FRANCIS P. CHINARD The Johns Hopkins School of Medicine, Baltimore, Maryland
AN EARLIER note on osmotic pressure1 was intended to encourage teachers of medical and premedical students to emphasize such fundamental concepts as would obviate or correct misconceptions current in many texts of physiology and of biochemistry. In this note an attempt is made to bring into an integrated system the several colligative properties of importance t o physiology and hiochemistry. These colligative properties are usually described with emphasis on the experimental methods and on the properties of solutes rather than on the properties of water. A thermodynamic approach is used here; some building of the background is necessary. This might he considered uneconomical were it not for the growing importance of thermodynamic concepts in hiochemistry, in physiology, and in certain clinical investigations. A full treatment of the concepts would, however, require more time than could be made available in the conventional medical curriculum. For this reason, development of the background is left to the instructor. An attempt has been made here, nonetheless, to convey something of the concept of chemical potentials in nonmathematical terms; it is recognized that such a description must he supplemented with further details. There follows the fundamental expression relating changes of chemical potential to changes of pressure, temperature, and chemical compositi&. From this fundamental expression are deduced, in a manner similar to that used some time aeo hv " Gehlem2 the limitine laws of osmotic pressure, of freezing-point depression, of hoiling-point elevation, and of vapor-pressure lowering.
-
of water is the chemical potential, p, of water. The chemical potential of water is simply the Gihhs free energy of water in a given volume divided by the number of moles of water in the same volume. At equilibrium with respect to water the chemical potentials of water in all the phases are the same. If equilibrium does not obtain with respect to water, there will be mass transfer of water between the phases provided a pathway is available; molecular exchanges occur a t unequal rates between the phases; the chemical potentials of water are not equal. The mass transfer occurs from the phase or phases where the average energy or chemical potential of water is higher to the phase or phases where the chemical potential of water is lower. Experimentally, it is found that mass transfer may occur when there are differences between the phases in the values of one of the variables, P, T, or chemical composition. These variables, therefore, determine the relative values of the chemical potential of water in the several phases. FUNDAMENTAL RELATIONSHIP
The relationship of the of the chemical potential of water to variation of pressure, temperature, and chemical composition is given by the following expressiona: d'
-
N.I. T
(1)
N is the mole fraction of water, f is the,mole fraction activity coefficient; the product Nf is therefore the activity of water; the subscripts to the parentheses indicate that these variables are held constant for the particular variation of p considered. The relationships which follow:
THE CONCEPT OF CHEMICAL POTENTIALS
Consider a system comprising several phases with water a constituent of each phase. Each phase can be characterized by its pressure P, its temperature T, its volume V, and its chemical composition (expressed in terms of the mole fractions of the several constituents). Equilibrium with respect to water obtains when there is no mass transfer of water among the several phases. Molecular exchanges of water take place in and out of each phase at equal rates though the values of P. T,
($1
=
are now substituted in equation (I) to give: d p = VdP
2
CAINARD, F. P.,J. CEEM.EDUC.31. 66 (1954). GEHLEN, H., Z. Elektrochem. 48, 110 (1942).
(2)
V is the partial molal volume of water (assumed to be - - . ~ ~ .~ oonventiond definition of activity in terms of chemical potenn~~
rates it may he considered that the average energy available Der molecule of water is the same in the different ~h~ average.energy per mole
+ RTd In Nf - idT
~
~
~~
~~~~
-~ ~
~
-
~
~ --r
~
-
- ~~~-~~ ~ -
~
~~~~
tials: pn,o - &,o = RT In an,o where p0e,o is the chemical potential in a standard state &which the activity of water, ae,o, in unity. Instead of this second term, Gehlen considers the variation of chemical potential with mole fraction and thus limits the applicability of his fundamental relationship to ideal dilute snlutions.
