HEATCAPACITY
Sept., 1958
OF PHOSPHORIC
ACIDS O L U T I O N S
1091
HEAT CAPACITY OF PHOSPHORIC ACID SOLUTIONS, 15 T O 80' BY
EDWARD P. E G A N , JR.,BASILB. LUFFAND ZACHARYT.
WAKEFIELD
Division of Chemical Development, Tennessee Valley Authority, Wilson Dam, Alabama Received M a y 3,1968
The heat capacities of phosphoric acid solutions containing 5 to 85% HSP04 were measured a t 15, 25, 40, 60, 70 and 80". Curves of heat capacity against temperature were inflected sharply above 60" for solutions containing more than 5% HaPOa. The calculated partial molal properties of the components are tabulated as functions of concentration and temperature.
As part of a broad study of the thermodynamic properties of phosphate systems, the specific heats of phosphoric acid solutions containing 5 to 85% Ig3P04were measured a t temperatures of 15 to 80". Partial molal heat capacities, heat coiitents, free energies and entropies of the solutions were calculated from the measured specific heats and known data1s2a t 25'. Earlier measureineiits of similar character seem to be limited to the specific heats a t 21' and the average specific heats between 21' and the boiling point^.^ These specific heats averaged about 0.3% higher than the present measurements. Phosphoric Acid Solutions.-Phosphoric acid hemihydrate, 2HaP04.H20, was twice crystallized from reagent grade phosphoric acid. The hemihydrate was mixed with the calculated amounts of distilled water to yield solutions with nominal concentrations of 5, 10, 15, 20, 30, 40, 50, 60, 70, 80 and 85% H3POa by weight. More dilute solutions involve relatively large inherent errors of measurement, whereas more concentrated solutions are so hygroscopic that they would require special apparatus. The compositions of the solutions were determined from measurements of their densities4 a t 25". The solutions stood 6 months before the first measurements of specific heat were made. At the start and near the end of the measurements, the densities of the 5, 40 and 85% solutions were redetermined; the greatest change in density was 0.04 mg. per ml. Construction and Operation of Calorimeter.-The calorimeter was used as a solution calorimeter in other work,6 and only the modifications to fit it for measurements of heat capacities of solutions are described here. The hollow stirrer shaft was plugged, and a snug-fitting neoprene washer was placed on the shaft below the bottom bearing in an attempt to seal the vapor space in the calorimeter from the atmosphere. The volume of the calorimeter vessel below the constricted neck was 850 m1.-of the vapor space in the neck, 160 ml. For the measurements a t 15 and 25", and for part of the measurements at 40°,the copper-manganin thermometer and the heater were wound on a thin-walled copper tube, :tiid the assembly was coated with Apiezon wax. The thermometer was calibrated frequently against an NBScertified platinum resistance thermometer. The assembly proved satisfactory a t 25". When cooled to 15" or heated to 40°,however, the wax occasionally was cracked by thernial shock, with resultant exposure of electrical connections to the acid solutions. Upon completion of about I d f the measurements at 10°, the heater was replaced with a 4-lead helical constantan heater that was mounted in a glass spiral and suspended from the calorimeter cover by 5-mm. glass tubes through which the leads were passed. The thermometer was replaced with a capsule-type platinum resistance thermometer which was immersed in oil in a small glass capsule suspended (1) T. D. Farr. Tennessee Valley Authority, Chem. Eng. Rept., No. 8 (1950). (2) K. L. Elmore, C. R4. Mason and J. H. Christensen, .7. Am. Chsm. Soc., 68,2538 (19413). (3) M.M . Popov, N. N. Feodos'ev and S. M . Skuratov, Tvans. Sci. Inst. Fertilizers ( U . S. S. E . ) , 110, 23 (1933). (4) J. H. Christensen and R. B. Reed, Ind. Eng. Chem., 47, 1277 (1955); E. P. Egan, Jr., and B. B. Luff, ibid., 47, 1280 (1955). (5) E. P.Egan, Jr., Z. T. Wakefield and K. L. Elmore, J . Am. Cksrn. Soc., 78, 1811 (1956).
