Thermodynamics and Chemistry

APPENDIX G FORCES, ENERGY, AND WORK

G.6 THELOCALFRAME ANDINTERNALENERGY 498

We expect that an observer in the local frame will find the laws of thermodynamics
are obeyed. For instance, the Clausius statement of the second law (Sec.4.2) is as valid
in a manned orbiting space laboratory as it is in an earth-fixed laboratory: nothing the
investigator can do will allow energy to be transferred by heat from a colder to a warmer
body through a device operating in a cycle. EquationG.6.7is a statement of the first law
of thermodynamics (box on page 56 ) in the local frame. Accordingly, we may assume that
the thermodynamic derivations and relations treated in the body of this book are valid in
any local frame, whether or not it is inertial, whenUandware defined by Eqs.G.6.5and
G.6.9.
In the body of the book,wis called thethermodynamic work, or simply the work. Note
the following features brought out by the derivation of the expression forw:

 The equationwD

P

R

FsurdR^0 has been derived for aclosedsystem.

 The equation shows how we can evaluate the thermodynamic workwdone on the

system. For each moving surface element of the system boundary at segmentof the

interaction layer, we need to know the contact forceFsurexerted by the surroundings

and the displacement dR^0 in the local frame.

 We could equally well calculatewfrom the force exerted by thesystemon the sur-

roundings. According to Newton’s third law, the forceFsysexerted by segmenthas

the same magnitude asFsurand the opposite direction:FsysDFsur.

 During a process, a point fixed in the system boundary at segmentis either station-

ary or traverses a path in three-dimensional space. At each intermediate stage of the

process, letsbe the length of the path that began in the initial state. We can write

the infinitesimal quantityFsurdR^0 in the formFsurcos (^) ds, whereFsuris the
magnitude of the force, dsis an infinitesimal change of the path length, and (^) is
the angle between the directions of the force and the displacement. We then obtain
the following integrated and differential forms of the work:
wD

X

Z

Fsurcos (^) ds ∂wD

X

Fsurcos (^) ds (G.6.10)
 If only one portion of the boundary moves in the local frame, and this portion has
linear motion parallel to thex^0 axis, we can replaceFsurdR^0 byFxsur 0 dx^0 , where
Fxsur 0 is thex^0 component of the force exerted by the surroundings on the moving
boundary and dx^0 is an infinitesimal displacement of the boundary. In this case we
can write the following integrated and differential forms of the work:
wD

Z

Fxsur 0 dx^0 ∂wDFxsur 0 dx^0 (G.6.11)

 The workwdoes not include work done internally by one part of the system on

another part.

 In the calculation of work with Eqs.G.6.9–G.6.11, we do not include forces from an

external field such as a gravitational field, or fictitious forcesFiaccelif present.