HEAT OF DILUTION OF ALCOHOL I N BENZENE BY WILLIS A. GIBBONS
I. Heat of Dilution and Its Relation t o Abnormal Molecular Weights The heat of dilution of alcohol in benzene was measured by Walker and Henderson1in an investigation including several other binary organic mixtures. Walker found a marked absorption of heat when solutions of alcohol in benzene were diluted. His observations were made at a temperature in the neighborhood of 16' C and he did not study the effect of temperature. Bancroft2 discussed the general bearing of the heat of dilution of solution on the observed molecular weight of the substance dissolved in the solvent in question. The points brought out in that paper are of such general interest and, a t the same time, so often overlooked that a brief r6sum6 may not be out of place. The equation of van't Hoff is
where P = osmotic pressure; V = volume through which a semipermeable' piston moves in squeezing out that amount of solvent in which one molecular weight of the solute is dissolved; R = the gas constant; T = the absolute temperature; N/N = gram molecules of solvent per gram molecule of solute (the molecular weights are those of the solvent in the vapor and of the solute in the solution) ; p o = the vapor pressure of the pure solvent at the temperature T ; p' = the partial pressure of the solvent in the solution a t the temperature T. This equation is independent of any assumption in regard to the heat of dilution. Trans. Roy. SOC.Canada, 8 111, IO.=, (1902). *Jour. Phys. Chem., IO, 319 (1906). Kgl. Svenska Vetenskaps Akademiens. Handlingar, phys. Chem., I , 483 (1887).
21,
3 (1886); Zeit.
Heat of Dilutiovl of Alcohol in Bevlzevle
49
If we assume that PV = RT, the preceding equation simplifies to n
N -
Po
=
log
F
which is the van’t Hoff-Raoult formula and is the basis of our methods of determining molecular weights in solution. The equation PV = R T holds only in case the heat of dilution is zero and in case the degree of association or dissociation of the solute does not change with varying concentration. It is customary to attribute all variations to an actual change in molecular weight and to ignore the effect due to the heat of dilution. I n Bancroft’s paper it was pointed out that for any given concentration we could apparently always postulate the relation
where Q is positive if heat is evolved when the solution is diluted. This equation can be integrated if we know Q as a function of T. When Q is zero, we have PlVl PZVZ -- - = const.
T2
TI
and experiments show that the particular solution PV = R T is the right one. When Q is independent of the temperature and is equal to the constant A we have the expression
Adopting the same solution as before we obtain the equation PV
=
RT+A
which can then be substituted in the original van’t Hoff equation. If we make the assumption that Q=A-BT
where A and B are constants, and if we solve as before we obtain the equation PV= RT+A+BTlogT.
50
Willis A. Gibbons
The heat of dilution which applies in these equations, and which we will call QH, is the heat effect observed when N/n molecular weights of the solvent are added to an infinitely large mass of the solution. This heat effect cannot be measured directly but may be calculated from the equation Q
- _ 6Q _N
H --
N’n
6n
where Q is the molecular heat of dilution obtained by diluting to the concentration N/n. It is positive when heat is evolved on dilution. Bancroft showed that the abnormal molecular weights for sodium in mercury and for sulphuric acid in water were eliminated if a correction was made for the heats of dilution which were independent of the temperature in these two cases. With alcohol in benzene, however, the heat absorbed when the solution is diluted is so great that the correction, when applied to the values found by Beckmann,l gives negative molecular weights, an absurd result. This may mean that a temperature coefficient must be taken into account, or that something has been overlooked in the deduction. 11. Theoretical Effect of Temperature on Heat of Dilution
It follows from the first law of thermodynamics that the temperature coeficient of the heat of dilution will be zero if the heat capacity of the mixture is the same as the sum of the heat capacities of its components taken separately, that is, if i t can be calculated linearly. In every other case the heat of dilution will vary with the temperature. This relation is given by Nernst.2 UT + t - UT K,-K= t
- dU - ZT where KO is the original heat capacity, K the heat capacity Zeit. phys. Chem., 2, 728 (1888). Theoretical Chemistry, 6th Ed., pp. g, 161.
