Fig. 1.12 Irreversible, isothermal compression of an ideal gas
One of the most amusing and astute explanations of the First law was given by the physicist Richard Feynman in his lectures to physics students at Cal Tech. Feynman's story is too long to be repeated here, but Van Ness (Ref. 2, pp. 2 - 8) recounts it in full, and is well worth reading.
It is an empirical observation, never refuted, that the change in a property labeled the internal energy of a closed system resulting from addition of heat and performance of work is given by:
AU = U(final) - U(initial) = change in system internal energy
Q = heat added to the system
W = work done by the system
Equation (1.4) is the most common form of the First Law of Thermodynamics. It applies to a process that takes a system from an initial to a final state. It neglects changes in the system's potential energy and kinetic energy. The signs of the terms on the right side are important. Q represents heat added to the system, so that if the process results in removal of heat, Q is negative. Similarly, work done by the system is considered to be positive, which accounts for the negative sign in Eq (1.4).
Actually, the left side of Eq (1.4) should be the change in total energy of the system, which includes kinetic and potential energies as well as internal energy. However, for commonly encountered closed systems, kinetic and potential energy changes are usually absent, so Eq (1.4) suffices. For open systems, where fluid flows across system boundaries, however, inclusion of these two terms is often necessary (see Chap. 3).
The First Law can also be written for a differential slice of the process:
The manner of expressing the differentials in Eq (1.5) rises from the fundamentally different natures of internal energy and heat or work. The differential dU represents an infinitesimal change in the property U. The differentials SQ and SW, on the other hand, denote infinitesimal quantities of heat and work exchanged between system and surroundings. Q and W are not thermodynamic properties, and so cannot "change" the way that U can.
The differential and integral forms of the First Law are directly related because: AU = f dU = U(final) - U(initial)
Jsq w=is and
Itegration proceeds from the initial state to the final state. The integral of dU is the difference between the values of U in these two states, but the integrals of Q and W cannot be similarly written because these quantities do not have "values" associated with the state of the system; they depend on the path taken in the process. By "path" is meant the variation of the system's state (e. g., p and V) as it moves from the initial to the final state.
Although Q and W individually depend on the path, the difference Q -W is independent of the path because it represents AU, a property change. For instance, in the examples in the preceding section of compressing an ideal gas reversibly (Fig. 1.12) and irreversibly (Fig. 1.13), AU = 0 because the initial and final temperatures are the same and the gas is ideal. However, the example at the end of the preceding section shows that the work done by the surroundings on the system in the irreversible process is approximately twice that of the reversible process. Consequently, Eq (1.4) requires that the heat released to the environment in the irreversible process must also be about twice that released in the reversible process.
Figure 1.14 shows the paths followed by the gas inside the cylinder in Figs. 1.12 and 1.13) in the two compression processes on a p - V coordinate diagram. According to Eq (1.3), the area under reversible curve is Wrev . The path of the irreversible compression process is shown schematically as a dashed curve; the p - V trajectory cannot be deduced from thermodynamics because the system is not in equilibrium except in the initial and final states.
An important consequence of the First Law expressed by Eq (1.4) is that increasing AU can be achieved either by adding heat to the system (Q > 0) with W = 0 or by performing work on the system (i.e., W< 0) in an adiabatic process (Q = 0). Work addition can be accomplished in a number of ways, including compression (i.e., pV work), or in the form of shaft work, typically represented by a paddle rotating in a fluid contained in a sealed insulated vessel. Friction of the moving paddle in the fluid degrades the work to heat, which causes the increase in internal energy. Irrespective of the form of work done on the system, as far as the First Law is concerned, heat and work produce identical effects, and are said to be equivalent. One of the early triumphs of thermodynamics was the quantitative measurement of the "mechanical equivalent of heat" as 4.184 J/calorie by the Scottish scientist Joule in the 19th century.
The above discussions of the First law dealt with one-way processes, which began with an initial state and ended in a final state. A second representation of the First Law (and historically the first) applies to a cyclical process, such as the one shown in Fig. 1.15.
Fig. 1.15 A two-step cyclical process between states 1 and 2
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