Thermodynamics and Chemistry

APPENDIX G FORCES, ENERGY, AND WORK

G.7 NONROTATINGLOCALFRAME 499

G.7 Nonrotating Local Frame

Consider the case of a nonrotating local frame whose origin moves in the lab frame but
whose Cartesian axesx^0 ,y^0 ,z^0 remain parallel to the axesx,y,zof the lab frame. In this
case the Cartesian components ofFisurfor particleiare the same in both frames, and so
also are the Cartesian components of the infinitesimal vector displacement dri. According
to Eqs.G.4.4andG.6.8, then, for an arbitrary process the value of the heatqin the local
frame is the same as the value of the heatqlabin the lab frame.
From Eqs.G.4.3andG.6.7withqlabset equal toq, we obtain the useful relation

ÅUÅEsysDwwlab (G.7.1)

This equation is not valid if the local frame has rotational motion with respect to the lab
frame.
The vectorR^0 has the same Cartesian components in the lab frame as in the nonrotating
local frame, so we can writeRR^0 DRlocwhereRlocis the position vector in the lab
frame of the origin of the local frame (see Fig.G.3). From Eqs.G.4.2andG.6.9, setting
.RR^0 /equal toRloc, we obtain the relation

wwlabD

X

Z

Fsurd.R^0 R/D

Z^ X

Fsur

!

dRloc (G.7.2)

The sum

P

F

suris the net contact force exerted on the system by the surroundings.
For example, suppose the system is a fluid in a gravitational field. Let the system boundary
be at the inner walls of the container, and let the local frame be fixed with respect to the
container and have negligible acceleration in the lab frame. At each surface element of the
boundary, the force exerted by the pressure of the fluid on the container wall is equal in
magnitude and opposite in direction to the contact force exerted by the surroundings on the
fluid. The horizontal components of the contact forces on opposite sides of the container
cancel, but the vertical components do not cancel because of the hydrostatic pressure. The
net contact force ismgez, wheremis the system mass andezis a unit vector in the vertical
Czdirection. For this example, Eq.G.7.2becomes

wwlabDmgÅzloc (G.7.3)

wherezlocis the elevation in the lab frame of the origin of the local frame.

G.8 Center-of-mass Local Frame

If we use acenter-of-mass frame(cm frame) for the local frame, the internal energy change
during a process is related in a particularly simple way to the system energy change mea-
sured in a lab frame. A cm frame has its origin at the center of mass of the system and its
Cartesian axes parallel to the Cartesian axes of a lab frame. This is a special case of the
nonrotating local frame discussed in Sec.G.7. Since the center of mass may accelerate in
the lab frame, a cm frame is not necessarily inertial.
The indicesiandjin this section refer only to the particles in thesystem.