Thermodynamics and Chemistry

CHAPTER 3 THE FIRST LAW

3.4 DEFORMATIONWORK 73

Fsur

Fsys

(a)

ds

As;

(b)

Figure 3.5 Deformation of an isotropic phase (shaded) confined by a wall.

(a) Equal and opposite forces exerted by the surroundings and system at surface ele-

ment(thick curve) of the system boundary.

(b) Change from initial volume (dotted curve) to a smaller volume.

The appearance of the symbolpin these equations, instead ofpb, implies that the equations
apply only to a process in which the system has at each instant a single uniform pressure.
As a general rule,an equation containing the symbol of an intensive property not assigned
to a specific phase is valid only if that property is uniform throughout the system, and this
will not be explicitly indicated as a condition of validity.

Some texts state that expansion work in a horizontal cylinder-and-piston device like

that shown in Fig.3.4should be calculated fromw D 

R

pextdV, wherepextis a

pressure in thesurroundingsthat exerts the external forceFexton the piston. However,

if the system is the gas the correct general expression is the one given by Eq.3.4.8:

w D 

R

pbdV. This is because it is the forceFgas DpbAsthat is exerted by the

system on the surroundings, whereas the forceFextDpextAsis exerted by one part of

the surroundings on another part of the surroundings.

In other words, if the integrals

R

Fgasdxand

R

Fextdxhave different values, it is the

first of these two integrals that should be used to evaluate the work:wD 

R

Fgasdx.

Both integrals are equal if the expansion or compression process is carried outre-

versibly. This is because in the limit of infinite slowness the piston has neither friction

(FfricD 0 ) nor acceleration (FnetD 0 ), and therefore according to Eq.3.4.3,FgasandFext

are equal throughout the process. Another situation in which the two integrals are

equal is when the piston is frictionless and is stationary in the initial and final states,

because then bothFfricand

R

Fnetdxare zero. (The integral

R

Fnetdxcan be shown to

be equal to the change in the kinetic energy of the piston, by a derivation similar to

that leading to Eq.G.1.5on page 488 .) In the general irreversible case, however, the

integrals

R

Fgasdxand

R

Fextdxarenotequal.^10

3.4.3 Expansion work of an isotropic phase

Expansion work does not require a cylinder-and-piston device. Suppose the system is an
isotropic fluid or solid phase, and various portions of its boundary undergo displacements in
different directions. Figure3.5shows an example of compression in a system of arbitrary
shape. The deformation is considered to be carried out slowly, so that the pressurepof

(^10) For an informative discussion of this topic see Ref. [ 6 ]; also comments in Refs. [ 29 ], [ 7 ], [ 95 ], [ 8 ], and [ 121 ];
also Ref. [ 92 ].