J. Phys. Chem. 1988, 92, 487-493
an excess electron is deposited on a smaller ZnO particle, the number of ZnO molecules affected is smaller, this effect leading to a smaller change in absorbance. On the other hand, one would expect that the optical changes should also become smaller in the case of very large particles (larger than used in the present work) as the electric field produced by the excess electron would be rather weak in parts of the particle. These two opposing effects make us believe that there might exist a definite particle size where the absorbance changes that accompany the deposition of one electron are most pronounced. The large bleaching coefficient of 1.1 X lo5 M-' cm-' , observed for 40-50 A particles, may be taken as an indication that this particle size is close to 40 A. However, further experiments with particles of different size are required to check the validity of this supposition. It has recently been proposed that the fluorescence of colloidal ZnO in aerated solution is brought about by a "shuttle" mechan i ~ m .An ~ electron generated by light absorption transfers to adsorbed O2to form 02-.The latter transfers the electron into a deeper trap on the surface. Recombination with a preexisting hole is accompanied by light emission. In the course of the present studies on solutions in which 0, is formed a few experiments were carried out to check this mechanism. In an experiment with the analyzing light beam shut off, fluorescence light should have been emitted, if 0, transferred an electron to ZnO and if recombination with a preexisting hole took place. No fluorescence was observed. We have to conclude that the more direct kinetic method of pulse radiolysis does not confirm the above mechanism. A shuttle mechanism in solutions containing methylviologen, MV2+, as additive has also been proposed, MV+ acting as the shuttle agent which is formed by electron transfer to MV2+and then transfers its electron into a lower surface state of ZIIO.~More recent pulse radiolysis experiments in our laboratory showed that MV+ does not react with ZnO particles. Again, it must be concluded that a shuttle mechanism is not operative.
shorter wavelengths (Figures 9 and 10). Upon application of the first pulse (Figure 10) the changes in absorbance were smaller than in the subsequent ones. This is attributed to traces of oxygen adsorbed on the colloidal particles which cannot be removed by bubbling the solution with another gas. The small radiation dose applied in the first pulse was sufficient to remove this residual oxygen. The rate constant of reaction of the radicals with ZnO particles was determined in a single-pulse experiment using a solution which had been preirradiated with two pulses. At a concentration of ZnO particles of 3.5 X 10" M the half-lifetime of the 340-nm bleaching, which obeyed pseudo-first-order kinetics, was about 2 ms. A rate constant of 1 X lo8 M-I s-l is calculated. This value is more than 10 times smaller than expected for a diffusion-controlled reaction. It has already been mentioned that the intensity of the analyzing light beam had to be kept low in the laser flash and electron pulse experiments using N20-saturated solutions in order to avoid storage of electrons on the ZnO particles before the laser flash or electron pulse arrived. In experiments with aerated solutions (Figures 9a and 11) the intensity of the analyzing light beam was as strong as in ordinary pulse radiolysis experiments with chemical systems that do not photolyze. The reaction of oxygen with stored electrons being rather slow as described above, illumination with the analyzing light beam led to a certain stationary concentration of stored electrons on the colloidal particles. The oxidizing radicals which were formed in the electron pulse via the processes sCH2OH 0 2 *02CH20H (1)
+
-
*O2CH2OH
HO2
+ CH2O
487
(2)
+
H 0 2 s H+ 02(3) thus reacted with ZnO particles already carrying electrons. Removal of an electron by an oxidizing radical was accompanied by the recovery of the bleaching (at 345 nm) and of the absorption (at 320 nm) which were originally caused by this electron. The result being that an absorption signal at 345 nm and a bleaching signal a t 320 nm were now observed (Figure 11). The mirror images of the difference spectra obtained after the attack of reducing and oxidizing radicals (Figure 9) are thus readily understood. From the data in Figure 11 it was calculated that the oxidizing radicals reacted with a specific rate of 3.2 X lo6 M-l s-l with the colloidal particles. The changes in absorbance were smaller when ZnO particles of smaller size were used in the experiments of Figure 9. When
Acknowledgment. We express our gratitude for the excellent assistance in the laboratory to Mrs. M. Weller, for helpful discussions to Mrs. L. Katsikas, and for cooperation in the electron microscopic investigations with Dr. W. Kunath, Dr. B. Tesche, and Mr. K. Weiss in the Fritz-Haber-Institut, Max-Planck-Gesellschaft. We also thank Dr. E. Janata for advice in the pulse radiolysis experiments. Registry No. ZnO, 13 14-13-2; CH,OH, 2597-43-5; 01, 7782-44-7;
formaldehyde, 50-00-0.
