The Foundations of Chemistry

This equation can also be written as

^13 NAv(2^12 mu^2 )RT

From physics we know that the kinetic energyof a particle of mass mmoving at speed uis
^12 mu^2. So we can write

^23 NAv(avg KEper molecule)RT

or

NAv(avg KEper molecule)^32 RT

This equation shows that the absolute temperature is directly proportional to the average
molecular kinetic energy, as postulated by the kinetic–molecular theory. Because there are
NAvmolecules in a mole, the left-hand side of this equation is equal to the total kinetic
energy of a mole of molecules.

Total kinetic energy per mole of gas^32 RT

With this interpretation, the total molecular–kinetic energy of a mole of gas depends only
on the temperature, and not on the mass of the molecules or the gas density.
We can also obtain some useful equations for molecular speeds from the previous
reasoning. Solving the equation

^13 NAvmu^2 RT

for root-mean-square speed, urmsu^2 , we obtain

urms

We recall that mis the mass of a single molecule. So NAvmis the mass of Avogadro’s number
of molecules, or one mole of substance; this is equal to the molecular weight, M,of the gas.

urms

3 RT



M

3 RT



NAvm

12-13 The Kinetic–Molecular Theory 469

EXAMPLE 12-22 Molecular Speed

Calculate the root-mean-square speed of H 2 molecules in meters per second at 20°C. Recall
that

1 J 1 

kg

s



2

m^2



Plan

We substitute the appropriate values into the equation relating urmsto temperature and mo-
lecular weight. Remember that Rmust be expressed in the appropriate units.

R8.314 

mo

J

lK

8.314 

m

k

o

g

l



K

m



2

s^2