Reversible cyclical process

Theorem.—The work done in any isothermal reversible cyclic process is zero (J. Moutier, 1875). 

For this purpose we suppose the following reversible cyclic process executed. This process, it will be seen, is not a Carnot s cyclic process, but is of another type. 

Figure 2.10 (a) A schematic Carnot cycle in which isotherms at empirical temperatures 6 and 62 alternate with adiabatics in a reversible closed path. The enclosed area gives the net work produced in the cycle, (b) The area enclosed by a reversible cyclic process can be approximated by the zig-zag closed path of the isothermal and adiabatic lines of many small Carnot cycles. 

Consider any reversible cyclic process that involves the exchange of heat and work. Again, the net area enclosed by the cycle on a p-V plot gives the work. This work can be approximated by taking the areas enclosed within a series of Carnot cycles that overlap the area enclosed by the cycle as closely as possible as shown in Figure 2.10b. For each of the Carnot cycles, the sum of the q/T terms... 

Equation (1.58) applies to any isothermal or adiabatic reversible cyclic processes. If there are many isothermal and adiabatic processes from high temperature TH to low temperature TL before the process goes back to its original state, any reversible cyclic process [Equation (1.58)] can be generalized as ... 

Further insight may be gained by study of an arbitrary reversible cyclic process, as represented schematically on a PV diagram in Fig. 5.4. We divide the entire closed area by a series of reversible adiabatic curves since such curves cannot intersect (see Prob. 5.1), they may be drawn arbitrarily close to one another. A few of these curves are shown on the figure as long dashed lines. We connect adjacent adiabatic curves by two short reversible isotherms which approximate the curve of the general cycle as closely as possible. The approxima-... 

At this point we can precisely define the terms reversible and irreversible in a thermodynamic sense. In a reversible, cyclic process both the system and the surroundings are returned exactly to their original conditions. As it turns out, this process is hypothetical. On the other hand, an irreversible process is one in which, even when the system is cycled (state 1 —> state 2 — state 1) and thus returned to its original state, the surroundings are changed in a permanent way. All real processes are irreversible. 

If these facts and their graphical representation are-treated from the thermo-dynamical side, the well-known reversible cyclic process ap-... 

The reversible cyclic process to be carried out consists in the conversion to and fro of the quantities occurring in the equation, i. e. of 2CI2 and 2H2O. That is to be accomplished by the cylinder c and piston d, which by means of a par-... 

Since in this reversible cyclic process the temperature remains constant, no heat is converted into work the sum of the separate amounts of work done is therefore zero. We win express this as... 

Figure 5.4 An arbitrary reversible cyclic process drawn on a PV diagram...

That is to say, the entropy of the thermodynamic system represented by die working gas returns to its original value when it passes through a Carnot cycle. Equation (9) is valid for any reversible cyclical process. 

Now, we need to define the relation between entropy and reversible cyclic processes. In 1824, Sadi Carnot proved that in any cyclic reversible process which takes place between two heat reservoirs one hot, QH, and one cold, Qc-... 

Now in such a reversible cyclic process a certain finite amount of external work will be supplied, which is given by the area contained between the four curves. Since no heat is taken up on the curves BC and DA, and since, by equation (1), no heat can be taken up over the curve CD because the latent heat disappears at the absolute zero, it follows that the external work must have been supplied at the cost of the heat removed from the heat reservoir at the temperature AT, which is very low perhaps, but still finite. But as this contradicts the Second Law, we arrive at the conclusion which was to be demonstrated. 

From the reversible cyclic process carried out in a closed homogeneous system by combination of infinite number of consecutive carnot cycles it follows that ... 

Rem. 16, the Second law (1.18) forbids for (infinite dimensional vector of) heat distribution of cyclic processes (or its densities) the region with absorbed heat only (q > 0,q =0). Moreover, using closeness and completeness of universe Ul, U2 with Carnot cycles, the heat distributions (or their densities) must fall into halfspace which does not meet the forbidden region (with corresponding boundary hyperplane of reversible cyclic processes). This may be similarly expressed through positive function /(H) > 0 of empirical temperature d by... 