Heat of Dilution of Alcohol in Benze?ze
51
after mixing, UT + and UT the heat effects obtained by mixt and T, respectively. For small ing a t the temperatures T temperature differences the right hand expression may be written dU,/dT, or using the notation employed elsewhere in this paper
+
This enables us to calculate the temperature coefficients of the heats of dilution of a pair of liquids if the heat capacities for various proportions are known. I n most calorimetric measurements the temperature is not kept constant throughout each experiment, nor from one experiment to another. It was thought that the measurement of the heat of dilution of alcohol in benzene a t different temperatures might afford interesting results as an illustration of the abovementioned effects of temperature on the heat of dilution, and also on account of the abnormal molecular weights of alcohol in benzene. 111. Apparatus for Measuring Heat of Dilution
at Various
Temperatures The apparatus used in the experiments was what might be called an isothermal calorimeter, similar in principle to those used by Heselius, Waterman, and Cady. The principle is to keep the calorimeter a t a fixed temperature during the mixing, by adding or withdrawing a known amount of heat, depending on whether mixing is accompanied by absorption or evolution of heat. In this way we minimize errors by radiation, particularly those due to changes in temperature on mixing, and we avoid a correction for water equivalent. Since we are working a t a fixed temperature, we eliminate the effect of changing temperature on the heat of dilution, and make unnecessary the determination of the specific heats of the various liquid mixtures. Since i t was desired to measure the heat Jour. de Physique, 7, 499 (1888). Phil. Mag., ( 5 ) 40,413 (1895); Phys. Rev., 4, 161 (1896). Jour. Phys. Chem., 2, 5 6 2 (1898).
52
Willis
A. Gibbons
of dilution a t I O O , 2 0 O and 30 O C, respectively, suitable thermostatic baths were provided to enclose both the calorimeter and the vessel containing one of the liquids to be mixed. The calorimeter vessel was a Dewar tube, A, about 4” in diameter and 8 ” deep, internal measurements. I n this a copper cup or tube, B, about I ~ / ~ ’in’ diameter and 5 ” deep was set. The cup rested upon a wooden block. A stirrer moving perpendicularly operated inside this cup, and another similar stirrer in the space between it and the glass wall. A thermometer reading directly to 0.01O C was held in the outside space between the copper tube and the wall of the Dewar. The calorimeter vessel was closed by two corks, one above the other. One cork was bored to hold the copper tube B, and both corkswere providedwith the necessary holes for the passage of the stirring apparatus, thermometer and solvent inlets. The upper cork was even with the mouth of the Dewar, and both corks were held together by wide glass tubes passing through the holes for stirrers, thermometer, etc. I n addition, a hole in the center of the upper cork allowed warm water to be run into the copper tube. The dilution vessel D consisted of a large separatory funnel, inclosed in an inverted bell-jar. A thermometer reading to 0 . 0 2 O C was held in the interior of the dilution vessel. The latter, with its water jacket, was held a t a fixed height above the table, but could be swung in position over the calorimeter vessel as desired. Both calorimeter and dilution vessel were surrounded by electrically controlled thermostatic baths of the type described by Morgan1 These allowed a regulation of temperature to within a few hundredths of a degree. The water bath E provided a supply of water a t a temperature slightly above that at which the experiment was to be run, the temperature of this bath being regulated by a gas burner connected to a thermo-regulator. The outlet of the water bath E was made as large as possible and held Jour. Am. Chem. SOC.,33, 344 (1911).
