CC BY
55
YflK 544.03
BecTHHK Cn6ry. Cep. 4. 2013. Bun. 1
N. G. Polikhronidi, I. M. Abdulagatov, R. G. Batyrova, G. V. Stepanov
QUASI-STATIC THERMO- AND BAROGRAMS TECHNIQUES FOR ACCURATE MEASUREMENTS OF THE PHASE BOUNDARY PROPERTIES OF THE COMPLEX FLUIDS AND FLUID MIXTURES NEAR THE CRITICAL POINT
1. Introduction. Isochoric heat capacity measurements near the critical point are very important for scientific applications, for example, for fundamental studies of the phase transition and critical phenomena in complex fluids and fluid mixtures. Isochoric heat capacity is a key thermodynamic property (very sensitive tool) for study of the phasetransition and critical phenomena. Its first density derivative, (dCv/dp)T, provides valuable information on temperature derivatives of the thermodynamic functions of compressed fluids. For example, the slopes of one-phase Cv —p isotherms are directly related with the second temperature derivative of pressure, (d2P/dT2)p = — (p2/T)(dCv/dp)T, which is cannot be accurately calculated by twice differentiating of the equation of state or directly measured, even very precise, PVT data are available. The slopes of two-phase (Cv2—V) isotherms are also gives information on the values of second temperature derivative of vapor pressure, T(d2PS/dT2), while the intercepts for V = 0 are related with —T(d2^/dT2) as (Yang—Yang relation)
Cv 2 _ d2\i vd2ps m
T ~ dT2 dT2 U
Accurate Cv and PVT data near the critical point are also needed to calculate the universal critical exponents (a, P, y, 6), universal ratios between the asymptotic (A-/A+, A+r+B2, Do^O"1, r+/r0 ) and non-asymptotic (A+ /r+,A+/Bi) critical amplitudes, to establish accuracy of the scaling theories and their predictions, and to check the crossover model parameters [1-9]. Accurate experimental isochoric heat capacity and singular diameter data near the critical point for pure fluids are also needed to confirm the conclusions and basic idea and physical bases of a "complete" scaling theory of critical phenomena [10-12]. A new, non-analytical contribution of liquid—gas asymmetry ("complete" scaling
term or "2P" anomaly) of the singular diameter and the strength of Yang—Yang anomaly, Rn, (contribution of the d2^./dT2 and d2PS/dT2 in singularity of two-phase Cv2 at the critical point) was found using the accurate isochoric heat capacity measurements [10-14]. In this work we developed the technique for simultaneously measurements of one- and two-phase isochoric heat capacity of pure fluids (methanol, ethanol, n-propanol, and DEE) along the critical isochore and liquid—gas coexistence curve, and liquid and vapor saturation densities near the critical point to check the basis "complete" scaling theory conclusions and Yang—Yang anomaly strength, RThe measured values of Cv along the critical isochore and (TS, pS, pS) at coexistence curve near the critical point were used to calculate the
N. G. Polikhronidi — Senior researcher, Institute of Physics of the Dagestan Scientific Center of the Russian Academy of Sciences, Makhachkala, Russia; e-mail: [email protected]
Ilmutdin Magomedovich Abdulagatov — professor, present address: National Institute of Standards and Technology, Colorado, USA; e-mail: [email protected]
R. G. Batyrova — Senior researcher, Institute of Physics of the Dagestan Scientific Center of the Russian Academy of Sciences, Makhachkala, Russia; e-mail: [email protected]
G. V. Stepanov — Division chief, Institute of Physics of the Dagestan Scientific Center of the Russian Academy of Sciences, Makhachkala, Russia; e-mail: [email protected]
© N. G. Polikhronidi, I. M. Abdulagatov, R. G. Batyrova, G. V. Stepanov, 2013
asymptotic critical amplitudes, (A0 , A+ , Bo), which are can be used to check the universality of the critical amplitude ratios.
2. Experimental. The experimental details (physical basis and the theory of the method, the apparatus, procedures of the measurements, and the uncertainty assessment) of the isochoric heat-capacity, PVT, and thermal-pressure coefficient measurements and its application to complex thermodynamic systems (with solid—liquid, liquid—liquid, liquid—gas, liquid—liquid—gas, solid—liquid—gas, and solid—liquid—liquid phase transitions) have been described in detail in our earlier publications [15-26]. In the present work, the previously developed adiabatic piezo-calorimeter was additionally supplied with a calibrated piezoelectrictransducer (barograms) to accurately and simultaneously measure of the PVT, Cv VT, and thermal-pressure coefficient, yv, along the liquid—gas phase-boundary curve. In this work we provided detailed description of the physical bases of the quasi-static thermo-and barograms techniques to accurately measure phase boundary properties.
2.1. Isochoric heat-capacity measurements. Isochoric heat-capacity (CvVT) measurements were performed with a high-temperature, high-pressure, nearly constant-volume adiabatic piezo-calorimeter. The piezo-calorimeter construction is shown in Fig. 1. If a layer of a semiconductor (Cu2O) is placed between two concentric spherical vessels, the system will behave as a highly sensitive thermo-element that can serve as a sensor, which is detecting the deviations from adiabatic conditions. Since cuprous oxide (Cu2O) has a very high thermoelectric power a (about 1150 ^V/K), the adiabatic conditions are reliably maintained in this calorimeter. This makes it possible to detect extremely small temperature differences (10°6 K) between the inner and outer spherical vessels and to eliminate possible heat transfer through the semiconductor layer. Therefore, the heat that is released by the micro-heater located in the calorimeter (inner vessel) is used only to heat the fluid located inside the inner thin-walled shell of the calorimeter and a thin layer of Cu2 O that directly adjoins the inner shell. Since Cu2O has a small thermal conductivity (X « 2.09 W-m/K), use of this semiconductor leads only to negligibly small heat losses. The out-of-balance signal from the integrating thermo-element (Cu2O) is applied first to the input of an amplifying microvoltmeter whose output feeds a high-precision temperature regulator (HPTR). In difference of vacuum calorimeter where adiabatic condition is controlling between the selected points (parts) of the inner and outer vessels (controlling only local heat losses), in this calorimeter the adiabatic conditions are controlled at the all parts (integrated adiabatic calorimeter) between the inner and outer vessels i. e. Q = j>S X gradTdS = 0. In the present calorimeter this condition is satisfied automatically.
The heat capacity is obtained from measurements of the mass of the fluid, m; electrical energy released by the inner heater, AQ; temperature change, AT, resulting from addition of an energy AQ; and the empty calorimeter heat capacity, C0. The final working equation for this method is
The sample under study was vigorously mixed using a stirrer, when the measurements were made near the critical point. The uncertainty in the temperature measurements was 15 mK. Based on a detailed analysis of all sources [20] of uncertainties likely to affect the determination of Cv, the expanded uncertainty (a coverage factor k = 2 and thus a two-standard-deviation estimate) of measuring the heat capacity was 2 % to 3 % in the near-critical region, 1.0 % to 1.5 % for the liquid isochores, and 3 % to 4 % for the vapor phase. The uncertainty in density measurements was within 0.02 % to 0.05 %.
