Page 266 - Physical Chemistry

lev38627_ch08.qxd 3/14/08 12:54 PM Page 247

247
Redlich–Kwong with the parameters a and b taken as functions of the mixture’s com- Section 8.3
position. For a mixture of two gases, 1 and 2, one often takes Condensation

2
2
a x a 2x x 1a a 2 1>2 x a and b x b x b (8.10)
1 1
2 2
1 2
2 2
1 2
1 1
where x and x are the mole fractions of the components. b is related to the molecu-
1
2
lar size, so b is taken as a weighted average of b and b . The parameter a is related to
2
1
intermolecular attractions. The quantity (a a ) 1/2 is an estimate of what the intermo-
1 2
lecular interaction between gas 1 and gas 2 molecules might be. In applying an equa-
tion of state to a mixture, V is interpreted as the mean molar volume of the system,
m
defined by
V V>n tot (8.11)
m
For the virial equation of state, the second virial coefficient for a mixture of two
2
2
gases is B x B 2x x B x B , where B is best determined from experimen-
12
1
2 2
1 2 12
1
1
tal data on the mixture, but can be crudely estimated as B (B B ).
2
1
2
12
The mixing rule (8.10) works well only if the molecules of gases 1 and 2 are similar (for
example, two hydrocarbons). To improve performance, a in (8.10) is often modified to a
2
2
x a 2x x (1 k )(a a ) 1/2 x a , where k is a constant whose value is found by
1 1
12
1 2
12
2
1 2
fitting experimental data for gases 1 and 2 and differs for different pairs of gases. Many
other mixing rules have been proposed [see P. Ghosh, Chem. Eng. Technol., 22, 379
(1999)].
8.3 CONDENSATION
Provided T is below the critical temperature, any real gas condenses to a liquid when
the pressure is increased sufficiently. Figure 8.3 plots several isotherms for H O on a
2
P-V diagram. (These isotherms correspond to vertical lines on the P-T phase diagram
of Fig. 7.1.) For temperatures below 374°C, the gas condenses to a liquid when P is
increased. Consider the 300°C isotherm. To go from R to S, we slowly push in the pis-
ton, decreasing V and V and increasing P, while keeping the gas in a constant-
m
temperature bath. Having reached S, we now observe that pushing the piston further
in causes some of the gas to liquefy. As the volume is further decreased, more of the
P/atm
H
H 2 O
Figure 8.3
218 Isotherms of H O (solid lines).
2
Y Not drawn to scale. The dashed
G line separates the two-phase
U T S 400°C region from one-phase regions.
85 W
M The critical point is at the top of
m
15 J L R 374°C the dashed line and has V 56
3
cm /mol. For the two-phase
Liquid N 300°C
Liquid + Vapor Vapor region, V V/n . (The dotted
m
tot
200°C curve shows the behavior of a van
der Waals or Redlich–Kwong
K isotherm in the two-phase region;
see Sec. 8.4.)
V m