Concise Physical Chemistry

c01 JWBS043-Rogers September 13, 2010 11:20 Printer Name: Yet to Come

14 IDEAL GAS LAWS

Solution 1.2 Consider a particle (molecule or atom) moving in thexdirection in

a cubic boxlunits on an edge, oriented so that the particle path is perpendicular to

one of its faces. Consider for now only collisions with one wall. The momentum of

the particle ismvon the way into the collision, and−mv(in the opposite direction)

on the way out of the collision. The change of momentum is 2mv. The number of

collisions per second (collision frequency) isv/ 2 lbecause the particle must travel

the length of the boxlto collide with the opposite wall andlonce again on the return

trip. Force is the change in momentum with respect to time,dp/dt(Newton’s second

law). Force per collision isF=ma=dp/dt, whereais the acceleration, a change

in speed over unit time (in seconds). The total force exerted by the particle on the

wall for many collisions is the force per collision times the number of collisions per

unit time (a frequency):

F= 2 mv

v

2 l

=

mv^2

l

Pressure is force per unit areaA=l^2 :

p=

mv^2

l

(

1

A

)

=

mv^2

l

(

1

l^2

)

=

mv^2

l^3

=

mv^2

V

where the volume of the box is the cube of one of its edges. Given that there are very

many molecules in the box, on average, only one-third of them are moving in thex

direction; or, better said, only one-third of all components of all velocity vectors are

oriented in thexdirection. (The other two-thirds are oriented in theyandzdirections.)

For Avogadro’s number,NA, of particles we find,

p=

1

3

NA

mv^2

V

wherevis an unbiased or average particle speed. This is the equation we sought.

We already know thatpVis an energy and that^12 mv^2 is the kinetic energy of a

moving massmat velocityv. One sees the proportionality in the preceding equation.

More explicitly, multiply and divide the preceding equation by 2. Now,

pV=

2

3

NA

1

2

mv ̄^2 x=

2

3

E ̄kinetic

where the overbar notation for ̄vxandE ̄kineticis added to stress that the speed (scalar

magnitude of the velocity component) is an average value over all of theNAparticles.