Thermodynamic Properties of Liquid Toluene L. Ter Minassian,* K. Bouzar, and C. Alba Laboratoire de Chimie Physique, 1 I rue P. et M . Curie, 75231 Paris, France (Received: May 13, 1987;
In Final Form: July 29, 1987)
Expansivity and compressibility measurements have been made for toluene in the temperature range 200-450 K up to 4 kbar. The experiments were performed by a modified piezothermal technique bringing out a self-consistent set of data for these quantities. An anomaly is observed in the behavior of the heat capacities at low temperature in the high-density region. This has been interpreted as a progressive cwversion from one type of molecular motion to another as volume increases. The qualitative aspects of this phenomenon are discussed.
Introduction In a previous study,' we have reported a set of empirical equations of the thermodynamic properties of the liquid phase based on a direct measurement of the expansivity a as a function
of pressure and temperature. Derived from such molecules as C 0 2 and n-butane, these equations exhibit simple forms: (i) As a function of pressure, the expansivity follows a square root law CY
(1)
Alba, C.; Ter Minassian, L.; Denis, A,; Soulard, A. J . Chem. Phys.
1985.82, 384.
0022-3654/88/2092-0487$01.50/0
= A / @ - pJ'I2
(1)
where A and p Aare pressure-independent coefficients. In spite 0 1988 American Chemical Society
488
The Journal of Physical Chemistry, Vol. 92, No. 2, 1988
Ter Minassian et al.
of its generality, this law has not yet received theoretical support. (ii) When extended to take into account the temperature dependence, eq l may be written in the following phenomenological form a =
.ob0
- Px)’/2/b - P P
(2)
n
where px = f( specifies the pseudocritical line of divergence of (Y and where the pair (ao,po) defines the coordinate of a single intersection point for all the isotherms. This feature is in agreement with and enhances the observations already performed by Bridgman at the beginning of the century.* Equation 2 may be applied to most of the range of existence of the liquid phase. Such a typical behavior is shown in Figure 1 in the case of nbutane. However, while all liquids investigated-pure and homogeneous mixtures-conform to eq 1, their temperature dependence may differ from that predicted from eq 2. Consequently, different behavior of the heat capacity C, along an isotherm may be expected as emphasized by the following equation (aCp/ap)T = -VT[a2
+ (aa/aT),]
I
500
I
loo0
I
1500
I
m p
1
bor
Figure 1. Typical behavior of the expansivity 01 as a function of pressure. Isotherms in the case of n-butane are within the range of temperatures from 169 to 41 1 K.
(3)
where the quantity ( a ~ u / a Tcontrols )~ the decrease (stabilization)-or increase-of C, as a function of pressure. As the present study on toluene will show, the driving parameter (aa/aT), may be used as a classification criterion for liquids: liquids following eq 2 will be referred to as “simple” as opposed to “complex” liquids. Also, this work has provided the opportunity to extend our observation field to additional thermodynamic quantities. The direct determination of the difference in the heat capacities (C, - C,) and of the compressibility K has been performed by a novel technique. This allowed the development of a self-consistent set of thermodynamic properties.
Apparatus and Experimental Method The determination of the expansivity by the piezothermal method is already w e l l - k n ~ w n . ~ - ~With regard to the new technique developed here, we recall that a quantity of heat liberated by a variation Ap of the pressure is measured isothermally. When V, is the internal volume of a vessel and a , a, are the expansivities of a sample and its container, respectively, we have
SQ = -(a
- a,)TVrAp
(4)
An alteration of this procedure gives new thermodynamic quantities: We consider two identical vessels communicating through a capillary tube of a negligible internal volume. One of them is positioned in a calorimeter, while the other is in a thermostat at the same temperature. The system is filled with a fluid and closed at a pressure p . The temperature of the thermostat is modified by an increment AT. (Typical increments are AT = 1 f 0.1 “C.)The ensuing overpressure in the whole system gives rise to a heat effect in the calorimeter according to eq 4. An elementary analysis gives the quantity of heat measured with the calorimeter (5)
where K is the compressibility of the sample and where F is a positive constant giving the deformation of the vessel under the constraint dp:
This type of measurement may be identified with a direct de(2) Bridgman, P. w. Proc. Am. Acad. Arrs Sci. 1913, 49, 3 . (3) Ter Minassian, L.; Petit, J. C.; Van Kiet Nguyen; Brunaud, C. J . Chim. Phys. Phys.-Chim. Biol. 1970, 67, 265. (4) Petit, J . C.; Ter Minassian, L. J . Chem. Thermodyn. 1974, 6, 1139. (5) Ter Minassian, L.; Pruzan, P. J . Chem. Thermodyn. 1977, 9, 375.
