Physical Chemistry Third Edition

14.2 Classical Mechanics 621

14.2 Classical Mechanics

According to classical mechanics, the state of a system is specified by giving the

position and velocity of every particle in the system. Consider a single particle without

any internal structure so that it cannot rotate or vibrate. If it can move in three dimen-

sions we can specify its position by the three Cartesian coordinatesx,y, andz. These

three coordinates are equivalent to aposition vectorthat reaches from the origin of

coordinates to the location of the particle. Any quantity that has direction as well as

magnitude is a vector. We denote a vector by a letter in boldface type and we denote this

position vector byr. It can also be denoted by a letter with an arrow above it, as in−→r,

by a letter with a wavy underscore, as in. The Cartesian coordinatesx,y, andzare

called theCartesian componentsof the position vectorr. A vector can also be denoted

by listing its three Cartesian components inside parentheses, as in (x,y,z). Any quan-

tity that has both magnitude and direction is a vector. A quantity that has no particular

direction but can be positive, negative, or zero, is called ascalar. Appendix B contains

an elementary introduction to vectors, and additional information can be found in the

references of note number 1.

The velocity of the particle is the time derivative of its position vector and is a vector

vwith the Cartesian componentsνx,νy, andνz:

νx

dx

dt

νy

dy

dt

νz

dz

dt

(14.2-1a)

These three equations can be represented by a single vector equation

v

dr

dt

(14.2-1b)

Thespeedof a particle is the magnitude of the velocity, given by

ν|v|

√

ν^2 x+ν^2 y+ν^2 z (14.2-2)

which is a three-dimensional version of the theorem of Pythagoras. The kinetic energy

of the particle is defined by

K ^12 mv^2 ^12 mv^2 (v^2 x+v^2 y+v^2 z) (14.2-3)

wheremis the mass of the particle. The kinetic energy is an example of a function of

the variables that specify the state of the particle. A function of these variables is called

astate function.

Units of Measurement

In using a formula such as that of Eq. (14.2-3), one must use consistent units or risk

getting the wrong answer. The set of units that chemists and physicists now use is the

international system of units,orSI units. The letters SI stand forSystème Internationale,

the French name for the set of units. In this system there are seven base units. The unit

of length is themeter(m). The unit of mass is thekilogram(kg). The unit of time is

thesecond(s). The unit of temperature is thekelvin(K). The unit of electric current

is theampere(A). The unit of luminous intensity is thecandela(cd). The unit for the

amount of a substance is themole(mol).