Heat of Dilutiow of Alcohol i.tz Benzewe
53
54
Willis A. Gibbons
a thermometer reading to 0.02' C. The glass delivery tube leading from the outlet containing the thermometer could be swung through a wide arc so as to deliver water of known and constant temperature through the cork into the copper tube B, or into the waste pipe as desired. The operation of the apparatus was as follows: The thermostatic baths around the dilution vessel and calorimeter were set a t the desired temperature. A weighed amount of benzene was introduced into the dilution vessel D, and a weighed amount of a known mixture of alcohol and benzene which was to be diluted was put in the Dewar bulb A. Copper tube B was weighed and put in place in the Dewar. The water in the bath E was heated and maintained a t a temperature a few degrees higher than that in the other vessels, and the water allowed to run out through the outlet in which the thermometer was placed through the delivery tube into the drain. The stirrers were set in motion and when the liquids in the dilution vessel and calorimeter were at exactly the same temperature (i. e . , the temperature of the baths surrounding these vessels), the benzene was run into the calorimeter mixing with the solution in the latter. At the same time the temperature of the water running from the water bath E was read and water was run into the copper tube B in the calorimeter, in sufficient amount to maintain practically constant temperature in the calorimeter. The tube B was then removed, its outside carefully dried, and the tube and water weighed. The weight of the water multiplied by the difference in temperature between the bath E and the calorimeter vessel gave the number of calories absorbed in diluting the solution in the calorimeter. For the most part the only correction needed was for the heat added by the stirring. This was found by observing the time required for the experiment, starting with the addition of benzene and ending with the final addition of water for the establishment of the original temperature. The heat effect of the stirrer was found by a blank experiment run on a mixture of known heat capacity.
Heat of Dilution of Alcohol i n Benzene
VI
&.
n
55
Willis A. Gibbons
56
" I, m
Heat of Dilutiow of Alcohol in Benzene
I3
m
Y.
~ wl Wl wl wl wI WI w 1W lw wI wI WI wl wI l
"
Q
0 0 0 0 0 0 0 0 0 0 0 0 0 0
m
57
58
Willis
A. Gibbons
In general, the results of two experiments on mixtures of the same concentration check within twenty calories. The results were calculated to molecular quantities and are given in Tables 1-111 inclusive. Working at 10' C as the temperature of the calorimeter it was found that more accurate results were obtained by maintaining the temperature of the bath C a little below IO' C. In this way the heat introduced by stirring, etc., in most cases was neutralized, making the correction very small. Working a t 30' C as the temperature of the mixing liquids it was found that there was a slight but constant loss by radiation, etc., in spite of the careful insulation of the calorimeter. The time of all the experiments in this series was 8 minutes, for which a correction of -30 calories was introduced, this correction being based on the results of blank experiments. The mixture in Expt. 2 a t 20' was slightly different from that resulting in Expt. I in this series. The corresponding value of Q was found by prolonging the curve from N/w = 0.52 to N/n = 0.59, the latter value representing the solution used in Expt. 2 . The correction due to this difference in concentration was 2 0 calories. While, as stated before, the values at high dilutions are likely to be inaccurate they are of interest in showing the considerable absorption of heat even with very dilute solutions. The bearing of this heat effect a t high dilutions in the case of electrolytes is discussed by Bancr0ft.l
IV. Form of Heat of Dilution Curve Julius Thomsen2 found that the heat of dilution of sulphuric acid in water was represented by the expression
a = - -1 7-8 6 0 ~ x + 1.8
where Q= heat of dilution in gram calories, x = gram molecules of solvent (water) per gram molecular weight of solute. Jour. Phys. Chem., IO, 319 (1906). Thermochemische Untersuchungen, 3, 34.
Heat of Dilution, of Alcohol in Benzene
59
An equation of this type does not represent the results given in Tables 1-111. A formula of the general type
+ 42
x = bQ2 Q--a
where a, b and c are constants, was suggested by Professor Bancroft and found to represent the experimental results quite satisfactorily . As we have already shown, when the heat capacity of the mixture is the same as that of the ingredients the heat of dilution for this case will be independent of temperature, or, in other words, the same for all temperatures. Whatever other conditions obtain a t finite dilutions, a t infinite dilution the heat capacities will be the same; that is, as infinite dilution is approached the temperature coefficient approaches zero. This means that the curves representing the heat of dilution at different temperatures will tend to unite a t infinite dilution. In the equation it will be noted that when Q = a, x = . Accordingly, we may replace the heretofore empirical constant a with the value of the heat at infinite dilution. Taking 3900 g calories for this value, which is probably not far from the truth, we have
where b and c are constants, different for each temperature. Since heat is absorbed, the minus sign is used before the right hand expression above for clearness, the values of the constants changing accordingly. The latter may be determined quite simply by substituting the values of x and Q for two points on each of the three curves and solving the simultaneous equations obtained for'each curve. The observed and calculated values of Q for different values of x at IO', 20' and 30' C are given in Table IV.
