Textbook Errors, 778 Richard C. Parker and David 5. Kristol Nework College of Engineering Newark, New Jersey 07102
Freezing Points, Triple P o ~ ~ s , and Phase Equilibria
There is widespread ambiguity concerning the theoretical definitions and everyday practical applications involving the terms: triple point, melting point, freezing point, and freezing point depression. The misuse of these terms has led to confusion among students and teachers alike. Recently, Katz (I) pointed out the state of confusion in introductory textbooks regarding the use of nonvolatile solutes with respect to these concepts and Swinton (2) indicated the ambiguity surrounding the familiar triple point of water. The concepts of triple point and freezing point are not identical. A melting point or freezing point' is a condition of temperature and pressure a t which a solid and a liquid phase of a system are in equilibrium. A triple point is a similar condition at which any three phases coexist in equilibrium. A triple point of a pure suhstance is the condition of temperature and pressure at which any three phases coexist in equilibrium in a system containing only that pure suhstance, i.e., the system contains one component only. This particular triple point should properly he called a true triple point (3). Much confusion arises hecause we generally think of the particular triple point involving solid, liquid, and vapor in equilibrium (the s-l-v triple point).l This particular triple point is a melting point which has the additional restriction t h a t the solid and liquid are also in equilibrium with the vapor. For example, the most commonly referred to true s-l-v triple point is that for water, a t 4.58 torr and 0.0099"C. Application of Gihbs phase rule shows that a pure snbstance at a true triple point has zero degrees of freedom (f = c -p 2 = 1 - 3 + 2 = 0). Thus, true triple points are characteristic properties of a pure suhstance since they are unique and completely specified by the nature of that substance. They are fixed and cannot be altered by either a change of pressure or temperature. They are as
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Suggestions of material suitable for this column and guest columns suitable for publication directly should be sent with as many details as possible, and particularly with reference to modem textbooks, to W. H. Eberhardt, School of Chemistry, Georgia Institute of Technology, Atlanta, Georgia 30332. Since the purpose of this column is to prevent the spread and continuation of errors and not the evaluation of individual texts, the sources of errors discussed will not he cited. In order to he presented, an error must occur in at least two independent recent standard books. 'The terms freezing point and melting !mint are frequently used interchangeably in elementary discussions. Obviously, the difference between the two is in the direction of approach to equilibrium. For a one-component system, thesetwo points coincide; for more complex systems they generally differ. The freezing point is reasonably welldefined experimentally; the melting point may be very diffuse. Careful studies establish an equilibrium and then determine composition of each phase. a A common misconception is that the triple point must occur only at the s-l-v equilibrium. A true triple point occurs when any three phases of a one-component system are in equilibrium. A suhstance which has more than one solid phase has more than one true triple point. In fact, H20 has seven true triple points. See Ref. (9). 658
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characteristic a property as the molecular weight of a pure substance. Confusion arises from the fact that the addition of a solute can lead to a new three phase equilibrium which is also represented as a triple point. However the triple point of this new three phase equilibrium is in a two-component system and is in fact one of a n infinite number of triple points whose temperature and pressure vary with the concentration of solute in the liquid phase in accordance with the phase rule which predicts one degreeoffreedom(f = 2 - 3 + 2 = 1). Returning to a one-component system, application of the phase rule shows that a suhstance a t a melting point has one degree of freedom (f = 1 - 2 + 2 = 1). This means that solid and liquid can coexist in equilibrium over a ranae of oressures and temoeratures: the temoerature at whiFh the equilibrium exists for a &en pressure is the "true meltina point" or "true freezina point" of the pure substance fo; chat pressure. True meking points are ihus distinguished by the fact that they refer to a system of only one component, whereas the general terms, melting point and freezing point, refer to a solid in equilibrium . for eutectics. with a multicomoonent liauid ~ h a s e Exceot the equilibrium temperatire or a mixture depends on the relative amounts of liauid and solid ohases of a oarticular sample and does not correspond to unique temperature even for a fixed pressure. The terminology is further complicated by the experimental conditions used to measure these properties. Melting points are frequently determined in open containers a t atmospheric pressure. Consequently, melting points as reported in the literature normally correspond to the temperature a t which solid, liquid, and vapor are in equilibrium a t a total external pressure of 1 atm but the liquid may or may not actually be saturated with air. The student is given the impression that the melting point thus obtained is the unique melting point for the suhstance. The temperature is, in fact, that of a triple point of a twocomponent system which is one of an infinite number of triple points for this two-component system. Of course, the difference between the true melting point a t one atmosphere of mechanical pressure and the observed triple point is very small, in fact, usually ignorahly small under
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Comments by the Editor
A number of individuals have pointed out to me the basic error addressed in this Column: freezing-point studies are normally carried out under atmospheric pressure whereas almost all texts describe this process in terms of the phase diagram of pure solvent and imply that it is the true triple point which is lowered. Drs. Parker and Kristol have presented a coherent treatment of this complicated experiment situation which we hope will clarify the pbenomenon even though it does not make elementary presentations any easier.
