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Unformatted text preview: last modified 10/19/2009 © M. S. Shell 2008 1/19 Phase equilibrium ChE210A It is a familiar fact that pure substances tend to exist in one of three distinct states: solid, liquid, and gas. Heat a crystal up, and at a specific temperature it suddenly melts into a liquid. Con- tinue heating that liquid, and eventually it will hit a temperature at which it spontaneously vaporizes into a gas. These transitions are discontinuous, i.e., they occur at specific state conditions—specific combinations of g and G . At those conditions, the system can exist in more than one form such that two (or more) phases are in equilibrium with each other. Although we are mostly familiar with phase behavior at atmospheric pressure, most substances experience a diverse set of phases over a broad range of pressures. Pure substances tend to have more than one crystal phase, depending on the pressure. In rare cases, pure substances can also have distinct liquid phases. When we consider multicomponent mixtures, rather than pure substances, there are even more possibilities for phase equilibrium: we can have equili- brium between two liquids of different compositions, or among multiple solid and liquid phases, for example. Here is a phase diagram of liquid water, for example: In this lecture, we will discuss two important features of phase equilibrium: • What are the thermodynamic conditions for phase equilibrium? last modified 10/19/2009 © M. S. Shell 2008 2/19 • Why do phases change discontinuously? For example, why does water have a definite boiling point temperature at ambient pressure? Derivation of phase equilibrium conditions We start by defining a phase, conceptually: A phase is homogeneous region of matter, i.e., there is no spatial variation in av- erage density, energy, or composition. By average, we mean what we obtain when we smooth out the discrete, kinetic nature of molecular motion. Phases can also be distinct in their molecular structure. For example, water has different phases of ice that differ in their crystallographic structure. A phase can be considered a distinct “system.” The notion of equilibrium between phases means that there are two or more phase “systems” present. These phases exist spontaneously on their own without the use of partitions or membranes or other interventions, and since they are in equilibrium with each other, they can exchange energy, volume, and particles. We have seen previously that, when such is the case, the conditions of equilibrium are given by maximization of the entropy subject to the con- straints of constant total energy, volume, and particles. For two phases, this becomes: maxgG ¡ ¢£ ¡ ,¤ ¡ ,¥ ¡ ¦ § G ¨ ¢£ ¨ ,¤ ¨ ,¥ ¨ ¦© In general, at a maximum, we have: ªG ¡ § ªG ¨ « 0 ¬ ªG ¡ ª£ ¡ ª£ ¡ § ¬ ªG ¡ ª¤ ¡ ª¤ ¡ § ¬ ªG ¡ ª¥ ¡ ª¥ ¡ § ¬ ªG ¨ ª£ ¨ ª£ ¨ § ¬ ªG ¨ ª¤ ¨ ª¤ ¨ § ¬ ªG ¨ ª¥ ¨ ª¥ ¨ « 0 1 ® ¡ ª£ ¡ § ¯ ¡...
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