Page 77 - Essentials of physical chemistry

The Kinetic Molecular Theory of Gases 39

2
n(KE) ¼ nRT. Thus, equating some experi-
3
using the phenomenological ideal gas law, PV ¼
3RT M v 2
. So, we have carried out a path
2 2
mental data to physical reasoning will be true if KE ¼ ¼
of reasoning ‘‘that is probably true’’ based on the ideal gas law, which we know is only true at low
pressure and high temperature, but let us see where this assumption takes us. First, solving this
equation for the velocity gives the result
r ffiffiffiffiffiffiffiffiffi
3RT p ffiffiffiffiffi
2
v :
M
v ¼ ¼
This derivation is not very satisfying because we used a velocity that was not really averaged over
all orientations and the result depends on the phenomenological ideal gas law. We also note that the
resulting form of ‘‘v’’ is a scalar as the square root of a vector squared using the dot product and we
call it ‘‘v rms .’’ It is good that we obtain a scalar, but what does it mean to have a ‘‘root-mean-square’’
speed? However, we can use it to estimate the speed as for N 2 gas at 258C to get some idea of
the KMTG velocities. Why not calculate this apparent speed in miles per hour (mph)? Note this is
a diatomic molecule and also the formula has no dependence on pressure, just temperature
dependence. We better check the units for R in this calculation.
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!

7
2
3(8:314 10 gcm =s = K mol)(298:15 K) 3600 s=h

2
¼ 1152:6 mph:
2 14:007 g=mol 1:6093 10 cm=mile
v ¼ 5
Some things to note are that we need to use either the cgs or mks value of R, remember to double the
atomic weight of nitrogen, add 273.15 to the 8C, and recall that 1 mile ¼ 1.6093 km. Let us do it
again in mks.
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !
2
3(8:314 kg m =s = K mol)(298:15 K) 3600 s=h

2
¼ 1152:6 mph:
v ¼ 3 3
2(0:014007 10 )kg=mol 1:6093 10 m=mile

5280 ft=mile
That is a very high speed, v ¼ (1153 miles=h) ¼ 1691 ft=s, which is faster than a
3600 s=h
small bore rifle bullet. The .22 LR Stinger is rated at 1435 ft=s. Two obvious questions arise. First, if
the molecules are that fast, why is the speed of sound much less at about 1125 ft=s or 768 mph? The
answer is that there are a lot of collisions between the gas atoms=molecules and some of the recoil
trajectories have backward components, thus slowing the average speed. The second has to do with
possible injury from ‘‘bullet molecules.’’ Fortunately, the actual mass of the ‘‘atomic bullets’’ is less
than 10 20 g, so it takes a lot of collisions to make the pressure variations we call ‘‘sound’’ that we
sense with our ears.
WEIGHTED AVERAGING: A VERY IMPORTANT CONCEPT

At this point, we would like to proceed to apply the KMTG to experimentally measurable quantities,
but we need a firmer foundation for the velocities and speeds of atoms=molecules in the gas phase.
The velocity based on the phenomenological ideal gas law is suspect because we know it may
not apply to high pressure and=or low temperature, so we need a more rigorous method. The
concept=principle of weighted averaging occurs in kinetics, statistical thermodynamics, and in
quantum mechanics, so we think this is more than just a ‘‘math interlude’’; it is a unifying principle.