Thermodynamics and Chemistry

CHAPTER 3 THE FIRST LAW

3.4 DEFORMATIONWORK 72

Thexcomponent of the force exerted by the system on the surroundings at this portion of
the boundary,Fxsys, is equal toFgas. (The other forces shown in Fig.3.4are within the
surroundings.) Applying the differential form of Eq.3.1.2, we have∂wDFgasdxwhich,
with the substitutionFgasDpbAs(from Eq.3.4.2), becomes

∂wDpbAsdx (3.4.6)

It will be convenient to change the work coordinate fromxtoV. The gas volume is
given byV DAsxso that an infinitesimal change dxchanges the volume by dV DAsdx.
The infinitesimal quantity of work for an infinitesimal volume change is then given by

∂wDpbdV (3.4.7)

(expansion work,

closed system)

and the finite work for a finite volume change, found by integrating from the initial to the
final volume, is

wD

ZV 2

V 1

pbdV (3.4.8)

(expansion work,

closed system)

During expansion (positive dV),∂wis negative and the system does work on the surround-
ings. During compression (negative dV),∂wis positive and the surroundings do work on
the system.

When carrying out dimensional analysis, you will find it helpful to remember that the

product of two quantities with dimensions of pressure and volume (such aspbdV) has

dimensions of energy, and that 1 Pa m^3 is equal to 1 J.

The integral on the right side of Eq.3.4.8is aline integral(Sec.E.4on page 480 ). In
order to evaluate the integral, one must be able to express the integrandpbas a function of
the integration variableV along the path of the expansion or compression process.
If the piston motion during expansion or compression is sufficiently slow, we can with
little error assume that the gas has a uniform pressurepthroughout, and that the work can
be calculated as if the process has reached its reversible limit. Under these conditions, Eq.
3.4.7becomes

∂wDpdV (3.4.9)

(reversible expansion

work, closed system)

and Eq.3.4.8becomes

wD

ZV 2

V 1

pdV (3.4.10)

(reversible expansion

work, closed system)