from a 3-mm. glass rod that extended through the hollow stirrer shaft. A small oil seal between the shaft and the rod separated the vapor space from the atmosphere. Several redeterminations of specific heats at 15, 25 and 40" shoyed that the measurements were unaffected by the radical changes in the heater and thermometer. The calorimeter bath was operated about 1" above the nominal temperature so that the corrections would be in one direction. A water-bath was used a t 15, 25 and 40°, and the calorimeter was sealed liquid-tight in the supporting frame with Apiezon wax. An oil-bath was used a t 60, 70 and 80", and the calorimeter was sealed with an epoxy resin (Carl H . Biggs, potting compound P-420). The Apiezon wax was soluble in oil, whereas the resin would not remain tight in water. The charge of phosphoric acid solution, in a special 850ml. volumetric flask. was adiusted to thermal eauilibrium and to volume in the bath iri which the measurgment was to be made. The charge was introduced into the calorimeter at a slightly lower temperature, the weight of the charge being determined by differential weighing of the flask a t room temperature, with buoyancy correction. For measurements at 60" and above, the cover of the calorimeter was preheated 5 t o 10" above the temperature of the bath to prevent condensation on the cover before the assembly was placed in the bath, a precaution unnecessary at the lower temperatures. The calorimeter system, with the stirrer active, was allowed 30 minutes to approach equilibrium with the bath. Foreperiod temperature readings were taken for 15 minutes, and then measured energy was introduced to raise the temperature of the solution 0.5". Temperatures were read for an intermediate period of 15 minutes, then measured energy was introduced to give a second temperature rise of 0.5". Temperatures were read for a final period of 15 minutes. The initial temperature was so adjusted that the mid-point temperature of the first heating period was 0.25' below the nominal temperature, and that of the second heating period was 0.25' above the nominal temperature. Essentially duplicate determinations of the specific heat thus were obtained for each concentration of solution a t each temperature. The space above the solution in the calorimeter was assumed to be isolated from the atmosphere. The heat required for its saturation with vapor entailed corrections based upon vapor pressures and heats of vaporization.' It was assumed that only water was evaporated. The correction for heat leak was made by the method of Dickinson,6 an arbitrary value of 0.60 of the interval being used for the corrected temperature rise. In runs a t thcs lower temperatures where the copper-manganin thermomcter-heat,er assembly was used, the temperature and the energy input could not be read during a heating period because of the thermal conduction of the copper tube. The heat leak correction required for this assembly was used, for consistency, in runs made with the independent, g1:tssenclosed heater and thermometer. The bridge unbalance on the copper-manganin thermonieter was read to the nearest 0.1 pv. on a Wenner potentiometer (low range). The temperature readings were reproducible within d~0.0002". The platinum resistance thermometer was read with a Mueller bridge, and temperature differences were read to four decimal places. The ice-point, resistance of the thermometer was checked periodically, and the Mueller bridge was calibrated against an NBS-certified 10-ohm resistor. The input energy was measured with a Wenner potentiometer (high range). The reference potential was six (G) H. C. Dickinson, Bull. Null. Bur. Standaxis, 11, 257 (1914).
EDWARD P. EGAN, JR.,BASIL B. LUFFAND ZACHARYT. WAKEFIELD
1092
Vol. 62
unsaturated standard calomel cells, certified by NBS and maintained at 25". Time was measured with a 1000-second timw that was graduated to 0.1 second and driven through a 60-cycle frequency standard. One defined calorie was taken as 4.1840 absolute joules-the ice point as 273.16'K. The average precision of the specific heat determinations was =k0.0004 cal. deg.-' g.-l of solution.
