J. Phys. Chem. 1994, 98, 4196-4204
4196
How Solutes Alter Water in Aqueous Solutions H. T. Hammel Department of Physiology and Biophysics, Biomedical Sciences Program, Meyers Hall, Indiana University, Bloomington, Indiana 47405 Received: November 16, 1993; In Final Form: January 27, I994@
Two theories, one by G . Hulett in 1903 and one by G . N. Lewis in 1908, describe how solutes alter water in a n aqueous solution. Hulett recognized that the solutes alter the vapor pressure of water in a solution as would a negative pressure applied to pure liquid water a t the same temperature, where the applied pressure equals the osmotic pressure of the water a t a free surface of the solution. H e proposed that the solutes exert an internal pressure so as to alter the internal tension of the water in the solution exactly as the applied negative pressure alters the internal tension in pure water and, thereby, alters all colligative properties of the water. Lewis defined a term, fugacity of a solvent, which equals the vapor pressure of a n ideal solvent and which approaches the vapor pressure of the real solvent only when the real solvent vapor pressure is vanishingly small. Lewis also defined solvent activity as the ratio of fugacity of solvent in the solution to fugacity of pure liquid solvent, and he attributed the lower chemical potential of the solution solvent to its diminished activity. Internal tension is an intensive property of the solvent whereas fugacity and activity are not real properties of a real solvent in a real solution. Neither Hulett’s nor Lewis’s theory incorporates a kinetic theory to account for the altered state of the solvent in a solution. At that time, the existence of atoms and molecules was not well established. Moreover, an equation of state for liquid solvent was not known, and without an equation of state, no rigorous theory could be stated. In this article, I compare Hulett’s and Lewis’s theories and derive a kinetic theory compatible with Hulett’s theory. Although Lewis’s theory is almost universally taught to students of physical chemistry and chemical thermodynamics, Hulett’s theory deserves attention on its merit.
I. Introduction
Many attempts have been made to describe how water in an aqueous solution is altered by a solute dissolved in it. Of course, the chemical potential of water in the solution is less than the chemical potential of pure water at the same temperature and applied pressure. But how does the solute lower the chemical potential of the water? In 1908 G. N. Lewis’ proposed that the solutelowered thefugacityand theactivityofthe water. However, these terms are mere inventions of convenience which allow an approximate treatment of water and its vapor as if water had the properties of an ideal solvent and water vapor had the properties of an ideal gas. Since no liquid and its vapor are ideal, in accord with Lewis’s definitions, he had to invent another term, activity coefficient, in order to treat a real solvent by the same thermodynamic relationship as he derived for the ideal solvent and its ideal vapor. Lewis’s account of the altered water in an aqueous solution became nearly universally accepted and taught to students of physical chemistry and chemical thermodynamics. In this article, I will revive and revise an older theory by G. Hulett.2 I will also review Lewis’s theory and then compare these two incompatible theories. Finally, I will attempt to derive a kinetic theory to account for the osmotic pressure of water in an aqueous solution, which is compatible with Hulett’s theory. Before adding solutes to water, let us consider some of the properties of water which will change when a solute is dissolved in it. Water exposed to air at sea level will be subjected to a pressureof 1 atm(expressedas760Torr, 1.013 25 bar,O.lOl 325 J ~ m - or ~ , 101.325 kPa). At temperature T, gases in air will dissolve in water by an amount depending on the partial pressures of the gases in air. Under these circumstances, liquid water in an enclosure with air will equilibrate with water vapor at a certain partial pressure which depends on Tand is designated,p,g( T,pc]), where pc] is 1 atm, the pressure applied to liquid water by air. To simplify consideration of water, remove all air from the enclosure and place the enclosure in a micro gravity field at temperature e
Abstract published in Advance ACS Abstracts, March 15, 1994.
0022-3654/94/2098-4196$04.50/0
T. Now the vapor pressure of the pure liquid water ispeg*(T,pi*), where the pressure applied to the liquid water is only its own vapor pressure, Le., pc]* = peg*; the asterisks indicate that the liquid water and vapor are pure. In a micro gravity field, the molar free energies of liquid and vapor water are the same throughout; the liquid and vapor have no weight and have the same potential energy in both phases. At T = 20 OC,pCg*(T,peg*) = 17.542 Torr. The melting temperature of a tiny crystal of ice in pure liquid water in the enclosure is 0.007 OC. As solutes dissolve in water, the vapor pressure of the liquid in equilibrium with its own vapor will diminish. The melting temperature of ice in equilibrium with the water will also decrease as solutes dissolve in the water. On the other hand, the boiling temperature of the water exposed to air at 1 atm will increase. Other chemical, physical, and thermodynamical properties of pure liquid water will change as solutes dissolve in it. These include: (1) the molar enthalpy of vaporization, Le., the amount of heat required to evaporate 1 mol (molecular weight = 18.0152 g) of water at T; (2) the molar volume of liquid, Le., the volume of 1 mol; (3) the molar internal energy of liquid water; and (4) the molar free energy of liquid water, also designated as its chemical potential, p ~ ~ o The l . question implied in the title can be restated: Can we recognize a single change in liquid water which will account for all the changes that take place when solutes dissolve in it? 11. Hulett’s Thought Experiment
During the last years of the nineteenth century, George Hulett, an American, was a graduate student in the laboratory of Wilhelm Ostwald. At that time, Ostwald (and also his contemporary, Ernst Mach) held that matter was continuous, and they did not believe in the atomic and molecular nature of matter. They had little regard for kinetic theory, which they viewed as inferior to thermodynamics as a science for understanding behavior of gaseous, liquid and solid matter. Nevertheless, in the year 1903, Hulett2 published an article in Zeitschrift f u r physikalische Chemie for which Ostwald was one of its editors. In this article, 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 15, 1994 4197
How Solutes Alter Water
r I.