the common laboratory conditions. However, it should he noted that melting points determined under abnormal conditions may differ considerably from those determined in a freshman chemistry l a h ~ r a t o r y . ~ As an example, we examine once again the phenomena associated with water. The normal melting point, defmed as the temperature for ice in equilihrium with air-saturated water under a total pressure of 1 atm, is defined as 0°C: the t m e t r i ~ l e~ o i n is t determined to he +0.0099'C. hi effects associated with increase in mechanical pressure and air solubility can he computed separately The Effect of the Pressure Increase: The pressure effect on the melting point is calculated using the Clapeyron equation. For the Hz0 (solid) = H z 0 (liquid) equilihrium dP AH, (1440 cal /moleX0.0413 1 atm/cal) (273"K)(-O.W163l/male) dT - TAV = - 134 atm / degree Thus, d T / d P = -0.0075 and the melting point of ice will decrease by 0.0075"C for each atmosphere of pressure increase. The Saturation of Liquid Water with Air: The solubilities of Nz and 0 2 in water a t O0C are 8.3 x 10-4 and 4.4 X lo-' m, respectively (4). The resulting freezing point depression is 4.4) X lo-' m = O.OO%"C. AT=k,m=? ('86")(83
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If the increase in pressure alone is considered, the effect is the same as if the air had been insoluble in the liquid ~ h a s eor if equilihrium is so slow that i t is not attained during the time of the measurement. The temperature a t which equilihrium would he attained is 0.0024"C below the s-I-vtriple point of pure water. If we assume that the effects of increase in pressure and solubility are independent and additive, the equilihrium temperature of ice and air-saturated water is (0:0024 + 0.0075) = 0.0099"C below the true t r i ~ l e~ o i n t The . effects due to Dressure are instantaneous and unavoidable in open systems; those due to soluhilitv are much slower and equilihrium is not always attained. A very elegant and classic experiment demonstrating these effects separately and combined was reported by Richards, Carver, and Schumh using benzene (5). Water is actually an unusual substance in that the presence of dissolved air and the increase in pressure both lower the freezing point. In most other substances, the pressure of the atmosphere increases the freezing point hut the presence of dissolved air lowers it, and thus these effects tend to cancel each other. Richards, Carver, and Schumh established experimentally the true triple point of benzene, then o ~ e n e dthe svstem to atmos~hericDressure and ohservedBn abrupt change in the temperatire due to the increase in pressure and a slow relaxation to the ultimate equilihrium value as air dissolved in the liquid. Introductory texts discuss the effect of a dissolved solute on the freezing point of a pure solvent by referring to the phase diagram for that solvent. In Figure 1, a typical schematic representation of the phase diagram for water is given; curves AB and AC are the vapor pressure of pure liquid solvent and pure solid solvent, respectively, as a function of temperature. The temperature a t which these two curves intersect is the true triple point (for water 0.0099"C and 4.58 torr), and it is also the true freezing point or true melting point of water a t this pressure, Po. Addition of a solute lowers the vapor pressure of water 3An interesting sidelight to this issue wad described by SwietasLawski (10). He pointed out that the solid, liquid, and vapor phases of a pure substance could not be maintained in equilibrium with each other at the true 8-1." equilibrium temperature by either isothermal or adiabatic processes due to the influence of gravity.
