give off heat, have valve leakage and have piston ring leakage. These deviations... 

Theorem.—A process yields the maximum amount of available energy when it is conducted reversibly.—Proof. If the change is isothermal, this is a consequence of Moutier s theorem, for the system could be brought back to the initial state by a reversible process, and, by the second law, no work must be obtained in the whole cycle. If it is non-isothermal, we may suppose it to be constructed of a very large number of very small isothermal and adiabatic processes, which may be combined with another corresponding set of perfectlyJ reversible isothermal and adiabatic processes, so that a complete cycle is formed out of a very large number of infinitesimal Carnot s cycles (Fig 11). 

As we have seen before, exact differentials correspond to the total differential of a state function, while inexact differentials are associated with quantities that are not state functions, but are path-dependent. Caratheodory proved a purely mathematical theorem, with no reference to physical systems, that establishes the condition for the existence of an integrating denominator for differential expressions of the form of equation (2.44). Called the Caratheodory theorem, it asserts that an integrating denominator exists for Pfaffian differentials, Sq, when there exist final states specified by ( V, ... x )j that are inaccessible from some initial state (.vj,.... v )in by a path for which Sq = 0. Such paths are called solution curves of the differential expression The connection from the purely mathematical realm to thermodynamic systems is established by recognizing that we can express the differential expressions for heat transfer during a reversible thermodynamic process, 6qrey as Pfaffian differentials of the form given by equation (2.44). Then, solution curves (for which Sqrev = 0) correspond to reversible adiabatic processes in which no heat is absorbed or released. 

For each state identified on the 9 isothermal as 1, 1, l",..., let us draw paths representing reversible adiabatic processes that intersect a second isotherm at 02. The intersections of the reversible adiabatic paths from states 1,1 and 1" on 9 with those on 02 are denoted by 2, 2 and 2", respectively. Along the three paths, 1-2, l -2, and l"-2", no heat is absorbed or liberated because the processes that connect these points are defined to be adiabatic. 

We wish to show that no points to the leftbb of 2 on the isotherm 62 are accessible from point 1 via any adiabatic path, reversible or irreversible. Suppose we assume that some adiabatic path does exist between 1 and 2. We represent this path as a dotted curve in Figure 2.11a. We then consider the cycle I —>2 —> 1 — 1. The net heat associated with this cycle would be that arising from the last step 1 — 1, since the other two steps are defined to be adiabatic. We have defined the direction 1 — 1 to correspond to an absorption of heat, which we will call qy. From the first law, the net work vv done in the cycle, is given by w = —q, since AU for the cycle is zero. Thus, for this process, iv is negative (and therefore performed by the system), since qy is positive, having been absorbed from the reservoir. The net effect of this cycle, then, is to completely convert heat absorbed at a high temperature reservoir into work. This is a phenomenon forbidden by the Kelvin-Planck statement of the Second Law. Hence, points to the left of 2 cannot be reached from point 1 by way of any adiabatic path. 

Presumably all points on the same surface can be connected by some solution curve (reversible adiabatic process). Flowever, states on surface S2, for example, cannot be connected to states on either Si or S3 by any reversible adiabatic path. Rather, if they can be connected, it must be through irreversible adiabatic paths for which dS 0. We represent two such paths in Figure 2.12 by dashed lines. 

The adiabatic expansion of a gas is an example of (b). In the reversible adiabatic expansion of one mole of an ideal monatomic gas, initially at 298.15 K, from a volume of 25 dm3 to a final volume of 50 dm3, 2343 J of energy are added into the surroundings from the work done in the expansion. Since no heat can be exchanged (in an adiabatic process, q = 0), the internal energy of the gas must decrease by 2343 J. As a result, the temperature of the gas falls to 188 K. 

It is useful to compare the reversible adiabatic and reversible isothermal expansions of the ideal gas. For an isothermal process, the ideal gas equation can be written... 