.
0 -
.
I
B
A
I
I5
I-
I I
--: -: -;
C
I E
Figure 1. Modified version of Hulett's original thought experiment is conducted in a micro gravity field (is., g = 0) and at T = 293.15 K. This experiment and Hulett's original experiment demonstrate that solutes alter solvent in a solution as pure solvent is altered by a pressure applied to it that is less than the pressure applied to the solution by the osmotic pressure of the solvent at the free surface of the solution at the same temperature. In part A, pure liquid water and its vapor are enclosed in a container with no other gas. The external pressure applied to the liquid water is its vapor pressure at T = 293.15 OK, namely, p i * = peg* = 0.002 338 8 J mol-' = 17.542Toi-r. Themolar volumeofthewatervapor is VH,og* = 1.04088 X lo6 cm3 mol-l, and its molar enthalpy is HH,og* = 2537.3 X 18.0152 = 45708.2 J mol-I. The molar enthalpy of the pure liquid water is HH,o~* = 83.8343 X 18.0152 = 1510.29 J mol-I, its molar = 18.0479 cm3 mol-I, and the free space available for volume is VH,O~* = 3.297 96 the motion of the centers of mass of its molecules is Vav~,ol* cm3 mol-'. In part B, a 1 m aqueous solution consisting of 1 mol of D-ghCOSe in 1000 g of water is in a cylinder enclosedin another container. The osmotic pressure of its water is T H , O ~ = 24.3306 bar. The vapor pressure of the water in the solution and the pressure applied to the liquid have been diminished a little t o p i =peg = 0.002 297 0 J mol-' = 17.2287 Torr. The molar volume of the water vapor is now a little more, V H , ~ = 1.05987 X lo6 cm3 mol-I. In part C , vapor was removed from the container until the resultant tension applied to the liquid water lowered its vapor pressure topeg = 0.002 297 0 J mol-' = 17.2287 Torr, the same as in part B. The molar volume of its water vapor is VH,O~* = 1.059 87 X IO6 cm3 mol-', and its molar enthalpy is HH,Og* = 2537.3 X 18.0152 = 45708.2 J mol-I, as in part B. The pressure applied to the liquid water is now p i = - 7 ~ ~=0(0.02297 ~ ) - 24.3306) bar, and the molar enthalpy of the pure liquid water is H~,ol*= 81.4891 X 18.0152 = 1468.04 J mol-'; its molar volume and its molar available volume are increased a little to V ~ ~ o=l *18.0682 cm3 mol-' and Vav~,0I*= 3.3107 cm3 mol-I. In parts D and E, the vapor pressures and molar volumes are as in parts B and C. (Parts A to E are modified versions of Figure A to E from Hammelg).
Hulett describes a gedanken (thought) experiment which answers the question we have asked. However, Hulett's article was neglected and soon forgotten. 111. Hulett's Thought Experiment in Orbiting Space Craft
Let us revive the essential idea expressed in Hulett's original experiment. However, let us conduct his thought experiment in a micro gravity field. The container described above is depicted in Figure 1A in which free fall reduces acceleration due to earth's gravity, g, to 0. One kilogram of pure liquid water is enclosed in the cylinder in the container at 20 OC, Le., at 293.15 K. It is also in equilibrium with its water vapor a t 17.542 Torr. The
molar volumes of water vapor and liquid are shown in Figure 1A as well as their molar enthalpies at 20 O C . In the same micro gravity field, 1 mol of D-glucose (180.16 g Of C6H1206) is dissolved in 1000 g (55.5087 mol) of liquid water. This 1 m solution is placed in an open cylinder and is enclosed in another container with no other molecules in it, Figure 1B. Only water is in the vapor phase at 0.022 969 6 bar (17.2287 Torr) a t 20 O C . The mole fraction of liquid water in the solution is XH,O' = 55.5087/56.5087 = 0.982 303. Its osmotic pressure is?rH2~I= 24.0124atm,24.3306 bar,or2.433 06 J~m-~,according to the conventional formulation (cf. eq 16). The anhydrous percentagesoluteis 15.2657%. Notethat the water vapor pressure in Figure 1B is slightly less and the molar volume of vapor is slightly greater than these values in Figure 1A. Hulett revealed why this must be. We can easily understand Hulett's explanation by considering Figure 1C and 1D. One kilogram of pure liquid water a t 20 OC is placed in a cylinder in a third container. The pure liquid in Figure 1C differs from the pure liquid in Figure 1A in that each end boundary is held by a rigid, porous matrix. As water vapor was removed from the container, the vapor pressure was diminished and some liquid water evaporated. As water evaporated from the cylinder, tension was applied to the remaining water since its boundaries were fixed. The resulting tension applied to liquid water lowers its vapor pressure. This process continued until the vapor pressure in Figure 1C became equal to the vapor pressure in Figure 1B. This applied tension (negative pressure applied to the liquid), which equates the vapor pressures in Figures 1C and lB, is 24.3306 bar, that is, exactly the osmotic pressure of the water in the solution in Figure 1B. Note that not only is the vapor pressure in Figure 1C less than it is in Figure 1A, due to the tension applied to the pure liquid water in Figure lC, but that the molar volumes of the liquid and vapor have increased slightly, the molar available volume of the liquid ( V a V ~ , o 1defined *, below) has also increased, and the molar enthalpy of the liquid water has diminished slightly, all because the applied tension is greater. In A and C of Figure 1,water molecules are distributed between the vapor and liquid phases in accord with Boltzmann's energy distribution principle.) Boltzmann's principle applies to any situation where molecules may be in two energy states, the molecules can reversibly pass between the two energy states, and they are in equilibrium. When molecules of liquid and vapor are in equilibrium, as in Figure 1A, the ratio of the number of moles of vapor per unit of available space in the vapor phase to the number of moles of liquid per unit of available space in the liquid phase is given by
