The Chemistry Maths Book, Second Edition

5.8 Pressure–volume work 157

(where the potential energy is zero). At an intermediate height,E 1 = 1 mgh 1 = 1 T(x) 1 + 1 V(x)

and the kinetic energy is

( 5. 63 )

If both the body and the surface are perfectly elastic, the velocity of the body is

reversed on contact with the surface, and the body returns to its original height at

x 1 = 1 hin an exact reversal of the falling motion. Thus, solving equation (5.63) for the

velocity,

and the velocity of the body is negative as it falls and positive as it rises. In the absence

of dissipative forces the bouncing motion is repeated indefinitely.

EXAMPLE 5.19Electrostatic potential energy

By Example 5.17, the work that must be done againstthe internal force to bring two

charges from infinite separation to separation xisW 1 = 1 q

1

q

2

24 πε

0

x. This is the same as

the work done bythe internal force in separating the charges:

The force depends only on the relative positions of the charges and is conservative, so

that a potential energy function V(x)exists such that F(x) 1 = 1 −dV 2 dxandW 1 = 1 V(x) 1 −

V(∞). It is conventional, and convenient, to choose the zero of potential energy of

interacting charges to be zero for infinite separation: V(∞) 1 = 10. Then

is the electrostatic potential energy of the system of two charges.

0 Exercises 54, 55

5.8 Pressure–volume work

Consider a fluid (gas or liquid) enclosed in a uniform cylindrical container, closed at

one end and fitted with a piston as shown in Figure 5.23. Let Abe the internal cross-

sectional area of the cylinder. A fluid with internal pressure pexerts a force of

magnitude 1 |F| 1 = 1 pAon the surface of the piston, and the piston moves in or out

Vx

qq

x

()=

12

0

4 πε

WFxdx

qq

x

x

==Z

∞

()

12

0

4 πε

v=± 2 gh x()−

Tx m mgh x()==−( )

1

2

2

v