(2)
910'
12
Q
11
14
13
Fig. 1. Schematic representation of the high-temperature and high-pressure nearly constant-volume adiabatic piezo-calorimeter supplied with a calibrated piezoelectrictransducer: 1 — PRT; 2 — semiconductor layer (Cu2O); 3 — outer shell; 4 — inner thin-walled spherical vessel; 5 — thermal screen; 6, 9 — air; 7 — perforated stirrer; 8 — inner heater; 10 — outer heaters; 11 — valve; 12 — pressure transducer; 13 — membrane separator (null indicator); 14 — dead-weight pressure gauge (MP-600)
The heat capacity was measured as a function of temperature at a nearly constant density. The calorimeter was filled at room temperature, sealed off, and heated along a quasi-isochore. Each run for the heat capacity was normally started in the two- or three- phase (L-V, L-L-V, or L-S-V) regions, dependence on complexity of the system, and completed in the two- or single-phase (L-L, L-S, L-V, L or V, depending on filling factor and T and P) regions. Between initial (L-V, L-L-V, or L-S-V) multi-phase and final one- or two-phase states, the system passed through the liquid—gas, liquid—liquid, liquid—solid phase transitions at temperatures, TS. This method (quasi-static thermo- and barograms techniques) enables one to determine with an uncertainty of 0.2 K, the phase transition temperatures, Ts, the jump in the heat capacity ACv, and Cv data in the single-, two-, and three-phase regions for each fixed quasi-isochore. The single- and two-phase vapor (CV1, C"2) and liquid (C'v 1, C'v2) heat capacities at saturation, the saturated temperature (TS), and saturated-liquid (pS) and saturated-vapor (pSS) densities can also be measured for near-critical fluids with this method as discussed below.
2.2. Quasi-static thermo- and barograms techniques. 2.2.1. Quasi-static thermograms method. Accurate measurements of the phase transition properties (location of the phase-boundary parameters, Ts, Ps, pS, PS) near the critical point and the critical parameters (Tc, pc) can be performed using the quasi-static thermograms (reading of PRT, T—t
plot at isochoric heating of the sample) and barograms (readings of the piezoelectrictrans-ducer, P—x plot) techniques. Quasi-static thermograms technique was successfully used before by Voronel et al. [27-30]. But, in difference with the method used by Voronel et al. [27-30], the method of quasi-static thermograms developed in this work is supplemented by recording readings of the adiabatic control sensor (see Fig. 2, b). In combination of the thermograms, these ensure sufficient information on the changes in the sample thermodynamic state, for example, presence of any local temperature gradients in the different parts (in the middle or near the inner surface) of the sample volume of the inner calorimetric vessel. In this method the amount of heat removed (or supplied) per unit time must be less than the heat-effect, otherwise, the heat-effect cannot be detected and the rate of heating (cooling) must be small, so that the condition dT/dx < AT/xrel, where trei/AT is the relative time necessary for the relaxation of the system (quasi-static condition). The values of xrel/AT « 104 min/K and the rate of temperature variation (the thermogram's slope) are about dT/dx < 6 • 10~3 K/h.
As was demonstrated in our previous publications, the quasi-static thermo- and baro-grams methods can also be used to accurate simultaneously measurements of vapor-pressure, PS, PVT, CvVT, and thermal-pressure coefficient, yv, if the calorimeter additionally sup-
a
10-7 V
U
10-5 V
U
-t, s
b
Fig. 2. Typical experimental thermo- (T—t or UT —t) and barograms (P—t or Up—t) for the sample's temperature and pressure variations (a) and record of the adiabatic condition (b) along the selected isochore:
1 — thermogram; 2 — barogram; Uts and Ups are the phase transition points on the thermo-and barograms; t is the time; Ts — phase transition temperature; break of the adiabatic condition for short time (peak at t = 11520 s) is the phase transition point (temperature)
plied with a high accurate strain gage (calibrated piezoelectrictransducer, piezo-calorimeter, Fig. 1). The method of quasi-static thermo-barograms can be used for application to complex thermodynamic systems with different types of phase transitions (pure fluids and binary solutions with L-V, L-L, L-S, V-S, L-S-V, and L-L-S phase-transitions). Synchronous recording of the resistance thermometer (T—t plot) and of the sensor of adiabatic control readings (see Fig. 2) follows the thermodynamic state of the sample as it approaches the phase-transition point. On intersecting liquid-vapor (or any type phase transitions, even very weak sign of the phase-transition phenomena) phase-transition point, the heat capacity is known to change discontinuously, leading to a sharp change in the thermogram's slope, dT/dT. Any jumps in a Cv —T plot or thermogram (T—t plot) breaks means that the number of phases in the system has changed, increasing or decreasing depending on increasing or decreasing the value of , after the phase transition point. Typical experimental thermo-(reading of PRT, T—t plot) and barograms (reading of piezotenzotransducer, P—t plot) for the DEE at selected near-critical liquid isochore 394.8 kg/m3 is shown in Fig. 3. Each Cv-jump point is the one point (TS, PS, pS, pS)) on the L-V (or L-L-V) coexistence curves. Since the isochoric heat capacity jump is infinity at the critical point, ACv <x (T — Tc)-a [31-33] (see Fig. 4 for DEE, also our previous experimental results [15-26]), therefore, the thermograms slope jump
AC,,
dx
dT
VTs-0
dx
dT
VTs + 0
« (T - Tc)-
(3)
Fig. 3. Typical experimental thermo-(reading of PET, T—t plot) and barograms (reading of tenzotrans-ducer, P—t plot) for the DEE at selected near-critical liquid isochore 394.8 kg/m3:
1 — thermogram; 2 — barograms; brake points are the phase transition points
0 40 80
120 160
t, s
200 240 280
Fig. 4- Experimental liquid (AC'v) isochoric heat capacity jump of DEE on the coexistence curve as a function of temperature near the critical point together with values calculated (solid line) with scaling relations
a
k
is a very large near the critical point (thermograms slopes changes are about 30-50 % and more). Therefore, experimentally is very easy to detect the jump points (even impossible to miss it). In Eq. (3) (dx/dT)vts-o and (dx/dT)vTs+0 are the slopes of the thermograms before and after the phase-transition point TS, respectively; k is a coefficient depending on the power of the heat released by the heater and the mass of the sample under study; and TS is the temperature of the phase transition corresponding to the fixed isochore, V. The slope of the thermograms in the two-phase region (before the phase-transition temperature), is lower than in the one-phase region, (dx/dT)vts-0 < (dx/dT)vTs+0, because the isochoric heat capacity of the fluid in the two-phase state is higher than in the one-phase state, Cv2 > Cv1. Thus, we can precisely determine the location (Ts, pS) of the phase transition point. This makes the method very sensitive location of phase transitions near a critical point (see Fig. 5-10). Therefore, the quasi-static thermogram method is very suitable to accurately determine the phase boundary properties (coexistence curve shape and the critical parameters) in the critical region. Fig. 2 and 3 shows the typical thermo- (T—x plot) and baro- (P—t plot) grams and record of the adiabatic condition along fixed density. Thus, this is easy to detect of the any types of phase transitions (even weak signs of a phase transition like structural changes) occurring in the sample near the critical point using this technique.