mid-point temperature of each heating period are listed in Table I. Equations for s, specific heat in cal. deg.-l g.-1 of solution, in terms of t 2 where f is temperature in "C., were fitted to the data in Table I, and the specific heat was calculated for each concentration of acid a t integral temperatures Specific Heats.-Corrected specific heats for each of 15, 25, 40 and 60". With the exception of one concentration of phosphoric acid solution at the anomalous but reproducible point for 50% H3P04 a t 40°, the specific heats were smooth functions of TABLE I temperature between 15 and 60". OBSERVED SPECIFIC HEATSOF PHOSPHORIC ACIDSOLUT~ON, Plots of s against temperature changed abruptly GAL. L ) E G . - ~G.-1 in slope between 60 and 70". The temperature4.4GX Hap01 9.35% Hap04 14.89% HaPOi t , oc. 8 specific heat relation for a given composition did not t , oc. 8 t , oc. ,s lend itself to convenient expression over the entire 14.735 0.9218 14.731 0.8800 14.748 0.9623 range 15 to 80". Specific heats a t integral tempera15.301 .9625 15.291 .9223 15.252 ,8804 tures of 70 and 80" were estimated by plotting s 24.704 .9631 24.704 .925d 24.753 ,8851 against t a t each observed temperature, drawing a 25.273 .9628 25.273 ,9258 .8848 25.285 straight line between the points, and reading the 39.517 .9650 39.745 .9298 39.746 ,8897 value of s on the line a t 70.0 or 80.0". Where 40.270 .9647 40.270 .9293 40.290 ,8908 replicate determinations had been made, the aver59.587 ,9329 59.762 .9686 59.736 ,8953 age of the intersections of the lines with the 70 or 60.293 ,9680 60.128 ,9328 60.269 ,8948 80" ordinate was taken as the most probable value 69.810 .9377 69.740 .9692 69.735 ,8989 of s. The series of measurements a t all compo70.329 ,9373 70.269 .9710 .8986 70.259 sitions a t 70" and the nominal composition a t 15% 79.735 ,9031 79.730 .9392 79.714 .969P were included after a number of the measurements 80.269 ,9712 80.262 .9382 80.285 ,9022 had been made to define more closely the behavior 19.217' HaPOi 29.33% Hap04 39.87% Hap04 1, o c . 8 t, o c . 8 t , "C. 8 of the specific heats above 60". 14.721 0.8472 14.739 0.7728 14.741 0.7027 The rapid change in specific heat with elevation 15.302 .7727 15.267 ,7030 15.298 .8476 of the temperature above 60" may reflect a change 24.723 ,8523 24.742 ,7788 24.740 ,7092 in the property of water. Zwicky7 showed that 25.311 .7790 25.311 ,7098 25.295 .8522 the compressibility of water changes markedly 39.745 ,788X 30.728 .720? 39.740 ,8590 above 60". He concluded that water in a solution 40.269 .788l 40.248 ,i l 9 9 40.264 .8592 under the influence of the internal electric field 59.747 ,8070 59.745 .7980 59.767 ,7312 generated by the dissolved ions is affected in much 60.262 .T095 60.318 ,7314 60,283 .867!) the same way as water under pressure and that the 69.741 ,7347 8!).738 ,8003 ciY.752 .8709 specific heats should increase significantly above 70.264 ,7354 70.26-i ,7998 70.291 .si02 GO". I n a few measurements on potassium chloride 79.738 .SO10 79.722 ,7403 79.736 .8699 solutions, Guckers found the same effect as was ob80.290 .SO10 80.266 ,7406 80.284 ,8693 served in the present study; the specific heat of a 59.54% Hap04 70.10% HzPOa 49.36% Hsp0.1 potassium chloride solution was smaller a t 80" t , oc. S t, oc. 8 t , "C. 8 than a t 20". The degree of hydration of the ions 14.7-12 0.6321 14.674 0.5668 14.779 0.5097 in phosphoric acid solutions is unknown but likely 15.428 ,5099 15.259 ,5658 15.3xx ,6329 is aff ect,ed considerably by changes in tempera24. $31 . ti394 24.739 .5754 24.730 .5158 ture. The specific heats of sulfuric acid and, to :L 25.311 .(i307 25.328 ,5756 25.326 ,5152 less extent, of acetic acid showg abrupt changes in 39. 