# 5.27: Untitled Page 88

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## Chapter 5

Two‐Phase Systems & Equilibrium Stages

In the previous chapter, we began our study of macroscopic mass and mole balances for multicomponent systems. There we encountered a variety of *measures of concentration* and here we summarize these measures as

mass of species *A*

(5‐1a)

*A*

per unit volume

*A * *N*

*, * total mass density

(5‐1b)

*A*

*A * 1

*, * mass fraction

(5‐1c)

*A*

*A*

*c*

*MW , * molar concentration

(5‐1d)

*A*

*A*

*A*

*A * *N*

*c *

*c , * total molar concentration

(5‐1e)

*A*

*A * 1

*y * or *x*

*c c , * mole fraction

(5‐1f)

*A*

*A*

*A*

In the analysis of gas‐phase systems it is often important to relate the concentration to the pressure and temperature. This is done by means of an equation of state, often known as a *p*‐ *V*‐ *T* relation. In this chapter we will make use of the *ideal gas* relations; however, many processes operate under conditions such that the ideal gas laws do not apply and one must make use of more general *p*‐ *V*‐ *T* relations. Non‐ideal gas behavior will be studied in a subsequent course on thermodynamics.

**5.1 Ideal Gas Behavior **

For an *N*‐component ideal gas mixture we have the following relations *p V*

*n RT , *

*A * 1 *, * 2

*A*

*A*

*, ..., N *

(5‐2)

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*Two‐Phase Systems* & *Equilibrium Stages* 157

Here *p * is the *partial pressure* of species *A* and *R* is the gas constant. Values of *A*

the gas constant in different units are given in Table 5‐1. If we sum Eq. 5‐2 over all *N* species we obtain

*pV*

*nRT *

(5‐3)

where

*A * *N*

*p *

*p*

(5‐4)

*A*

*A * 1

*A * *N*

*n *

*n*

(5‐5)

*A*

*A * 1

Equations 5‐2 through 5‐5 are sometimes referred to as *Daltonʹs Laws. *

*Table 5‐1*. Numerical values of the gas constant, *R*

Numerical

Units

value

8.314

m3 Pa/ mol K

8.314

J/mol K

0.08314

L bar/mol K

82.06

atm‐cm3 /mol K

1.986

cal/mol K

In Figure 5‐1 we have illustrated a *constant pressure, isothermal* mixing process. In the compartment containing species *A* illustrated in Figure 5‐1a, we can use Eq. 5‐3 to obtain

*pV*

*n RT *

(5‐6)

*A*

*A*

while in the compartment containing species *B* we have *pV*

*n RT *

(5‐7)

*B*

*B*

Here we have made use of the fact that *p *

*A*

*pB p * for this particular process.

Upon removal of the partition and mixing, we have the situation illustrated in Figure 5‐1b. For that condition, we can use Eq. 5‐3 to obtain *pV*

( *n * *n *) *RT *

(5‐8)

*A*

*B*

where the volume is given by

158