In Chapter 2 (Section 2.2a) we qualitatively described the Carnot cycle, but were not able to quantitatively represent the process on a p— V diagram because we did not know the pressure-volume relationship for a reversible adiabatic process. We now know this relationship (see section 3.3c), and in Figure 3.3, we compare a series of p-V adiabats with different starting temperatures for an... 

When a process is isentropic, q - F a reversible process is isentropic when q = 0, that is a reversible adiabatic process is isentropic. 

The above relations apply for an ideal gas to a reversible adiabatic process which, as already shown, is isentropic. 

Besides the reversible and irreversible processes, there are other processes. Changes implemented at constant pressure are called isobaric process, while those occurring at constant temperature are known as isothermal processes. When a process is carried out under such conditions that heat can neither leave the system nor enter it, one has what is called an adiabatic process. A vacuum flask provides an excellent example a practical adiabatic wall. When a system, after going through a number of changes, reverts to its initial state, it is said to have passed through a cyclic process. 

In a system undergoing a reversible adiabatic process, there is no change in its entropy. This is so because by definition, no heat is absorbed in such a process. A reversible adiabatic process, therefore, proceeds at constant entropy and may be described as isentropic. The entropy, however, is not constant in an irreversible adiabatic process. 

Silverstein, J. L. et al., Loss Prev., 1981,14, 78 Nitrobenzene was washed with dilute (5%) sulfuric acid to remove amines, and became contaminated with some tarry emulsion that had formed. After distillation, the hot tarry acidic residue attacked the iron vessel with hydrogen evolution, and an explosion eventually occurred. It was later found that addition of the nitrobenzene to the diluted acid did not give emulsions, while the reversed addition did. A final wash with sodium carbonate solution was added to the process [1]. During hazard evaluation of a continuous adiabatic process for manufacture of nitrobenzene, it was found that the latter with 85% sulfuric acid gave a violent exotherm above 200° C, and with 69% acid a mild exotherm at 150- 170°C [2],... 

If there is no heat transfer or energy dissipated in the gas when going from state 1 to state 2, the process is adiabatic and reversible, i.e., isentropic. For an ideal gas under these conditions,... 

To determine the entropy change in this irreversible adiabatic process, it is necessary to find a reversible path from a to b. An infinite number of reversible paths are possible, and two are illustrated by the dashed lines in Figure 6.7. 

To calculate the change in entropy in this irreversible flow, it is necessary to consider a corresponding reversible process. One process would be to allow an ideal gas to absorb reversibly the quantity of heat Q at the temperature T2. The gas then can be expanded adiabatically and reversibly (therefore with no change in entropy) until it reaches the temperature Ti. At Ti the gas is compressed reversibly and evolves the quantity of heat Q. During this reversible process, the reservoir at T2 loses heat and undergoes the entropy change... 

As in the complete cycle (irreversible adiabatic process from atob followed by the three reversible steps)... 

In the fourth step, which is adiabatic and reversible, no entropy change occurs, but the temperature increases to the initial value T2- As the process is cyclic... 

For an isolated (adiabatic) system, AS > 0 for any natural (spontaneous) process from State a to State b, as was proved in Section 6.8. An alternative and probably simpler proof of this proposition can be obtained if we use a temperature-entropy diagram (Fig. 6.13) instead of Figure 6.8. In Figure 6.13, a reversible adiabatic process is represented as a vertical line because AS = 0 for this process. In terms of Figure 6.13, we can state our proposition as follows For an isolated system, a spontaneous process from a to b must lie to the right of the reversible one, because AS = Sb Sa> 0. 

Second Law of Thermodynamics. There have been numerous statements of the second law. To paraphrase Clausius It is impossible to devise an engine or process which, working in a cycle, will produce no effect other than the transfer of heat from a colder to a warmer body. According to Caratheodory, the Second Law can be stated as follows Arbitrarily close to any given state of any closed system, there exists an unlimited number of other states which it is impossible to reach from a given state as a result of any adiabatic process, whether reversible or not . 

No entropy change occurs. This usually requires i that the process is adiabatic and reversible. 

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