738 ,6517 39.679 ,5870 39.698 .5260 slope above 60". 401341 ,6525 40.210 .5860 40.154 ,5248 The specific heats of the solutions a t each in59.711 .5366 59.783 ,0625 59.723 ,5997 tegral temperature were expressed as ? cubic 60.223 .5377 60.261 .6000 60.372 ,6617 power series of the concentration w, in weight %, (59.748 .5438 69.745 ,6688 69.850 .6044 and a smooth curve was fitted to the deviations 70.279 .5435 70.376 .6053 70.253 ,6696 from each equation. Each equation then was 79.725 ,6073 79.725 .5460 79.728 ,6714 solved for s a t 1% intervals of w, the calculated 80.272 .5476 80.286 .6072 80.264 ,6718 values were corrected from the deviation curve, 84.81% HaPo4 79.97% HsPOa t, oc. 8 t , "C. 8 and the corrected values were extrapolated to 100% H3P04. Although cubic equations often will not 14.702 0.4391 14.760 0.4595 extrapolate smoothly, the present extrapolations 1-1.817 .4600 15.307 .4397 were smooth. The results appear to be reasonable 24.708 .4440 24.724 ,4671 and perhaps will be acceptable until measured 25.319 ,4442 25.302 .4666 values are available. The smoothed specific heats 39.872 ,4517 39.736 ,474ti a t integral concentrations and temperatures are 40.406 .4529 40.252 .4757 listed in Table 11. The specific heats of water are 59.753 .4862 59.749 .4624 from Osborne and co-workers.10 60.271 .4626 60.295 .4863 '
69.751 70.286 79.726 80.276
.4918 .4923 ,4937 .4947
69.736 70.272 79.737 80.290
.4671 ,4675 ,469ti .4707
(7) F. Zwicky, Physik.
Z.,27, 271 (1926).
(8) F. T. Guoker, Jr., J . A m . Cham. Soc., SO, 1005 (1928). (4) I,andolt-Biirnsto~n,"Phy~ikalisch-chetnischeTabellcn," Supylciiioiit 3, 5 t h ed., Julius Springer, Berlin, 1936, p. 2283.
HEATCAPACITY OF PHOSPHORIC ACID SOLUTIONS
Sept., 1058
1093
As a check on the thermal corrections at the high temperatures, measurements were made of the specific heat at 80" of a solution with a composition corresponding to KC1.25H20. The results, 0.8418 and 0.8409 cal. deg.-l g.-l, are to be compared with 0.8415 cal. deg.-' g.-l as calculated from a value reported by Guckers for the ratio of the specific heat of the solution to that of water a t 80" and Osborne's'0 value for the specific heat of water.
and 15 molal H3P04. Similar abrupt changes at these concentrations have been observed in the density, conductivity, pH and activity of phosphoric acid solutions at 25". The break at 2.2 m H3P04 was the most pronounced, and a break a t this concentration occurs in all the other properties. The breaks correspond perhaps t o changes in the number of species present in the solution or to marked changes in the relative concentrations of the species. An elucidation of the structure of TABLE I1 phosphoric acid solutions would aid materially in SMOOTHED SPECIFICHEATSOF PHOSPHORIC ACIDSOLUTIONS, explaining the breaks. At concentrations above CAL. D E G . - ~G.-1 15 m, the curves were fairly smooth and almost Wt. % parallel a t all temperatures. HsPOn 15' 25' 40' 600 70° 80 The value of COP,, the partial molal heat capacity 0 1.0004 0.9990 0.9987 1.0010 1.0013 1.0030 5 0.9576 .9591 ,9613 0.9638 0.9654 0.9666 of H3P04a t infinite dilution, must be determined is to be used in calculating the temperature 10 .9172 .9208 .9251 .9282 .9330 .9347 if 15 .8794 ,8839 .8895 ,8946 .8980 .9017 coefficients of the thermodynamic properties -of 20 .8414 .8465 ,8534 .8622 ,8652 .8639 phosphoric acid solutions, The shape of the Cp,
e,,
cpz
25 .8042 30 ,7679 35 .7322 40 .6971 45 .6622 50 .6281 55 .5952 60 ,5639 65 .5361 70 .509G 75 .4844 80 .4606 85 .4380 90" .4174 95" .3994 100" .3839 a Extrapolat;ed
.8102
.7743 .7387 .7036 .6689 .6353 .6030 .5725 ,5439 .5166 .4960 .4662 .4434 .4228 ,4045 .3889
.
