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5.27: Untitled Page 88

  • Page ID
    18221
  • 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 pVT 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 pVT 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)

    156

    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

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    index-167_2.png

    index-167_3.png

    index-167_4.png

    index-167_5.png

    index-167_6.png

    158