where RT is the product of Boltzmann's constant, Avagadro's constant, and the absolute temperature and where UH,O~* and V~,ol*are the molar internal energies of water vapor and liquid water, respectively. nH20g*is the moles of water vapor per unit volume of available space, and its reciprocal is the molar volume available for the motion of centers of mass of water vapor molecules, V a v ~ & * .Likewise, nH,ol*is the moles of liquid water per unit volume of available space, and its reciprocal is the molar volume available for the motion of centers of mass of liquid water mOleCUleS, VaVav~,o1*. Thus, nH20g*/nH201* = VavH201*/V a v ~ , o g * , which equals V a v ~ 2 0 1 * / [ V ~ zbH208*], ~ g * where V H ~ Ois~the * molar volume of water vapor and bH,og* is the second constant in the van der Waals equation of state. Using Boltzmann's energy distribution principle, the molar volume available for the motion of centers of mass of molecules in 1 mol of liquid water, V a v ~ , ~ l * , was calculated.3 Comparing A and C of Figure 1 shows that VaV~,0I* increased by 0.358% whereas VH,O~* increased by only 0.1 12%.
Hammel
4198 The Journal of Physical Chemistry, Vol. 98, No. 15, 1994
With this situation in mind, let us place the cylinders in Figures
1B and 1C in a fourth container in a micro gravity field, as in Figure 1D. The solutions in the cylinders in Figure 1B and 1D are identical, and the pure liquid water in the cylinders in Figure 1C and 1D is subjected to the same applied negative pressure (i.e., tension), namely, the water vapor pressure minus the osmotic pressure of the water in the solution. This negative pressure is (0.0229696- 24.3306) or -24.3076 bar. The vapor pressure in the container in Figure 1D is 17.2287 Torr as in Figure 1B and 1C. This is the only vapor pressure that can be in equilibrium with both the water in the solution and with water in which its applied pressure was decreased by 24.3310bar. Suppose in Figure 1D that a small amount of pure liquid water was removed from the cylinder containing pure water and then added to the solution. Tension applied to the remaining pure water would increase and, thereby, lower its vapor pressure. Also, the osmotic pressure of the water in the solution would decrease and, thereby, increase the vapor pressure of its water. Liquid water would then evaporate from the solution and vapor would condense into the pure water under tension until both vapor pressures became equal as before, Le., the condition described in Figure 1D. Two important conclusions can be drawn from this thought experiment. First, the water in the solution at 20 OC (or any temperature at which the water is liquid) is altered by the solute exactly as pure liquid is altered by an applied pressure equal to the negative of the osmotic pressure of the water in the solution at the same temperature. This is Hulett’s conclusion. Hulett did not indicate how the solute did it; he only revealed that this effect of the solute on the water accounts exactly for the lower chemical potentialof the water in the solution as well as its altered colligative properties, including the osmotic pressure of the water in the solution. The second conclusion is as follows: whatever the solute does to the water in an aqueous solution, it has already done to the water in solution in Figure l B , lD,and 1E. Let us consider a final container a t 20 OC,Figure lE,enclosing the same cylinders as in Figure 1D. Note that the net pressure on the walls of the cylinders is the same in Figure 1E as in Figure lD,Le., 17.2287 - 17.2287 = 0 Torr in the cylinder containing solution and -18249.4Torr in the cylinder containing pure water under tension. When these cylinders were juxtaposed initially, the common wall between the two cylinders was intact. A portion of this wall was then perforated with small openings which allowed only water to pass between the two cylinders. In this experiment, the water in the two cylinders is separated by vapor as before. However, the liquid water in the two cylinders is also coupled directly through a semipermeable membrane. In this case, there will be no net flow of water through the membrane; the liquid waters in the two cylinders have the same chemical potential. It is exactly as if the solute exerted an additional internal pressure of 24.3306 bar on the water in the solution, and this renders its internal tension equal to the internal tension in the pure water within the other cylinder, so there can be no net movement of water through the membrane. If for any reason some liquid water were to pass through the membrane, that same amount would