above the solution, as represented by curve EF. The intersection of E F with AC is the freezing point of the solution under the pressure, P. It is evident that the freezing point is depressed by the addition of solute. One is led to infer from Figure 1 that curves EC, EF, and EG simply represent another phase diagram for the one-component system water. Actually, this figure is a simplification of a more complex graphjcal representation fuido infrn~. Figure 2 represents schematically a portion of the phase diagram for a two-component system of water and nonvolatile solute in which Xz is the mole fraction of solute in the solution phase. Let us assume that ice and liquid H z 0 are in equilihrium a t the triple point A , 0.0099T under 4.58 tort pressure. Now consider the effect of mechanically increasing the pressure to 1 atm. The vapor phase will disappear, and the melting point, that is the temperature a t which ice and liquid will remain in equilihrium, will change along the curve A D until a t point D (1 atm) the temperature of 0.0024"C is attained. This temperature could he called the true normal melting point or true normal freezing point. Consider independently the effect of adding solute to H z 0 a t 0.0099'C and 4.58 tom. As solute is added, the solvent vapor pressure will decrease, and there will have to he a corresponding decrease in temperature along the curve AA' in order to maintain the ice-solution-vapr equilihrium. The result is a decrease in the freezing point. If we assume that A' is the freezing point of a solution obtained hy adding 1.27 X W3 mole of a nonvolatile solutejkg HzO, then this p i n t is the freezing
T
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TEMPERATURE -+ Figure 1. A schematic representation of the phase diagram for H20. The dashed curves EFand EG show the effect of adding a solute.
TEMPERATURE Figure 2. A schematic representation of a portion of the phase diagram for a two-component system H z 0 salute.
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point of a solution corresponding to the concentration of air dissolved in water a t 1 atm pressure, that is 0.0075"C. The result of the effect of pressure and solubility of air a t 1 atm on the ice-solvent equilibrium a t the triple point is given by the point D' on the diagram and is the sum of the two independent effects. Thus the one-component icesolvent-vapor equilihrium a t 4.58 t o n pressure is changed to a two-component ice-solution equilihrium a t a total mechanical pressure of 1 atm with a corresponding change in melting point for the ice from 0.0099 to 0°C. Now it is apparent why Figure 1 is misleading. Curves EC, EF, EG, and point E in this figure are actually the projection of curves A'C', A'B', A'D', and point A', respectively, onto the T,P plane (the XZ = 0 plane) in Figure 2. That is, EC, EF, EG, and E do not belong to the one-component system Hz0 a t all hut rather to a twocomponent system of Hz0 and solute. In Figure 1 point A is both a true triple point and a true freezing point for water. Most texts fail to state that the freezing point of the solution (E) is one of many triple points which depend upon the concentration of the solute, and are distinct by definition from the true solid-liquidvapor triple point of the one-component system. Thus, one incorrectly concludes that the true triple point has been depressed. What concerns us most. however. is that discussions found in these textbooks db not cor&spond to the reality which the student faces in the laboratow. In a twical freezing point depression experiment, thk student -fnst measures the normal s-1-u triple point for the two component system, solvent-air. Then he adds a small quantity of
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solute. thus intrnducine" a thinl comnonent. Havine fixed the atmospheric pressure and solute concentration, accordiue to the ~ h a s erule (3 - 3 + 2 = 2) no demees of freedom remain for the new s-I-v equilihrium; ohly one possible equilihrium temperature exists. It is the normal freezing point of a three component s-I-v equilihrium. Fortunately it matters only very little in the molecular weight determination by freezing point depression whether the solvent is initially a t its true triple point or its normal freezing point. What does matter is that the A T used in the calculation must correspond only to the addition of the solute whose molecular weight is to be determined with all other experimental conditions remaining as close to constant as possihle. Really precise cryoscopic studies do require consideration of the many effects pointed out in this paper, not only the solubility of the gas, hut also its change with temperature and the concentration of dissolved salts. The reader is referred to t h e literature for discussion of these more complex problems (6-8). Literature Cited (1) Kata, L.,J. CHEM. EDUC., M. 282 (1967). (2) Swinton, F.L., J. CHEM.EOUC.,44,567 (19671. (3) Ricei, J. E., "The Pheae Rule and Hetemgeneava Equilibrium," 0. Van Noatrand co.. NelxYork. 1951. p.37. (4) hternationel Critical Tables. Vol. m,McGrav-Hill Book Co.. New York, 1928. pp. 2567. (5) Richads, J. W., Carver,E. K.,and Schumb, W. C.,J Amer Cham. Soe., 41,2019