.8184 .7833 .7479 .7134 .6803 .6477 .6149 .5838 .5545 .5264 .4998
.4746 .4511 .4294 .4098 .3918
,8287 .7845 .7601 .7256 .6911 .6580 .6274 .5970 .5673 .5385 .5114 .4861 .4614 .4389 .4187 .4008
.8303 .7956 .7623 .7294 .6969 .6649 ,6334 ,6021 .5724 ,5440 ,5172 ,4919 .4666 .4429 .a12 ,4013
,8297 .7968 ,7658 .7342 .'io08 .6874 ,6350 ,6042 .5754 ,5474 .5202 .4941 .4692 .4454 .4227 .4010
Derived Thermal Data.-The conventional exwere modified pressionsll for +c, Cpaand for use with data on a basis of weight %
(epl eopl)
+c =
c,,
IlOOs - (100
- w)s~]Mz
W
= +c
+ w(100100- w)C*bwb
where s and so are the respective specific heats of solution and of water a t the same temperature, w is weight yo H3P04,and the other terms have their usual significance. The apparent molal heat capacity, dc, was calculated a t 1% intervals of concentration. The slope, bdclbw, was calculated with use of 7-point first-derivative coefficients.12 The partial molal heat capacity of Hap04 in phosphoric acid solutions, Cp2,was calculated from dc and bdc/aw. Plots of Cpa against w were not smooth curves; abrupt changes in slope occurred at concentrations corresponding to 0.1, 2.2, 4, 8 '
(IO) N. S. Osborne, H. F. Stimson and D. C. Ginnings, J. Research Natl. Bur. Standards, 23, 197 (1939). (11) H. 8. Harned and B. B. Owen, "The Physical Chemistry of Electrolytia Solutions," Reinhold Publ. Corp., New York, N. Y., 1943, p. 242. (12) H. E. Salaer, "Tables
of Coefficients for Obtaining the First Derivative Without Differences," Natl. Bur. Standarda, Applied Math. Ser. 1948, 2.
TABLE I11 PARTIAL MOLALHEATCAPACITY OF H,P04 IN PHOSPHORIC
cp,
ACIDSOLUTIONS, Wt. % Hap01
15'
250
40'
60'
70'
80"
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
16.23 23.95 22.86 23.30 24.78 25.94 26.71 26.92 27.70 28.46 29.97 32.06 33.96 34.85 35.43 36.07 38.45 37.13 37.50 37.63
21.90 24.84 23.73 25.05 26.25 26.80 27.41 27.62 28.68 29.87 31.40 33.10 34.11 34.94 35.70 36.46 37.03 37,61 37.98 38.11
26.51 27.20 27.13 28.02 28.64 28.02 28.67 30.45 31.26 31.32 32.04 33.70 34.63 35.52 36.28 36.97 37.56 38.01 38.28 38.39
26.97 30.09 34.96 32.64 30.76 30.77 30.59 30.58 30.56 33.29 34.60 34.93 35.49 36.27 37.42 37.74 38.27 38.84 39.17 39.28
32.70 30.24 32.85 30.60 30.18 32.18 32.57 33.00 33.50 33.99 34.41 35.10 36.17 37.09 37.91 38.26 38.45 38.97 39.24 39.33
33.68 34.48 30.39 29.05 31.88 35.43 35.18 34.69 32.27 33.14 34.17 36.35 36.86 37.40 37.93 38.41 38.82 39.09 39.24 39.33
T A B LIV~ RELATIVE PARTIAL MOLALHEAT CAPACITY OF WATERIN PHOSPHORIC ACID SOLUTIONS, ~opl)
-(epl-
Wt. %
€Tap04 5 10 15 20 25
30 35 40 45 50 55 60 65 70 75 80 85 90 95
100