return to its source by way of the vapor phase, if not directly back through the membrane. Note that the solute has already affected the water in the solution (as if the solute altered the solvent in the same way and thereby lowered its vapor pressure). Thus, when the water in the solution is coupled with the pure water under applied tension, there is no net flow of water. Hulett’s concept is that solvent in a liquid solution is altered as is pure liquid solvent at the same T when the pressure applied to the pure solvent is less than the pressure applied to the solution by an amount equal to the osmotic pressure, ?rmlvenJ,of the solvent a t the free surface of the solution. From this insight Hulett concluded that the solute alters the solvent in the same way that a negative pressure of -7rmIven) alters pure solvent. Hulett’s original
h m e
e m
e e
Z
0
Figure 2. Hulett’s original thought experiment with three columns above z = 0. Column I is the column of water vapor above the liquid water for which the vapor density decreases exponentially in acceptance with the law of the atmosphere. Column I1 is the liquid water supported by a matrix at z = h in which its vapor pressure decreases exponentially due to ita weight between 0 and h. Column 111is the solution between 0 and h supported at h by the solutes at the surface of the solution.
thought experiment was perceived to be made in the earth’s gravity field. He compared a free standing column of solution and a column of pure solvent held by a rigid, porous matrix at the same height, h, as the free surface of the solution in the gravity field, Figure 2. Both columns were continuous with pure solvent at their lower ends where the solution was separated from pure solvent by a membrane permeable only to the solvent. Again, Hulett did not indicate a mechanism by which the solute altered the solvent water in an aqueous solution such that its vapor pressure at its free surface equaled the vapor pressure of the pure water at its upper surface, where the pressure applied to it was negative an amount equal to the osmotic pressure of the water in the solution. H e did indicate that this alteration of the solution water lowered its chemical potential, lowered the vapor pressure of the water, raised its boiling temperature, and lowered the melting temperature of a small ice crystal surrounded by the solution at its surface. As we shall see, Hulett’s theory of osmotic pressure is compatible with a kinetic theory for calculating the magnitude of the osmotic pressure, Le., T H ~ O I when ~ B mol I of solute B are dissolved in n~,01mol of water. Another insight gleaned from Hulett’s original experiment further validates his conclusion. A column of water vapor stands above the pure liquid water. At its lowest level, z = 0, the vapor pressure is determined by T and by this vapor pressure applied to the water at that level. However, thevapor pressure diminishes exponentially above this level in a gravity field in accord with Boltzmann’s energy distribution principle (also known as the law of the atmosphere when used in this context). Thus, at z = h, the vapor pressure is less than that at z = 0, according to Boltzmann’s principle. With increasing height, the vapor pressure continues to diminish exponentially. At the top of the column of liquid water, at z = h, the liquid water has tension appliedLo it due to the weight of the water between h and 0. This applied negative pressure lowers the vapor pressure of the liquid water at h to equal exactly the vapor pressure at h. If these vapor pressures were not equal, there would be perpetual flow of water molecules, a situation forbidden by the second law of thermodynamics. Likewise, thevapor pressureof the water in thesolution
How Solutes Alter Water at its free surface at h must equal exactly the vapor pressure of pure liquid water under the applied tension at h and must equal the vapor pressure at h above the pure water at 0 level. Hulett's conclusion is inescapable: the solute in the solution exerts a pressure on the water in the solution in such a way as to render its internal tension the same as the internal tension in the pure liquid water column at all z's from 0 to h. The tension applied to the pure water at h is, of course, the osmotic pressure of the water in the solution at its free surface at h. Moreover, the vapor pressures above each column continue to diminish exponentially as does the vapor pressure above h and above pure liquid water at z = 0.4 Solute molecules also exert a pressure on the membrane in Figure 2 as they are reflected by and change momentum a t the membrane (cf. Fermis). This pressure, which is proportional to the solute concentration at the membrane, may differ from ,the osmotic pressureofthewater in thesolution. Theosmotic pressure is proportional to the solute concentration a t the free surface. In a gravity field, sedimentation of solute may occur in the solution so that the concentrations ofsolute at the freesurface and at the membrane differ, causing membrane pressure and osmotic pressure of the water to differ. Again, Hulett's insight makes it clear that solute molecules do nothing to the water at the membrane to affect the osmotic pressure of the water in the solution.