Willis' A. Gibbons
60
TABLEI V CALCULATION OF HEAT O F DILUTION CQ By formula of type x = bQ2
+
Q--a Heat of Dilution at 10' C O.OOI~Q' - 15.49 x=-
Q - 3900
N/lz = x
0.52 1.59 3.80 5.85 12.25 25.00 50.70 102.00
Robserved
I
0.52
-200
1.59 2.59 5.79 8.97
-499 -69 -1084 -1399 -2240 -2798 -2893
23.30
47.30 95.30
x = -
0.52 I .63 3.86 8.32 17.25 35 ' 1 0 70.80
-105 -336 -697 -965 -1568
-144 -367 -709 -936 -1423 -2099 -2775 -3277
-39 -3
+29 I45
+ +111
-2210
-277.5 -3190
I
-185 -465 -663 -1 128 -1432 -2190 -2855 -3167
1
-12
0
I
-87
-15 -34 -28 +44 33 -50 +57 +274
+
0 . 0 0 2 2 9 ~- 8 . 2 7 9
-213 -5 7 0 -1018 -1619 -2189 >2671 -3206
Q - 3900 -216 -571 -1010
-1611 -22
IO
-2
762
-3
2 00
$3 +I
-8 -8 +21
+91 -6
Heat of Dilution of Alcohol in Bewxene
61
As-will be seen from these curves, the variations in Q on-either side of the 20' curve are roughly equal. This indicates that the heat of dilution varies linearly with the temperature,-a t-any rate over the range included. As stated in an earlier part of this paper, the effect of temperature on the heat of dilution has been shown to depend on the heat capacities of the liquids before and after mixing. If the heat capacities are the same, i. e., if the specific heat of a
Fig. 2 Heat of Dilution of Alcohol in Benzene a t IO', zoo and 30' C (Omitting highest dilutions)
mixture may be calculated additively from the specific heats of itsTcomponents, the heat of dilution is independent of the temperature. Sulphuric acid in water is an example of this. Walker and Henderson1 have measured the specific heat of mixtures of alcohol and benzene. It may be of interest to calculate from their results what the temperature coefficient would be at various dilutions, and to compare it with the resultsTof the experiments discussed in this paper. The data are given in Tables V-VI. Trans. Roy. SOC. Canada, 8 111, 105 (1902).
Willis A. Gibbons
62
TABLB V Calculation of Temp. Coefficient of Heat of Dilution, from Specific Heats of Alcohol and Benzene (Measured by Walker)
! I
N/?z = x
0.0 I .03
1.585 2.71 6.2 16.8 33.7 57.2 86.2 172 00
Sp. heat observed
0.609 0.528 0.511 0.497 0.460 0.428 0.417 0.414 0.410 0.406 0.405
lHeat - ? ; :A I
Sp. heat calculated'
0.609 0.479 0.460 0.441 0.423 0.412 0.408 0.407 0.406 0,4057 0.405
capac. Ieat capac ity calculated'
28.06 67.04 86.89 101.00
243.4 576.5 1112.6 1864 2774 5472
Difference in heat capacities = AQX
AT
28.06 60 :86 78.28 113.5 223.6 554.9 1092 I833 2750 5468
-6.18 -8. 16 -12.5 -19.8 -21.6 -20.6 -31 -24 -4
00
0
00
0
TABLE VI CaJculation of Heat of Dilution a t 10' and 30' C Based on Experimental Results Qat30'C
Observed
0.52 1.59 2.59 5.79 8.97 23.12
47. I9
-200
-499 -69 -1084 -1399 -2240 -2798
I
-3.2 -9.0 -12.4 -17.4 -19.8 -23.2
-23.5
-212
-566 -791 -1308 -1645 -2470 -3140
I
Qat1o"C
Calculated
Observed
Calculated
-232 -589 -81 j -12 j 8 -1597 -2472 -303 3
-144 -367 -539 -915
-168 -409 -567 -910
-1200 -2000
-1201
-2450
-2008 -2563
If we plot the results, Fig. 3, in the last column, (V), we find that they show a fairly smooth curve with the exception of one or two points, which may be out of place due to experimental error. Considering that extreme accuracy was not sought and that we are multiplying their actual results to a much larger figure, particularly with more dilute solutions, Calculated additively.
