Thermodynamics and Chemistry

APPENDIX G FORCES, ENERGY, AND WORK

G.1 FORCES BETWEENPARTICLES 489

Next we look in detail at the force that particlejexerts on particlei. This force depends
on the nature of the two particles and on the distance between them. For instance, Newton’s
law of universal gravitation gives the magnitude of agravitationalforce asGmimj=rij^2 ,
whereGis the gravitational constant. Coulomb’s law gives the magnitude of anelectrical
force between stationary charged particles asQiQj=.4 0 rij^2 /, whereQiandQjare the
charges and 0 is the electric constant (or permittivity of vacuum). These two kinds of forces
are central forces that obey Newton’s third law of action and reaction, namely, that the forces
exerted by two particles on one another are equal in magnitude and opposite in direction and
are directed along the line joining the two particles. (In contrast, theelectromagneticforce
between charged particles in relative motion doesnotobey Newton’s third law.)
We will assume the forceFijexerted on particleiby particlejhas a magnitude that
depends only on the interparticle distancerijand is directed along the line betweeniand
j, as is true of gravitational and electrostatic forces and on intermolecular forces in general.
Then we can define apotential function,ij, for this force that will be a contribution to
the potential energy. To see howijis related toFij, we look at Eq.G.1.7. The left-most
expression,

R

FijdriC

R

Fjidrj, is the change in the kinetic energies of particlesiand
jduring a time interval (see Eq.G.1.4). If these were the only particles, their total energy
should be constant for conservation of energy; thusÅijshould have the same magnitude
and the opposite sign of the kinetic energy change:

ÅijD

Z

.Fijeij/drij (G.1.9)

The value ofijat any interparticle distancerijis fully defined by Eq.G.1.9and the choice
of an arbitrary zero. The quantity.Fijeij/is simply the component of the force along the
line between the particles, and is negative for an attractive force (one in whichFijpoints
fromitoj) and positive for a repulsive force. If the force is attractive, the value ofij
increases with increasingrij; if the force is repulsive,ijdecreases with increasingrij.
Sinceijis a function only ofrij, it is independent of the choice of reference frame.
EquationsG.1.8andG.1.9can be combined to give

WtotD

X

i

X

j>i

ÅijDÅ

0

@

X

i

X

j>i

ij

1

A (G.1.10)

By equating the expressions forWtotgiven by Eqs.G.1.5andG.1.10and rearranging, we
obtain

Å

X

i

1

2 miv

2

i

!

CÅ

0

@

X

i

X

j>i

ij

1

AD 0 (G.1.11)

This equation shows that the quantity

EtotD

X

i

1

2 miv

2

iC

X

i

X

j>i

ij (G.1.12)

is constant over time as the particles move in response to the forces acting on them. The
first term on the right side of Eq.G.1.12is the total kinetic energy of the particles. The
second term is the pairwise sum of particle–particle potential functions; this term is called