The Journal of Physical Chemistry, Vol. 98, No. 15, 1994 4199
Equation 3 is difficult toevaluateexactly because, like all solvents, water is compressible, meaning that VH,O~* is a complex function of pressure and temperature. Nevertheless, the chemical potential of the pure water in Figure 1C is less than the chemical potential of the pure water in Figure 1A by approximately VH,O~*XH,O~. T H , O ~is also not known exactly as there is no exact equation for assessing the osmotic pressure of water in an aqueous solution when only the number of moles of solute and the mass of pure liquid water are known. The value of A H ~ Oindicated ~ in Figure 1B is only approximate and was calculated by a conventional, empirical and approximate equation (eq 16). Another assessment of p&*( T,pe]*-a~,o')- p ~ , o I * (T,p:*) can be made because the difference between chemical potentials of the pure vapor in Figure 1C and 1A is exactly the same value, i .e., pH,O1* ( T,pi*-rH,Ol) -PHlo'* ( T@,'*)= kH,Og* ( T@* ( T$,'*1~~~01) - pHpog*(T,peg*(T,pJ*)). Since the temperatures are constant and the same in Figure 1C and lA, the integral of eq 2 becomes
IV. Why Hulett's Theory W a s Abandoned Anyone familiar with classical thermodynamics can appreciate its beauty. Although not mechanistic, thermodynamics is an elegant science accounting for energy changes due to changes in temperature, pressure, and composition. It is an appropriate science for describing the formation of solutions even though its description alone is banal, as we shall see. Consider, first, the effects of temperature and pressure on pure water. A very small change in temperature, dT, and a very small change in the external applied pressure, dp:*, changes the chemical potential (molar free energy) of liquid water by an amount, dpH,ol*( T,pL*), according to a simple differential equation,
where S~,ol*(T,p,]*) is the molar entropy of pure liquid water and where V~,ol*( T,p!*) is its molar volume. Both are functions of temperature and pressure applied to the water. Likewise, a very small change in temperature, dT, and a very small change in external pressure applied to pure water vapor, dp,g*, change the chemical potential of water vapor by a small amount, dpH,og*( T,pcg*),according to the differential equation
where SHzOg*(T,peg*) is the molar entropy of pure water vapor and where vH,og*(T,p,g*) is its molar volume. Both are functions of temperature and pressure applied to the water vapor. In the application of these equations that concerns us, the temperature is held constant and the pressure applied to the liquid water is decreased from p:* top:* - TH,OI. In this case, eq 1 can be integrated
so that the difference between the chemical potentials of the pure
Thus, when the vapor pressures of pure water in Figure 1C and 1A can be measured accurately (and if the gas law for water vapor is known accurately), then the chemical potential difference of the pure water vapor and of the pure liquid in Figure 1C and 1A can be accurately assessed. Likewise, the difference in chemical potentials between the water in the solution in Figure 1B and the pure water in Figure 1A can be assessed with equal accuracy, Le.:
In eq 5 , peg*(T,p,g*-rH,OI) also equals peg( T,peg,n~,ol,ng1); i.e., vapor pressures of the pure water under applied tension and the water in solution in Figure 1D and 1E are identical. If a small amount of solute B, dnel, is added to n~,oImol of water and ne1mol of solute, then the mole fraction of water, XH,O~ = ~ H , o ~ / ( ~+ H ,ne1), o ~ decreases a small amount, dXH,OI = -xH201dnB1,and the chemical potential of the water in the solution has been diminished a small amount. The thermodynamic statement for the change in chemical potential of water in solution caused by the changes dT, dp:, and dxH,Ol becomes
where SH,OI( T,p,I,x~,o~)and VH,OI(T,p,I,x~,o~) are the partial molar entropy and partial molar volume of the water in the solution, respectively. The last term in this statement accounts for an addition of a small amount of solute. It states that the change in the chemical potential of water (keeping applied temperature and pressure constant) is simply the change in chemical potential of water resulting from a change in the mole fraction of water multiplied by the change in the mole fraction of water. This is, indeed, a banal statement; it would be valid whatever meaning was assigned to the term ~ ~ ~ This 0 1statement .
4200 The Journal of Physical Chemistry, Vol. 98, No. 15, 1994 and thermodynamics do not and cannot reveal the physical nature of the effect of the solute on the solvent water. For this reason, G. N. Lewisl*6proposeda scheme which became widely accepted. His scheme also resulted in neglect of Hulett's theory of the altered state of water in a solution. Lewis defined an ideal solution as one for which its solvent vapor has two properties: (1) The solvent vapor of the ideal solution obeys the ideal gas law, namely, p V = nRT, where p , V, and T are the pressure, volume, and temperature of the gas; n is the moles of gas; and R is the gas constant. In other words, atoms or molecules of the ideal gas have no hard core volume; Le., their centers of mass can move in all the space within the volume V in which they are constrained. Moreover, there is zero force of attraction between ideal atoms or molecules. Clearly, for both reasons, no real gas is ideal. (2) The vapor of an ideal solvent obeys Raoult's law, namely, the vapor pressure of the solution solvent divided by the vapor pressure of pure liquid solvent at the same temperature and applied pressure equals the mole fraction of the solvent in the solution. For example, if Raoult's law applied to water in an aqueous solution, then
However, neither the vapor of water nor the vapor of any solvent obeys Raoult's law; Le., the vapor pressure of solvent in solution divided by the vapor pressure of pure solvent a t the same Tis only approximately equal to xsolvcn) and never equals the mole fraction exactly. Since neither the vapor of water nor the vapor of any solvent obeys either the ideal gas law or Raoult's law exactly, eq 5 is not easily solvable and is solvable only when the gas law for thevapor is known exactly. Thus, even when the vapor pressures of the solution solvent and pure liquid solvent are known exactly at the same T, the extent to which the chemical potential of water in an aqueous solution differs from the chemical potential of pure water at the same temperature cannot be determined easily by an integration of eq 5. For the ideal solvent in an ideal solution and its ideal vapor,
so that the integral in eq 5 becomes
Hammel
The fugacity of pure liquid solvent is identical with the vapor pressure of pure liquid solvent only a t the limit where the latter goes to zero, i.e.