Conventional P—T isochoric break point (or P—p isothermal break point) technique [34-39] is less sensitive to a phase transition in the critical region because the slope of the P—T and P—p curves is change very slightly due to the small difference between the densities of the phases before and after transition, which are became identical at the critical point (see Fig. 11-13). Therefore, quasi-static thermograms method is more suitable for accurate determinations of the phase boundary properties and the critical parameters than P—T and P—p break point techniques in the critical region. Far from the critical point, P—T and P—p break point techniques (quasi-static barogram method) are more accurate than calorimetric (quasi-static thermogram ) technique because the heat capacity jump ACv is small far away of the critical point (therefore changes in thermogram slopes also are small, see Fig. 2 and 3), while the slopes of the isochores P—T and isotherms P—p change more clearly (see Fig. 11-13). Therefore, this technique is more suitable for the determination of the phase transition properties far from the critical point. For near-critical isochores most accurate method is quasi-static thermograms (see above sec. 2.2.1).
2.2.2. Quasi-static barograms method. PVT and thermal-pressure coefficient yv, and phase boundary properties measurements of fluids and fluid mixtures can be performed using the same apparatus as was employed for the isochoric heat capacity, Cv VT, measurements using a high-temperature and high-pressure nearly constant-volume adiabatic piezo-calorimeter (see above, sec. 2.1). The pressure (P) and the thermal-pressure coefficient, , were measured with a calibrated piezoelectrictransducer. The measurements of pressure in the piezo-calorimeter were performed at constant temperature before the iso-choric heat-capacity measurements. Then, after turning on the heater, both temperature changes (thermograms, T—x, reading of the platinum resistance thermometer PRT, UT —x) and pressure changes (barograms, P—x, readings of the piezoelectrictransducer, UP —x, see Fig. 2) were synchronously recorded with strip-chart recorders. Using the records of the thermo- and barograms, the changes in temperature AT and in pressures AP, therefore, the temperature derivative (dP/dT)vx = limAT(AP/AT)vx, at any fixed time, were obtained. Each of the measured T—x and P—x isochores was fitted to linear equations, T = c + dx and P = a + bx. Therefore, the thermal-pressure coefficient, yv can be estimated as yv = (dP/dT)V = b/a.
Fig. 5. Measured two- and one-phase isochoric heat capacities of methanol, ethanol, n-propanol, and DEE as a function of temperature near the critical point along their critical isochores: solid circles — experiment; solid lines — crossover model
The measurements were made on isochoric heating of the system at quasi-equilibrium conditions. The rate of the temperature changes was less than 5 • 10-4 K/s. At these conditions, the measured quantities, (AQ/AT)V and (AP/AT)V, can be replaced by the partial derivatives, (dU/dT)V and (dP/dT)V, respectively. Thus, measured values of derivatives, (dU/dT)v (isochoric heat capacity) and (dP/dT)V (thermal-pressure coefficient), together with PVT measurements in the same experiment can be used to determine the values of the internal pressure by using the relation,
and the isochoric heat capacity Cv = (dU/dT)V. Therefore, in the same experiment we can simultaneously measure partial derivatives of the internal energy Cv (caloric property) and Pint (thermal property). The uncertainties in measurements of temperature changes AT and pressure changes AP are more accurate than measurements of their absolute values (T and P). Therefore, the uncertainty in measurements yv is within 0.12 % to 1.5 % depending on the temperature increment (AT changes within 0.02 K to 0.10 K). The uncertainty in pressure measurements is about 0.05 %.
The typical barograms (P—т plot) together with thermograms (T—т plot) along the selected liquid isochore for DEE are presented in Fig. 2, a and 3. As Fig. 2, a demonstrate for liquid isochore far from the critical point the changes of the barogram slope much more noticeable than thermogram slope changes, therefore barogram technique for liquid phase
12 10
S?
eg 8
3 6
J*
4
p = 265.77 kg/m3
485 14 12 )10
510
535
560
kg 8
O'
p = 277.49 kg/m3
460
490
520 T, K
550
580
11 £ 9 7 5
g k
TS = 560.87 K
p = 290.78 kg/m
g I
J
3l_
550
557
564
571
578
p = 269.50 kg/m3
TS = 561.40 K
550
557
564 T, K
571
578
12 10 8 6 4
p = 258.07 kg/m3
585 465 490 515 540 565
12 10 8 6 4
2
p = 274.87 kg/m3
460 490
11 9 7 5
3
530
11r
520 550 T, K
580
p = 252.00 kg/m3
Ts = 561.97 K
540
550
560
570
- p = 342.10 kg/m:
TS = 559.48 K
530 540
550
560 570
580
T, K
Fig. 6. Experimental two- and one-phase isochoric heat-capacities of pure methanol (a) and methanol+water (b) mixture as a function of temperature along the selected near-critical isochores
a
6
4
b
! 8 k 6 ' 4
p = 295.90 kg/m3
I
T„ = 514.430 K
V-------
p = 287.24 kg/m3
500 506
512
518 524
9 7 5
506 508 510 512 514 516
Ts = 514.438 K V s %»•
10
ад
3
4
510
10
S? 8
I
M
6
4_
517.5
p = 282.33 kg/m3
T„ = 514.438 K
513
516 T, K
519
p = 234.8 kg/m3
T„ = 520.244 K
518.5
519.5
520.5
10 8 6 4
510
10
4l_
518
p = 272.25 kg/m3
T„ = 514.433 K
513 516 519 T, K
p = 294.9 kg/m3
Te = 520.744 K
522
519
520
521
522
11 9
31-
518.2
p = 269.4 kg/m3
Ts = 520.75 K
519.2 520.2
T, K
10 8 6 4
p = 245.3 kg/m3
Ts = 520.5 K
521.2 518.5 519.1 519.7 520.3 520.9
T, K
Fig. 7. Measured two- and one-phase isochoric heat capacities of pure ethanol (a) and ethanol+water (b) mixure along the various liquid and vapor isochores as a function of temperature near the phase transition points
a
b
8
6
7
5
Fig. 8. Measured one- and two-phase isochoric heat capacities of 0.7393H2 O + 0.2607NH3 mixture along the selected near-critical isochores as a function of temperature near the phase transition points
transition measurements more accurate, while for near critical isochore thermogram slope changes considerable accurate than barograms method. Also, as we can note from Fig. 2, a and 3, both thermo- and barogram changes the slopes at the same time i. e. exactly at the same phase transition temperature TS for the each fixed density pS (isochore, liquid or vapor depending on filling factor). Thus, in these methods it is impossible to miss the phase transition point even weak sign of the phase transition phenomena. These techniques very useful to study of the phase transition phenomena in complex thermodynamic systems where visual method does not work or less reliable (large uncertainty). This method of determination of the phase transition points has certain advantages over other methods. In particular, it has a low uncertainty and high reliability. The most widely used experimental method of determining parameters of the coexistence curve by the visual observation of the meniscus disappearance lacks objectivity. Moreover, approaching the critical point, where the difference between the liquid and vapor phases vanishes, and the visual determination of the moment at which the phase transition occurs becomes ever less reliable. In addition, the observations are impeded by the development of critical opalescence. Therefore, the region of temperatures near the critical point (within 1 K of Tc) becomes virtually unattainable for investigation. The method under consideration makes it possible to obtain reliable data within 0.02 K of Tc. These are very good techniques to determine the phase changes in complicated multicomponent mixtures (for example, for mixtures with limited mutual solubility, L-L-V or L-S-V) where it is difficult to determine the number of phase changes at constant volume heating. The calorimeter is filled with the substance to be studied
22 18 14
32 10 6
643
647
p = 334.78 kg/m3
646
649 T, K
652
655
Fig. 9. Measured one- and two-phase isochoric heat capacities of 0.5H2O + 0.5D2O mixture along the selected near-critical isochores as a function of temperature near the phase transition points
22 ^ 18 14
H 10 6
J.
