Thermodynamics and Chemistry

APPENDIX G FORCES, ENERGY, AND WORK

G.2 THESYSTEM ANDSURROUNDINGS 491

For conservation of energy, the potential energy change in the time interval should have the
same magnitude and the opposite sign:

ÅifieldD

Z

Fifielddri (G.2.2)

Only if the integral

R

Fifielddrihas the same value for all paths between the initial and
final positions of the particle does a conservative force field exist; otherwise the concept of
a potential energyifieldis not valid.
Taking a gravitational field as an example of a conservative external field, we replace
FifieldandifieldbyFigravandigrav:ÅigravD

R

Figravdri. The gravitational force on
particleiis, from Newton’s second law, the product of the particle mass and its acceleration
gezin the gravitational field: FigravD migezwheregis the acceleration of free fall
andezis a unit vector in the vertical (upward)zdirection. The change in the gravitational
potential energy given by Eq.G.2.2is

ÅigravDmig

Z

ezdriDmig Åzi (G.2.3)

(The range of elevations of the system particles is assumed to be small compared with the
earth’s radius, so that each system particle experiences essentially the same constant value
ofg.) Thus we can define the gravitational potential energy of particlei, which is a function
only of the particle’s vertical position in the lab frame, byigravDmigziCCiwhereCiis
an arbitrary constant.
Returning to Eq.G.2.1for the total energy, we can now write the third term on the right
side in the form X

i

X

k

ikD

X

i

X

k^0

ik 0 C

X

i

ifield (G.2.4)

To divide the expression for the total energy into meaningful parts, we substitute Eq.G.2.4
in Eq.G.2.1and rearrange in the form

EtotD

2

4

X

i

1

2 miv

2

iC

X

i

X

j>i

ijC

X

i

ifield

3

5

C

"

X

i

X

k^0

ik 0

C

"

X

k

1

2 mkv

2

kC

X

k

X

l>k

kl

(G.2.5)

The terms on the right side of this equation are shown grouped with brackets into three
quantities. The first quantity depends only on the speeds and positions of the particles in
thesystem, and thus represents the energy of the system:

EsysD

X

i

1

2 miv

2

iC

X

i

X

j>i

ijC

X

i

ifield (G.2.6)

The three terms in this expression forEsysare, respectively, the kinetic energy of the system
particles relative to the lab frame, the potential energy of interaction among the system
particles, and the total potential energy of the system in the external field.