Thermodynamics and Chemistry

APPENDIX G FORCES, ENERGY, AND WORK

G.6 THELOCALFRAME ANDINTERNALENERGY 497

the same magnitude as the gravitational force but directed in the opposite direction, so that
theeffectiveforce on the object as a whole is zero.
The reasoning used to derive the equations in Secs.G.1–G.4can be applied to an arbi-
trary local frame. To carry out the derivations we replaceFibyFieff,ribyri^0 , andvibyv^0 i,
and use the local frame to measure the Cartesian components of all vectors. We need two
new potential energy functions for the local frame, defined by the relations

Å

(^0) field
i
defD

Z

Fifielddri^0 (G.6.3)

Åiaccel defD 

Z

Fiacceldri^0 (G.6.4)

Both

(^0) field
i and
accelmust be time-independent functions of the position of particleiin
the local frame in order to be valid potential functions. (If the local frame is inertial,Fiaccel
andiaccelare zero.)
The detailed line of reasoning in Secs.G.1–G.4will not be repeated here, but the reader
can verify the following results. Thetotal energyof the system and surroundings measured
in thelocalframe is given byEtot^0 DUC

P

i

P

k^0 ik^0 CE

0

surrwhere the indexk

(^0) is for
particles in the surroundings that are not the source of an external field for the system. The
energy of thesystem(the internal energy) is given by
UD

X

i

1

2 mi.v

0

i/

(^2) CX
i

X

j>i

ijC

X

i



(^0) field
i C

X

i

iaccel (G.6.5)

where the indicesiandjare for system particles. The energy of thesurroundingsmeasured
in the local frame is

Esurr^0 D

X

k

1

2 mk.v

0

k/

(^2) CX
k

X

l>i

klC

X

k

kaccel (G.6.6)

wherekandlare indices for particles in the surroundings. The value ofEtot^0 is found to
be constant over time, meaning that energy is conserved in the local frame. The internal
energy change during a process is the sum of the heatqmeasured in the local frame and the
macroscopic workwin this frame:

ÅUDqCw (G.6.7)

The expressions forqandw, analogous to Eqs.G.4.4andG.4.2, are found to be

qD

X

X

i

Z

ïiFisurdri (G.6.8)

wD

X

Z

FsurdR^0  (G.6.9)

In these equationsR^0 is a vector from the origin of the local frame to a point fixed in the
system boundary at segment, andriis a vector from this point to particlei(see Fig.
G.3).