COLLIGATIVE PROPERTIES

The colligative properties of dilute ideal solutions are those properties which depend only on the number of solute molecules present (number of moles or mole fraction) and not on the size or molecular weight of the molecules. As we shall see presently, these properties include vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure. As pointed out above, these properties of aqueous solutions are very important for proper design of food processing operations.

FIGURE1.3 Temperature-composition diagram of the benzene-toluene system.

A. Vapor Pressure Lowering

Assume we have a dilute aqueous solution containing a nonvolatile solute, for example, sucrose in water. Since the vapor pressure of sucrose in small (1 £ 10223atm), we can neglect it and apply Raoult’s law to the solvent.

Pw¼ xwPw8 ð76Þ

where w refers to water. Since xw¼ 1 2 xs, where xs is the mole fraction of

sucrose, then

Pw¼ 1 2 xð sÞPw8 ð77Þ

Rearrangement gives

Pw8 2 Pw¼ DPw¼ xsPw8 ð78Þ

where DPwis the vapor pressure lowering of the presence of xsmole fraction of

sucrose.

B. Boiling Point Elevation

The boiling point of a solution is the temperature at which its vapor pressure is equal to the external pressure. From the previous analysis, adding a nonvolatile solute to water lowers the vapor pressure resulting in an elevated boiling point.

The boiling point elevation can be estimated by writing the Clausius- Clapeyron equation [Eq. (68)] for the solution

ln Po P ¼

DHvapðT2 ToÞ

RTTo

ð79Þ where Poand Toare the vapor pressure and temperature of water at its boiling

point and P and T are the corresponding solution properties. DH is the molar heat of vaporization of water from the solution and, for dilute solutions, may be taken as the molar heat of vaporization of the pure water. In addition, since the boiling point elevation is generally small for a dilute solution T Toø T2o.

From Raoult’s law,

P ¼ xwPo and ln

P Po

¼ ln xw ð80Þ

Since ln xw¼ ln 1 2 xð sÞ; and for xs# 0:2;

ln 1ð 2 xsÞ ¼ x2 s 2 2 x3 s 3 2 · · · ø 2xs

Then ln Po

P ¼ xs ð81Þ

for xs# 0.2. Substituting Eq. (81) into Eq. (79) gives

xs¼ DHvapðT2 ToÞ RT2 o ¼DHvapDTbp RT2 o ð82Þ

where DTbpis boiling point elevation. Solving Eq. (82) for boiling point elevation

gives DTbp ¼ RT2 oxs DHvap ð83Þ Generally, this equation is written in terms of molality by noting that

xs¼ ns nwþ ns , ns nw ¼ WsMs WwMw ð84Þ for nw .. ns, where Wsand Wware the mass of solute and water, respectively,

and Msand Mware the respective molecular weights. Then

DTbp ¼ 18RT2 o 1000 DHvap 1000Ws WsMs   ð85Þ All the quantities in the first term are constants for a given solute-water system and can be expressed as Kb. The second term is the molality (Mw) of the solution.

Thus,

DTbp ¼ KbMw ð86Þ

For dilute aqueous solutions, Kb¼ 0.528C.

C. Freezing Point Depression

In an analogous thermodynamic analysis, freezing point depression for dilute aqueous solution leads to

DTbp ¼ 2

18RT2 oMw

1000DHfus

or

DTfp¼ 2KfMw ð88Þ

For aqueous solutions Kf¼ 1.868C.

Example for Freezing Point Depression: Calculate the temperature at which ice formation is initiated in an ice cream mix with the following composition: 10% butterfat, 12% solids-not-fat (54.5% lactose), 15% sucrose, and 0.22% stabilizer.

Solution: We will assume that the only constituent influencing the freez- ing point is the dissolved sucrose and lactose and that the solution is sufficiently dilute so that Eq. (88) applies.

DTfp¼ 21:8m ðwhere m ¼ molalityÞ

Molality ¼moles solute 1000 g water¼

g solute Ms 1000 g water

Mass fraction sucrose þ lactose ¼ 0:15 þ 0:12ð0:545Þ ¼ 0:2154 g gproduct 0:2154 g gp 1 g product   0:6728 g water

  ¼ 0:3431g solute_{g water} ¼ 343:1 g solute 1000 water 343:1 solute

342 water mol ¼ 1:003 molality DTfp¼ 21:86 1:003ð Þ ¼ 21:868C

Therefore, initial freezing begins at21.868C or 271.14K

D. Osmotic Pressure

The concept of osmotic pressure can be illustrated by considering two compartments, one containing solvent, e.g., water, and the other containing a solution separated by a semipermeable membrane (allows water to pass through but not solute). Practically speaking this system has two different phases, and at equilibrium, sufficient water will have migrated through the semipermeable membrane so that the hydrostatic head of the solution exactly counterbalances the difference in chemical potential between the pure water and the solution.

From thermodynamic principles, it can be shown that

p¼ c

M2

RT ð89Þ

where p is the osmotic pressure, c is concentration of solute in gl21of solution and M2is molecular weight of the solute.

One of the many uses of Eq. (89) was determination of molecular weight. In that connection, osmotic pressure was measured at several different concentrations and extrapolated to zero concentration to reduce deviation from the assumption of ideal behavior.

SYMBOLS

U internal energy (kJ) W work (kJ)

Q heat energy (kJ) V volume (m3)

P pressure (bars or atm) H enthalpy (kJ/kg) T temperature (K)

CV heat capacity at constant volume (kJ/kg K)

CP heat capacity at constant pressure (kJ/kg K)

R universal gas law constant (8.314 J/mol K) n moles (mol)

S entropy (kJ/K or entropy unit) G Gibbs free energy (kJ) A Helmholtz free energy (kJ) M chemical potential (kJ/kmol) G8 standard free energy (kJ) l absolute activity (dimensionless) a relative activity (dimensionless) g activity coefficient (dimensionless) x mole fraction (dimensionless)

k Henry’s law constant (atm mol21kg21)

f fugacity (bars) or degrees of freedom (dimensionless) c number of components (dimensionless)

p number of phases (dimensionless) Kb boiling point elevation constant (K or8C)

Kf freezing point depression constant (K or8C)

p osmotic pressure (bars or atm) M molecular weight (kg/kmol)

REFERENCES

Bainanu I. Elements of thermophysics and chemical thermodynamics: basic concepts. In Baianu IC, ed. Physical Chemistry of Food Processes: Vol. I. Fundamental Aspects. New York: Van Nostrand Reinhold, 1992, Chapter 2, pp. 32 – 44.

Chang R. Physical Chemistry with Applications to Biological Systems. New York: MacMillan, 1977.

Hass JL. Fugacity of H2O from 08 to 3508C at liquid-vapor equilibrium and at 1

atmosphere. Geochim Cosmochim Acta 34:929 – 934, 1970.

Othmer DF. Correlating vapor pressure and latent heat data. Ind Eng Chem 32:841, 1940. Rizvi SSH. Thermodynamic properties of foods in dehydration. In M.A. Rao and S.S.H. Rizvi, eds. Engineering Properties of Foods. New York: Marcel Dekker, 1986, Chapter 4, pp. 133 – 214.

Tinoco I Jr, Sauer K, Wang JC, and Puglisi JD. Physical Chemistry: Principles and Applications in Biological Systems, 4th ed. Upper Saddle River, NJ: Prentice Hall, 2002, pp. 1 – 170.

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