Figure 2. (a) Usual piezothermal device for the measurement of the expansivity with a flux calorimeter 1 and pressure gauge 2. (b) The modified piezothermal device with its thermostat 3 and high-pressure valve 4.
termination of the quantity (Cp- CJ: for the limits where a, = 0 and r = 0, the right-hand side of eq 5 reduces to where V is the molar volume. In fact, the coefficient r is not negligible and should be calibrated in practice. The experimental procedure consists of the following two steps: A first set of measurements is performed up to the highest pressure in order to determine the left side of eq 5 . This is completed by a second set of measurements in order to determine the quantity ( a - CY,)by the usual piezothermal method. The compressibility coefficient K is therefore determined provided the coefficient r is known. The calorimeters and the high-pressure part of the experimental device were described The device for the measurement of K is shown schematically in Figure 2b, which may be compared with our usual piezothermal device in Figure 2a. The vessels are high-pressure cells of heat-treated stainless steel. They are fitted with a Bridgman closure on one extremity while the other is connected to the high-pressure system through a capillary tube. Their shape (6- and 17-mm internal and external diameters, respectively; V, = 1.262 cm3) and the mechanical properties of the material allow estimation of the quantity r: r = 2.5 X 10“ bar-’. Nevertheless, we could not take into account the supple stack of gaskets of the Bridgman closure. Consequently, a calibration was performed with water as a reference. In order to have a constant a as a function of pressure, the temperature was 49 OC.’ The experiments performed in a (6) Ter Minassian, L.; Milliou, F. J . Phys. E 1983, 16, 450. (7) Bridgman, P. W. Proc. Am. Acad. Arts Sci. 1912, 48, 309. Ter Minassian, L.; Pruzan, P.; Soulard, A. 1981, 75, 3064.
The Journal of Physical Chemistry, Vol. 92, No. 2, 1988 489
Thermodynamic Properties of Liquid Toluene pressure range up to 3 kbar gave via eq 5 an average value of r = 5.8 bar-’ (*lo%). Processing of the Data. The set of experimental points for a = fi),K = g(p) as a function of pressure on different isotherms are processed through the following steps S: ( S l ) A general equation fitting the experimental points as a function of pressure and temperature is determined f f
10
(8)
=f(P,T)
(S2) The difference in compressibility between temperature T and a reference temperature T = Tr is computed algebraically:
-ST,( d f f / a P ) ~d T T
AK(T#)
K(T#) - K(Tr,P) =
(9)
(S3) Each of the points of the compressibility coefficient K,, = g(Tn,p,) (on isotherm T,,at pressurep,) is computed in order to determine a corresponding point K, = g(Tr,pm)on a reference isotherm at T = Tr: K(TrrPm) =
Knm
- AK(Tn,Pm)
3 1
I
I
I
I
2000
1000
I
I
1
I
LOO0
3009
p
bar
(10)
(S4) The set of points obtained from step S 3 is fitted with an empirical equation by the least-squares method K(Tr*P)= h(Tr,P)
I
tB\
(11)
(S5) Equations 8 and 11 are integrated algebraically in order to determine the volume
and 3 1
The figure for V ( T r p = 0 )is found from the literature. (S6) A numerical integration establishes the difference in heat capacity C, between pressures p and p = 0
where a and V are given by eq 8, 12a, and 12b. (S7) The difference in heat capacities (Cp- C,) is computed from eq 8, 9, 11, 12a, and 12b
cp- c, =
( Y 2 U / K
(S9)The absolute value of C, is then computed C,(T,P) = cp(T,P=o) iAcp- (c, - CU)T,P (16) where C,(T,p=O) is obtained from the literature from the melting to the boiling points.*
Results I . Expansiuity. The experiments were performed along 16 isotherms up to a pressure of 4 kbar, within the temperature range 200-450 K. The results are plotted in Figure 3a,b, arbitrarily separated into “low”- and ”high”-temperature regions. This selection allows observation of the intersections of the isotherm at low pressure in the “low”-temperature region, pointing to higher pressures as the temperature increases (Figure 3a). In the “high”-temperature region, the intersection takes place around 2 kbar (Figure 3b). (8) Scott, D. W.; Guthrie, G. B.; et al. J . Phys. Chem. 1962, 66, 911.
