PART II. PRESERVATION PROCESSES
IV. PHASE EQUILIBRIA
One of the most important applications of the concept of thermodynamics is phase equilibria. A phase is a homogeneous region of a system and we most frequently think of phases as solid, liquid, and vapor. Figure 1 is a phase diagram for water. A phase diagram represents the relationship between pressure and temperature for the phases of a system. For pure water, region S represents solid ice, L represents liquid water, and V represents water vapor. The solid lines represent the conditions under which two phases coexist in equilibrium. The point where all three phases coexist is called the triple point and is at T ¼ 273.16K and P ¼ 0.00603 atm. Finally, there exists a pressure and temperature, called the critical point, above which the liquid and vapor phases are indistinguishable (for water at 547.6K and 219.5 atm). Properties of fluids above the critical point take on characteristics distinct from either the liquid or vapor and consequently may be used uniquely in processing. An example is the use of supercritical carbon dioxide for separation of food components.
TABLE1.3 Fugacity and Fugacity Coefficients of Water Vapor in Equilibrium with Liquid at Saturation and at 1 Atmosphere Pressure
Temperature
100.00 1.0132 0.99856 0.9855 0.9986
110.00 1.4326 1.4065 0.9818 1.0004
120.00 1.9853 1.9407 0.9775 1.0019
130.00 2.7011 2.6271 0.9726 1.0031
140.00 3.6135 3.4943 0.9670 1.0042
150.00 4.7596 4.5726 0.9607 1.0051
160.00 6.1804 5.8940 0.9537 1.0059
170.00 7.9202 7.4917 0.9459 1.0066
180.00 10.027 9.3993 0.9374 1.0072
190.00 12.552 11.650 0.9282 1.0077
200.00 15.550 14.278 0.9182 1.0081
210.00 19.079 17.316 0.9076 1.0085
220.00 23.201 20.793 0.8962 1.0089
230.00 27.978 24.793 0.8842 1.0092
240.00 33.480 29.181 0.8716 1.0095
250.00 33.775 34.141 0.8584 1.0098
260.00 46.940 39.063 0.8445 1.0100
270.00 55.051 45.702 0.8302 1.0103
280.00 64.191 52.335 0.8153 1.0105
290.00 74.448 59.554 0.7999 1.0106
300.00 85.916 67.367 0.7841 1.0108
aIce-liquid-vapor triple point.
bbar ¼ 0.9869 atm.
Source: Hass (1970).
A. Liquid-Vapor Equilibria
For phase equilibria between liquid and vapor, the free energy per mole must be the same in both phases. If this were not the case, then the system would spontaneously adjust to achieve equal free energy in both phases. Thus at some P and T,
GlA ¼GAg ð62Þ
If T and P are both shifted some infinitesimal amount so that the molar free energies become GLþ dGLand GVþ dGV, then the equilibrium condition will still hold and
GLþ dGL¼ GV þ dGV ð63Þ
From Eq. (62), dGL¼ dGV, and from Eq. (30)
dGL¼ VL dP2 SL dT ¼ dGV ¼ VV dP2 SV dT ð64Þ or
dP
dT¼ SV2 SL
VV2 VL
¼D Svap
DVvap ð65Þ
FIGURE.1.1 Phase diagram of water. The liquid-vapor curve stops at x, the critical point (547.6K and 219.5 atm).
where DSvap and DVvap are the molar entropy and molar volume change of vaporization, respectively. Since DGvap¼ D Hvap2 T D Svap¼ 0 for a reversible process, then
dP
dT¼ D Hvap
T DVvap ð66Þ
where D Hvap is the molar enthalpy of vaporization. Equation (66) is extremely important and is a form of the Clausius-Chapeyron equation. Its usefulness can be seen by considering phase equilibria for water. At moderate temperature and over a limited range, DHvap may be assumed constant, and since VV .. VL, DVvap , Vv. For atmospheric conditions and applying the ideal gas law, Vv¼ RT/P, then Thus, a plot of ln P versus 1/T gives a straight line with slope DHvap/R.
Equation (68) is the most popular form of the Clausius-Clapeyron equation.
B. Solid-Liquid Equilibria
Similarly, analysis of phase equilibria between ice and liquid water leads to dP
dT¼DHfus DVfus
ð69Þ Water is unique in that the molar volume of liquid is less than that of ice DVfus ¼ DVL2 DVS, 0 which results in a negative slope on the P-T diagram for the ice-liquid equilibrium line. Furthermore, since DVLand DVSare similar in magnitude (at 273.15K, VL¼ 0:01802 L and DVV ¼ 0:01963L), the slope is very large.