15' 0.0214 .1527 .1261 .1451 .2241 ,3046 .3741 .3897 ,4974 .6223 ,9298 1.4653 2.0226 2.3718 2.6566 3.0615 3.3829 4.2544 5.1070
25O 0.0203 ,0711 .a319 .1258 ,1428 ,1817 ,2365 .2538 ,4006 .5984 .9103 1.3402 1.6466 1.9802 2.3350 2.8133 3.3125 4.0608 4.9046 0.0704 6,0061
40° 0.0196 ,0293
.0202 .0611 .a878 .a394 ,1005 ,3021 .4114 .4136 .5734 .9921 1.2754 1.6143 1.9896 2.4214 2.9380 3.5217 4.1712 5.0758
60D 0,0001 ,0521 .1904 ,1008 -0.0029 -0,0064 -0.0216 -0.0235 -0.0247 0.4309 ,6768 ,7604 .9321 1.2395 1.8024 1.9814 2.4691 3.1911 3.9555 4.8534
70' 0.0481 -0.0185 0.0717 -0.0368 -0,0564 0,1007 .1348 .1823 .2499 .3310 ,4138 .6007 ,9243 1.2773 1.6742 1.8661 2.0640 2.7221 3.3217 3.9728
80' 0.0638 .0633 -0.0569 -0.0511 0.0933 ,3532 .3268 .2750 -0.0320 0.1140 .3234 .8789 1.0377 1.2395 1.4994 1.7990 2.1543 2.5047 2.8522 3.1271
EDWARD P. EGAN,JR.,BASILB . LUFF.4ND ZACHARYT. WAKEFIELD
1094
Vol. 62
TABLE V must be integrated with respect to temperature. RELATIVE PARTIAL MOLALHEATCONTENT OF PHOSPHORICFor lack. of a_ satisfactory method of determining COP2,(HZ - HOZ)for 0.1 H3P04a t 25" was taken as ACIDSOLUTIONS, (R2- BO2) ACIDIN PHOSPHORIC Wt. H a 3
15'
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
534 756 953 1074 1139 1261 1545 1977 2308 2607 2917 3247 3574 3945 4321 4676 5034 5306 5510 5569
"
25
559 831 1017 1147 1227 1359 1648 208 1 2422 2731 3057 3405 3747 4126 4509 4871 5234 5512 5720 5780
40'
583 875 1050 1199 1293 1424 1723 2169 2525 2845 3189 3562 3917 4308 4703 5076 5449 5734 5947 6009
656 982 1191 1337 1422 1539 1847 2320 2686 3024 3387 3764 4146 4554 4969 5352 5735 6030 6249 6313
a reference point, so that
70°
80'
708 1029 1283 1403 1478 1605 1915 2392 2762 3112 3485 3879 4257 4675 5099 5487 5874 6172 6393 6458
800 1113 1360 1462 1545 1699 2010 2489 2852 3209 3587 3994 4381 4808 5237 5629 6019 6320 6544 6610
60'
(Rz - Roz), - (Rz - R0Z)O.l = ( i l z - R*z), The value of Cpzthen was taken a t the same reference point, 0.1 molal H3P04,and values of ( C P 2 ) r n - ( c P 2 ) O . l = ( G z - C*,,)m were calculated for all temperatures and concentrations of solution, On integration with respect to temperature of (Cp2- C*p2),there was obtained (Rz - R*&
=
(ilz- R*,),,+
J: (cpz
-
e",,)
In the integration of (Cp2 with respect to temperature, values for each concentration were plotted against temperature, and an arbitrary smooth curxe was drawn through the points. Values of (Cpz - f*p2) were read from the curve at 0.5" intervals between 15 and 40" and at 1" intervals between 40 and 80". The shorter intervals below 40" were necessary to obtain enough points for a satisfactory tabular integration. The inte-
TABLEVI RELATIVE PARTIAL MOLAL HEATCONTEXT OF WATERI X PHOSPHORIC ACID SOLUTIONS, -(Rx-
wt.
