# 5.32: Untitled Page 94

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## Chapter 5

coefficients in Antoine’s equation are obtained from Table A3 of Appendix A and they are given by

*A * 8 *. * 07246 *, *

*B * 1574 *. * 99 *, *

238 *. * 86

Once again we note that Antoine’s equation is a dimensionally incorrect empiricism, and one must follow the rules of application that are given above and in Table A3 of Appendix A. Substitution of the values for *A*, *B*

and into Eq. 5‐22 gives

1 *, * 574 *. * 99

log *p*

8 07246

*. *

2 10342

*. *

M *,vap*

238 86

*. *

25

and the vapor pressure of methanol at 25 C is *p*

= 126.9 mmHg =

M *,vap*

16,912 Pa. The result computed using the Clausius‐Clapeyron equation was *p*

= 19,112 Pa, thus the two results differ by 11%. The results of M *,vap*

this example clearly indicate that it is misleading to represent calculated values of the vapor pressure to five significant figures.

5.3.1 *Mixtures*

The behavior of vapor‐liquid systems having more than one component can be quite complex; however, some mixtures can be treated as *ideal*. In an ideal vapor‐liquid multi‐component system the partial pressure of species *A* in the gas phase is given by

Equilibrium relation:

*p*

*p*

*x *

(5‐23)

*A*

*A,vap*

*A*

Here *p * is the partial pressure of species *A* in the *gas phase*, *x * is the mole *A*

*A*

fraction of species *A* in the *liquid phase*, and *p* is the vapor pressure of species

*A,vap*

*A* at the temperature under consideration. It is important to remember that Eq. 5‐23 is an *equilibrium relation*; however *, * when the condition of *local* *thermodynamic equilibrium * is valid Eq. 5‐23 can be used to calculate values of *p* *A*

for dynamic processes.

The equilibrium relation given by Eq. 5‐23 represents a special case of a more general relation that is described in many texts (Gibbs, 1928; Prigogine and Defay, 1954) and will be studied in a course on thermodynamics. The general equilibrium relation is based on the *partial molar Gibbs free energy*, or the chemical potential, and it takes the form

*Two‐Phase Systems* & *Equilibrium Stages* 169

Equilibrium relation:

( )

( )

*, * at the gas‐liquid interface (5‐24)

*A gas*

*A liquid*

Here we have used to represent the *chemical potential* of species *A* that *A*

depends on the temperature (strongly), the pressure (weakly), and the composition of the phase under consideration. The matter of extracting Eq. 5‐23

from Eq. 5‐24 will be taken up in a subsequent course on thermodynamics.

For an ideal gas, use of Eq. 5‐2 along with Eq. 5‐3 indicates that the gas‐phase mole fraction can be expressed as

*y*

*p p *

(5‐25)

*A*

*A*

This result can be used with Eq. 5‐23 to obtain a relation between the gas and liquid‐phase mole fractions that is given by

Equilibrium relation:

*y*

*x*

*p*

*p *

(5‐26)

*A*

*A * *A,vap*

This equilibrium relation is sometimes referred to as *Raoult’s law*. For a two-component system we can use Eq. 5‐26 along with the constraint on the mole fractions

*x * *x*

1 *, *

*y * *y*

1

(5‐27)

*A*

*B*

*A*

*B*

to obtain the following expression for the mole fraction of species *A* in the gas phase:

*x*

Equilibrium relation:

*AB*

*A*

*y*

*A*

1 *x *(

(5‐28)

1)

*A*

*AB*

Here

is the *relative volatility* defined by

*AB*

*p*

*A,vap*

(5‐29)

*AB*

*pB,vap*

For a dilute binary solution of species *A*, one can express Eq. 5‐28 as *y*

*x , *

for *x *(

1) 1

(5‐30)

*A*

*AB*

*A*

*A*

*AB*

and this special form of Raoult’s law is often referred to as *Henry’s law*. For an *N*‐component system, one can express Henry’s law as

Henry’s Law:

*y*

*K*

*x *

(5‐31)

*A*

*eq,A*

*A*