lim peg*
RT In
Xlideal solvent
Therefore, the chemical potential of the ideal solvent in solution differs from the chemical potential of pure ideal solvent by an amount
In order to relate the chemical potential of a real solvent in a real solution to the chemical potential of its pure solvent, Lewis assigned a non-real property to the vapor of the solvent which he called fugacity, fc'( T,peg,xmlvcnJ), and he defined this fictional property by an equation similar in form to eq 6
0
Thus, the fugacities of pure solvent and solution solvent vapors differ from the vapor pressures of these solvent vapors. To appease or distract the mind of a thoughtful student, authors of textbooks of chemical thermodynamics will often describe fugacity as the tendency for vapor molecules to escape the liquid solvent when pure or in the solution. Of course, a t a given temperature, the equilibrium vapor pressure of the solvent is an exact measure of the tendency of the molecules to escape. The discerning student will wonder, no doubt, why the tendency to escape, fcg( T,peg), should differ from an exact measure of the tendency to escape, peg(T,peB). Some authors state that the fugacity is a "corrected" pressure of the solvent vapor. In fact, vapor pressure needs no correction. It is a physical and measurable property of the vapor; the term fugacity, for the sake of expediency, obfuscates the meaning of vapor pressure, the real property. Next, Lewis assigned another fictional property to the solvent in a solution, namely, solvent activity, and he defined this as
With eq 8, eq 7 becomes
Noting that eqs 6 and 9 are similar in form, Lewis defined another term for the solution solvent, solvent activity coefficient, rsolvenJ(T,peS,xwlvcn)), relating solvent activity and mole fraction of solvent in the solution, namely, a'soIvent(T,pt,X'solvent)
or, according to Raoult's law,
-
E
Y 1solvent(T*peB,x lsolvcnt)Xlsolvent
( 10)
and eq 10 renders eq 9 even more similar to eq 6. For the real solvent in a real solution, its chemical potential and the chemical potential of its pure liquid solvent a t the same T differ by I solvent(T,peB,X'solvent)
- I.CI* solvent(T,pt*) = RT 1' ylsolvcnt(T,Pt,X'wlvent)X'solvent ( 111
As indicated by eq 10 for water, YH$~(T,~~~,XH~OI) is a function of temperature, pressure, and mole fraction of water in the solution, as is aH201( T,pe',XHlO'). Therefore, in order to assess the chemical potential of water in a solution, its activity coefficient must be assessed at the T, p2, and xHpl appropriate for that solution. It is surprising that eqs 9 and 11, derived by Lewis,' are considered to be fundamental thermodynamic relationships in many standard textbooks of physical chemistry and chemical thermodynamics. Equation 11 is merely a mathematical device (a gimmick) to calculate changes in chemical potential of solvent in a solution. Equation 9 is not a fundamental relationship even though it and Lewis's concept of solvent activity have become the
How Solutes Alter Water accepted paradigm for describing the altered state of water in an aqueous solution. It is more surprising that Hulett’s theory2 of the altered state of water in an aqueous solution, published in 1903, was overwhelmed by Lewis’s theory, published in the year 1908.’ An understanding of the fundamental nature of osmosis has been obscured by this unfortunate twist of history. Physical chemists and chemical thermodynamicists continue to ignore and even ridicule Hulett’s theory when it is brought to their attention.