643 22 18 14
J?
32 10 6
2 —
640
21 17
s?
eg13
^
3 9
J*
p = 338.98 kg/m3
W-
_1_
_1_
646 649
652 655
p = 318.07 kg/m3
643 646 649 652 655 T, K
640 644 T, K
until an appropriate density is achieved. Then, the apparatus is brought into the working range of temperatures and is held under adiabatic conditions for a sufficiently long time. After this, thermograms are recorded. At each isochore, thermograms are recorded several times during both heating and cooling. The retardation of the temperature run at Tmax corresponds to a maximum in Cv, whereas a break at TS corresponds to the intersection of the phase transition curve, where the heat capacity decreases discontinuously. In order to pass to another isochore, part of the sample is extracted from the calorimeter into a measuring vessel, and the mass of the substance extracted is measured.
2.2.3. Critical parameters measurements. If the critical density of the fluid is well known, the values of the critical temperature and the critical pressure can be measured very accurately (within 0.02 K and 0.05 %, respectively) by heating the calorimetric vessel with the critical filling densities (see above, sec. 2.2.1). If the critical density of fluid unknown, the value of the critical density can be determine by measuring the phase transition temperature for the series of near-critical isochores (around the expected critical density), and then the value of the critical density can be accurately extracted by analytically evaluating the TS—pS
12
11
ég м.
M
10
p = 372.8 kg/m
14 13
Ts = 501.19 K 12
11 10
T„ = 477.06 K 9
p = 344.7 kg/m3
TS = 509.71 K
p = 408.8 kg/m:
Ts = 464.13 K
340
395
450 T, K
505
560
350
400
450 T, K
500
550
Fig. 10. Measured one- and two-phase isochoric heat capacities of CO2 + decane mixture along the selected near-critical isochores as a function of temperature near the phase transition points
24 20 16
s Рм
Я 12 ^ 8 4 0
25 20
£15
6.152 MPa 513.91 K
18 15 12
s Рм
Я 9 Сц
6 3 0
p = 386 kg/m3
Ps = 5.988 MPa
- Ts = 511.71 K . i , , i : i i
370
420
470 T, K
520
570
p = 338 kg/m3 P„ = 6.268 MPa
Ts = 514.61 K
" 10
480 530 T, K
Fig. 11. Measured PVT data of pure ethanol [51] along the selected isochores in the two- (vapor-pressure) and one-phase regions near the phase transition points (isochoric break technique)
9
8
8
data in the immediate vicinity of the critical point, for example, using the "complete" scaling theory of the singular coexistence curve diameter (see below, sec. 3.1). For pure fluids the maximum measured phase-transition temperature on the coexistence curve can be excepted as a critical temperature, which is correspond to the critical density (Tc and pc). Because the isothermal compressibility of the fluid is infinite (KT ^ at the critical point,
it is difficult to accurately measure the critical density directly. Conventional rectilinear
18 15 12
c3
Pm
s 9
18 15 12
c3
Pm
s 9
18 15 12 9 6 3 0
390
430
470
350 7 6 5 4 3
485 505 T, K
Fig. 12. Measured PVT data of pure DEE along the selected isochores in the two-(vapor-pressure) and one-phase regions derived from barograms method
23 21 19 17 15
580 590 12
10
8
6
4
510
•
_ p = 671.23 kg/m3 •
»
Ps = 8.9 MPa ,
Ts = 507.0 K
• /
16 12
s
, 8 4 0
310 360 410 460 510 560 6.2 ■
610
600 610 620 630 640
495 T, K
605 618 T, K
Fig. 13. Measured PVT and Cv,V,T data of 0.7393H2O + 0.2607NH3 mixture along the liquid and vapor isochores in the two- (vapor-pressure) and one-phase regions
diameter technique is not accurate enough. The rectilinear diameter exhibits a curvature (see below sec. 3.1) in the immediate vicinity of the critical point. The data exhibit some deviations from a straight line in the temperature range for which t < 10-2. Usually, the value of the critical density determined from the singular diameter law differs from the value that is obtained from linear extrapolation (from the rectilinear law) by 3 to 5 %. For an accurate estimate of the critical pressure, the measured vapor-pressure data can be analytically extrapolated to the critical temperature using scaling relations.
3. Results and discussion. The experimental results of temperature and density at saturation (liquid and vapor) for some pure fluids (methanol, ethanol, n-propanol, and DEE) and binary mixtures (H2O+methanol, H2O+ethanol, H2O+ammonia, H2O+D2O, CO2 +n-decane) derived using the method described above are presented in Tables 1-9 and shown in Fig. 14-19. Derived from the calorimetric measurements values of the critical parameters for pure fluids and binary mixtures (critical curve data) are presented in Table 10. These data were measured using the quasi-static thermo- and barograms techniques as described above. Fig. 14-19 also contain the data reported by other authors and calculated with various published correlation equations. As one can see the agreement between the present and reported data are in good agreement enough. Tables 1-4 also contain the one-and two-phase isochoric heat capacity data for pure fluids along the critical isochore near the critical point. Measured saturated liquid and vapor densities together with isochoric heat capacity data along the critical isochore were used (see below) to calculate the asymptotic critical amplitudes (Bo, A-, A+) and coexistence curve diameter asymmetric parameters a3 and b2.
