Physical Chemistry , 1st ed.

some initial velocity labeled vi, and after the collision it has some final veloc-
ity vf. The change in the velocity, which is a vector quantity, over the period of
time it takes for the collision to occur means that the particle has accelerated.
In terms of the velocities shown in Figure 19.1:

acceleration 



v

t

vf





t

vi

If Figure 19.1 represented the infinitesimal change in velocity, the acceleration
would be expressed in derivative form:

acceleration 

d

d

v

t



The force involved in the process illustrated by Figure 19.1 is given by Newton’s
second law:

Fma

where Fis the force,mis the mass of the object, and ais the acceleration. Using
the derivative form for acceleration, this becomes

Fm

d

d

v

t

 (19.3)

for a single particle. Newton’s third law says that for every force, there must
be an equal and opposite force, so if a gas particle collides with a wall and a
force on the particle is produced, an equal (and opposite) force is produced
on the wall.F, then, refers to the force on the particle as well as the force on
the wall.
In the above expressions,Frepresents the force exerted by onegas particle
making onecollision on the wall of the container. For a macroscopic sample,
many gas particles are colliding with the walls of the container over an ex-
tended period of time. Therefore, the force on the wall can vary with time, and
it is better to consider F(t), force as a function of time. The average force, Favg,
is the total force over a certain time period divided by the total time period. If
we break time into tiny intervals, this becomes



no. of time

F(t)

Favg

Of course, if the intervals are so small that they are infinitesimal, then the sum-
mation can be replaced with an integral:

Favg

to



tal

F(

ti

t

m

)

e



tim

1

e

F(t) dt

We can substitute from equation 19.3 to get

Favg

tim

1

e

m

d

d

v

t

dt

We are dropping the “total” descriptor on the time variable. The dtterms can-
cel, so we get

Favg

tim

1

e

m dv

intervals F(t)

total time

19.2 Postulates and Pressure 653