Thermodynamics and Chemistry

APPENDIX G FORCES, ENERGY, AND WORK

G.10 EARTH-FIXEDREFERENCEFRAME 503

x^0 axis of the local frame. There is a gravitational force in thezdirection; this force is
responsible for the only external field, whose potential energy change in the local frame
during a process isÅ

(^0) grav
i Dmig Åzi(Eq.G.2.3).
The contribution to the effective force acting on particleidue to acceleration when!
is constant can be shown to be given by^8
FiaccelDFicentrCFiCor (G.9.1)
whereFicentris the so-calledcentrifugal forceandFiCoris called theCoriolis force.
The centrifugal force acting on particleiis given by
FicentrDmi!^2 riei (G.9.2)
Hereriis the radial distance of the particle from the axis of rotation, andeiis a unit vector
pointing from the particle in the direction away from the axis of rotation (see Fig.G.5). The
direction ofeiin the local frame changes as the particle moves in this frame.
The Coriolis force acting on particleiarises only when the particle is moving relative
to the rotating frame. This force has magnitude2mi!vi^0 and is directed perpendicular to
bothv^0 iand the axis of rotation.
In a rotating local frame, the work during a process is not the same as that measured in
a lab frame. The heatsqandqlabare not equal to one another as they are when the local
frame is nonrotating, nor can general expressions using macroscopic quantities be written
forÅUÅEsysandwwlab.

G.10 Earth-Fixed Reference Frame

In the preceding sections of AppendixG, we assumed that a lab frame whose coordinate
axes are fixed relative to the earth’s surface is an inertial frame. This is not exactly true, be-
cause the earth spins about its axis and circles the sun. Small correction terms, a centrifugal
force and a Coriolis force, are needed to obtain the effective net force acting on particlei
that allows Newton’s second law to be obeyed exactly in the lab frame.^9
The earth’s movement around the sun makes only a minor contribution to these correc-
tion terms. The Coriolis force, which occurs only if the particle is moving in the lab frame,
is usually so small that it can be neglected.
This leaves as the only significant correction the centrifugal force on the particle from
the earth’s spin about its axis. This force is directed perpendicular to the earth’s axis and has
magnitudemi!^2 ri, where!is the earth’s angular velocity,miis the particle’s mass, and
riis the radial distance of the particle from the earth’s axis. The correction can be treated
as a small modification of the gravitational force acting on the particle that is at most, at the
equator, only about 0.3% of the actual gravitational force. Not only is the correction small,
but it is completely taken into account in the lab frame when we calculate the effective
gravitational force fromFigravD migez, wheregis the acceleration of free fall andez
is a unit vector in theCz(upward) direction. The value ofgis an experimental quantity
that includes the effect ofFicentr, and thus depends on latitude as well as elevation above the
earth’s surface. SinceFigravdepends only on position, we can treat gravity as a conservative
force field in the earth-fixed lab frame.

(^8) The derivation, using a different notation, can be found in Ref. [ 108 ], Chap. 10.
(^9) Ref. [ 68 ], Sec. 4–9.