Table 1
Experimental values of one- and two-phase isochoric heat capacity (Cv) along the critical isochore (p = 265.77 kg/m3), temperature (Ts), saturated liquid (pS) and vapor (pS) density of pure methanol from the calorimetric measurements (quasi-static thermograms method)
Ts, K Ps, kg/m3 T, K Cv, kJ/(kg-K)
337.188 750.08 494.909 6.837
371.082 714.24 495.195 6.849
382.150 701.85 495.481 6.875
391.970 689.56 503.212 7.511
423.468 647.59 503.447 7.549
456.095 592.10 503.719 7.584
474.975 548.37 503.922 7.619
490.418 499.75 507.603 8.017
508.525 398.92 507.791 8.153
- - 507.979 8.203
- - 508.168 8.151
- - 512.046 9.803
- - 512.234 10.364
- - 512.422 10.586
- - 512.515 10.830
- - 512.609 11.135
- - 512.702 11.673
- - 512.775 12.250
Ts, K kg/m3 T, K Cv, kJ/(kg-K)
512.775 265.77 512.775 6.410
Ts, K pg, kg/m3 T, K C„, kJ/(kg-K)
512.758 258.07 512.796 6.369
512.665 244.33 512.890 6.254
512.633 241.06 512.983 6.098
512.005 212.16 513.077 5.980
511.772 206.16 513.186 5.887
504.531 136.14 513.288 5.797
- - 513.375 5.710
- - 516.602 5.077
- - 516.788 4.861
- - 516.882 4.729
- - 517.068 4.740
- - 517.162 4.677
- - 522.571 4.432
- - 522.760 4.396
- - 522.949 4.388
- - 523.233 4.393
Table 2
Experimental values of one- and two-phase isochoric heat capacity (Cv) along the critical isochore (p = 282.33 kg/m3), temperature (TS), saturated liquid (pS) and vapor (pS) density of pure ethanol from the calorimetric measurements (quasi-static thermograms method)
Ps, kg/m3 Ts, K T, K Cv, kJ/(kg-K)
365.50 512.67 510.63 6.646
347.97 513.50 510.82 6.672
338.24 513.85 511.15 6.741
331.54 514.05 512.02 6.930
321.71 514.24 512.21 6.984
311.41 514.36 512.39 7.031
304.86 514.39 512.58 7.142
302.65 514.42 512.77 7.200
298.13 514.42 512.95 7.245
295.90 514.43 513.15 7.321
287.24 514.44 513.24 7.379
- - 513.39 7.452
- - 513.52 7.550
- - 513.70 7.817
- - 513.80 7.908
- - 513.99 8.030
- - 514.08 8.133
- - 514.17 8.486
- - 514.27 8.770
- - 514.36 9.166
- - 514.44 10.170
pg, kg/m3 Ts, K T, K Cv, kJ/(kg-K)
282.33 514.44 514.44 5.542
272.25 514.43 514.46 5.498
264.27 514.42 514.54 5.400
pg, kg/m3 Ts, K T, K Cv, kJ/(kg-K)
250.53 514.35 514.64 5.239
247.22 514.30 515.30 4.384
232.12 514.11 515.39 4.420
224.13 513.95 515.58 4.413
219.33 513.85 515.76 4.456
207.85 513.40 515.85 4.413
196.26 512.79 517.07 4.103
193.32 512.55 517.16 4.057
- - 517.35 4.104
- - 517.54 4.128
- - 520.80 3.939
- - 520.89 3.988
- - 521.08 3.868
- - 521.17 3.910
- - 521.26 3.852
- - 521.35 3.911
Table 3
Experimental values of one- and two-phase isochoric heat capacity (Cv) along the critical isochore (p = 272.90 kg/m3), temperature (Ts), saturated liquid (pS) and vapor (pS) density of pure n-propanol from the calorimetric measurements (quasi-static thermograms method)
Ps, kg/m3 Ts, K T, K Cv, kJ/(kg-K)
283.15 811.69 530.12 5.534
293.15 803.86 532.05 6.088
303.15 795.54 535.23 6.918
313.15 788.02 536.24 7.650
323.15 779.42 536.28 7.820
333.15 771.01 536.32 7.920
343.15 762.19 536.36 8.150
353.15 753.01 536.40 8.550
363.15 743.49 536.44 8.830
373.15 733.68 536.48 9.450
383.15 723.07 536.52 10.610
393.15 712.25 - -
403.15 700.77 - -
413.15 688.70 - -
423.15 675.68 - -
433.15 661.81 - -
443.15 647.25 - -
453.15 631.31 - -
463.15 611.25 - -
473.15 594.88 - -
483.15 574.05 - -
493.15 550.37 - -
503.15 523.01 - -
513.15 490.68 - -
523.15 449.03 - -
Ps, kg/ill3 Ts, K T, K Cv, kJ/(kg-K)
533.15 380.52 - -
534.15 368.05 - -
535.15 351.62 - -
536.15 322.58 - -
536.56 272.85 - -
pg, kg/m3 Ts, K T, K Cv, kJ/(kg-K)
503.15 70.220 536.60 5.330
513.15 89.290 536.64 4.970
523.15 118.34 536.68 4.820
533.15 172.41 536.72 4.780
534.15 182.82 536.76 4.655
535.15 197.24 537.15 4.260
536.15 224.21 437.34 4.140
536.56 272.85 537.53 4.120
- - 537.72 4.055
- - 537.91 3.995
- - 538.10 3.970
- - 538.90 3.870
- - 539.08 3.845
- - 539.48 3.825
- - 541.91 3.660
- - 542.10 3.650
- - 542.29 3.665
- - 542.67 3.650
- - 542.86 3.630
- - 543.05 3.620
- - 553.22 3.360
- - 553.41 3.365
- - 553.60 3.360
- - 553.98 3.350
- - 563.88 3.230
- - 564.07 3.230
- - 564.43 3.240
- - 564.80 3.220
- - 564.99 3.220
Table 4
Experimental values of one- and two-phase isochoric heat capacity (Cv) along the critical isochore (p = 265.10 kg/m3), temperature (TS), saturated liquid (pS) and vapor (pS) density of DEE from the calorimetric measurements (quasi-static
thermograms method)
Ts, K Ps, kg/m3 T, K Cv, kJ/(kg-K)
416.096 534.6 453.523 3.746
425.190 517.3 453.625 3.760
437.138 485.2 453.728 3.775
450.603 441.5 453.830 3.750
450.766 440.8 453.933 3.774
459.608 394.8 454.035 3.833
Ts, K Ps, kg/m3 T, K Cv, kJ/(kg-K)
464.785 348.2 454.137 3.800
466.474 310.8 454.240 3.797
466.755 296.7 454.342 3.797
466.795 288.3 460.827 4.180
466.805 278.7 460.930 4.220
466.805 275.7 461.032 4.270
- - 461.131 4.256
- - 461.234 4.240
- - 461.336 4.250
- - 461.437 4.270
- - 461.538 4.281
- - 461.638 4.280
- - 466.150 5.141
- - 466.251 5.215
- - 466.352 5.289
- - 466.453 5.375
- - 466.495 5.386
- - 466.534 5.426
- - 466.575 5.468
- - 466.615 5.535
- - 466.654 5.577
- - 466.694 5.619
- - 466.735 5.747
- - 466.775 5.866
- - 466.816 6.049
- - 466.845 6.249
Ts, K kg/m3 T, K Cv, kJ/(kg-K)
466.845 261.2 466.845 3.565
466.835 260.7 466.856 3.421
466.825 255.0 466.896 3.243
466.659 233.7 466.937 3.116
466.122 212.6 466.977 3.137
- - 467.018 3.059
- - 467.059 3.045
- - 467.159 2.984
- - 467.261 2.957
- - 467.765 2.869
- - 467.866 2.847
- - 467.967 2.887
- - 468.068 2.856
- - 468.169 2.808
- - 468.270 2.850
- - 468.371 2.803
- - 479.634 2.634
- - 479.733 2.565
- - 479.833 2.619
- - 479.932 2.597
- - 480.032 2.566
Ts, K Ps, kg/m3 T, K Cv, kJ/(kg-K)
- - 480.131 2.619
- - 480.231 2.604
- - 480.330 2.590
- - 480.430 2.611
Table 5
Experimental values of temperature (TS) and saturated liquid (pS) and vapor (pS) density of equimolar 0.5H2O + 0.5D2O from the calorimetric measurements (quasi-static thermograms method)
Ts, K Ps, kg/m3 Ts, K p'é, kg/m3
644.773 274.05 645.455 338.98
645.275 299.58 645.424 352.11
645.417 318.07 645.295 385.36
645.449 334.78 - -
Table 6
Experimental values of temperature (TS) and saturated liquid (pS) and vapor (pS) density of H2O + CH3OH mixtures from the calorimetric measurements (quasi-static thermograms method)
Ts, K Ps, kg/m3 Ts, K Ps, kg/m3
x = 0.5004 mole fraction of CH3OH
556.816 394.80 561.709 235.00
559.480 342.10 561.494 214.42
560.874 290.78 - -
561.403 269.50 - -
561.673 259.50 - -
561.797 251.44 - -
x = 0.5014 mole fraction of CH3OH
561.966 252.54 - -
561.944 244.77 - -
Table 7
Experimental values of temperature (TS) and saturated liquid (pS) and vapor (pS) density of H2O + C2H5OH mixtures (mole fraction of C2H5OH x = 0.8554) from the calorimetric measurements (quasi-static thermograms method)
Ps, kg/m3 Ts, K pS, kg/m3 Ts, K
391.1 517.518 269.4 520.746
351.8 519.958 256.8 520.655
345.7 520.166 245.3 520.506
338.4 520.346 234.8 520.244
330.3 520.453 - -
328.8 520.488 - -
328.4 520.495 - -
318.0 520.599 - -
313.6 520.644 - -
310.2 520.664 - -
294.9 520.744 - -
282.0 520.759 - -
Experimental values of temperature (TS) and saturated liquid (pS) and vapor (pS) density of NH3 + H2O mixtures (mole fraction of NH3 x = 0.2607) from the calorimetric measurements (quasi-static thermograms method)
TS, K Ps> kg/ma TS, K kg/ma
508.07 671.23 599.67* 310.95
519.77 649.65 599.78 308.35
545.62 594.32 599.89 307.35
561.03 552.58 600.02 305.44
565.36 540.25 600.17 303.47
580.18 476.26 600.29 301.94
584.95 456.58 600.48 299.79
595.16 376.89 600.63 297.60
597.56 342.19 600.81 295.31
598.33 329.00 600.93 292.97
598.96 320.33 601.05 291.16
599.42 312.91 601.34 285.71
599.57 311.19 602.05 271.09
- - 602.45 260.02
- - 602.91 243.98
- - 602.96 230.31
- - 602.97" 227.54
- - 602.96 224.53
- - 602.85 218.92
- - 602.74 214.83
- - 602.26 206.00
- - 601.75 195.99
- - 601.16 188.05
- - 597.36 160.07
- - 594.14 147.43
- - 585.31 120.03
* Critical point. ** Maxcondentherm point.