317
~ ~ ~ ~ ~ ~
378
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independent of pressure in liquid water), R is the gas constant, and j. is the partial molal entropy of water. For sufficiently small variations and for systems of two condensed phases or two gaseous phases, the difference of the chemical potentials of water in the two phases may be written (from equation (2)) as:
FREEZING-POINT DEPRESSION
The freezing point of an aqueous solution may be defined as the temperature a t which the chemical potential of water in the solution is equal to the chemical potential of pure water as ice at the same temperature. Experimentally, it is found that a pure ice crystal will melt in an aqueous solution held a t the p" - p , = F(Pv - P') + RT In NMfX/N'j'- S(T" - T ' ) ( 3 ) freezing point of pure water. There is mass transfer The single primes are used to refer to variables or propof water from the ice phase to the solution. If the temerties in a pure-water phase; the double primes are perature of the solution is gradually lowered there will used to refer to variables or properties in a phase be found a temperature a t which an ice crystal a t the containing other substances in addition to water. same temperature will neither gain nor lose water; there is no mass transfer of water and equilibrium obtains. OSMOTIC PRESSURE Below this temperature, there will be mass transfer of Consider an isothermal system consisting of two water from the solution to the ice phase. There is aqueous phases separated by a membrane permeable sought an expression relating the composition of the only t o water. In one of the phases a dissolved sub- solution to the difference of the freezing points of the stance is present in addition to water. Experimentally, solution and of pure water in a system a t uniform presit is found that a pressure difference must be imposed sure. The difference of the chemical potentials of water in across the membrane in order to establish equilibrium with respect to water and to prevent mass transfer of the solution and of pure water a t the freezing point of water from the pure-water phase to the phase con- pure water is (from expression (3)) : taining the solute; it is also found that the greater presw ' t , ~~ pni,ro= RT, in a'z,m/anr,r. (8) sure must he applied to the phase containing the solute. At a given pressure difference the chemical potentials of The subscript I denotes water as a liquid; To is the water in the two compartments are equal; hence, in an freezing point of pure water. Activity is denoted by a. At the freezing point of pure water: isothermal system, equation (3) reduces to: T ( P n - P')
=
-RT in N"f"/N'f'
(9)
B ' Z ~ T=~ P:,T~
(4)
and at the freezing point T" of the solution:
On rearrangement, this may be written:
(lo)
p'&f" = p ' i , ~ "
The subscript i denotes water as ice. Again from (3) : where ?r is the osmotic pressure. The product Nf is the activity; the convention that the activity of pure water in the liquid state is unity is now used. Expression (5) reduces to:
p ' i , ~~p'b~" =
-%(To
- TX)
(11)
By subtraction of (12) from (11) there is obtained: For a sufficiently dilute solution N"f" approaches N" in numerical value. The mole fraction of the solute is denoted by NtV. I n the present simplified system N" Nz" = 1. The product N T in (6) may therefore be replaced by 1 - Nz". For approximate purposes, In (1 - N2") may he replaced by -N2". By definition, Nz" = nz"/(nv nzN)where n" and nz" d e not,e the number of moles, in this phase, of water and of the solute respectively. However, because the solution is dilute, n u is much greater than n,". We may then write:
+
+
But nXV = V where V is the volume of the phase. For n S X / Vwe may write C2" where C2" is the molal concentration of the solute. Expression (6) then becomes: T
=
CnrRT
(7)
This is Van't Hoff's limiting law of osmotic pressure.
P ' ~ , T~ P ' ~ , T "- CI'I,T~
+
p'b~" = (St
- $6)
(T,
- T U ) (13)
and by substitution of (8) and (9) in (13): From this there is obtained: *',,YO
- N",,"," = ( S t - 3 ; ) (7." - T " )
(15)
(3, - it) is equal to LJT0 where I,, is the latent heat of fusion. (i, - 3JT0 is t,he latent heat of transition of water to ice a t T,. By appropriate substitution there is obtained:
By definition, all,, = 1; To - T" is AT, the freezingpoint lowering. Expression (1G) then becomes:
By simplifications similar to those used for osmot,ic pressure there is obtained:
JULY, 1955
This is the limiting law for freezing-point depression. BOILING-POINT ELEVATION
Consider a system consisting of an aqueous phase and a gas phase and maintained a t a constant pressure of one atmosphere. The solute is assumed to have a negligible vapor pressure. The hoiling point of the solution may be defined as that temperature t o which the solution must be raised so that the chemical potentials of water in the solution and in the gas phase be equal when water is the sole constituent of the gas phase (i. e . , when the vapor pressure is equal to the total pressure of one atmosphere). At this temperature, there will be no mass transfer of water from one phase to another; if the temperature of the system is lowered. there will be mass transfer of water from the gas phase to the solution. There is sought an expression relating the composition of the solution to the difference of the boiling points of the solution and of pure water when a t the same total pressure. The derivation is similar t o that for freezing-point depression except that here we deal with So - 5, = L,/T,. The subscript g denotes the gas phase, L, is the latent heat of evaporation of water, and T,is the boiling point of pure water. There is obtained the expression:
