Page 53 - Essentials of physical chemistry

Ideal and Real Gas Behavior 15

to condense into a liquid or even a solid, the volume reduces to a smaller but finite volume. This
small volume will be proportional to the number of moles of gas as in

? 2 / n ¼ bn:

Thus, a modified gas law was proposed by a Dutch physicist Johannes Diderik van der Waals
(1837–1923) in 1873 in his doctoral thesis as a way to simulate the condensation of gases to liquids.
He received the Nobel Prize for this work in 1910.

2
n
P þ a (V nb) ¼ nRT
V 2

Chemical engineering students in this class will be aware of more accurate and complicated
equations of state, but for this text we will be content to use the van der Waals equation as a useful
treatment of real gases. Thus, parameters are available in terms of values of ‘‘a’’ and ‘‘b’’ for a
number of gases and each gas has separate parameters, see Table 1.3. We may find some tables of
these parameters in older units but lists are available in SI units as well.
In the original 1873 doctoral thesis of van der Waals, the goal was to explain the process of
condensation of gases to liquids in a smooth way. The ideal gas equation that predicts the PV product
at any fixed temperature should be one branch of a hyperbola. Such ‘‘isotherm’’ curves are indeed
found on a plot of pressure versus volume of a fixed amount of gas at constant temperature, for low
pressures above the boiling point of a given material. However, as one lowers the temperature a small
bump in the isotherm will be observed, which is at first an ‘‘inflection point’’ and then at still lower
temperatures enters a region where there is a fog or mist of liquid condensate droplets. It is easy to
2
show that the van der Waals equation is cubic in V by multiplying through by V .
2
n
2 2 3 2 2 3
V P þ a (V nb) ¼ nRTV or PV n(bP þ RT)V þ n aV n ab ¼ 0:
V 2

TABLE 1.3
van der Waals Constants for Common Gases
2
2
Gas a (L bar=mol ) b (L=mol)
He 0.0346 0.0238
Ne 0.208 0.0167
0.2452 0.0265
H 2
Ar 1.355 0.0320
1.382 0.0319
O 2
N 2 1.370 0.0387
CO 1.472 0.0395
CH 4 2.303 0.0431
3.658 0.0429
CO 2
4.225 0.0371
NH 3
Source: Lide, D.R. Ed., CRC Handbook of Chemistry
and Physics, 87th Edn., CRC Press, Boca Raton,
FL, 2006–2007, p. 6–34.