Cyclic integrals

The difference between the cyclic integrals jdW and f d 1 2 is the work associated with entrapment of mercury and can be evaluated from the area between curves A and B. 

The important point is that the final value of the integral depends only on the two endpoints, i.e., the value of the function z at (x, yi) and (x2, y2), but not the chosen path of integration (as illustrated in Sidebar 1.4). Moreover, in the special case of a cyclic integral (denoted ), where initial and final limits coincide, the integral (1.15) necessarily vanishes for an exact differential, independent of how the cyclic path is chosen. We can therefore state the following integral criterion for exactness ... 

It was recognized by Carnot that cyclic integrals of the form (4.24c) must actually vanish for all reversible cycles ... 

We now divide the cyclic integral (4.44) into its outbound irreversible (A —> B) and inbound reversible (B —> A) steps ... 

From mathematics we recognize that the quantity (dQ + dW) is an exact differential, because its cyclic integral is zero for all paths. Then, some function of the variables that describe the state of the system exists. This function is called the energy function, or more loosely the energy. We therefore have the definition... 

However, the cyclic integral of an exact differential is zero and therefore QJT is an exact differential of some function. The notation dQ, is used to emphasize that the process is reversible. The new function is called the entropy function and is defined in terms of its differential, so... 

It is clear that the left-hand side of Equation (109) is simply the sum over the cycle of the quantity (Q/T). Thus, it can be written as the cyclic integral of the differential quantity for reversible systems ... 

The work produced in a cyclic transformation is the sum of the small quantities of work 4W produced at each stage of the cycle. Similarly, the heat withdrawn from the surroundings in a cyclic transformation is the sum of the small quantities of heat dQ withdrawn at each stage of the cycle. These sums are symbolized by the cyclic integrals of 4W and dQ -... 

In contrast, note that if we sum the differential of any state property of the system over any cycle the total difference, the cyclic integral, must be zero. Since in any cycle the system returns at the end to its initial state, the total difference in value of any state property must be zero. Conversely, if we find a differential quantity dy such that... 

An exact differential integrates to a finite difference, Jf dU = U2 — U, which is independent of the path of integration. An inexact differential integrates to a total quantity, li 6 = 6> which depends on the path of integration. The cyclic integral of an exact differential is zero for any cycle, Eq. (7.7). The cyclic integral of an inexact differential is usually not zero. 

We have shown that 4Qr y/T has a cyclic integral equal to zero only for cycles that involve only two temperatures. The result can be generalized to any cycle. 

Consider the following cycle A system is transformed irreversibly from state 1 to state 2, then restored reversibly from state 2 to state 1. The cyclic integral is... 

How can J vanish when integrated around a cycle while the cyclical integral of dQy y... 

Cyclic integrals of q and w are not zero as they are path dependent functions. 8.6.3 Mechanical Work... 

In both cases the cyclic integral measures the area enclosed by the four steps in Figures 9.12 and 9.13. 

A cyclic process is a process in which the state of the system changes and then returns to the initial state. In this case the integral of dX is written with a cycUc integral sign dX. Since a state function X has the same initial and final values in a cycUc process, X2 is equal to Xi and the cyclic integral of dX is zero ... 

Keep in mind that the value of the cyclic integral i dq/T y depends only on the path of the experimental cycle, that this process can be reversible or irreversible, and that Tres is a positive constant. 

This relation is known as the Clausius inequality. It is valid only if the integration is taken around a cyclic path in a direction with nothing but reversible and irreversible changes—the path must not include an impossible change, such as the reverse of an irreversible change. The Clausius inequahty says that if a cyclic path meets this specification, it is impossible for the cyclic integral (dg/Tb) to be positive. 

It physically states that the cyclic integral or the sum of heat transfer in all processes is equal to fhe cyclic integral or the sum of work done in all... 

This is not an exact differential. We see in the next example that its cyclic integral does not necessarily vanish. Since dwrev is not exact, we write... 

The cyclic integral does not vanish because dwrev is an inexact differential. 

Exercise 9.5. A two-phase system contains both liquid and gaseous water, so its equilibrium pressure is determined by the temperature. Calculate the cyclic integral of dwrev for the following process The volume of the system is changed from 10.001 to 20.001 at a constant temperature of 25.00° C, at which the pressure is 24.756 torr. The system is then heated to a temperature of 100.0° C at a constant volume of 20.001. The system is then compressed to a volume of 10.001 at a temperature of 100.0° C, at which the pressure is 760.0 torr. The system is then cooled from 100.0° C to a temperature of 25.00° C at a constant volume of 10.001. Remember to use consistent units. 

Other cyclic integrals might not be zero but a state variable will sum to zero that is the main characteristic of a state variable. The circle on the integral sign means that the process is carried out over a cyclic process. We can offer a simple insight in that state variables are universal variables and characteristics of the universe while q and w are subjective variables, which depend on how a process is carried out. 

Such an integral is called a cyclic integral. Equation (6-52) can be thought of as the consummate test for a state function. 

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