Table 9 Experimental values of temperature (TS) and saturated liquid (pS) and vapor (pS) density of CO2 + n-decane mixtures from the calorimetric measurements (quasi-static thermograms method)
TS, K Ps> kg/m3 TS, K p£, kg/nT1
x = 0.2633 mole fraction of n-decane
452.53 520.6 514.45 328.9
475.10 466.9 519.34 312.6
490.87 409.8 525.69 298.7
501.19 372.8 - -
505.32 358.9 - -
509.71 344.7 - -
x = 0.095 mole fraction of n-decane
383.363 534.8 339.759 658.8
393.006 509.4 352.093 622.7
397.344 498.0 368.528 574.1
Ts, K Ps, kg/m3 Ts, K pg, kg/m3
404.551 477.6 - -
433.991 398.3 - -
442.459 374.7 - -
450.729 349.4 - -
460.096 320.2 - -
471.107 281.4 - -
481.001 241.0 - -
485.999 201.8 - -
488.104 176.7 - -
487.052 155.0 - -
481.831 131.5 - -
475.488 110.8 - -
463.061 87.03 - -
x = 0.178 mole fraction of n-decane
464.130 408.8 447.317 457.50
477.058 375.0 451.390 445.04
488.424 339.4 457.767 426.08
497.073 310.8 460.150 420.00
504.829 284.3 - -
522.198 164.2 - -
519.436 117.8 - -
515.001 83.31 - -
Table 10
The critical parameters of pure fluids and binary mixtures from the present calorimetric measurements (quasi-static thermograms method)
Fluid or fluid mixture x (mole fraction) pc, kg/m3 T, K Pc, MPa
Methanol - 276.74 ± 2 512.79 ±0.2 8.105 ±0.01
Ethanol - 279.12 ±2 514.44 ±0.2 6.155 ±0.01
n-Propanol - 272.85 ± 2 536.56 ±0.2 5.175 ±0.01
DEE - 265.1 ±2 466.85 ±0.2 3.605 ±0.01
H20+D20 0.5000 338.98 ± 2 645.45 ±0.2 21.871 ±0.01
H20+Methanol 0.5004 280.00 ± 2 562.51 ±0.2 12.340 ±0.01
H20+Methanol 0.5014 338.98 ± 2 563.45 ±0.2 12.35 ±0.01
H20+Ethanol 0.8554 275.51 ±2 520.80 ±0.2 7.085 ±0.01
H2O+Ammonia 0.2610 309.95 ± 2 599.67 ±0.2 21.408 ±0.01
H2O+Ammonia 0.2591 310.87 ±2 600.17 ±0.2 21.397 ±0.01
CO2 + n-Decane 0.0950 551.00 ±2 377.50 ±0.2 16.25 ±0.01
CO2 + n-Decane 0.1780 420.00 ± 2 460.15 ±0.2 18.23 ±0.01
CO2 + n-Decane 0.2633 344.70 ± 2 509.71 ±0.2 15.37 ±0.01
3.1. Singular coexistence-curve diameter and Yang—Yang anomaly strength from isochoric heat-capacity measurements. Measured saturated density data for pure fluids (methanol, ethanol, n-propanol, and DEE) have been interpreted in terms of "complete' and "incomplete" scaling theory. The "complete" scaling theory developed by Fisher and Orkoulas [10] showed that the strength of the Yang—Yang anomaly defined as
600
575
550
525
500
475
450íí
425
400
0
200
400 600
p, kg/m3
800
1000
575
515
455
395
335
275
H2O
170 340 510 680
p, kg/m3
850 1020
Fig. 14. Saturated liquid and vapor densities of light and heavy water derived from isochoric heat capacity measurements (quasi-static thermo- and barograms methods) together with the data reported by other authors
= }Ínn Г
t^ü C,
C„
Л,
vp
+ Cv
v^
Лр + Лц
-TcP^/dT2, Лц and Лр are the asymptotic amplitudes
where Cvp = VcTd2Ps/dT2, CЩ1 =
of the singularity of —Td2^/dT2 « Лцt-a and d2PS/dT2 « Apt-a, can be determined as Rц = a3/(1 + a3), where a3 is the system dependent asymmetry coefficient of the liquid—gas coexistence curve diameter, which is determined by details of intermolecular interactions (a3 = 0). Therefore, Дц = 0 for real fluids, if a3 = 0. As one can see from the definition of the Yang—Yang anomaly strength, the value of the parameter Rц is define the contribution of d2PS/dT2 and d2^/dT2 on the divergence of Cv2 (see Eq. (1)). For previous "incomplete" scaling models, a3 = 0 or Лц = 0, therefore, the divergence of the two-phase heat capacity Cv2 in Eq. (1) near the critical point is caused only by the divergence of Td?PS/dT2, while —Td2^/dT2 remains finite as "incomplete" scaling predict. If R^ = 0, i. e., Лц = 0, the divergence of the two-phase isochoric heat capacity Cv2 in Eq. (1) is caused by divergence of both derivatives or only just by —TcPft/dT2. The Yang—Yang anomaly strength R^ related with the asymptotic critical amplitude B0 of the coexistence curve and "complete" scaling term amplitude [11] B4 as R — ц = B4/B2 (see below). Therefore, Yang—Yang anomaly strength directly related with the singular diameter criticality as discussed below and can be also estimate from the asymmetry coefficients in the coexistence curve or diameter singularity. The measured in calorimetric experiment (quasi-static thermo- and barograms techniques) saturated liquid and vapor densities (Ts, pS, pS) near the critical point (Tables 1-4) were used to accurately determine Yang—Yang anomaly strength parameter, Ril. The present simultaneously measured saturated density and isochoric heat capacity measurements provides
0
500 460 ,420 * 380 340 300
0 110 220 330 440 550 660 770
p, kg/m3
Fig. 15. Saturated liquid and vapor densities of ethanol, methanol, and DEE, derived from isochoric heat capacity measurements (quasi-static thermo- and baro-grams methods) together with the data reported by other authors
500 480 ^ 460 440
420 -1
400
465 -
455 -
445 -
435 -,
425
415
95 190 285 380 475 570 665
p, kg/m3
250 350
p, kg/m3
550
0
Fig. 16. Saturated liquid and vapor densities of 1-propanol and 2-propanol derived from isochoric heat capacity measurements (quasi-static thermo- and barograms methods) together with the data reported by other authors
Fig. 17. Measured saturated liquid and vapor densities of H2O + methanol (a), H2O + ethanol (b), H2O + NH3 (c) and H2O + D2O (d) mixtures derived from isochoric heat capacity measurements (quasi-static thermo- and barograms methods) together with the data reported by other authors