c*,,) dT
no,)
% HsPOi
15'
25'
40'
600
70'
80'
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
1.942 5.051 10.16 13.84 17.74 26.83 52.32 100.1 144,6 194.4 257.8 340.1 441.1 580.6 766.0 993.6 1294 1661 2128
2.150 6.141 10.82 15.22 19.56 29.21 55.23 103.2 149.1 200.6 267.0 354.2 459.4 602.1 790.9 1023 1328 1703 2178
2.448 4.616 11.14 16.63 21.19 30.84 57.73 107.4 155.2 208.2 278.5 371.9 481.2 628.8 823.1 1062 1375 1760 2247
2.700 5.447 13.42 18.34 22.27 30.66 58.15 110.7 159,9 216.6 291.21 389.4 502.8 657.1 860 9 1106 1429 1828 2328
2.936 5.678 14.90 18.36 21.94 31.04 58.70 111.4 161.4 220,6 296.6 396.1 512.0 669.7 878.3 1125 1452 1857 2364
3.511 5.938 14.82 17.90 22.28 33.18 60.96 113.8 162.8 222.8 300.1 403.4 521.5 682.3 894.2 1144 1473 1883 2395
curves, however, precludes the use of conventional methods'l for the extrapolation to infinite dilution. Gucker and Schminke13 were skeptical about the extrapolation of Cpzto m = O in any event. The partial molal heat capacities of H3P04,Cp2, are listed in Table 111-the relative partial molal heat capacities of water in phosphoric acid solutions, (Cpl - Cop,),i; Table-IV. Values of (HZ - Hoz) for phosphoric acid solutions at 25" have bee; published.' For the present paper, however, (Hz - HOz) a t 25" (Table V) was recalculated from unpublished heats of dilution, ( d h - $ O I ~ ) , of phosphoric acid. For calculations of (Hz - ITz)a t other temperatures, the - COP,), relative partial molal heat capacity, (fpz (13) F. T.Gucker, Jr., and K. H. Sohminke, J . Am. Chem. SOC.,64, 1358 (1932).
,
gration mas made with &point Lagrangian integration coefficients, l4 The value of (Hz - I&) for 0.01 m phosphoric acid a t 25", as calculated from the heat ofdilution data, was 41 cal. The value of - C*,,) for 0.01 m acid a t each temperature was calculated. Then, at each temperature (I72 - Roz) = ( I 7 2 - R*?) + [(Rz- R*Z)O.I - (Rz - R*2)0.011 41 This calculation entfiils the assumption that (Rz HOz)o.ol- (Hz - HOz)o= 41 cal. a t all temperatures. Values_ of (Rz - ROJTwere recalculated with ' (R2 - HOZ)a t 25" and C, a t To,both values for
(ep$
+
(14) Works Progress Administration, Mathematical Tables Proje c t , "Tables of Lagrangian Interpolation Coefficients," Columbia
University Press, New York, N. Y.. 1944.
