J. Phys. Chem. 1995,99, 8392-8400
8392
Boltzmann's Principle Depicts Distribution of Water Molecules between Vapor and Liquid for Pure Liquid and for Aqueous Solutions H. T. Hammel Department of Physiology and Biophysics, Medical Sciences Program, Myers Hall 263, Indiana University, Bloomington, Indiana 47405 Received: October 7, 1994; In Final Form: March 2, 1995@
Boltzmann's energy distribution principle can be applied to liquid water and its vapor. However, for Boltzmann's principle to be successful, the molar hard core volume of water Gc*H20(T,p:*), Le., the volume not available for the motion of the centers of mass of molecules in a mole of water in phase a at temperature T and applied pressure p,"*,must be excluded from Boltzmann's principle as well as from the equations of state of liquid water and its vapor. The equation of state for water in phase a is z$,(T,p;*) = p f - RT/ a* where zH20 is the intemal tension in the cohesive force bonding water molecules in phase a. c;H20(T,p,"*), For pure liquid water, its molar volume is designated ?;,,(T,pf) so that its molar volume available for the motion of the centers of mass of its molecules is ?g~H2,(Tp'*) = V';,,(T,pr) - ~ ~ c H 2 0 ( T , pAt~ 20 ) . "C and 'e, p p = 0.023 388 bar, the vapor pressure of water at 20 "C, ?!,H,,(293.15 K, 0.023 388 bar) = 3.329 796 cm3 mol-' so that r:;,(293.15 K, 0.023 388 bar) = -7390.51 bar. At 20 "C and p f = [0.022 97 - 24.33 061 bar, i.e., reducing the pressure applied to pure liquid water by an amount equal to the osmotic pressure of water in a solution of 1 mol of solute in 1 kg of water, its vapor pressure is reduced to 0.022 969 7 bar, the same as the vapor pressure of water in the solution. At 20 "C and -24.3076 bar, ?a:H2,(293.15 K, -24.3076 bar) = 3.310 72 cm3 mol-' so that 2],*,,(293.15 K, -24.3076 bar) = -7386.36 bar. The applied negative pressure of -24.3076 bar lessens the intemal tension by 4.15 bar. Always, when a negative pressure (tension) is applied to liquid water, it lessens (not enhances) the intemal tension in the water because the increase in the molar volume of available space lessens the internal tension more than the applied tension increases it. These values exemplify successful applications of Boltzmann's energy distribution principle and of the equation of state to pure water. The same treatment applies to water in an aqueous solution. As a result, Boltzmann's principle yields the same equation for the osmotic pressure of water in a solution, dH2,, as obtained from-a kinetic treatment of Hulett's theory of osmosis.
I. Introduction In a recent article Hammel' applied Boltzmann's energy distribution principle to phase changes of pure species i when the molecules of i in two phases were in equilibrium at the same temperature and applied pressure. For example, when the vapor and liquid of species i are in equilibrium at temperature T and when the pressure applied to the vapor of i at Tis p:*(T),then the molecules of i are distributed between the two phases according to Boltzmann's principle, namely,
multiplied by T); it must include the pressure x volume work performed against the applied, extemal pressure pf*(T) as the liquid molecules increase in volume; and it must also include the pressure x volume work due to the fact that the internal tension between molecules of liquid greatly exceeds the internal tension between molecules of vapor. Internal tension is a negative pressure and, in phase a, it is designated zu* for pure species i. In any phase it combines with the intemal thermal pressure pa* exerted by molecules of i so as to equal the externally applied pressure p:*. In the vapor phase, rf*is small when T is much less than the critical temperature and, according to van der Waals' equation, is -a?*/[F*]* so that the difference between the pressure applied by the wall of the vessel containing the molecules and the thermal pressure exerted by the vapor molecules as they are reflected at their boundary is small. In the vapor phase, = pf* - pf* or pf* = pf* a f * / [ T * l 2 .In the liquid phase, the thermal pressure of its molecules is very large whereas the applied pressure may be zero or it may be the equilibrium vapor pressure, in which case, .f* = p',* - pf*, where peI* - pee, When a mole of pure liquid evaporates at T and applied pressure pf*(T'), the product of its internal tension and molar volume, becomes the product of the intemal tension and molar volume of its vapor, zf*v*. The work performed by the system during vaporization is the negative of the difference, that is, -[rf*v*or - p t * [ v * - vf*] p f * v * pf*vf*.At the same time, heat is added to increase the entropy
e*
where q*and vf* are the molar internal energies of pure vapor and liquid i, respectively, and where iip* and iiif are the moles per unit volume in the vapor and liquid phases, respectively. However, Boltzmann's principle applies to these two energy states only if (1) the unit volume in which the molecules are distributed excludes the hard core volume of the *:A mole of molecules i in phase a, Le., the unit volume must be available to the motion of the center of mass of each molecule in the liquid phase as well as in the vapor phase, and ( 2 ) the energy required to evaporate 1 mol of pure liquid i includes the molar enthalpy of vaporization (which at constant temperature and applied pressure is the molar entropy of vaporization @Abstractpublished in Advance ACS Absrracrs, April 15, 1995.
e*
+
.f*t*,
d*r/l*]
0022-365419512099-8392$09.00/0 0 1995 American Chemical Society
+
Boltzmann's Principle and Hulett's Theory Yield Same dH2, of the molecules. Thus, the energy required to evaporate 1mol of liquid i is (L$* = AI$*+&*- pE*[y* pgxy*- pf*q*,where AI$*+&*is the molar enthalpy of vaporization at T and pe*(r), namely,
q*)
q*] +
[ q * ( T , P : * ( m- Ht*(TP:*(T))I =
m;*(TP:*(m
- $*(T,p:*(r))l
In this treatment, it was necessary to exclude the molar hard core volume from the unit volume of molecules in both phases when applying Boltzmann's principle. It was also necessary to exclude the molar hard core volume when assessing the thermal pressure in the vapor phase as well as in the liquid phase. Accordingly, pf* = RT/e:i and p!* = RT/V',*,,, where the molar available space for the motion of the vapor molecules is