This is the limiting law of boiling-point elevation. VAPOR-PRESSURE LOWERING
Consider two systems each of which consists of an aqueous phase and a gas phase. I n the first system, water is considered to be the only constituent of the aqueous phase; in the second system, one or more nonvolatile solutes are present. The total pressures and the temperatures of the two systems are equal. Experimentally, i t is found that the amount of water pel unit volume of gas phase (and the mole fraction of water in the gas phase) in each system is constant. I n each system molecular exchanges take place a t such rates in and out of each phase that there is no mass transfer of water. Equilibrium with respect to water obtains in each system between aqueous and gas phases; in each system the chemical potentials of water in the aqueous and gas phases are equal. I t is also found, however, that the amount of water per unit volume of gas phase is greater in the first system than in the second system. If the two systems are allowed t o communicate through their gas phases, it will be found that mass transfer of water occurs from the first system to the second system. Equilibrium with respect to water does not obtain between the two phases; the chemical potential of water in the first system is greater than the chemical potential of water in the second system. As a matter of convenience and convention, the amounts of water per unit volume of gas phase will be considered
379
indirectly in terms of partial pressures or vapor pressures. We wish to determine first the difference of the vapor pressures of water in the two systems and to relate this difference to the concentration of solute in the aqueous phase of the second system when the total pressures and temperatures of the two systems are equal. Equilibrium exists with respect t o water between the aqueous and gas phases in each system but not between the two systems. From (3) for equal total pressures, the difference of the chemical potentials in the aqueous phases is given by: fir" - PI' = RT In arn/at' (19) The difference of the chemical potentials of water in the gas phases is: &" - &' = RT in a,"/a.' (20) = p,' we must have: Since ,,," = ,,," and
'+" - &+=, ,p - &',,
(21)
If the water in the gas phase behaves as an ideal gas we may substitute p X / p ' for a,'/a,' where p is the vapor pressure of water. On making this substitution and on making obvious simplifications there are obtained the following equalities: p"/p' = a~'/at'
= frrNz"/fi'Nz'.
(22)
jdenotes the mole fraction activitycoefficient. Assume f," = j,'. By definition, N,' = 1. Accordingly, we have: pW/p' = Nz" (23) From this there is obtained:
where N2"is the mole fraction of solute. Finally there is obtained: Pr
- pw = p ' ~ I "
(25)
This relationship indicates that the lowering of the vapor pressure of water in a solution is proportional t o the mole fraction of the solute. This is Raoult's law, the limiting law of vapor-pressure lowering. We wish now to determine the effect of differences of total pressure on the vapor pressure; we wish aLso to determine the difference of total pressure that must be imposed in order that equilibrium obtain with respect to water between the two systems. In the individual systems we have:
On taking the appropriate ratios there is obtained: %"/ao' = a ~ " / a ~ '
(28)
If the water vapor behaves as an ideal gas, we may write:
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concentration of the solutes. This is in contrast to In order t o establish equilibrium with respect to water current trends in some of the physiological literature between the liquid phases we must have a pressure where, for example, "osmolar concentrations," "os~ even "renal osmolar clearances" are condifference between the liquid phases. Accordingly, m o l ~ , 'and sidered. for p,' to he equal to p,' we must have, from (3): Water is the major constituent of most biological RT in atr/at' = - v(PW- P') (30) systems. Changes, of physiological and clinical sigBy appropriate substitution and rearrangement there nificance, in the distribution of water among the several body compartments occur because of differences of the is obtained the relationship: chemical potentials of water. The changes in distriRT In p'/pN = P(PX- P ' ) (31) bution of water are determined in part by differences If p" is less than p', then the total pressure on the in the concentrations of other constituents of the solution must be greater than the pressure on the pure compartments, hut only in so far as the concentration water. This pressure differenceis exactly equal to the differences mediate differences in the chemical poosmotic pressure though there is no membrane involved tentials of water. here. ACKNOWLEDGMENT DISCUSSION
The deduction of the limiting laws from a single fundamental expression serves to point up the interrelationships of these several colligative properties and of the limiting laws. The properties of water are emphasized rather than the solutes; attention is focused on the chemical potentials of water and not on the
The author is indebted to the John and Mary R. Markle Foundation for a scholamhip in the medical sciences during the tenure of which this article was written, and t o the Life Insurance Medical Research Fund for support of work that has made evident the advantages of a more uniform treatment of colligative properties.
