Page 270 - Physical Chemistry

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For the Redlich–Kwong equation, a similar treatment gives (the algebra is com- Section 8.4
plicated, so the derivation is omitted) Critical Data and Equations of State

5>2
2
5>2
2
a R T >912 1>3 12P 0.42748R T >P c (8.20)
c
c
c
b 12 1>3 12RT >3P 0.08664RT >P c (8.21)
c
c
c
1
PV >RT 0.333 (8.22)
c m,c
c
3
To use a two-parameter equation of state, we need to know the substance’s
critical pressure and temperature, so as to evaluate the parameters. If P and T are
c c
unknown, they can be estimated to within a few percent by group-contribution
methods (Sec. 5.10); see Poling, Prausnitz, and O’Connell, sec. 2-2.
Since there is a continuity between the liquid and gaseous states, it should be pos-
sible to develop an equation of state that would apply to liquids as well as gases. The
van der Waals equation fails to reproduce the isotherms in the liquid region of Fig. 8.3.
The Redlich–Kwong equation does work fairly well in the liquid region for some liq-
uids. Of course, this equation does not reproduce the horizontal portion of isotherms
in the two-phase region of Fig. 8.3. The slope ( P/ V ) is discontinuous at points S
m T
and W in the figure. A simple algebraic expression like the Redlich–Kwong equation
will not have such discontinuities in ( P/ V ) . What happens is that a Redlich–
m T
Kwong isotherm oscillates in the two-phase region (Fig. 8.3). The Peng–Robinson and
the Soave–Redlich–Kwong equations of state (Probs. 8.15 and 8.16) are improvements
on the Redlich–Kwong equation and work well for liquids as well as gases.
Hundreds of equations of state have been proposed in recent years, especially by
chemical engineers. Many of these are modifications of the Redlich–Kwong equation.
An equation that is superior for predicting P-V-T behavior of gases may be inferior for
predicting vapor–liquid equilibrium behavior, so it is difficult to identify one equation
of state as the best overall. For reviews of equations of state and mixing rules, see J. O.
Valderrama, Ind. Eng. Chem. Res., 42, 1603 (2003); Y. S. Wei and R. J. Sadus, AIChE
J., 46, 169 (2000); J. V. Sengers et al. (eds.), Equations of State for Fluids and Fluid
Mixtures, Elsevier, 2000.
The van der Waals and Redlich–Kwong equations are cubic equations of state,
meaning that when they are cleared of fractions, V is present in terms proportional to
m
2
3
V , V , and V only. A cubic algebraic equation always has three roots. Hence when
m m m
a cubic equation of state (eos) is solved for V at a fixed T and P, three values of V
m m
will satisfy the equation. At a temperature above the critical temperature T , two of the
c
roots will be complex numbers and one will be a real number so there is a single real
V that satisfies the eos. At T , the eos has three equal real roots. Below T , there will
m c c
be three unequal real roots. A cubic eos isotherm in the two-phase region below T will
c
resemble the dotted line in Fig. 8.3, which has three values of V that satisfy the eos
m
at the fixed condensation pressure, namely, the V values at points J, L, and N. The V
m m
values at J and N correspond to V of the liquid and V of the gas, respectively, that
m m
are in equilibrium with each other. The value of V at L has no physical significance.
m
The portion of the eos dotted-line isotherm from J to the minimum at K corre-
sponds to liquid that is at 200°C but is at a pressure lower than the 200°C vapor
pressure of 15 atm. Such a point lies below the liquid–vapor equilibrium line in
Fig. 7.1 and hence the liquid is in a metastable superheated state (Sec. 7.4) at points
between J and K. Likewise, the dotted-line isotherm portion NM corresponds to
supercooled vapor. The isotherm portion KLM has ( P/ V ) 0. As noted after
m T
Eq. (1.44), ( V / P) 1/( P/ V ) must be negative, so the portion KLM has no
m T m T
physical significance.
At some temperatures, part of the JK Redlich–Kwong or van der Waals isotherm
goes below P 0, indicating negative pressures for the superheated liquid. This is
nothing to be alarmed about. In fact, liquids can exist in a metastable state under