The Journal of Physical Chemistry, Vol. 98, No. 15, 1994 4201
Mechanisms of Osmosis
V. Does It Make a Difference Which Theory Is Promulgated?
Solutes dissolved in a solvent lower the chemical potential of the solvent in the solution relative to the chemical potential of the pure solvent. Hulett correctly recognized that this diminished chemical potential must be attributed to the solutes exerting an internal pressure on the solvent and altering its internal tension. Hulett gave no explanation as to how the solute does this to the solvent. We may speculate that he could not have given an explanation, as that would have required reference to the kinetic theory of matter, a theory that was not favored by his mentor and editor, Wilhelm Ostwald. About the same time, Lewis proposed an alternate treatment of the solvent in a solution. Lewis also did not indicate how solutes lower the fugacity and activity of the solvent in a solution. He also would have had to invoke kinetic theory to provide an explanation of his proposal. Unfortunately, Lewis’s theory came to dominate the treatment of osmosis and the altered state of the solvent in a solution. More unfortunately, subsequent students of the subject began to speculate about an explanation based on Lewis’s teaching. One popular explanation (among biologists) is that solutes lower the concentration of the solvent in the solution. The solvent is perceived as diffusing from pure solvent, where its concentration is alleged to be high, into solution, where it is alleged to be lower. This process is perceived as continuing until the pressure in the solution solvent increases to exceed the pure solvent pressure by the osmotic pressure. Moreover, this perception associates the flow of solvent into the solution with the concentration of the solvent at thesemipermeable membrane through which the solvent flows. These perceptions are diametrically opposed to Hulett’s theory, which requires that internal solvent tensions equalize a t the membrane when the solution solvent and pure solvent are in equilibrium. Moreover, Hulett’s theory equates the osmotic pressure of the solvent in the solution to the pressure exerted by the solutes a t the free surface of the solution, and this pressure relates to the concentrations of the solutes and solvent at the free surface, not at the membrane. Clearly, a reconciliation of Lewis’s theory and Hulett’s theory is not possible, as pointed out by Mysels.’ More arguments opposing the view that solvent concentration is the sole cause of osmosis are in the Appendix. Figure 3 illustrates the features of the differing theories when two circumstances prevail: (on the left) when an aqueous solution and pure liquid water are separated initially by a semipermeable membrane and (on the right) after water has entered the solution and solution water and pure liquid water are in equilibrium. Initially, the conventional view is that water pressures are the same on both sides of the membrane and water diffuses into the solution because water activity (and concentration) is less in the solution. A small pressure gradient moves water through the pores of the membrane. On the other hand, the Hulett-Dixon view (cf. Dixon’s theory below) is that water molecules in the solution pull pure water molecules (with which they are coupled) through the pores down a steep pressure gradient and into the solution as would pure liquid water to which was applied a negative pressure, -?TH~o~. In equilibrium, the conventional view is that the water pressure on the solution side of the membrane rises abruptly to the osmotic pressure in the solution. This alleged pressure increase raises the chemical potential of the water to equal that of the pure water, so water no longer flows across the
FINAL
INITIAL
Pressure Profiles Across Membrane
li Conventional View
d
-
Hulett Dixon View
p’
lF(h I
- z)
d
Figure 3. Comparing the differencesbetween the conventional (Lewis) theory and Hulett’s (and Dixon’s) theory.
membrane. On the other hand, the Hulett-Dixon view is that water no longer flows across the membrane because there is no longer an internal tension difference between pure water and solution water. Again, it is clear that Hulett’s theory is not compatible with any theory that attributes movement of solvent across the membrane to concentrations of solute or solvent at the membrane or to solvent activity at the membrane. An account of the pressures within the cytoplasm of leaf cells depends on to which theory the author subscribes. Authors of modern textbooks of plant physiology will universally attribute turgor pressure within the cell to a positive pressure in the cytoplasmic water. They account for this by stating that solutes in the cytoplasm lower the activity (concentration) of its water which, in turn, induces (attracts or draws) extracellular water to diffuse through the plasma membrane into the cytoplasm. Water enters and is alleged to enter until its pressure in the cell builds up sufficiently to oppose this tendency to enter the cell. The extracellular water (which is in xylem vessels and cell wall matrix and is nearly free of solutes) is in tension due to gravity and to flow resistance. Usually, the osmotic pressure of the cytoplasm (measured after removal from the cell) is greater than the external water tension; water enters and osmotic pressure in excess of the opposing external tension becomes the turgor pressure within the cell. Turgor pressure is universally attributed to the pressure in the cell water in excess of the pressure in extracellular water. This account is contrary to Hullet’s theory, and it is also not the account offered in 1914 by the pre-eminent plant physiologist of his time, Henry Dixon, professor of botany, Trinity College, Dublin, and author of the widely accepted cohesion theory. Dixon’s theory of solute pressure is in his classic monograph entitled Transpiration and the Ascent of Sap in Plants.8 Dixon attributed the internal pressure on the membrane to solutes reflected by the membrane. The membrane, supported by the matrix of the cell wall, opposes all this solute pressure. Should the solute pressure exceed the tension in the extracellular water, the excess solute
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The Journal of Physical Chemistry, Vol. 98, No. 15. 1994
pressure distends the membrane and its supporting cell wall as water enters the cell through the membrane. Water enters until the internal water tensions inside and outside the membrane are equal. The cell wall opposes the excess solute pressure, and this excess solutepressure is the turgor pressure within the cell. Internal water tension in the cell cannot differ from internal water tension outside when there is no flow. Dixon's account of these pressures is in accord with Hulett's theory and is contrary to a common interpretation of Lewis' theory. Dixon does not cite either theory in his monograph, indicating that his understanding of the role of solutes in producing turgor pressure was independently perceived. The answer to our question posed for this section is that it makes an important differenceas to which theory is promulgated. VI. Kinetic Theory Applied to Hulett's Concept of Osmosis
The combination of solute and water molecules in an aqueous solution exerts an internal force on a unit area of a free surface of the solution at (T,p,l) so as to distend the water in the solution. To be in accord with Hulett's theory, the water in the solution must bedistended as much as pure water is distended by a pressure of (pel- T H , O ~ )applied to it a t the same T. The internal pressure, which distends the water in the solution, pH201(T,p~,nH,ol,nBl), must exceed the internal pressure which distends pure liquid water, ~ H , o ' *T,peI-~~,ol), ( by an amount TH,O~( T,p,'). Equilibrium between water in the solution and pure water is assured when
The total internal force on a unit area of free surface of pure liquid water at (T,p,l) is (cf. Hamme13,9)
where VaVH,ol*(T,p,l) is the molar volume available for motion of the centers of mass of water molecules at (T,pi). This total internal force per unit area distends the pure liquid water. The fraction of the free surface of pure liquid water which is water is
solute molecules is the volume of the solution minus the hard core volume of the solute in the solution, i.e.