I
I
2000
3000
I
LOO0
p bar
Figure 3. Expansivity a as a function of pressure in the case of toluene. (a, top) A selection of isotherms on the “low”-temperatureregion: A, 304.5 K A, 263.0 K 0 , 2 4 1 . 4 K; 0,218.9 K +, 202.0 K (-) the fit from eq 17. (b, bottom) A selection of isotherms on the “high”-temperature region: A, 458.3 K; 0,401.0 K; 0 , 335.5 K; (-) the fit from eq 17. TABLE I: Values of the Constant Coefficients as They Appear (A) in Eq 17s-17c of the Expansivity Coefficient and (B) in Eq A4 of the Compressibility Isotherm at the Reference Temperature” (A) a0 = 1.5735 X lo-’
(14)
(S8) The difference of the heat capacity C,between pressures p and p = 0 is determined from the tabulated values of C, and eq 14: AC, E C,(T,p) - C,(T,p=O) = Acp - [(cp- c v ) T , p - (cp- cu)T.p-Ol (15)
I
1wo
MO
(B) K-’
= 150.82
A , = 470.0 At = 3026.6
a, = -3.1882 X lom2K-’
a2 = -2.0498 X lo-)
K-’
T, = 591.77 K pc = 42.15 bar “The coefficients agree with the pressure in bar units. Proceeding now according to step S1,the following equation fits our experimental points a = ao(mo- mJ2/(m
- mA)l/*
(17a)
with
mA = (a,t + 1 ) / ( a 2 t + 1)
(17b)
where ao,mo, a , , and a2 are constant coefficients and where the variables are reduced according to the critical coordinates (Tc,pc). t = T, - T, m = P/P,, mA = P A / A (17~) The values of the coefficients are given in Table I, and the general aspects of the isotherms of a may be seen in Figure 4. 2. Compressibility Coefficient K ( T,p). The experiments are performed along four isotherms at “high” temperatures, Le., in the range from 326 to 432 K, in order to minimize errors (see next paragraph). The results are plotted in Figure 5 as a set of K,, experimental points on isotherm T, and pressure p,. Pro-
Ter Minassian et al.
490 The Journal of Physical Chemistry, Vol. 92, No. 2, 1988
TABLE I 1 Difference AC, and AC, of the Heat Capacities at Constant Pressure and Constant Volume, Respectively" T = 180K T = 200 K T = 220 K T = 240 K T = 260 K T = 280 K T = 300 K ACp 4Cv 4cp 4C" 4cp 4C" 4cp 4C" ACp AC" ACp ACu Pfbar ACp AC" 3.67 -6.13 0.97 -4.02 -1.19 -2.16 -3.05 -0.60 7.35 -8.44 12.94 -10.85 4000 22.74 -13.16 3.22 -6.03 0.62 -3.93 -1.46 -2.08 -3.26 -0.51 6.75 -8.32 12.10 -10.69 3800 21.45 -12.93 2.78 -5.80 0.28 -3.73 -1.73 -1.89 -3.48 -0.34 6.15 -8.04 11.25 -10.33 3600 20.16 -12.43 5.56 -7.62 2.34 -5.46 -0.05 -3.43 -1.99 -1.63 -3.68 -0.10 10.42 -9.80 18.87 -11.72 3400 0.20 1.91 -5.03 -0.38 -3.06 -2.25 -1.30 -3.88 4.98 -7.11 9.59 -9.15 17.59 -10.85 3200 0.54 1.49 -4.54 -0.70 -2.64 -2.49 -0.93 -4.07 4.40 -6.52 8.76 -8.42 16.31 -9.88 3000 1.07 -4.01 -1.01 -2.19 -2.72 -0.54 -4.25 0.89 3.84 -5.88 7.95 -7.63 -8.85 2800 15.04 1.25 3.28 -5.23 0.67 -3.46 -1.30 -1.73 -2.94 -0.13 -4.41 7.14 -6.82 -7.80 2600 13.77 0.27 -4.55 1.61 2.74 -4.57 0.29 -2.92 -1.58 -1.25 -3.14 6.35 -6.01 -6.77 2400 12.52 0.65 -4.67 1.94 2.22 -3.93 -0.07 -2.39 -1.84 -0.82 -3.32 5.58 -5.22 -5.79 2200 11.28 2.24 1.72 -3.32 -0.41 -1.89 -2.07 -0.40 -3.48 1.00 -4.76 4.82 -4.47 -4.87 2000 10.06 1.31 -4.82 2.48 1.25 -2.75 -0.72 -1.42 -2.27 -0.03 -3.60 4.09 -3.78 8.86 -4.03 1800 2.68 0.30 -3.68 1.56 -4.84 0.81 -2.23 -1.00 -1.01 -2.43 3.38 -3.14 7.68 -3.28 1600 1.74 -4.80 2.80 0.41 -1.77 -1.23 -0.65 -2.55 0.56 -3.71 2.72 -2.57 6.53 -2.62 1400 0.76 -3.67 1.85 -4.69 2.84 0.06 -1.36 -1.41 -0.36 -2.60 2.09 -2.06 5.41 -2.04 1200 2.78 1.87 -4.49 0.88 -3.55 1.52 -1.61 -0.23 -1.02 -1.52 -0.12 -2.58 4.34 -1.56 1000 1.80 -4.16 2.61 0.92 -3.31 0.04 -2.46 1.01 -1.22 -0.45 -0.73 -1.54 3.32 -1.14 800 2.29 0.14 -2.20 0.86 -2.92 1.60 -3.65 0.58 -0.88 -0.56 -0.49 -1.44 2.36 -0.80 600 1.26 -2.89 1.79 0.71 -2.31 0.17 -1.76 0.25 -0.57 -0.56 -0.29 -1.19 1.47 -0.50 400 1.05 0.43 -1.39 0.75 -1.75 0.12 -1.06 0.05 -0.28 -0.39 -0.13 -0.74 0.68 -0.25 200 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0 0.00 0.00 "AC,
C,(p,T) - C"(p=O,T). Values are in J
CP@,T) - C,(p=O,T); AC,
2
T = 320 K 4C" -4.75 0.66 -4.92 0.75 -5.09 0.92 -5.26 1.15 -5.41 1.42 -5.55 1.73 -5.68 2.06 -5.80 2.39 -5.90 2.71 -5.97 3.00 -6.01 3.26 -6.02 3.47 -5.99 3.62 -5.89 3.69 -5.72 3.68 -5.45 3.55 -5.04 3.29 -4.43 2.87 -3.52 2.24 -2.16 1.32
4cp
0.00
0.00
K-'mol-' units. 30
K -1
t
R 20!?+
I
I
I
1000
2000
3000
?'
I
J-
4000
0
bar
Figure 4. Expansivity a as a function of pressure computed from eq 17. The general aspect of its behavior on isotherms from 200 to 480 K.
ceeding according to step S2, the quantity A K ( T , ~given ) as eq A1 of the Appendix is computed. Then, by step S3, the K, points are processed in order to define the reference isotherm with the corresponding set of K , points. This set may be observed in Figure 5, along the reference isotherm at the temperature T , = 178.2 K. Finally, by step S4, eq A4 of the Appendix is determined with the relevant constants given in Table I. The agreement of the experimental points with the computed isotherms resulting from the inverse procedure may be observed in Figure 5 . This is achieved with eq A1 and A4 and K ( T , ~=) 4 T r , p ) + AK(T,P) (18) Remark: the reference T,is the melting temperature of toluene at p = 0, so that the compressibility isotherm may be considered as virtual or in an 'overcompressed" state. Finally, Figure 6a,b illustrates the general behavior of K as a function of temperature and pressure, respectively. 3. Volume V(T,p). The integrals (eq 12a, 12b) of step S5 are given in the Appendix (eq A5, A6). They determine the molar volume in the plane (p,T) provided the quantity V(T,,p=O) is known. The literature gives the following v a l u e ~ : ~V, J ~= 95.73 cm3, T, = 178.2 K. (9) Bosio, L.;Defrain, A.; Folcher, G. J . Chim. Phys. Phys.-Chim. Biof.
I
-I
1000
--
1
I
2000
-
-.
I
3000
1
J
LOO0
bar
Figure 5. Compressibility coefficient K as a function of pressure on four isotherms: +, 431.9 K; A, 382.0 K; X, 335.4 K; 0, 326.0 K. On the lower curve, the set of points 0 results from the processing of the data according to step S3. It determines the reference isotherm at T, = 178.2 K; (-) computed according eq 18.
