Physical Chemistry Third Edition

2.1 Work and the State of a System 41

For a finite process we can divide the displacement into small steps such that the

force is nearly constant in each step. This division is shown schematically in Figure 2.2.

The work done on the object in the entire process is obtained by adding the work

done in all of the steps. In the limit that the sizes of the steps approach zero the sum

becomes an integral. If the displacement and force are both in thezdirection this

integral is

w

∫

dw

∫

Fzdz (2.1-4)

where we denote an amount of work done in a finite process byw. In order to calculate

wfor an actual process, we must know howFzdepends onz.

dr

(an infinitesimal

displacement)

r

(a finite displacement)

Figure 2.2 A Displacement Divided

into Small Steps.

EXAMPLE 2.1

An object is pushed in thezdirection by a force given byFzaz+b, where the constanta

has the value 300.0 N m−^1 and the constantbhas the value 500.0 N. Find the work done in

moving the object fromz0toz 10 .0m.

Solution

Since the force is in thezdirection, Eq. (2.1-4) applies:

w

∫ 10 .0m

0

Fzdz

∫ 10 .0m

0

(az+b)dz

(

az^2

2

+bz

)∣∣

∣∣

∣

10 .0m

0



(300.0N m−^1 )(10.0m)^2

2

+(500.0 N)(10.0m) 2. 00 × 104 J

The Work Done on a Closed Fluid System

If a force is transmitted between a system and its surroundings and if the volume of the

system changes there is work done by the surroundings on the system or by the system

on the surroundings. Consider a gas confined in a cylinder with a piston as depicted

in Figure 1.1. We assume that any friction is negligible. Let an external forceFextbe

exerted downward on the system by the piston. If the external force is greater than the

force exerted on the piston by the system, the piston will accelerate downward and the

surroundings perform work on the system. If the external force is smaller, the piston

will accelerate upward and the system performs work on the surroundings. If the height

of the piston,z, increases by an infinitesimal amountdz, the amount of work done on

the surroundings is given by Eq. (2.1-1):

dwsurrF(transmitted)dz (2.1-5)

whereF(transmitted) is the force that is actually transmitted to the surroundings.

Reversible Processes

At equilibrium, the force exerted on the piston by the gas is equal to

FPA (2.1-6)