Heat of Dilution of Alcohol in Benzene
63
the result is very satisfactory. By drawing a smooth curve, as shown, we obtain a basis for calculating the effect of varying temperature on the heat of dilution. I
I
I
I
,
I
Fig. 3 Temperature Coefficient of Heat of Dilution of Alcohol in Benzene. Calculated from Specific Heats Measured by Walker and Henderson Actual Observations-o
I n Table VI, our experimental results at zoo C are taken as a basis and the values at 30' and 10' C, respectively, are calculated and compared with the observed. The results, are, of course, only approximate, but show that the variations in heat of dilution with temperature are considerable and that the experimental differences are of the same magnitude as would be expected from the theory. VI. The Influence of t h e Heat of Dilution on t h e Lowering of the Freezing Point The molecular lowering of the freezing point for any solvent, which forms the basis of the molecular weight determination by the freezing-point method, is ordinarily taken as a constant quantity for any solvent. For dilute solutions, when there is no heat of dilution and the variation of the heat of fusion with temperature is negligible, van't Hoff gave the following relation :
Willis A. Gibbons
64
where E is the constant for lowering of freezing point and is equivalent to the number of degrees lowering of freezing point produced by dissolving a gram molecule of the solute in IOO grams of the solvent; T is the absolute temperature of the freezing point; R = gas constant = 2 calories; w = heat of fusion of I gram of the solvent. The heat of fusion of the solvent in the presence of a solution is different from that in the presence of the pure solvent, if there is any developm-ent of heat on diluting the solution. This question was discussed by Ewanl who gave the following equation for the relation between the relative vapor pressures of solvent and solution, the heat of fusion, the heat of dilution, and the lowering of the freezing point. lo&
P
=
c U O
T-T’ TOT’
dQ 7”-T
+ &‘y-’ 2 (T’TTo)2 T T-’I” T-’I” T-T’ -.(- - -T- 7” ‘I’ ) I
’
z(
I]
where po = vapor pressure of pure solvent a t To; p’ = partial R = gas constant in pressure of solvent in solution a t To; calories; To = freezing point of pure solvent; T’ = freezing point of solution, TI = temperature a t which heat of dilution is determined; T = temperature a t which osmotic pressure is to be determined; w, is the heat of fusion of one gram molecule, of the pure solvent at To; c = co - cl; co = heat capacity of one gram molecule of liquid solvent; c1 = heat capacity of one gram molecule of solid solvent; X = N/%, N being the molecular weight of the solvent as vapor; dQ/dX = heat of dilution; y = NCL. The value of CL is given by the equation
where W = grams of solvent per gram solute; C1 = specific heat of one gram of solution, and C , = specific heat of one gram of solvent. I n recalculating the molecular weights, using Beckmann’s Zeit. phys. Chem., 14,409 (1894).
Heat o j Dilution of Alcohol in Benzene
65
observations1 for the depression (A) of the freezing point of benzene at different concentrations, we have used the following values:
c, = 0.405 =
T = T0
wo = 2360
278.5
The values of QH, determined by experiments described = 283. in this paper, were taken.at the temperature --TI The values of C’are taken by interpolation from Walker’s2 results, which as herein shown compared well with our results for the heat of dilution a t different temperatures when reduced to the same basis.
