Microsoft Word - Cengel and Boles TOC _2-03-05_.doc

from ideal-gas behavior. One way of doing that is to use more exact equa-

tions of state (van der Waals, Beattie–Bridgeman, Benedict–Webb–Rubin,

etc.) instead of the ideal-gas equation of state. Another way is to use the

compressibility factor (Fig. 13–8) as

(13–9)

The compressibility factor of the mixture Zmcan be expressed in terms of the

compressibility factors of the individual gases Ziby applying Eq. 13–9 to both

sides of Dalton’s law or Amagat’s law expression and simplifying. We obtain

(13–10)

where Ziis determined either at Tmand Vm(Dalton’s law) or at Tmand Pm

(Amagat’s law) for each individual gas. It may seem that using either law

gives the same result, but it does not.

The compressibility-factor approach, in general, gives more accurate results

when the Zi’s in Eq. 13–10 are evaluated by using Amagat’s law instead of

Dalton’s law. This is because Amagat’s law involves the use of mixture

pressure Pm, which accounts for the influence of intermolecular forces

between the molecules of different gases. Dalton’s law disregards the influ-

ence of dissimilar molecules in a mixture on each other. As a result, it tends

to underpredict the pressure of a gas mixture for a given Vmand Tm. There-

fore, Dalton’s law is more appropriate for gas mixtures at low pressures.

Amagat’s law is more appropriate at high pressures.

Note that there is a significant difference between using the compressibil-

ity factor for a single gas and for a mixture of gases. The compressibility fac-

tor predicts the P-v-Tbehavior of single gases rather accurately, as discussed

in Chapter 3, but not for mixtures of gases. When we use compressibility

factors for the components of a gas mixture, we account for the influence of

like molecules on each other; the influence of dissimilar molecules remains

largely unaccounted for. Consequently, a property value predicted by this

approach may be considerably different from the experimentally determined

value.

Another approach for predicting the P-v-Tbehavior of a gas mixture is

to treat the gas mixture as a pseudopure substance (Fig. 13–9). One such

method, proposed by W. B. Kay in 1936 and called Kay’s rule,involves the

use of a pseudocritical pressure Pcr,mand pseudocritical temperature Tcr,m

for the mixture, defined in terms of the critical pressures and temperatures

of the mixture components as

(13–11a, b)

The compressibility factor of the mixture Zmis then easily determined by

using these pseudocritical properties. The result obtained by using Kay’s

rule is accurate to within about 10 percent over a wide range of tempera-

tures and pressures, which is acceptable for most engineering purposes.

Another way of treating a gas mixture as a pseudopure substance is to

use a more accurate equation of state such as the van der Waals, Beattie–

Bridgeman, or Benedict–Webb–Rubin equation for the mixture, and to deter-

mine the constant coefficients in terms of the coefficients of the components.

Pcr,¿m
a

k

i
 1

yiPcr,i¬and¬T¿cr,m
a

k

i
 1

yiTcr,i

Zm
a

k

i
 1

yiZi

PV
ZNRuT

686 | Thermodynamics

Pm Vm = Z = Zm Nm Ru Tm

Zm m = = ∑ y yi Z Zi

k

i = i = 1

FIGURE 13–8

One way of predicting the P-v-T

behavior of a real-gas mixture is

to use compressibility factor.

Pseudopure substance

P'cr,m = ∑ yi Pcr,i

k

i = 1

T'cr,m = ∑ yi Tcr,i

k

i = 1

FIGURE 13–9

Another way of predicting the P-v-T

behavior of a real-gas mixture is to

treat it as a pseudopure substance with

critical properties Pcrand Tcr.

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