Thermodynamics and Chemistry

APPENDIX G FORCES, ENERGY, AND WORK

G.3 SYSTEMENERGYCHANGE 492

The last bracketed quantity on the right side of Eq.G.2.5depends only on the speeds
and positions of all the particles in thesurroundings, so that this quantity is the energy of
the surroundings,Esurr. Thus, an abbreviated form of Eq.G.2.5is

EtotDEsysC

X

i

X

k^0

ik 0 CEsurr (G.2.7)

The quantity

P

i

P

k^0 ik^0 represents potential energy shared by both the system and sur-
roundings on account of forces acting across the system boundary, other than gravita-
tional forces or forces from other external fields. The forces responsible for the quantityP

i

P

k^0 ik^0 are generally significant only between particles in the immediate vicinity of
the system boundary, and will presently turn out to be the crucial forces for evaluating
thermodynamic work.

G.3 System Energy Change

This section derives an important relation between the changeÅEsysof the energy of the
system measured in a lab frame, and the forces exerted by the surroundings on the system
particles. The indicesiandjwill refer to only the particles in thesystem.
We write the net force on particleiin the form

FiD

X

j§i

FijCFifieldCFisur (G.3.1)

whereFijis the force exerted on particleiby particlej, both particles being in the system,
andFisurD

P

k^0 Fik^0 is the net force exerted on particleiby the particles in the surround-
ings that are not the source of an external field. During a given period of time, the work
done by forces acting on only the system particles is

X

i

Z

FidriD

X

i

X

j§i

Z

FijdriC

X

i

Z

FifielddriC

X

i

Z

Fisurdri (G.3.2)

We can replace the first three sums in this equation with new expressions. Using Eq.
G.1.4, we have
X

i

Z

FidriDÅ

X

i

1

2 miv

2

i

!

(G.3.3)

From Eqs.G.1.8andG.1.9we obtain

X

i

X

j§i

Z

FijdriDÅ

0

@

X

i

X

j>i

ij

1

A (G.3.4)

where the sums are over the system particles. From Eq.G.2.2we can write

X

i

Z

FifielddriDÅ

X

i

ifield

!

(G.3.5)