HEATCAPACITY OF PHOSPHORIC ACIDSOLIJTIONS
Sept., 1958
1095
TABLE VI1 RELATIVE PARTIAL MOLALFREEENERGY OF PHOSPHORIC ACIDIN PHOSPHORIC ACIDSOLUTIONS, ( RWt. % Hap04
5 10
15 20 25
so
35 40 45 50 55 60 65 70 75 80 85 90 95 100
15'
25'
-424.8 89.22 424.4 700.8 953.4 1203 1464 1746 2044 2347 2674 3030 3402 3789 4181 4565 4939 5269 5594 5708
-458.5 64.67 404.9 686.6 945.4 1199 1459 1736 2035 2336 2663 3020 3393 3780 4173 4558 4932 5264 5593 5709
40°
5.0 m acid, as the reference points. This calculation contributed nothing mathematically, but it did test the consistency of the d&a. The two reference points gave values of (Hz - Hoz) that differed by an average of 10 cal. The two sets of values were averaged, and the average values were used in suhsequknt calculations. The averaged values of (H2 - Ho2) as a function of temperature are shown in Table V. The calcu-Jation of (271 - E&)T from (I71 2701)25 and ,(& - @DI)T Was straightforward. The values of (HI.- Rol) as a function of temperature are shown in Table VI. Values of al and a2 for phosphoric acid solutions also have been publishede2 The results of unpublished work show that a,, as used in reference 2, is the same-as a2. Values'of ai were calculated from (Hi - Hoi) a t each temperature by the rela-
ep2
c:
,.-
LilUll
In (ai)= - In (u& =
60'
-510.2 24.86 372.9 661.8 929.1 1189 1447 1716 2012 2313 2640 2996 3371 3758 4151 4537 4911 5246 5581 5699
-
The calculated values of a] and a2 at each temperature were converted to partial molal free energies by the relation (Pi - P i ) = RT In ai Values of (Fz - Fez) as functions of temperature are shown in Table VII, and values of (F1- PO1) are shown in Table VIII. The published vapor messure data' serve as an indirect check on the measured heat capacities. Values of al are converted to vapor pressures by the re1a t'1011
70'
-620.1 -63.86 298.6 600.9 885.7 1159 1412 1663 1952 2250 2573 2928 3302 3688 4079 4466 4840 5178 5524 5649
-582.2 -32.70 326.0 623.3 902.1 1171 1426 1683 1975 2273 2598 2954 3328 3715 4107 4494 4868 5205 5548 5670
FOz)
80'
-659.9 -96.79 268.8 576.7 867.5 1144 1396 1640
1927 2223 2580 2899 3272 3657 4048 4434 4807 5147 5497 5623
TABLE VI11 RELATIVE P A R ~ A L MOLALFREEENERGY OF WATER PHOSPHORIC ACIDSOLUTIONS, -( Fl - Fal)
IN
wt.%
1E
Hapod
15 20 25 30 35 40 45 50
E:
65 70 75
150
6.31 13.52 22,25 32.98 46.48 63.90 87.06 118.2 158.4 208.9
250 6.46 13,7g 22.66 33.62 47.44 65.14 88.21 118.8 158.8 209.3
400 6.67 6.9: 14,20 14,,8 23,24 23.86 34.51 35.60 48.80 50.52 66.90 69.20 89.80 91.82 119.5 120.4 159.1 159.2 209.5 209.4
i:::: i:: ii::: 477,8 478,7 627.3 818.0
628.5 819.4
700 8oo 7.05 7.17 15,06 15,33 24.25 24,52 36.11 36.64 51.37 52.23 70.35 71.47 92.83 93.80 120.8 121.0 159.2 159.1 209.1 208.7
iii:; iii:: iii:: 477,4 476,3
479,1 629.2 820.0
478,3 628.2 818.6
627.2 817.1
625.8 815.0
i1822 : iii: iig": iiii :i i1813 :; iig"; 1824 1822 1818 1824
90 95 100
2543
...
2557 3365
2574
...
2592
...
2600
...
2607
...
a1 = P / P O
The calculated vapor pressures agree with those in reference 1 within 10j,-the limit of acouracy of the vapor pressure data. The 5% intervals of concentration in the tables are spacesaving but are too broad to show adequately the trends of the data. Partial molal entropies may be calculated from the relation
(Si - Pi) = (Ri - ROi) T- (Fi -
P i )
