Lecture_05

Lecture_05 - CHAPTER 5 LECTURE NOTES Phases and Solutions...

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CHAPTER 5 LECTURE NOTES Phases and Solutions Phase diagrams for two one component systems, CO 2 and H 2 O, are shown below. The main items to note are the following: ± The lines represent equilibria between two phases. ± The regions &interior± to these lines represent regions where only one phase is present. ± The triple point is a unique, invariant point for each system where three phases are simultaneously in equilibrium. Thermodynamic condition for equilibrium between two phases a and b is G a = G b , so that during an equilibrium phase change, G ab = G a ² G b = 0. Note that the slope of the line separating the solid from the liqid region for CO 2 has a positive slope. This is typical of most substances and is indicative of the fact that the solid is denser than the liquid (how?). On the other hand, the slope of the solid-liquid boundary for water is negative, indicating that the liquid is denser than the solid. This is by far the exception . 1
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The Clapeyron Equation If we confine ourselves to conditions where two phases, say vapor and liquid, are in equilibrium, we effectively limit ourselves to a one-dimensional world defined by the line separating the liquid region from the vapor region. Under those conditions, the Maxwell relation (3.124) æ è ¹ P ¹ T ö ø V = æ è ¹ S ¹ V ö ø T , (3.124) can be modified as follows: (5.8) dP dT = dS dV j S m , v S m , l V m , v V m , l = , vap S m , vap V m Substituting the definition of entropy change for a phase change, we obtain (5.9) dP dT = , vap H m T , vap V m . This is called the Clapeyron equation . Similar equations can be written for the solid-vapor boundary (enthalpy of sublimation!) and the solid-liquid boundary (enthalpy of fusion!). Note that, since enthalpy is a state function, sub H m = fus H m + vap H m . (5.10) The Clausius-Clapeyron Equation: Eq. (5.9) may be written as (5.11) dP dT = , vap H m T æ è V m , g V m , l ö ø l , vap H m TV m , g j , vap H m RT 2 P , where we have assumed that the vapor behaves ideally to get the last equality. This equation may be rearranged and integrated to give or (5.14) ln P u = , vap H m RT + C , (5.16) ln æ è P 2 P 1 ö ø = , vap H m R æ è 1 T 1 1 T 2 ö ø . The Clausius-Clapeyron equation is applicable to solid-vapor equilibrium as well. The pressure plotted in the phase diagrams and used in these relationships is the vapor pressure of the substance under consideration, not the total pressure . Eq. (5.14) gives the P - T relationship required to plot the lines in the phase diagrams. Therefore, Eq. (5.14) for the liquid-vapor and the solid-vapor equilibria can be used to find the triple point for a one-component system. Examples 5.2, 5.3, Problems 5.1&5.4, 5.7&5.10, 5.12, 5.15, 5.18, 5.20. 2
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The Gibbs Equation: By definition, dG = VdP & SdT . At constant temperature, the second term vanishes and we get dG = VdP . For the liquid vapor equilibrium considered earlier, under these circumstances, we have dG m,v = dG m,l or V m,v dP = V m,l dP t , where the pressure term on the vapor side is the vapor pressure of the substance while on the liquid side, it is the total pressure acting on the liquid surface. Now,
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Lecture_05 - CHAPTER 5 LECTURE NOTES Phases and Solutions...

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