Page 50 - Modern physical chemistry

3.2 Elements of Ideal-Gas Theory 39

Averaging Xi 2 ,
".i;? -
_£..,, __ =.i; 2 [3.5]
t
N '
and substituting the result into equation (3.4) gives us

L I1Pi = Nm.i;2 . [3.6]
I1t l
From Newton's second law, the rate of change of momentum equals the force causing
the change. So the sum on the left side here equals the net force exerted by the wall Pl2:

N · 2
Pl2=~ [3.7]
l
Since eequals the volume of the gas V, equations (3.7) rearranges to

PV=Nm.i;2. [3.8]
By synunetry,
PV=Nmy 2 , [3.9]

PV=Nmz2. [3.10]
Sununing (3.1) over all N molecules and dividing by N yields

LUT L.i;T LYT LZT
--=--+--+-- [3.11 ]
N N N N
or
U2 =.i;2 +y2 +Z2. [3.12]

Finally, combining (3.8), (3.9), (3.10) , and (3.12) gives us
1 -
2
PV=-Nmu • [3.13]
3
Since the translational energy of the ith molecule is 1I2mu/, the total translational energy
ElI of the N molecules is
,,1 2 1 2
E tr = £..,,-mUi =-Nmu . [3.14]
2 2
And (3.13) can be written in the meaningful form
2
PV =-E tr . [3.15]
3
In the laboratory, atmospheriC pressure is determined with a mercury barometer. The
height of the mercury column balancing the air pressure is measured in millimeters. In
accurate work, this is corrected to 0° C and to a standard gravity of 9.80665 m S·2. Instead
of reporting this in mm (of Hg), it is generally given in torrs, 1 torr being 1 mm (of Hg)
under the standard conditions.

Example 3. 1

Recall Boyle's law and interpret the ElI in (3.15).