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SECTION 2·9 THB RBVllRSIBLB PROCESS

The phase-rule variables are temperature, pressure, and the phase compositions.

If

any two of

these variables are arbitrarily specified, the others must have unique values in order for the

system as described to exist. The composition variables considered in connection with the

phase rule are either the weight or mole fractions of the species in a phase. It should be noted

that for each phase the sum ofthe weight or mole fractions must be unity. Thus fixingthe mole

fraction of water in the liquid phase of the system under consideration automatically fixes

the mole fraction of the alcohol in the liquid phase. These two compositions cannot both be

arbitrarily specified.

2 ·9 T he R ev er sib le P r oc es s

An essential concept of thermodynamics is revealed through examination of the

simple process that results when a gas expands against a piston in a cylinder. The

question to be answered is: How can the process be carried out so that the expansion

process yields the maximum work? We adopt a pair of simplifying assumptions in

order to focus attention on the essentials of the problem:

1 . The cylinder and piston neither absorb nor transmit heat; thus the effectsof heat

transfer need not be considered.

2. The piston moves within the cylinder without friction; thus the possibility of

mechanical inefficiencies is removed.

The apparatus isshown in Fig. 2·4. The system is chosen so as to include only

the gas. All else constitutes the surroundings. The object is to allow the process to

take place in such a way as to raise a weight, thereby accomplishing useful work in

the surroundings.

Consider first Fig.

2 ·4a .

The piston is shown inserted far into the cylinder,

where it confines a gas at a pressure just sufficient to balance the weight ofthe piston,

the atmosphere, and the weight

w.

This is a condition of equilibrium, for the system

has no tendency to change. The weight must be removed if the piston is to rise.

Imagine in this first alternative that the weight is suddenly slid from the piston to a

shelf that is placed close by and at the same level. The piston rises and eventually

reaches a new condition of equilibrium, such that the pressure exerted by the gas is

just sufficient to balance the weight of the piston and atmosphere. The system has

undergone a change; the pressure of the gas is lower, and the piston has been

elevated. However, no useful work has been accomplished, for the weight is still at

the same level as before. Work has been done in elevating the piston and in pushing

back the atmosphere, but this does not satisfy our definition of useful work.

In an improved process shown in Fig.

2 ·4b

the weight w is divided into two

parts. Initially the piston isdepressed as before, but this time only half ofthe weight is

slid over to the shelf. The piston rises roughly half as far as before, and eventually

assumes an equilibrium position. The second half of the weight is then slid from the

piston to another conveniently located shelf as shown, and the piston completes its

stroke. This process is an improvement over the first one, since part of the weight

ha s

4 1



4 2

CHAPTER 2 THE FIRSf LAW AND OTIiER BASIC CONCIlPTS

G

a b

1

c

Fig. 2 -4 I rr ev er sib le a nd r ev er si bl e e xpan si on p ro ce ss es .

been raised roughly half the distance of the piston's travel and useful work has been

accomplished.

A further division of the weight into parts results in even greater improvement ,

The ultimate extension of this idea requires a weight made

up

of differential elements.

This condition can be approached if the weight is replaced by a pile of fine sand as

shown in Fig.

2-4c.

The grains of sand are removed one at a time and piled

in

such a

way as to remain at the level at which they are removed from the piston. Various

stages of the process are represented in the figure. The removal of a grain of sand

causes very little change, and equilibrium is restored by only a minute adjustment of

the system after each grain is removed. The system is never more than slightly out of

balance, never more

than

slightly removed from equilibrium. A s a result, the pile

of sand reaches almost the full height of the piston's travel. On average, all the weight

has been raised somewhat less than half the distance of the piston's stroke. The only

further improvement possible would be obtained

if

the grains of sand were made

infinitesimal in size. We now consider the implications of this optimum process,

accomplished in differential steps.

The process carried out in this manner yields the maximum possible work that

can be obtained under the conditions imposed on the system. In no other way can the



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Thermo Notes3

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