very useful information about the qualitative behavior of the liquid—gas coexistence curve shape near the critical point, which is very important to correct interpretation coexistence curve diameter singularity, asymmetry of the coexistence curve, and precisely determine the value of the Yang—Yang anomaly strength parameter. This is also very important to correctly determine the value of the critical density of fluids from saturated density measurements near the critical point. The coexistence curve diameter exhibits a curvature (see below) in the immediate vicinity of the critical point. According to the renormalization group theory [40] of liquid—gas critical phenomena, the first temperature derivative of the coexistence-curve diameter, (dpd/dT), where pd = (p^ + p^)/2pc, diverges as the isochoric heat capacity t-a [41-44]. The current theory of "complete" scaling [10, 11] predicts a "2|" anomaly of the singular diameter of the coexistence curve, dpd/dT <x t21-1. Thus, according to the "complete" scaling theory [10, 11], densities of the liquid—gas coexistence curve can be represented by
Ap = ±Bot! ± Bit!+A + B2t1-a - B3t + B4t2|î, (5)
where Bi (i = 0,4) are the adjustable system-dependent critical amplitudes. In Eq. (5), ±B0t! is the asymptotic (symmetric) term, ±B1t|+A is the non-asymptotic (symmetric Wegner's correction) term, B2t1-a is the "singular diameter" (the first non-analytical contribution to the liquid—gas asymmetry predicted by "incomplete" scaling), B4t2| is the new non-analytical contribution of the liquid—gas asymmetry (new "complete" scaling term),
p, kg/m3
Fig. 18. Detailed view of the measured saturated liquid and vapor densities of 0.2607NH3 + O.7393H2O together with the values reported by other authors near the critical point of mixture derived from isochoric heat capacity measurements (quasi-static thermo- and barograms methods) together with the values calculated
from fundamental EOS
Fig. 19. Measured saturated liquid and vapor densities of CO2 + n-decane mixture derived from isochoric heat capacity measurements (quasi-static thermo- and barograms methods)
and B3t is the rectilinear diameter. As one can see from Eq. (5), the effect of the Yang—Yang anomaly strength, on the coexistence curve diameter is given by [10, 11]
Pd = 1 + B2t1-a - Bst + B4t2^, (6)
where "complete" scaling asymmetry parameter B4 x A^/Ap. A Yang—Yang anomaly implies a leading correction, pd x B4t2$, ("complete" scaling theory correction) would dominate the previously expected correction pd x B2t1-a [10, 11] ("incomplete" scaling theory correction). Therefore, the first temperature derivative of the coexistence-curve diameter diverges as the isochoric heat capacity dpd/dT x t-a and as dpd/dT x t2^-1 (2^ — 1 « -0.352), i. e., the divergence of the liquid—gas coexistence curve diameter is shared between the two terms, "incomplete" scaling term B2t1-a and "complete" scaling term B4t2^. The contribution of bot terms is defined by the Yang—Yang anomaly strength parameter. As was discussed in works [13, 14], the coefficients , BCI, B2, and B3 in Eqs. (5), (6), and (7)
(7)
where kB is Boltzmann's constant, are not independent. As it was shown in works [10-14], the asymmetric coexistence curve "incomplete" scaling parameter B2 and rectilinear parameter £>3 are directly related with the isochoric heat capacity asymptotic critical amplitude Aq and fluctuation induced "critical background" parameter BCI, respectively, in Eq. (5). The explicit relations (see below) between the coefficients Aq , BCI, B2, B3, B4, and Bo were provided previously by Wang and Anisimov [14]. The present two-phase experimental results for pure fluids (methanol, ethanol, n-propanol, and DEE) along the critical isochore (Tables 1-4) were fitted to Eq. (7). The derived values of the asymptotic critical amplitude are given in Table 11 together with the reported values for other fluids. It is very difficult to accurately estimate the value of the fluctuation induced "critical background" parameter Bcr/kB by using the fitting procedure because empirical way determination of the fitting parameter strongly depends on input data, for example, on the fitting temperature range, weight of the experimental data, etc. In general the regular part of the scaling relation (7) contains analytical classical term, ideal gas contribution, and fluctuation induced regular parts, which are very difficult empirically identified (Sengers [45]). Sengers [45] proposed analytic extrapolation technique of Tfit — Tc ^ 0, where Tfit is the fitting temperature range. Anisimov and Wang [13] and Wang and Anisimov [14] proposed theoretically estimate the value of "critical background" parameter Bcr/kB as
Bcr = ^
kB i?o(l — w)a/A' K)
where Rq = 0.7 is a universal constant [46, 47], u is an effective coupling constant which is depend on the cutoff wave number A of the critical fluctuations. At A ^ to, the parameter u, therefore the relation (8) is universal. For many fluids the values of u are within 0.4 to 0.5. According universal relation between the critical amplitudes (A+/A-), the critical amplitude A- = 1.916A+ or can be determine directly from the one-phase isochoric heat capacity data along the critical isochore, T ^ Tc. The value of calculated from Eq. (8) is in satisfactory agreement with the value directly calculated from the present experimental data. We have fitted our experimental isochoric heat capacity data for methanol, ethanol, n-propanol, and DEE along the critical isochore (Tables 1-4, see Fig. 13) to Eq. (7). The results are given in Table 11.
The asymptotic critical amplitudes, asymmetric coexistence curve parameters (a3, b2), and Yang—Yang anomaly strength (R^) of pure fluids calculated from the present calorimetric measurements near the critical points
Fluid Bo Si b2 0,3 Ry.