6 c1 The values of 6~ were obtained
graphically from the curve representing Walker’s results. Such data as are available indicate that there is no great difference between the specific heats of solid and liqtlid benzene, so the term involving these quantities, and representing the change of heat of fusion with temperature is neglected. A little inspection of Ewan’s equation will show that the net effect of the heat of dilution is as follows: If the solvent melts in the presence of the solution, and heat is evolved on dilution of this solution, this heat will appear in addition to the heat of fusion, and d l make the apparent heat of fusion abnormally large. This, in turn, will make the constant, E, abnormally small, and, since we have the relation
used by Beckmann, the molecular weight calculated thus will be too low; that is, if heat is evolved on dilution, the molecular weight calculated from the freezing-point lowering will be too low, on account of the heat of fusion being different in the case where there is a heat of dilution. When heat is absorbed on dilution the opposite is true throughout: the heat of fusion is too low, and the molecular weight too large. Beckmann: Zeit. phys. Chem.,
2, 728
(1888).
* Trans. Roy. SOC.Canada, 8 111, 105 (1902).
66
Willis A. Gibbons
Heat of Dilution o j Alcohol in Benzexe
67
The heat of dilution referred to here, in connection with the heat of fusion, is that caused by adding a gram of the solvent to a very large mass of solution of the concentration in question and is represented by
g,
where W
=
the
number of grams of solvent to one gram of solute. If we calculate the value of log P
P’ ’
according to the
equation of Ewan, we get a result similar to that discussed above with reference to the effect of heat of dilution on the molecular weight, calculated from the van’t Hoff-Raoult formula : The results of this calculation of the molecular weight allowing for the heat of dilution are given in Table VII. As will be noted, the results differ from those obtained by Beckmann (Table VII, Col. 9) for the following reasons: ( I ) The presence of a negative heat of dilution tends to make Beckmann’s values too high for the reasons above stated. (2) The experimental value for E, used by Beckmann is 49, whereas the theoretical value calculated in the equation E = RT2/10oW is 51. This will tend to make Beckmann’s values too low by the same percentage throughout. ( 3 ) The value E = R T 2 / ~ o o W is only true strictly speaking a t infinite dilution where not only the heat of dilution but also the lowering of the freezing point are negligible as regards the total value of the absolute temperatures of the freezing point of the pure solvent and of the solut‘on. As Ewan points out the true expression for E is E = - RT, T’ IOOW’
Tobeing the absolute temperature at which the solvent freezes and T’ the absolute temperature a t which the solution freezes. For very dilute solutions we have, of course, T = To = T’, therefore, RT2 E = -IO0 W’
68
Willis A. Gibbows
At more concentrated solutions where To>T’, we have
T2>TOT’ and the true value of E will tend to be less. This effect tends to make Beckmann’s values too high. We see that factors ( I ) and (3) tend to make Beckmann’s values too high, the error being proportional to the concentration, and becoming practically zero in each case a t infinite dilution. The second factor (2) tends to make all of his results somewhat low, but by the same percentage. This composite correction produces the result noted in Column IO, Table VII, namely, that at high dilution the second effect is predominant, and our corrected values are higher than Beckmann’s. For instance, at x = 359, he obtained 45.9, and the corrected value is 47.5, of course. dQ/dW is negligible and T = To = T’, practically. On the other hand a t high concentrations factors ( I ) and (3) predominate, and make the corrected molecular weight lower than that given by Beckmann. This accounts for the apparent discrepancy between Columns 9 and IO, causing one to be lower than the other in some cases and higher in others. Since we are not 6nly interested in these corrected values, but also in noting the effect of heat of dilution, we have calculated molecular weights using the formula given by Ewan for log p 0 / p ’ , but making the deliberate assumption that Q and y are equal to zero. These values, found in Column I I , compared with those in Column I O show the error caused by the effect of heat of dilution on the molecular lowering of the freezing point. It must be noted that in all of the above calculations for the molecular weight, where we make allowance for the heat of dilution it is only as i t affects the heat of fusion, and we make the explicit assumption that PV = RT, which is necessary for the relation n = log 7 P O
N
P
Heat of Dihtion of Alcohol in Benzene to be true.
