Thermodynamics and Chemistry

APPENDIX G FORCES, ENERGY, AND WORK

G.1 FORCES BETWEENPARTICLES 487

G.1 Forces between Particles

Amaterial particleis a body that has mass and is so small that it behaves as a point,
without rotational energy or internal structure. We assume each particle has a constant mass,
ignoring relativistic effects that are important only when the particle moves at a speed close
to the speed of light.
Consider a collection of an arbitrary number of material particles that have interactions
only among themselves and with no other particles. Later we will consider some of the
particles to constitute a thermodynamicsystemand the others to be thesurroundings.
Newton’s laws of motion are obeyed only in aninertialreference frame. A reference
frame that is fixed or moving at a constant velocity relative to local stars is practically an
inertial reference frame. To a good approximation, a reference frame fixed relative to the
earth’s surface is also an inertial system (the necessary corrections are discussed in Sec.
G.10). This reference frame will be called simply thelab frame, and treated as an inertial
frame in order that we may apply Newton’s laws.
It will be assumed that the Cartesian components of all vector quantities appearing in
SectionsG.1–G.4are measured in an inertial lab frame.
Classical mechanics is based on the idea that one material particle acts on another by
means of aforcethat is independent of the reference frame. Let the vectorFijdenote the
force exerted on particleiby particlej.^1 ThenetforceFiacting on particleiis the vector
sum of the individual forces exerted on it by the other particles:^2

FiD

X

j§i

Fij (G.1.1)

(The term in whichjequalsihas to be omitted because a particle does not act on itself.)
According to Newton’s second law of motion, the net forceFiacting on particleiis equal
to the product of its massmiand its acceleration:

FiDmi

dvi

dt

(G.1.2)

Hereviis the particle’s velocity in the lab frame andtis time.
A nonzero net force causes particlei to accelerate and its velocity and position to
change. Theworkdone by the net force acting on the particle in a given time interval
is defined by the integral^3

WiD

Z

Fidri (G.1.3)

whereriis the position vector of the particle—a vector from the origin of the lab frame to
the position of the particle.

(^1) This and the next two footnotes are included for readers who are not familiar with vector notation. The quantity
Fijis printed in boldface to indicate it is avectorhaving both magnitude and direction.
(^2) The rule for adding vectors, as in the summation shown here, is that the sum is a vector whose component
along each axis of a Cartesian coordinate system is the sum of the components along that axis of the vectors
being added. For example, the vectorC DACBhas componentsCxDAxCBx,CyDAyCBy, and
CzDAzCBz.
(^3) The dot between the vectors in the integrand indicates a scalar product or dot product, which is anonvector
quantity. The general definition of the scalar product of two vectors,AandB, isABDABcos whereA
andBare the magnitudes of the two vectors and is the angle between their positive directions.