Physical Chemistry Third Edition

9.2 A Model System to Represent a Dilute Gas 391

vz

kvz

ivx

0

ivx 1 jvy

v 5 ivx 1 jvy 1 kvz

kvz

jvz

vy

vx

Figure 9.4 A Velocity Vector in Velocity Space.

The velocity of particle numberiis specified by the velocity vectorvi:

viivix+jviy+kviz (9.2-9)

The components of the velocityviare the rates of change ofxi,yi, andzi:

vix

dxi

dt

, viy

dyi

dt

, viz

dzi

dt

(9.2-10)

The velocity vector can be represented geometrically as in Figure 9.4. The mathematical

space of this figure is calledvelocity space, in which distances on the axes represent

velocity components. Cartesian components are shown, and these can be handled in

the same way as components in ordinary space.

The direction of the velocity vector is the direction in which the particle is moving.

Thespeedis the magnitude of the velocity, given by a three-dimensional version of the

theorem of Pythagoras:

|vi|vi

(

v^2 ix+v^2 iy+v^2 iz

) 1 / 2

(9.2-11)

We use either of the two notations in Eq. (9.2-11) to denote the magnitude of a vector:

the boldface letter within vertical bars or the letter in plain type. The magnitude of a

vector is always non-negative (positive or zero).

Exercise 9.5

a.Use the theorem of Pythagoras to verify Eq. (9.2-11). (It must be used twice.)

b.Find the speed of a particle with the velocity components:

vx400 m s−^1 , vy−600 m s−^1 , vz750 m s−^1