Physical Chemistry , 1st ed.



[g

x

v

(v

x

x)]g

y(vy) gz(vz) 

[

v

(

x

v)]





[g

x

v

(v

x

x)]g

y(vy) gz(vz) 

[

(

v

v)]



v

v

x

 (19.18)

The right side of the second equation shows the influence of the chain rule.
The left side of the equations above show that the derivative applies only to
the gxfunction, since it is the only one of the three functions that depends
on vx.
It turns out that we can determine an expression for v/ vxusing equation
19.17 and determining the total derivative ofv:

v^2 vx^2 vy^2 vz^2

d(v^2 ) d(vx^2 vy^2 vz^2 )

2 v dv 2 vxdvx 2 vydvy 2 vzdvz

In terms of partial derivatives, since all other variables are kept constant, we
have dvydvz0; therefore,

2 v dv 2 vxdvx



v

v

x



v

v

x (19.19)

This is the desired result. Substituting into equation 19.18, we have

^ [g

x

v

(v

x

x)]g

y(vy) gz(vz) 

[

(

v

v)]v

v

x

Using standard calculus notation, we will use a prime to indicate a derivative
with respect to a variable. The above equation becomes, more succinctly,

gx (vx) gy(vy) gz(vz)    (v) v

v

x

If we divide the above equation by gx(vx) gy(vy) gz(vz)  (v), we can can-
cel terms on the left and rearrange the variables to get



v

1

x



g

g

x

x

(

(

v

v

x

x

)

)



1

v



(

(

v

v

)

)

 (19.20)

If we did the same analysis for gyor gz, we would get a similar expression, only
with different subscripts on the left side.
Equation 19.20 is interesting. All of the terms in vxare on one side of the
equation, and all of the terms in vare on the other side of the equation. If
one side of the equation is independent of one variable and the other side of
the equation is independent of the other variable, then neither side depends
on either variable; that is, the expression on each side equals a constant. Using
Kto represent this constant, we have



v

1

x

g

g

x

x

(

(

v

v

x

x

)

)^1

v

^

(

(

v

v

)

)K (19.21)

Understand that this does not mean that vand vxdo not vary, just that the
particular combination of the functions on each side of equation 19.20 does
not vary. Understand also that if we performed this analysis for the other two

19.3 Definitions and Distributions of Velocities of Gas Particles 659