Simple Nature - Light and Matter

traveling a distance 2Lparallel to thexaxis between collisions with

that wall. The time between collisions is ∆t= 2L/vx, and in each

collision thexcomponent of the atom’s momentum is reversed from

−mvxtomvx. The total force on the wall is

F=

∑∆px,i

∆ti

[monoatomic ideal gas],

where the index irefers to the individual atoms. Substituting

∆px,i= 2mvx,iand ∆ti= 2L/vx,i, we have

F=

1

L

∑

mv^2 x,i [monoatomic ideal gas].

The quantitymvx^2 ,iis twice the contribution to the kinetic energy

from the part of the atoms’ center of mass motion that is parallel to

thexaxis. Since we’re assuming a monoatomic gas, center of mass

motion is the only type of motion that gives rise to kinetic energy.

(A more complex molecule could rotate and vibrate as well.) If the

quantity inside the sum included theyandzcomponents, the sum

would be twice the total kinetic energy of all the molecules. Since we

expect the energy to be equally shared amongx,y, andzmotion,^1

the quantity inside the sum must therefore equal 2/3 of the total

kinetic energy, so

F=

2 Ktotal

3 L

[monoatomic ideal gas].

Dividing byAand usingAL=V, we have

P=

2 Ktotal

3 V

[monoatomic ideal gas].

This can be connected to the empirical relationPV ∝nT if we

multiply byV on both sides and rewriteKtotalasnK ̄, whereK ̄ is

the average kinetic energy per molecule:

PV =

2

3

nK ̄ [monoatomic ideal gas].

For the first time we have an interpretation of temperature based on

a microscopic description of matter: in a monoatomic ideal gas, the

temperature is a measure of the average kinetic energy per molecule.

The proportionality between the two isK ̄ = (3/2)kT, where the

constant of proportionalityk, known as Boltzmann’s constant, has

a numerical value of 1.38× 10 −^23 J/K.

The Boltzmann constant has the value it does because the cel-

sius and kelvin scales were defined before the microscopic picture

of thermodynamics had been discovered. For some calculations, it

is more convenient to work in more natural units wherek= 1 by

definition, and then the units of temperature and energy are the

(^1) This equal sharing will be justified more rigorously on page 333.
318 Chapter 5 Thermodynamics