A New Derivation of the Approximate Phase Change Formula AT = Kern
Maria Grazia Bonicelli and Francesco di Giacomo lstituto di Chimica della Facolta di lngegneria dell'universita di Roma, Rome, ltaly Mario Emilio Cardinali lstituto di Ricerca per la Protezione ldrogeologica nell'ltalia Centrale. CNR. Perugia, ltaly ltalo Carelli Centro di Studio per la Elettrochimica e la Chimica Fisica delle lnterfasi, CNR. Rome, ltaly
The usual approximate formula for the freezing point depression or the boiling point elevation of a dilute solution of a nonelectrolytic solute a t constant pressure which does not associate is AT = Km
(1)
where AT can he the change in freezing or boiling point, m is the solute molality, and K is the solvent cryoscopic or ehullioscopic constant. A new derivation of eqn. (1) starting from the exact formula is presented and discussed here, hut we first outline the two slightly different paths that have hitherto been given to get to (1). Referring in particular to the cryoscopic case, the starting point is the exact equation (1) - In x = (Lo,-dR)(lIT - 1/T0)
(2)
which gives-for an ideal dilute solution of solute B in a given solvent and under conditions of constant total pressure-the temperature T a t which the solution is in equilibrium with the pure solid solvent as a function of mole fraction x of the solvent in solution, R, To and Lo,-,being the gas constant, the melting (or freezing) temperature of the pure solvent, and the latent heat of melting of pure solvent a t To, respectively. Equation (2) may be rewritten in the form - ln(1- x 3 = (Lo.-dR)(ATITT")
-
(3)
where XB is the mole fraction of solute B and AT To - T is the freezing point depression. Equation (1) is usually ohtained (1-7) from eqn. (3) by means of the three following approximations: 1) the logarithm is expanded in the form
In a more direct derivation (S), steps (i) and (ii) are com. pounded. In fact - In r = ln(1 + ngln) and as l n ( l + Y ) = Y - Y Z / ~ + Y ~ /. .~. and therefore -In x = ng/n - ( n ~ l n ) ~+/ 2( n ~ l n ) ~-/ 3. . .
-
it is seen that retaining oniy the first term in the expansion, the separate assumption x~ ne/n is no longer necessary and so can he removed. Through either one of the ahove paths, eqn. ( 2 ) gives and subsequent conversion of mole rationaln to molality m brings eqn. (4) to the familiar form of eqn. (1). In our derivation we start from eqn. (2) written in terms of molality. Equation (2) is first recast in the form ATITO = -K'(ln x)l(l - K'ln x)
(5)
where K' = RTOIL0.-I is a dimensionless quantity. Moreover, as -1nz = l n ( l + M m ) where M is the molar mass of the solvent, eqn. (5)becomes AT/TD= 1- l/[l + K' ln(1 + Mm)]
(6)
which is eqn. (2) written in terms of molality. We now expand eqn. (6) in a Taylor series:
+
f(m) = f(mo) f'(m&(m - mo)
+ (l/df"(mo).(m- m d 2 +
for mo = 0 (McLaurin expansion). We get: ATIT* = K'Mm
and only the first term is retained, i.e., 2) the mole fraction of solute B is written
-
by neglecting the amount of B with respect to that of the solvent; 3) in the product TTo it is assumed T To so that TT" s (To)=
- K'(1 + 2K')(Mm)Z/2+ . ..
derived fully in the Appendix; i.e., the usual approximate formula (1)is ohtained when the ahove expansion of eqn. ( 6 ) is truncated after the first-order derivative (which is K'MT" = K). The advantages of our derivation through an "all molality" route are 1) Only one step is needed in going from the exact formula (6) for
AT/To to the approximate formula (I),i.e., our mathematical path is more straightforward.
Volume 61
Number 5 May 1984
423
2) The form (eqn. (6)) we adopted for the exact equation is immediately comparable with the approximate formula (eqn. (1)) and so it is easy to compute the relative error 100(ATappAT,,)/AT., (where AT,, is from eqn. (1) and AT,, from eqn. (6)) made in the computation of ATITO when using the approximate formula (1). For example, for a hypothetically ideal solution 1 mold of a solute in water (To = 273.16 K; L".-l = 6020 Jlmol; M = 0.0180 kglmol) the error is equal toahout 2%. If we take benzene as solvent (To = 278.66 K; Lo.-) = 10,280 Jlmol; M = 0.0781 kglmol) the same computation gives a value of about 5.5%. Lastlv. if we cheek chloroform (To = 209.66 K: Lo.-, = 9340 ~ l m o l=: 0.1194 ~ kelmol) weeet'avalue of about .. 8.59r. l'hr t i d u r r %show that the &or rhnngrs when urmg dif. f e r e n t sol rent^ nnd mrremes wnh the mdar mws of the a h t n t , an effect generally overlooked
-.
We note also t h a t eqn. (1) gives acceptably accurate results even a t rather high values of molality; our derivation gives a firm mathematical foundation t o this "good quality" of t h e attractively simple eqn. (1).
Literature Cited (1) Denbigh, K.. "The Principle
ofChemieal Equilibrium,"3dd..CambridgeUniversity Plena. London. 1971, p. 261. Cliffs, 1972, (2) Mmm, W. J.,"PhyaicslChemi~try."4thod.,Pmntiee-Hsll,Ine.,Englewood p. 249. (3) Bsnmv, G., "Physical Chemistry: 3rd ed., MeOram~Hill.he.. New York, 1978, p. 603. ( 4 ) Mortirner. C. E., "Chemistry A Coneeptusi Approach." 3rd d., 0.Van Nonhand Co., New York, 1975. p. 2%.
424
Journal of Chemical Education
(5) Brescia, F,Arents. J., Msislich, H., and Turk,A , 'Tundamentals of Chcmiatq." 3rd ed., Academic Presn, Inc., New York, 1975, p. 184. 161 . . Sienko. M. J.. and Plane. R. A,. "Chemical Princioles and Prooerties." 2nd ed.. ~ c d r a w - ~ i Inc, i i , N e w ~ o r k 1'974, . p. 234. (7) Mahan,B.H.,"UniversityChemishy,)'3ded.,Addison-WasleyPuhSshmgCa.,Resding. MA I*?'." I R A ....., .... ,-. . ... (8) Sehuhert. F. E.,J. CHBM.EDVC..56,259 (1979). (9)"Standard MathematicalTsbies: CRC Pness. Ciedand, 1976, p. 397.
.
.~~ ~.
Appendix f(m) = 1 -
1
1
+ K' l n ( l + Mm)
from which successively:
f'(m) =
1
(E) + Mm
(1 + K'In(l+ Mm))% 1 P(0) = K'M
