Page 52 - Essentials of physical chemistry

14 Essentials of Physical Chemistry

1.2

1

0.8
Z= PV/RT 0.6 Series 1

0.4 Series 2
Series 3

0.2

0
0 200 400 600 800 1000 1200
o
T ( K)
FIGURE 1.6 Compressibility factor of air (a mixture of N 2 ,O 2 ,CO 2 , Ar, etc.) at different pressures. Series 1
(diamonds) ¼ 1 bar pressure, Series 2 (rectangles) ¼ 10 bar pressure, and Series 3 (triangles) ¼ 250 bar pressure.
(Data from Perry’s Chemical Engineers’ Handbook, 6th Edn., McGraw-Hill, 1984. table 3-162. Z-values are
calculated from values of pressure, volume (or density), and temperature in Vassernan, Kazavchinskii, and
Rabinovich, Thermophysical properties of air and air components, Moscow, Nauka, 1966, and NBS-NSF Trans.
TT 70-50095, 1971 and Vassernan and Rabinovich, Thermophysical properties of liquid air and its component,
Moscow, 1968, and NBS-NSF Trans. 69-55092, 1970. Courtesy of Mr. Ian C. Roman of Air Liquide, Delaware
Research and Technology Center, Newark, DE.)

molecules flying around with a lot of space between them (recall Dalton’s law). That idea should
include collisions of molecules within the volume. Consider a collision of an auto with a fixed wall
compared to a head-on collision with another similar auto. There will be a change of momentum
force in the collision with the wall but the force of the head-on collision will be double that with a
fixed wall! Thus, the real pressure within the gas volume is actually higher than the pressure we
measure, although the collisions are relatively infrequent due to the large space between molecules
of the gas. An additional consideration is that there may be some weak short-range attractive forces
(now known as van der Waals interactions) between the gas molecules that causes them to approach
each other more forcefully at short range; thus, leading us to expect a plus sign for this small
correction. In fact we can use the idea of concentrations as (moles=volume) for the number of gas
molecules in the volume. The pressure correction is small but proportional to the square of the
concentration. Of course this idea includes collisions of some molecules with themselves but
N molecules can only collide with (N 1) other molecules. However, when N is of the order of
23
10 then (N 1) is essentially the same as N. We are reminded that when a proportionality is
observed phenomenologically we can use a ‘‘proportionality constant’’ to create an equation with
a constant as in

n n n
2
¼ a :
V V V
? 1 / 2
Another factor included in this term is any bimolecular electronic interactions. Thus, the ‘‘a’’
parameter absorbs a number of interaction terms as well as an amount of the bimolecular
collision pressure.
Next we need to consider that while the molecules are very small, their volume is not really zero;
they have a finite volume and when you have a mole of molecules at very low temperatures tending