Herbert R. Ellison Whealon College Norton. Massachusetts 02766
Binary Solid-Liquid Phase Equilibria
Most physical chemistry laboratory textbooks describe an experimental procedure to construct a binary solid-liquid phase diagram ( 1 4 ,but few go much beyond determination of the eutectic temperature and the corresponding eutectic composition. It is the purpose of this article to point out some of the other information that mav he ohtained from this experiment and to descrihe a computer program that will plot an ideal phase diagram to which the experimental one may he compared. The relationship between solubility and freezing point of a componrnt in an ideal binary solution is given h i t h e ideal sulubilitv or freerinr-. point esuntion (7). (Only pure solvent freezes out.) AH dln~='.'~ (1) RT2 This equation gives the variation in the mole fraction of the component which is considered to he the solvent as the temperature is varied. Equation (1) may refer to either component of the system and is clearly independent of the other component. It should he noted that the distinction between solvent and solute in any binary mixture is purely arbitrary. Equation (1) could readily be integrated if AHr, the heat of fusion, were not temperature dependent. However, this is usually not the case and so AHimust be expressed as a function of temperature. For example, the following equations are valid over a wide temperature range for napthalene and p dichlorohenzene ( 8 ) ~~
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CloHs: AHr = 4265 - 26.31 T + 0.1525 T2- 0.000214 T3 (2) p-CGH4C12: AHf = -10250 + 94.07 T - 0.1511 T2 (3) Substituting eqns. (2) and (3) into eqn. (1)and integrating yields ClaHs (solvent A) ' (4) -4265 26.31 In T 0 1525 T 0.000214 P lnN~=-+-- R RT R 2R p-CsH&ln (solvent B) 10250 94 07 In T 0.1511 T (5) lnNe=-+------+I' R R RT -~ ..When N = 1 (i.e., the pure substance) and hence InN = 0, T is equal to the absolute melting point; inserting these values into eqns. (4) and (5) permits evaluation of the integration constants. Doing this and rearranging yields +
CloHs (solvent A): -214625 T2 In NA= ----- 13.240 1n T + 0.07674 T - 5.3845 X T +63.3578 (6)
p-CsH& (solvent B): InNg=- 5158.04 T
+ 47.338 In T - 0.07604 T - 264.9806
(7)
These eauations mav he used to calculate the mole fraction of the component which is to he considered the soluent from the observed freezine point of the solution. It should he noted that these calculati&s must he done with a calculator or computer since the final answers are small numbers which are the sums and differences of several large numbers. Returning to eqn. (1)it can readily be seen that if AHi is not a function of temperature, then AH
1
I~N=-Lx-+I" R T 406 1 Journal of Chemical Education
(8)
Approximate Amounts of Compounds C& Sample #
1 2 3 4 5 6 7 8 9 10
(A1
p C e H K h (81
(g)
(el
0.00 1.00 2.00 3.00 4.00 6.00 . 7.00 9.00 10.00 12.00
12.00 12.00 11.00 10.00 10.00 8.00 6.00 4.00 2.00 0.00
where I" is the integmtiun constant. Plotting InN versus 1/T from the meltinr point of thernmponrnt considered to be the solvent to the eiiectic will thus produce an average value of AH$for that component. 'I'hr freeeing pilint lowering constant, Kr, of thesnlvent can hr: ohtained from the experimental data hy making use of the equation which results from integrating eqn. (1) between the limits of the pure solvent and a dilute solution. (9)
where N , is the mole fraction of the solvent in the dilute solution, T1 is the solution's freezing point, and Tf is the freezing point of the oure solvent. Since N F = 1for the Dure solvent. then lnNr = 0 and eqn. (9) can be rewritten as T~- T~= - ---RTITf x in NI AH.
--
-..,
(10)
For dilute solutions TI Ti, In N 1 = -Nz (the mole fraction of the solute) and defining AT = Tf - T I enables eqn. (10) to be written as -..,
The slooe of a d o t of T versus Nq is RTFlAHpand corresoonds . . to the slupr of ihc tangent drawn to the exprrimentnl curve where it intrrserts the vertical axis. Now the frrreinr point lowering constant is defined by (7) K*=- RT: A H.,,.,., -.