where v( T,pJ,nH201,nBl) = nHzOIvH201(T#L~nH2Ol,nBl)+ nBIVB1(T,pi,?IH201,nBl),VhcBl is the partial molar hard core volume of solute B in the solution, and VH,O~and Vel are the partial molar volumes of the water and the solute in the solution, respectively. We shall designate the total volume available for motion of the centers of mass of solute molecules B as VavB( T,p,l,nHl0',nB1)so that VaVB( T,p,',nH,ol,nBl) v( T,pi,nH201,nB') - nBIVhcBI' (T,p,],nH,ol,nB1). The internal force per unit area of solution surface due to reflection of solute molecules is nBIRT/vavB(T,pJ,nHpl,n& whenall are reflected. Likewise, the total volume available for the motion of the centers of mass of water molecules, V'avHzO( T,p~,n~~o',ngl), is the volume of the solution minus the hard core volume of the water in the solution. That is
where &HIOl is the partial molar hard core volume of water in the solution. The internal force per unit area of solution surface due to reflection of water molecules is nH,olRT/ VavH20( T,pi,n~~~l,ng~) when all are reflected. Thus, the internal pressure due to the reflection of all molecules is the sum of these pressures, namely
When all solute and water molecules at the boundary are reflected and no molecules escape., the internal pressure distending a unit area ofwater a t a free surface of thesolution,pH,o'( T,pJ,nH2o1,nBl), is $( T,p,',n~~ol,n~l) divided by the fraction of water in the unit area of solution surface, so that
so the internal force on a unit surface area distending pure water is
If the pressure applied to the pure water is decreased an amount lrH201, then the internal force on a unit area distending pure water is
where V a B v ~ , ~TI,*p(e 1 - ~ ~is, ~the I ) molar volume available to the centers of mass of the water molecules at (T,pL-?r~,o~). The total internal force exerted on a unit area of free surface of solution, pl(T,pel,nHz0l,ngl), is the internal force due to the reflection of solute molecules plus the internal force due to the reflection of water molecules per unit area of solution surface. The available volume for the motion of the centers of mass of
Thus
The difference between eqs 14 and 13 is, according to eq 12, the osmotic pressure of water in the solution a t (T,pL),
How Solutes Alter Water
The Journal of Physical Chemistry, Vol. 98, No. 15, 1994 4203 pressure of the water in a solution. As Figure ID,E reveals, this vapor pressure for the solution is identical with the vapor pressure of pure liquid water to which a negative pressure -THJ has been applied. That is, peg(T,peg,n~201,n~1) = peg*( T,p,g*?TH~OI( T,peg,n~2~1,n~l)). A table ofpeg*(T,peg**ApL*) versus Api* can be generated experimentally by direct measurement of peg*(T,pcg*&Ap2*)for each Ape]* from 0 to 100 bar applied to pure liquid water at, say, 45 OC. From these data, a polynomial regression of ~~g*(45~C,p~g*fApe]*) as a function of ApJ* can be obtained. Thus, when the vapor pressure of the water in a solution is measured a t 45 OC,the pressure applied to pure liquid water a t 45 O C which yields this vapor pressure can be obtained from the polynomial regression. M. Conclusion
The complexities of the volumes not available for the motion of the centers of mass of the solvent and of the solute molecules (their hard corevolumes) as a function of temperature and pressure render any simple formula for the osmotic pressure of the solvent an approximation. Although the phenomenon of osmosis may be easily understood, its evaluation cannot be simplified and remains rigorous. Disregarding the difficulties of calculating or directly measuring the osmotic pressure of water in an aqueous solution, Note also that solute molecules reflected by the semipermeable there is no doubt that an understanding of the effects of solute membrane in Figure 1E exert a pressure against the membrane on the water were foreseen most clearly by George Hulett. which is p,, = nelRT/ [ V(T,pL,n~p',neI) - ~B'VIIB~(T@!,~H~~'J~B')~. Hulett's theory deserves to be restored to its rightful status in all When ne1