There is another notable point. The temperature in the system need not be temporally nor spatially constant. The transfer of a quantity of entropy St from a hot to a cold subarea by use of an auxiliary body that repeatedly undergoes a reversible cyclic process (see Fig. 24.2) delivers useful work to be stored in the system because all the possible paths for energy outflow have been blocked. Energy U remains constant in the process. Transferring the same amount of entropy 5t by... 

State function In thermodynamics a variable is a state function if, when all the thermodynamic variables are specified, it has a unique value. As a result, the change in any state function in a reversible cyclic process must be zero. 

The change in entropy, dS, is just QIT, where T is the temperature. The factor l/T is an integrating factor that transforms SQ into an exact differential just as 1/w transforms vdu — udv into the exact differential d(u/v). Because the change in entropy, dS = SQ/T,is exact differential, the change in entropy in a reversible cyclic process is zero. The entropy of a thermodynamic state is a well-defined single-valued function and the entropy is said to be a state function. An equivalent statement of the second law of thermodynamics is... 

Incidentally, the fact that for a reversible cyclic process, the sum... 

We now show that Eq. (3.2-2) is valid for the reversible cyclic process of Figure 3.4a. Steps 1, 3, and 5 are isothermal steps, and steps 2, 4, and 6 are adiabatic steps. Let point 7 lie on the curve from state 6 to state 1, at the same temperature as states 3 and 4, as shown in Figure 3.4b. We now carry out the reversible cyclic process 1 2 3 7 1, which is a Carnot cycle and for which the line integral... 

To show that two reversible adiabats cannot cross for other systems we assume the opposite of what we want to prove and then show that this assumption leads to a contradiction with fact and therefore must be false. Assume that there are two different reversible adiabats in the state space of a closed simple system and that the curves coincide at state number 1, as depicted in Figure 3.6. We choose a state on each reversible adiabat, labeled state number 2 and state number 3 such that the reversible process leading from state 2 to state 3 has q > 0. Now consider a reversible cyclic process 1 2 3 1. Since steps 1 and 3 are adiabatic. 

As the designation indicates, a reversible process can be reversed by an infinitesimal change of the influence exerted on the system by the surroundings. In a reversible cyclic process returning a system to its initial state, no permanent change of the thermodynamic universe will occur. 

The Carnot cycle is a reversible cyclic process during which a thermodynamic system cycles between two heat reservoirs of different temperatures. A Carnot process with an ideal gas will be described here as an illustration, but as will be shown later, there is no restriction on the nature of the system. 

Assume a closed system of 1 mol of an ideal gas, which is confined in a cylinder below a non-frictional piston. In its surroundings there are two heat reservoirs, one with constant, high temperature Ti, and one with constant, low temperature Ts. In its initial state (a), the system is in contact with the heat reservoir the gas temperature is T and the volume is 14. Now the following reversible cyclic process is performed... 

After this sequence of processes, the initial state (a) is restored therefore, taken as a whole, the processes constitute a reversible cyclic process. Prom the first law, it is now possible to set up a total energy account for the cyclic process... 

The derived expressions (4.9) and (4.10) can be shown to be general for reversible cyclic processes. Further, it can be shown that the reversible Carnot cycle with an ideal gas is the most efficient process possible for transforming heat... 

Figure 4.11. Any reversible cyclic process can be described as the sum of reversible Carnot cycles an arbitrarily good approximation of the process can be obtained by the choice of division of the process.

The expression shows that in a simple reversible cychc process, the quantity Qrev/T = 0. Since any reversible cyclic process in principle can be decomposed into an arbitrary number of sub-processes fulfilling this term, it can be shown... 

It can be proved that this result is not subject to the gas behaving ideally. It holds for real gases. It also applies to any other reversible cyclical process because one can subdivide any reversible cyclical process of expansions and compressions into a sequence of individual Carnot cycles. 

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