C. Solid-Vapor Equilibria
For solid-vapor equilibria, the Clausius-Clapeyron equation is dP
dT¼ DHsub
TDVsub
ð70Þ where the subscript denotes sublimation. According to Hess’s law, DHsub
may be calculated as the sum of DHvapþ DHfus. Consequently, the slope
of the equilibrium line between solid and vapor is greater than that between liquid and vapor because of the additional heat of fusion.
An example of the application of the Clausius-Clapeyron equation is calculation of the heat of vaporization of food products as proposed by Othmer (1940). The water vapor pressure of the product (or equilibrium relative humidity) is measured as a function of temperature. From Eq. (31) and knowing the vapor pressure of water at corresponding temperatures, the following applies:
DHvap;w
where w refers to water and p refers to product.
Example: The equilibrium relative humidity for precooked freeze-dried beef containing 10% moisture was 0.065, 0.1, 0.38, and 0.44 at 3, 9, 21, and 368C, respectively. Compute the heat of vaporization for the product at 308C.
Solution:
1. Using Eq. (35), plot logarithm of product vapor pressure versus logarithm of water vapor pressure to obtain the ratio of heat of vaporization (Othmer plot).
A useful rule regarding phase equilibria was derived by Gibbs and is called the phase rule:
where c is the number of components, p is the number of phases present, and f gives the degree of freedom on number of variables (pressure, temperature, composition), which must be fixed to completely describe the system. For example, for pure water vapor (v), c ¼ 1 and p ¼ 1 so that f ¼ 2. Thus knowing two out of the three P, V, and T, the system is completely described.
The third variable (P, V, or T ) can be calculated from the equation of state (gas law). In any pure-phase region of the phase diagram, f ¼ 2. Along any boundary, however, p ¼ 2 and f ¼ 1. Thus, for any value of P there is only one unique value of T at which both phases exist in equilibrium. At the triple point, p ¼ 3 and f ¼ 0, indicating that there is only one unique pressure-temperature combination where all three phases exist simultaneously. Some caution should be used when applying the phase rule because of additional constraints which may be applied to the relationships between components.
Extra relationships include, for example, balanced chemical relationships and electrical neutrality in ionic solutions.
E. Pressure-Composition Diagram
One of the most important processes in the food industry is the removal of water in either evaporation or in drying with minimum removal of flavor or aroma compounds. In addition, the food flavor and fragrance industry uses fractional distillation to separate components creating desired food additives that enhance flavor and taste. These processes are all based on the difference of vapor pressure of components at a given temperature. In order to design processes, it is useful to construct diagrams that show the vapor pressure of a solution as a function of mole fractions as well as composition of the vapor in equilibrium with the solution.
From Eqs. (46) and (47) and Dalton’s law, we can derive the following:
P1¼ x1P81; P2¼ x2P82
where xv1and xv2are the mole fractions of components 1 and 2 in the vapor phase, then
Equation (75) allows us to calculate the composition of liquid and vapor when the pressure is known. The pressure-composition diagram for benzene-toluene is given in Fig. 1.2. At point a on the liquid curve, the mole fractions are xbenzene¼ 0.2 and xtoluene¼ 0.8. The composition of the vapor in equilibrium with the solution (point b) is xvbenzene¼ 0.5 and xvtolene¼ 0.5. Thus the vapor is richer in benzene than the liquid. If we condense the vapor (b-c) and re-evaporate the liquid (c-d), the mole fraction of benzene will be even higher in the vapor phase. By repeating this process of condensing – re-evaporating in several stages, eventually the two components will be separated to the desired level.
The previous example is for a constant temperature process. In practice, distillations are usually carried out at constant temperature. Since the relation between temperature and composition is complex, the temperature-composition or boiling point diagram is determined experimentally. Figure 1.3 is the temperature-composition diagram for benzene-toluene. From Fig. 1.3, it can be seen that benzene, the component with the higher vapor pressure, has the lower boiling point. The process can be tracked as it was on the P-C diagram through a series of evaporation-condensation stages at constant pressure to produce a liquid
FIGURE1.2 Pressure-composition diagram of the benzene-toluene system. (From Chang, 1977, p. 264.)
phase rich in toluene and poor in benzene. Eventually the two components will be completely separated.
Since most solutions are nonideal, the experimentally determined T-C diagrams will be more complex. If the system deviates in a positive direction from Raoult’s law, the curve will show a minimum boiling point, whereas a negative deviation will produce a maximum boiling point. Under these circumstances, eventually the vapor and liquid will have the same composition, and further separation by distillation is not possible. The resulting equilibrium mixture is called an azeotrope. In most mixtures of importance to food processing, deviation is positive and minimum boiling mixtures are produced.
Examples include ethanol-water and n-proponal – water.