where n l is the number of moles in 1kg of solvent; so comparison of eqns. (11) and (12) shows that Kf=--slope - slope X MI nj 1000 where M 1is the molecular weight of the solvent. Experimental The table lists the approximate proportions of napthalene and p-dichlorobenzene that are suitable for this experiment. Both compounds may be purified by distillation,if desired. The melting point of pure CloHsis 80.55T and the melting ponit of pure p-CsH&12 is 53.1°C. Weigh the compounds, to -t0.01g, into 8 X l-in. teat tubes and stopper until used. The samples are melted by placing the test tubes in a beaker of hot water maintained above 80%. but this should be done onlv shortlv
ring stirrer. Attach a second thermometer (graduated in degrees) t o the first with small rubber bands. This second thermometer will he used to find the mean stem temperature in order to calculate the corrected freezing temperature.' To determine the freezing point of a samnle remove the molten material from the heatine - bath and arranm thermometer and stirrer ~~~a~~~~~~ in nlace. Immerse the test tube in .-.-.the .~-. another beaker of hot water (8kSQ'C if starting with pure napthalene; 60-65°C if starting with pure p-diehlorobenzene) and allow to cool slowly, stirrina the outer bath a t least once every 30 s. The sample is also stirred slowly while cooling (once every 2-3 s), and the temperature a t which the first crystah appear is taken as the freezing point. If the exact ooint of crvstallization is missed the sarnnle mav be rewnrmed nnd'the nrecedken .o r d u r e reneated. For ace&& work the trmperattvedrfference k t w r ~ nthe pnmple and the water h t l l should not dffer hy more than a few degrees 11necessary, a hot plate may be placed under the water bath to retard the rate of cooling. The above procedure is used to obtain the freezing point of each sample. When the freezing point is below 40°C i t is advisable toadd small chins of ice t o the water bath as the rate of cwling.of the large heaker is'too slow. ~
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Calculations Calculate the mole fraction of napthalene (A) present in each sample. The mole fraction of the second component (B)is 1- NA. Prepare a plat of the corrected freezing point temperature versus the mole fraction, with the latter quantity along the horizontal axis. Carefully draw the best smooth curve through the points and extraoolate the two curves to the intersection which is the eutectic mint. The rutrrric tmnprmture is repurted to br J0.2'C nnd the mole fraction of napthnlenent theeuwtti< I*,beO.R7h(R).St1rd~nll~hould dircusr the significance i,f the terms rutccrir point and eutectic temperature and comment on possible reasons for discrepancies betwrm rheir results and the literature values. l'rrparr plots of In 3'1 and In Nu V W S U ~IIT and from the dopes of t h ~ ~ u w e s d e t p r m i the n e aVWaeQvalur- of M a and MR. Studenu should comnare their results wiih those that can be cale&ted from eqns. ( 2 ) and t:rl using the meltinc pointh of the pure components. h a w tangent4 to the curves (rf freeling temp?rartlrP versus mole fraction wh1.r~they intersect the Mmperaturt.axes"'se eqn. (13) to calculate an experimental freezing point constant for each component. Compare with literature values or with values calculated from eqn. (12) using the literature values for AHrobtained from eqns. (2) and (3). The ba5ic wmputrr progrnm PHASE ran be used to prepare on a teletype wrminal the ideal hinnry aol~d-liquidphasr diagram for the iy*tem under study:' Input dnur are the melting p o i n t s of thr two
.
The corrected temprmturr. I,,.,, is found by wing the aquntion = r,, T 1 . 6 x 10-411r(lo- I , ) , where 3 . r the ~ lengthrxprer.ied indegrees of the exposed mercury column, to is the observed temperature, and tm is the mean stem temperature as recorded by the second thermometer. A section of a glass rod or a flat-edged mirror helps in drawing these tangents. A listing of this program is available upon request. Please specify whether an output formatted for 72 or 132 characters per line is desired 1
I,.,
.
.
comnonents and the coefficients ofeons. (2) . and (3). This nromam .. usrr'rquatmns ofthe fwm eqns ( 6 1 and ( ? I to &lculare thc ideal sulubilitirsand plotr the results ineithcra 15 X 13 cm or 26 X 24 cm furmat !thclnrter I* for terminalsthat can print a 132 character l:nr,. Students may then plot their experimental results directly on the computer print-out and can determine the ideality (or nonideality) of the sys&m. Alternatively, the ideal solubility of napthalene (A) in p-dichlorohenzene (B) may he calculated fram eqn. (7) using the experimentally determined temperatures (in "K) between the freezing point of pure p-dichlorobellzene and the eutedic temperature. Note that eqn. (I)actually gives N B ; the desired quantity NAmay he found by suhtraeting NBfram unity. Likewise eqn. (6) and the experimentally determined temperatures between the freezing point of pure napthalene and the eutectic may he used to calculate the ideal solubility of p-dichlorohenzene in napthalene. Plot these ideal points on the same graph containing the experimental temperature versus mole fraction curves. ~
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Additional Experiments Several other binary systems are given in the references. I t is possible to calculate heats of fusion as functions of temperature for those eompounds which have no equations given hy employing heat capacities of the solid and liquid forms and the heat of fusion a t the melting point. For the equilibrium process A(s) z= A(I) we can write = H(1) - H(s)
(14)
Differentiating with respect t o temperature a t constant pressure results in
the melting point of the pure compound, and Integrating between TI, some other temperature T produces
Appropriate values or expressions for the temperature dependency of the heat capacities of many eompounds may he found in the International Critical Tables (9).
Literature Cited (1) Shoemaker. D. P., G a r h d . C. W., and Steinfeld, J. 1.. "Experiment8 in Physical . Chemistry," 3rd Ed., MeCrew-Hill Baok Co., New York, 1 9 7 4 , ~ 237. (2) Daniels, F., Williams. J. W., Bender, P., Alberty. R. A , and Cornwall. C. D.. "Experimental Physical Chemistry," 6th Ed., McGraw-Hill Bmk Co.. New York, 1962, p. 116. (3) While,J. M.,"PhysicalChemistryLabaratoryExperiment8,"Prentia-Hallhe.. New Jersey, 1975, p. 188. (4) Wjlson,J. M.. Nwmmhe, R.J., Denar0.A. R.. and Riekatt.R. M. W.."Expsrimsnt8 m Phy8ical Chemistry," 2nd Ed., Pergumon Presa, New York, 1968. P. 46. (5) Bettelheim, F. A.."Ezpprimental Physical Chemistry,"W. B. S a u d d d Co.,PhIladdIphia, 1971.p. 241. (6) Oelke, W. C.. "LaboratoryP h y s i d Chemistry,"Van NalrandReinhold Co.,New York, 1969. p 303. (7) Found in various physicalchemistry terfs, such as Moore, W. J., "Physical Chemistry," 4th Ed., Prentim-Hsll, Inc., New Jersey, 1972.p. 249. (8) Monis. R. E., and Cmk, W. A,. J. Am. Chem. Sor., 57,2403 (1935). (9) "International Critical Table.: Vol. V, McCraw-Hill Bmk Co., New York. 1929, pp. 101 and 132.
Volume 55, Number 6, June 1976 1 407
