# chapter12 - Chapter 12 Phase Equilibria Involving Dissolved...

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Chapter 12 Phase Equilibria Involving Dissolved Solids in Liquids Crystallization is the formation of solid particles from within a homogeneous phase and it is one of the unit operations used to separate substances in pure solid form from a liquid mixture. Therefore, reliable data on the solubility of solids in liquid solvents are needed in designing crystallizers. This chapter fi rst provides equations to estimate solid solubility in solvents. Then, how boiling and freezing points of liquids are a ff ected by the presence of dissolved solids will be discussed. 12.1 EQUILIBRIUM BETWEEN A PURE SOLID AND A LIQUID MIXTURE Let substance i be present as a pure solid and also as a component in a liquid solution at constant temperature and pressure. The condition of equilibrium states that f S i ( T, P ) = b f L i ( T, P, x i ) = f L i ( T, P ) γ i ( T, P, x i ) x i (12.1-1) Rearrangement of Eq. (12.1-1) gives ln ( γ i x i ) = ln " f S i ( T, P ) f L i ( T, P ) # (12.1-2) The fugacities of pure solid and liquid are related to the molar Gibbs energies by Eq. (5.2-1), i.e., ln " f S i ( T, P ) P # = e G S i ( T, P ) e G IG i ( T, P ) RT (12.1-3) ln " f L i ( T, P ) P # = e G L i ( T, P ) e G IG i ( T, P ) RT (12.1-4) Thus, ln " f S i ( T, P ) f L i ( T, P ) # = e G S i ( T, P ) e G L i ( T, P ) RT = e G fus i ( T, P ) RT = e H fus i ( T, P ) T e S fus i ( T, P ) RT (12.1-5) 365

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The use of Eq. (12.1-5) in Eq. (12.1-2) leads to ln ( γ i x i ) = e H fus i ( T, P ) T e S fus i ( T, P ) RT (12.1-6) Since the solution temperature, T , is usually much lower than the normal melting point temperature, T m , of pure solid, a pure substance does not exist in the liquid form at the solution temperature. Thus, extrapolation of the liquid-vapor pressure curve from the triple point temperature to the solid region is necessary until the temperature reaches the solution temperature as shown in Figure 12.1. vap solid P vap liquid subcooled P Pressure Temperature T t T VAPOR LIQUID SOLID Figure 12.1 Extrapolation of vapor-liquid pressure curve to the solution temperature, T . Systematic way of extrapolation can be carried out by devising a following path as shown in Figure 12.2. Step 1: Solid is heated at constant pressure from solution temperature, T , to triple point temperature, T t . Step 2: Solid is melted to form a liquid ( e H fus i and e S fus i at T t are known). Step 3: Liquid is subcooled without solidi fi cation from T t to T . Temperature t T T Solid Subcooled Liquid Phase Figure 12.2 Calculation path for e H fus i ( T ) and e S fus i ( T ) . For steps 1 and 3, the changes in enthalpy and entropy can be calculated from Eqs. (3.3-1) and (3.4-2), respectively. Since pressure remains constant, the changes in enthalpy and entropy for the overall process are expressed as e H fus i ( T ) = Z T t T e C S P i dT + e H fus i ( T t ) + Z T T t e C L P i dT (12.1-7) 366
e S fus i ( T ) = Z T t T e C S P i T dT + e S fus i ( T t ) + Z T T t e C L P i T dT (12.1-8) At the triple point, solid and liquid phases of pure component i are in equilibrium with each other, i.e., e G fus i ( T t ) = e H fus i ( T t ) T t e S fus i ( T t ) = 0 e S fus i ( T t ) = e H fus i ( T t ) T t (12.1-9) The use of Eq. (12.1-9) in Eq. (12.1-8) leads to

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