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5.32: Untitled Page 94

  • Page ID
    18227
  • 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

    170