4. The Other Thermodynamic Quantities, ACp and AC,. The parameters a,K, and Vas determined from eq 17a-l7c, eq Al-A4, and eq A5, A6 form a self-consistent set of thermodynamic quantities. This consistency derives from the procedure, so that we may continue through steps S6-S8 on firm grounds. The differences ACp and AC, are given on Table 11, and Figure 7 illustrates the behavior of AC, as a function of pressure on different isotherms. With these results, the determination of the thermodynamic properties issued from the piezothermal method is complete. Criticism of the Experimental Results. It is difficult to appreciate the degree of accuracy of all the thermodynamic functions because of their mutual dependence arising from the interplay of derivations and integrations. We shall therefore concentrate our discussion on the primary experimental quantities a and (Cp - CUI. The accuracy on the expansivity a is of the order of 0.5% to 1%, while the fit with eq 17a and 17b determines an average accuracy of f 1.5% with respect to all the experimental points. Comparison of our data with existing values, at "high" temperatures and p = 0, reveals a significant discrepancy."J* This is
1976, 73, 813.
(10) Anderson, M.; Bosio, L.; Bruneaux-Poulle, J.; Fourme, R. J . Chim. Phys. Phys.-Chim. Biol. 1977, 74, 69.
(1 1) Marshall, J. G.; Staveley, L. A.; Hart, K. R. Trans. Faraday SOC. 1956, 52, 19.
The Journal of Physical Chemistry, Vol. 92, No. 2, 1988 491
Thermodynamic Properties of Liquid Toluene T = 340 K ACp AC, -6.39 -6.53 -6.67 -6.79 -6.91 -7.02 -7.1 1 -7.19 -7.25 -7.28 -7.29 -7.25 -7.17 -7.03 -6.8 1 -6.48 -5.99 -5.28 -4.23 -2.62 0.00
1.61 1.71 1.87 2.09 2.36 2.65 2.95 3.26 3.55 3.82 4.05 4.23 4.35 4.38 4.32 4.15 3.83 3.33 2.60 1.54
0.00
T = 360 K ACp AC, -8.05 -8.16 -8.27 -8.36 -8.45 -8.53 -8.59 -8.63 -8.66 -8.66 -8.63 -8.56 -8.44 -8.26 -7.99 -7.60 -7.05 -6.24 -5.05 -3.18 0.00
2.27 2.37 2.54 2.75 3.01 3.29 3.58 3.87 4.15 4.39 4.60 4.76 4.85 4.86 4.78 4.57 4.21 3.66 2.85 1.69 0.00
T = 380 K ACp AC, -9.80 -9.89 -9.96 -10.03 -10.09 -10.14 -10.18 -10.19 -10.19 -10.16 -10.10 -10.00 -9.85 -9.63 -9.32 -8.88 -8.25 -7.36 -6.01 -3.86 0.00
2.65 2.76 2.93 3.14 3.40 3.67 3.95 4.23 4.49 4.73 4.93 5.07 5.15 5.14 5.04 4.81 4.43 3.85 3.00 1.78
0.00
T = 400 K ACp AC, 2.76 2.89 3.06 3.29 3.54 3.81 4.09 4.36 4.62 4.85 5.04 5.18 5.25 5.23 5.12 4.89 4.49 3.90 3.04 1.81 0.00
-1 1.72 -1 1.78 -1 1.83 -1 1.88 -11.91 -1 1.94 -1 1.95 -1 1.94 -11.91 -1 1.85 -1 1.76 -1 1.63 -1 1.45 -11.19 -10.84 -10.35 -9.67 -8.67 -7.17 -4.70 0.00
-13.92 -13.95 -13.97 -13.99 -14.00 -14.00 -13.98 -13.95 -13.90 -13.81 -13.70 -13.54 -13.33 -13.04 -12.65 -12.11 -11.36 -10.27 -8.61 -5.78 0.00
-r
Ld
T = 420 K ACp ACu 2.64 2.78 2.97 3.20 3.45 3.73 4.01 4.28 4.54 4.77 4.96 5.09 5.16 5.15 5.04 4.80 4.42 3.83 2.98 1.77 0.00
T = 440 K ACp ACu -16.54 -16.54 -16.54 -16.52 -16.50 -16.47 -16.43 -16.37 -16.29 -16.18 -16.03 -15.85 -15.60 -15.28 -14.85 -14.27 -13.46 -12.27 -10.43 -7.21 0.00
2.26 2.43 2.63 2.88 3.15 3.44 3.73 4.01 4.27 4.50 4.70 4.84 4.91 4.9 1 4.80 4.58 4.21 3.64 2.83 1.67 0.00
T = 460 K ACp ACo -19.87 -19.83 -19.78 -19.72 -19.66 -19.59 -19.51 -19.41 -19.30 -19.15 -18.98 -18.76 -18.48 -18.12 -17.65 -17.02 -16.14 -14.86 -12.83 -9.15 0.00
