Physical Chemistry Third Edition

406 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium

isφ, the angle between the positivevxaxis and the line segment in thevx–vyplane that

lies directly under the velocity vector.

An infinitesimal volume element in velocity space corresponding to an infinites-

imal increment in each velocity component is crudely depicted in Figure 9.10. The

volume element is an infinitesimal box with length in thevdirection equal todv.

The length of an arc of a circle is the radius of the circle times the measure in

radians of the angle subtended by the arc, so the length of the box in theθdirection

is equal tovdθ. The length of the line segment in thevx–vyplane that lies directly

under the velocity vector isvsin(θ), so that the length of the box in theφdirection

is equal tovsin(θ)dφ. Since the increments are infinitesimal the volume element is

rectangular and its volume is equal tov^2 sin(θ)dvdθdφ. We abbreviate the volume

element by the symbold^3 v:

d^3 vv^2 sin(θ)dvdθdφ (9.4-1)

Using the same symbold^3 vfor the volume element in Cartesian coordinates and in

spherical polar coordinates enables us to write some equations that apply to either set

of coordinates.

The probability that the velocity of a randomly chosen molecule lies in the volume

elementv^2 sin(θ)dvdθdφis

(

probability that

vlies ind^3 v

)

g(v)d^3 vg(v)v^2 sin(θ)dvdθdφ (9.4-2)

vz

vx

(^0) vy
dv v d

d

d



v
vsin(

)
d
Figure 9.10 Spherical Polar Coordinates in Velocity Space, with a Volume Element.