Physical Chemistry Third Edition

390 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium

The Microscopic States of Our Model System

According to classical mechanics, the microstate of our model system is specified by

giving the position and velocity of every particle. We number the particles in our model

system from 1 toN. The position vector of particle numberican be written as a vector

sum of three terms:

riixi+jyi+kzi (9.2-8)

Each term in Eq. (9.2-8) is a product of a scalar quantity (the component) and a unit

vector. Ascalarquantity can be positive, negative, or zero but has no specific direction

in space. The unit vectoripoints in the direction of the positivexaxis, the unit vectorj

points in the direction of the positiveyaxis, and the unit vectorkpoints in the direction

of the positivezaxis. Figure 9.2 shows the vectorri, the Cartesian axes, the unit vectors,

and the Cartesian components of the vector.

Value of z

Value of y

Value of x

x

r

k

i j

z

y

Figure 9.2 A Position Vector r in
Three-Dimensional Space.The vectorr
specifies the position of the particle when
its tail is at the origin. The unit vectorsi,j,
andkare also shown.

The product of a positive scalar quantity and a unit vector is a vector with the same

direction as the unit vector and length equal to the scalar quantity. If the scalar quantity

is negative, the product is a vector that has length equal to the magnitude of the scalar

quantity and is in the opposite direction from the unit vector. Thesum of two vectors

can be represented geometrically by moving the second vector without rotating it so

that its tail is at the head of the first vector. The sum vector has its tail at the tail of

the first vector and its head at the head of the second vector. A third vector is added

to the sum of the first two vectors in the same way. Figure 9.3 shows how the three

components times their unit vectors add to equal the position vectorri.

z

kz

0

r 5 ix 1 jy 1 kz

iy 1 ky

x

ix

kz

jy

jy

y

Figure 9.3 The Addition of the Components of a Position Vector.