The Chemistry Maths Book, Second Edition

154 Chapter 5Integration

This work is positive for like charges. The same formula applies to the case of unlike

charges, but the work is then negative.

0 Exercise 53

Work and energy

When work is done on a system by an external force, the energy of the system is

increased by the amount of the work done. Conversely, when a system does work

against an external force, the energy of the system is decreased by the amount of the

work done. The energy of a system is usually expressed as the sum of two parts. For a

simple system, with no internal structure, these parts are (i) the kinetic energy, or

translational energy, arising from the motion of the system in space, and (ii) the

potential energy, arising from the position of the system in space and from the forces

acting on the system at that position. In the case of a system with internal structure,

the kinetic energy is the sum of the kinetic energies of its constituent parts, and the

potential energy is the sum of the potential energies of its parts.

(i) Kinetic energy

The work done on a body by an external forceF(x)as the body travels from point A

to point Bis

(5.51)

Becausev 1 = 1 dx 2 dt, the element of length dxcan be replaced by the differential v 1 dt, so

that

(5.52)

in which use has been made of d(v

2

) 2 dt 1 = 12 vdv 2 dt, and the integration limits now

refer to the times at Aand B. It follows that the work done between Aand Bis

(5.53)

wherev

A

andv

B

are the velocities of the body at Aand Brespectively. The quantity

is called the kinetic energyof the body and is usually denoted by the symbol T

(or K). The work done on the body is therefore equal to the change in kinetic energy:

W

AB

1 = 1 T

B

1 − 1 T

A

(5.54)

We note that the kinetic energy of a body at rest is defined to be zero.

(ii) Potential energy and total energy

Let the force acting on a body depend only on the position of the body, so that

F 1 = 1 F(x). This condition excludes time-dependent forces and, more importantly,

1

2

2

mv

Wm

AB BA

=−













1

2

22

vv

Wm

d

dt

dt m

d

dt

dt m

AB

== =













ZZ

A

B

A

B

v

vvv

1

2

1

2

22

()

AA

B

WFxdxm

d

dt

dx

AB

==ZZ

A

B

A

B

()

v