1.58 1.78 2.03 2.30 2.60 2.91 3.22 3.52 3.80 4.05 4.26 4.42 4.5 1 4.52 4.43 4.23 3.88 3.35 2.59 1.51 0.00
T = 480 K ACp ACu -24.47 -24.35 -24.22 -24.10 -23.96 -23.83 -23.68 -23.52 -23.35 -23.15 -22.92 -22.66 -22.33 -21.93 -21.41 -20.73 -19.78 -18.39 -16.16 -1 1.96 0.00
0.40 0.68 0.99 1.34 1.69 2.06 2.42 2.77 3.09 3.38 3.62 3.81 3.93 3.97 3.92 3.75 3.44 2.95 2.26 1.29 0.00
A Cv
J
X Ji
bu-’
3-
2-
-8
i
7
0 240
280
320
360
400
440
480
520
I
-121
K
I
I
I
lo00
2000
3000
I
-
4030
bof
Figure 7. Difference of heat capacity AC, as a function of pressure. From 1 to 10: Isotherms from 180 to 360 K by steps of 20 K.
of 2%but is of about 10%for the quantity r. Neglecting the error in the quantity ( a - q),we can estimate the accuracy of the quantity a 2 / K , when the measured quantity is w = a 2 / ( K + r). The result is
where the A refers to absolute errors. In the “high”-temperature region where the experiments are performed, the ratio r / K = 114 at the highest pressures. In the most unfavorable condition we have
bor
Figure 6. (a, top) Compressibility coefficient as a function of temperature. From 1 to 7: Isobars at the following pressure in bar units: -500, -250, 42.15, 250, 500, 1000, and 4000. (b, bottom) Compressibility coefficient as a function of pressure. From 1 to 5 : Isotherms at the following temperatures in kelvin units: 200, 300, 400, 500, and 591.8.
not considered as conclusive of the existence of a systematic error. In fact, our measurements attain their full precision at pressures higher than 200 bar. The primary quantity measured for the compressibility is the left side of eq 5 . The accuracy of this quantity is of the order (12) Tyrer,
D.J . Chem. SOC.1913,103, 1675.
The same figure applies to the quantities A K / Kand A(Cp- C,)/(Cp - C”).
Discussion As we could already foresee, toluene is a “complex” liquid. The isotherms of a follow the square root rule as a function of pressure (eq 2) but violate the rule of a unique crossing point. This feature draws attention to the problem of the heat capacity C,computed through step S9. Figure 8 gives the general aspect of C, as a function of temperature on three different isobars. The curves merge on the high-temperature side while they separate at low temperature, indicating a lowering of C, as pressure increases.
492
Ter Minassian et al.
The Journal of Physical Chemistry, Vol. 92, No. 2, 1988
CV
V
90 I
I
I
2M)
220
I
240
I
I
I
260
280
300
I
I
320 340
1
90
K
Figure 8. Heat capacity C, as a function of temperature. From 1 to 3: Isobars at 200, 2000, and 4000 bar, respectively. 1301 A 1 2 6J K - ' ~ o ~ - ~-
122
-
98
%
102
106
110 V
cm3 mot-' Figure 10. The "ad hoc" mechanism of conversion of motion. The qualitative aspects of the exws heat capacity due to this effect according to eq 24-26: R = 4 , # = 2000 K, V, = 86 cm3mol-', = 92 cm3mol-', p = -5 X cm3 mol-' K-I.
The generality of eq 20 is questionable, regardless of the basic assumption of conversion of motion. Nevertheless, we may bring out the main features of the model and estimate the contribution of f ( T , I / ) to the overall partition function of the liquid. We determine the internal energy E by the appropriate derivative of eq 20 and cast the result in the following form
118114 110106-
10298 94 90
/ ,
I
90
.
