Thermodynamics and Chemistry

APPENDIX G FORCES, ENERGY, AND WORK

G.1 FORCES BETWEENPARTICLES 488

The integral on the right side of Eq.G.1.3is an example of aline integral. It indicates

that the scalar product of the net force acting on the particle and the particle’s displace-

ment is to be integrated over time during the time interval. The integral can be written

without vectors in the form

R

Ficos.ds=dt/dtwhereFiis the magnitude of the net

force, ds=dtis the magnitude of the velocity of the particle along its path in three-

dimensional space, and is the angle between the force and velocity vectors. The

three quantitiesFi, cos , and ds=dtare all functions of time,t, and the integration is

carried out with time as the integration variable.

By substituting the expression forFi(Eq.G.1.2) in Eq.G.1.3, we obtain

WiDmi

Z

dvi

dt

driDmi

Z

dri

dt

dviDmi

Z

vidviDmi

Z

vidvi

DÅ

1

2 miv

2

i

(G.1.4)

whereviis the magnitude of the velocity.
The quantity^12 miv^2 i is called thekinetic energyof particlei. This kinetic energy de-
pends only on the magnitude of the velocity (i.e., on the speed) and not on the particle’s
position.
ThetotalworkWtotdone by all forces acting on all particles during the time interval is
the sum ofWifor all particles:WtotD

P

iWi.

(^4) EquationG.1.4then gives us
WtotD

X

i

Å

1

2 miv

2

i

DÅ

X

i

1

2 miv

2

i

!

(G.1.5)

EquationG.1.5shows that the total work during a time interval is equal to the change in
the total kinetic energy in this interval. This result is called the “work-energy principle” by
physicists.^5
From Eqs.G.1.1andG.1.3we obtain a second expression forWtot:

WtotD

X

i

WiD

X

i

ZX

j§i

FijdriD

X

i

X

j§i

Z

Fijdri (G.1.6)

The double sum in the right-most expression can be written as a sum over pairs of particles,
the term for the pairiandjbeing
Z
FijdriC

Z

FjidrjD

Z

Fijdri

Z

Fijdrj

D

Z

Fijd.rirj/D

Z

.Fijeij/drij (G.1.7)

Here we have used the relationsFjiD Fij(from Newton’s third law) and.rirj/D
eijrij, whereeijis a unit vector pointing fromjtoiandrijis the distance between the
particles. EquationG.1.6becomes

WtotD

X

i

X

j§i

Z

FijdriD

X

i

X

j>i

Z

.Fijeij/drij (G.1.8)

(^4) The workWtotdefined here is not the same as the thermodynamic work appearing in the first law of thermo-
dynamics.
(^5) Ref. [ 151 ], p. 95.