69
However, as already stated PV = RT only when
QH equals zero; otherwise a still further correction is needed
before we arrive at the true molecular weight, as will be shown presently. VII. The Effect of Heat of Dilution on Molecular Weight As has already been shown in this paper, when QH = a constant, A, and is independent of the temperature, we have for any fixed concentration the relation PV = RT * A. The sign * is used to avoid confusion, the heat effect if heat is evolved on dilution. I n the parbeing taken as ticular case of alcohol and benzene in which we are now interested, heat is absorbed on dilution, so we have, in reality, PV = RT -A, if we adopt the same form of function as that already applied with success to sulphuric acid in water. That is, the observed value of PV is lowered if we apply the correction for the heat of dilution; in other words, the true osmotic work is greater than it appears to be if the heat of dilution is neglected. This means that our observed molecular weights are too high in the ratio of RT : RT - A. However, in the case of alcohol and benzene, the value
+
is so great that for most concentrations we obtain very large negative corrections, in most cases so large that the resulting “corrected” molecular weight is absurdly low, or even negative, as shown in Table VIII. These values are calculated from our results a t IO’ C, the values for 6Q/6xlOo being found graphically except for the two highest dilutions, for which they were calculated by differentiafion of the equation given in Table IV. Although, as will be seen, the slope is relatively small at these dilutions the correction is so considerable that even in these cases the resulting corrected values are much lower than the observed, which are here close to the true value. The data are given in Table VI11 showing the molecular weights at different dilutions as observed by Beck-
Willis A. Gibbozzs
70
mann, corrected as discussed in the preceding section of this paper for the change in the heat of fusion of a solvent in the presence of a solution when there is a heat of dilution.
VIII. Effect of Temperature We have already discussed in general the form of the equation when Q varies with the temperature. If we apply this correction adopting the same form of function, we obtain still more improbable results than those given in Table VI11 ; the correction causes an increase in the observed molecular weight, which is already too high. As we have shown earlier in this paper, a t any fixed concentration, we have: , TABLE VI11 Application of Heat of Dilution to Observed Molecular Weights of Alcohol in Benzene, assuming Q Independent of T dol. weights observed N/n = : :Beckmann) Revised (Tab. VII,
Col. I
1.82 2.61 4.02 6.66 ’IO. I 1
16.92 25.8 54.2 119.5 359
T
RT
9 I0”X
“Mol. wt.” corrected
3
4
6
7
273.5 264.5 275 275.8 276.3 276.8 277.1 277.6
547 549 5 50 551.6 552.6 553.6 554.2 555.2 556 556.6
11)
2
308 256 207
158 131 IO0
84.5 62.9 54.6 47.5
278
278.3
-179 -1.53 -124 -89.6 -67.2 -43.5 -30 -13.2 -3. I -0.49
-326 -400 -498 -596 -679 -736 -774 -716 -370 -176
132
69.5 19.3 -19.3 -30. I -32.9 -33.5 -20.8 18.3 32.4 (1)
Let us assume that the same solution applies as in the case of sodium in mercilry, and sulphuric acid in water ; also that the heat of dilution varies with the absdute temperature and is expressed by the following relation :
QH = * A * BT,
(2)
Heat of Dilution of Alcohol in Benzene
71
as before, choosing our signs with the understanding that QH is positive if heat is evolved on dilution. A and B are constants and T the absolute temperature. By integration of ( I ) we obtain PV = RT * A * B T l o g T (3) To inquire into the meaning and value of the constants A and B, we make use of the rule already discussed in which we saw that the temperature coefficient of the heat of dilution is equal to the difference in heat capacity of the liquids before and after mixing. This is discussed by n'ernstl and Ewan;2 the latter's expression for this, changing symbols and concentration basis to harmonize with our present requirements is: QT = QT1 [(ms ~ L ~ ) C( Z~ 4 % m~xCw)I[TI--] (4), where Q T = molecular heat of dilution a t absolute temperature T a t which osmotic pressure is to be determined; QT, = molecular heat of dilution a t absolute temperature T1, at which heat of dilution is measured. vns = molecular weight of solute; vnz = molecular weight of solvent, as gas; C1 = specific heat of solution resulting from mixing; C s = specific heat of
+
+
+
solute; Cw = specific heat of solvent; x =
N
1z
- gram
molecules of solvent per gram molecule of solute. Since in differentiate
QT
(2)
we are interested in QH
=
dx
x, we
with respect to x, and multiply by x:
In order to obtain this in the form of have only to simplify the above:
gT x = ~-' Q Tx + DTlx - DTx 6X
6X
Theoretical Chemistry, 6th Ed., pp. 9, 161. Zeit. phys. Chem., 14,409 (1894).