I
94
L
l
98
I
1
102
.
*
I
I
106
110
v
I
114
cm3 mol' Figure 9. Heat capacity C, as a function of volume. From 1 to 8: Isotherms from 180 to 320 K by steps of 20 K (1 and 2 are extrapola-
tions). A different aspect of the same phenomenon may be observed when C, is plotted as a function of volume along different isotherms S constant as a function of volume (Figure 9). The C ~ remain within the limits of experimental errors. However, at low temperatures a significant increase of C, with the volume may be observed. This moderate effect attains its maximum amplitude on the isotherm a t 240 K where the increase is of 6 J K-I mol-'. This type of behavior may be explained on the basis of some conversion from one type of mechanical motion into another:I3 The system explores another energetic degree of freedom, and the particular dynamics by which it explores that degree of freedom is not necessarily being revealed. It is reasonable that the degree of freedom could be one which involves the reorientation of a molecule in a sense which was forbidden earlier because of packing restrictions. The partition function relevant to the problem relies upon the concept of a volume-dependent barrier e(V. It has been written in the following formI4J5
+
where E , and Eo are the energies corresponding to the partition functions u l ( T ) and uo( T), respectively, and N is the Avogadro number. We note the existence of an extra term in eq 21 and 22; Le., the quantity E is not a weighted average between the quantities E l and Eo and is volume dependent through the potential energy t(v). Supposing now a situation where E , = Eo = kT and performing the derivative of E with respect to temperature give the heat capacity C,. We have from eq 21 and 22
Equation 23 emphasizes an interesting aspect: At constant temperature the heat capacity C, mQy be modified by the pressure without any additional degree of freedom provided R # 1. This is achieved through the function c(V) not yet defined. Tentatively, we may guess such a function by trial and error and bring out a general behavior in general accord with the observations. However, we may not assume that e will remain a temperatureindependent quantity. In this context, eq 23 should be replaced by the following equation:
-c,Nk
1=
[---I ,g
T
80 dT
2
( R - l)e-e/T a20 - ( R - 1)T (24) ( ( R - l)e"lT 1)2 R - 1 + eo/Ta P
+
f(T,V) = vo(T)(l - e-f/kq ul(T)e-'ikT (20) where uo( 7 ) is the classical partition function for an oscillator and vl( T ) may be that for a translational or rotational motion.
where a temperature dependence of e has been assumed and where B = t(V,T)/k. Now, performing an alteration to an equation proposed e l s e ~ h e r e , ' ~we~ 'find ~
(13) Pauling, L. Phys. Reu. 1930, 36, 430. (14) McLaughlin, D. R.; Eyring, H. Proc. Narl. Acad. Sci. U.S.A. 1966, 55, 1031. (15) Eyring, H.; Mu Shik Jhon Significant Liquid Structures; Wiley: London, 1969.
where V, and $ are constant coefficients and where Vois a temperature-dependent reference volume with constant coefficients (Acl):
J . Phys. Chem. 1988, 92,493-496
The results are given in Figure 10. When compared with the experimental results in Figure 9, it gives a rough idea of the reliability of the mechanism of conversion from one type of molecular motion to another.
Conclusion The thermodynamic properties of the liquid phase of toluene have given us access to two sets of information. The first follows from the examination of the expansivity coefficient. Its behavior as a function of pressure conforms to the general rule, but its behavior as a function temperature classifies toluene in the "complex" liquid category. The nature of this "complexity" may be examined through the behavior of the heat capacity C, as a function of volume, giving us the second set of information. A discussion on the qualitative aspects of the phenomenon leads us to conclude that there is a measurable conversion from one type of molecular motion to another taking place at low temperature in the high-density region. Acknowledgment. It is a pleasure to acknowledge useful discussions with Gaston Berthier and Alfred Maquet. We thank Dr. E. J. Land for his comments on the manuscript. Appendix I. The Compressibility h ( T , p ) . Following step S2, the integral of eq 9 is determined where a(T,p) is given by eq 17a-17c. Putting down w = a 1 / a 2 and il =
ao(1 - w )
a2 we oDerate the change of variable A? = (mo- w ) / ( w- mA);the result is
493
where the function f takes one of the following two forms depending on the value of m: case 1: m
>w
c= (mo - w)lI2 ( m - W ) I / ~
1
+ ( m - mA)II2
( w - mA)'/2
(A2) case 2: m