*A *
BT, we
Willis A. Gibbons
72
I n the case of alcohol and benzene where we have heat absorbed, QT is negative, likewise SQT - x. 6X
Where T1>T and the heat capacity after mixing is greater than before, as occurs with alcohol and benzene, the entire second member of (4) is positive in value. Ordinarily, QT will, therefore, be numerically less than the heat of dilution a t the temperature where this value is measured, but it will be of the same sign, namely, minus. Since, however, we assume for lack of better information, that D is always positive, where T is sufficiently lower than TI,the second member of (6) may be so much larger than the other two members of the right hand side of this equation which are in this case of opposite sign to that of the second member that SQT -. x will be positive. 6X
This, however, is not
likely to be met with in the experiment since we are ordinarily limited in our values for T by the freezing point of benzene solutions. The same considerations hold on differentiation. We now have (6) the heat of dilution expressed as a function of the absolute temperature and in the form QH
=
*A
* BT - that
is
(2)
A t any fixed concentration,
+
-~' Q T DT, 6X
is a constant, which we shall call A1, to distinguish from the constant A used previously when the heat of dilution was independent of the temperature. Likewise D is a constant which to harmonize with ( 2 ) we shall call B. From this we have
Before going any further and substituting in the integrated
Heat of Dildion of Alcohol in Benzene
73
expression ( 3 ) , for clearness the nature of the signs can still further be fixed for ordinary conditions. I n the case of alcohol and benzene ordinarily T1> T and
~ Q T -x 6X
Q T ~ and ~x are nega6X
tive, and D is positive. Therefore, in this particular case the express'on assumes the form (8) 6X =
A1 - BT,
where A l = - X 6QT1 x B
=
+ DTlx
Dx.
As we have already seen, if QH = * A1
* BT,
adopting our original solution, we have PV = RT * A1 * B T log T. (3) Substituting (8) the expression QH = AI - BT which fits this particular case, we have PV = RT A1 B T log T, (9 ) where, as before.
+ +
A1
=
-
B
=
Dx
9+
DTlx
or PV
RT - l~-
Q T
6X
x
+ DTlx + DTx.
(10)
If we try to apply this expression to correct the observed molecular weights of alcohol in benzene, we get a still more absurd result, namely, the correction raises the molecular weight since DTlx and DTx are both positive and being much
greater than 8QT -' x will ordinarily increase rather than di6X
minish the numerical value of the right hand s'de of the equation ( I O ) , indicating that the observed osmotic work is greater than the true value or that the observed molecular
Willis A. Gibbons
74
weight is lower than the true molecular weight. As we have already seen, the observed molecular weights are already much larger than would be expected, so it seems evident that the form of function used in previous cases cannot be used here, and the correction for the heat of dilution cannot be satisfactorily applied to the observed molecular weights of alcohol in benzene. There seems to be no question but that the observed molecular weight is always too high when heat is absorbed on dilution, and always too low when heat is evolved on dilution. Consequently a correction must be applied ; but the mathematical theory of the subject has not been developed to such a point that we know how to apply the correction. If we take the simple hypothetical case in which the heat of dilution is negative and independent of the temperature the formula PV = R T - A should apply to the case. If we make the further assumption that A > R T we come out with an absurd result. Until this hypothetical case has been treated satisfactorily, it is hardly worth while to do more experiments. The first question to be decided is whether the error is in the
or whether it is in the particular solution PV = R T - A. In conclusion I wish to express my hearty thanks to Professor W. D. Bancroft for the suggestions and encouragement which he has given me in this work. Cornell University