Methanol (this work) 54.35 1.944 0.8774 0.0412 0.2455 0.197
Ethanol (this work) 62.14 1.903 0.2543 -0.0338 0.0252 0.024
Butanol (this work) 32.27 1.890 0.0403 0.0155 0.8907 0.474
tert-Butanol [24] 17.73 1.915 0.2151 -0.1041 -0.0343 -0.036
sec-Butanol [26] 25.13 1.744 0.7200 -0.1146 -0.3344 -0.502
Isobutanol [25] 33.02 1.870 0.5656 -0.0199 0.0766 0.071
n-Propanol (this work) 56.92 2.000 0.2198 0.0023 0.1713 0.146
DEE (this work) 32.00 1.628 1.5319 0.0429 0.4513 0.311
CH4 [14] 17.18 1.551 - -0.0730 -0.0238 -0.024
C2H6 [14, 51] 22.18 1.649 - -0.0603 0.0014 -0.004
C2H4 [14, 51] 21.27 1.642 - -0.0745 -0.0035 -0.004
C3H8 [10, 51] 24.75 - - - - 0.560
n-CgHia [14, 51] 34.58 1.776 - 0.0207 0.0110 0.011
n-C7Hi6 [14, 51] 31.30 1.843 - 0.0941 0.3690 0.269
SF6 [14, 51] 29.88 1.733 - 0.0351 0.1810 0.153
C02 [10, 11, 51] 27.67 - - - - -0.400
H20 [14, 51] 31.63 2.035 - -0.0482 0.0618 0.058
Ne [14, 51] 19.11 1.497 - -0.0683 -0.0177 -0.018
N2 [14, 51] 23.57 1.565 - -0.0701 -0.0177 -0.018
R-113 [51] 27.04 1.841 - 0.0483 0.2180 0.179
According to the "complete" scaling theory the coexistence curve diameter (Eq. (6)) can be represented as [14]
Pd
l + a3 °
1 — a kB
(9)
where a3 and b2 are the system dependent asymmetry coefficients, which are determined by the details of intermolecular interactions [14]. The values of the coexistence curve parameters (B2, B3, and B4, Eq. (5)) are related with the isochoric heat capacity parameters (A^ and
BCT, Eq. (7)) as [14]
£>2 — —b2 0
Bc
1 ; B3 — -b2-1 — a kB
B4 =
a3
1 + a3
B2
(10)
As one can see from Eq. (10), the critical amplitude of the two-phase isochoric heat capacity along the critical isochore (Bcr) and fluctuation induced "critical background" term ( ) and coexistence curve amplitudes (B0, B2, B3, B4) are depend each other. As was mentioned above, the critical amplitudes of the coexistence curve diameter ("complete" scaling term, B4) and asymptotic coexistence curve amplitude (Bo) are related with the Yang—Yang anomaly strength R^ = a3/(1 + a3) or R^ = B4/B2 [10, 11]. As one can see from Eqs. (5) and (10), the pressure mixing coefficient a3 can be directly determined experimentally by measuring both side of the coexistence curve (saturated-liquid and -vapor densities) near the critical point. Equation (5) together with restrictions (10) for the parameters B2, B3, and B4 with fixed values of A- and Bcr from measurements was applied to the present
saturated-liquid and -vapor densities (TS, p?, p?) for methanol, ethanol, n-propanol, and DEE in the critical region derived from the present Cv measurements (Tables 1-4). The results are presented in Table 11. As one can see, in general, the singular liquid—gas coexistence curve diameter (Eq. (9)) contains only two adjustable parameters a3 and b2. As we can note from Table 11, both asymmetry coefficients are positive for all fluids, a3 > 0 and b2 > 0, except ethanol (b2 = —0.0338). This means that both non-analytical contributions (t1-a and t2^) in Eq. (9) for methanol, n-propanol, and DEE are compensate each other, producing an imitation of a rectilinear diameter [14], where "complete" scaling term is positive, a3/(1 + a3)B¡2t2$ > 0, while "incomplete" scaling term is negative, — b2A-/(1 — a)t1-a < 0). This compensation can explain why the coexistence curve diameter for some fluid close to rectilinear [48, 49], depending on the magnitude of the amplitudes of these terms. For methanol, n-propanol, and DEE the singularity of the coexistence curve diameter is shared between the terms t1-a and t2$, however, the contribution of "complete" scaling term t2$ is slightly dominant (for methanol, ethanol, and n-propanol) i. e. singular diameter behavior basically controlling by the "complete" scaling term (see Eq. (9)). For ethanol coexistence curve diameter is exhibit a small deviation from the rectilinear diameter in the immediate vicinity of the critical point because of the opposite sign of the 'complete' and "incomplete" scaling terms (see Eq. (9)), both non-analytic terms compensate each other. The same positive values of both asymmetric parameters a3 and b2 of the coexistence curve diameter were found also for other fluids in previous publications (n-pentane, n-heptane, SF6, and R-113, see Table 11) [32]. For these fluids the contribution of the "complete" and "incomplete" terms is opposite (see Eq. (9)), namely, t2^ is positive, t1-a is negative. Opposite signs, the negative contribution of t2^ and positive contribution of t1-a terms, was found for neon, methane, nitrogen, ethane, and hard-core square-well fluids [14]. For some fluids (ethane, water, RPM, and HCSW) fluids, (see, for example [13, 14]), depending on the effect of their physical-chemical nature (nature of the intermolecular interactions) on coexistence curve asymmetry, the contribution of "complete"
t2U
and "incomplete" t1 a scaling terms is positive, while the contribution of t1-a is negative (water, RPM, and HCSW fluids). For n-pentane, n-heptane, SF6, and Freon-113 both non-analytical contributions are opposite signs [14], while for water, HCSW, and RPM fluids both non-analytical contributions are opposite sign [14]. For DEE the value of the asymmetric parameter a3 is unusual large (0.450). So far, the maximum value of a3 = 0.37, was found for n-heptane [14]. Classical equation of states (Lattice gas, van der Waals, Redlich—Kwong, Peng—Robinson type EOS) predict the values for asymmetric parameters within from 0 to 0.493 for a3 and from 0 to 0.089 for b2.
The value of Yang—Yang anomaly strength R^ = a3/(1 + a3), where a3, for four pure fluids calculated from the derived critical amplitudes of the coexistence curve diameter are given in Table 11. As we can note from Table 11, R^ for DEE is about 0.45, which means that two-phase isochoric heat capacity anomaly in Eq. (1) almost equivalently shared between the vapor-pressure and chemical potential terms. For ethanol the value of R^ is very small (0.024), which means that the contribution of the chemical potential d2^/dT2 to Cv2 divergence is almost zero (singularity of Cv2 caused by vapor pressure term, d2PS/dT2). For methanol and n-propanol also R^ is relative small (0.245 and 0.171, respectively), therefore the singularity of the Cv2 basically caused by vapor-pressure term. The values of Yang—Yang anomaly strength can be derived from the direct isochoric heat capacity measurements i. e. from d2PS/dT2 and d?\x/dT2, (see Refs. [24-26] and Eq. 1). The values of R^ derived from heat capacity measurements in good agreement with the values derived from the coexistence curve density or singular diameter data. This is confirming the thermo-
dynamic consistence of the simultaneously measured isochoric heat capacity and saturated density data near the critical point and reliability of the quasi-static thermograms technique. Therefore, the value of the Yang—Yang anomaly strength parameter estimated from the second temperature derivatives, d2PS/dT2 and cP^/dT2, ([55-57]) and singular diameter data is in good agreement. Positive value of R^ means that mixing coefficient is positive, a3 (and < 1) and amplitude of the "complete" scaling term is also positive, B4 = a3/(1 + a3)B2 > 0. Therefore, the singular diameter asymptotically curves towards the vapor phase (see Eq. (9) and Fig. 12). The negative value of the Yang—Yang anomaly strength was found previously for CO2 (-0.35) [10, 11], Ne (-0.018), CH4 (-0.024), N2 (-0.018), C2H4 (-0.0035) [14], and tert-butanol (-0.03) [24]. Numerical MC simulation [12, 50] of the hard-core square-well fluid also indicates that R^ is small and negative (close to zero).
Since the system dependent asymmetry coefficients of the liquid—gas coexistence curve diameter, a3 and b2, are determine by the details of intermolecular interactions (a3 and b2 =0 for real fluids) we found correlation between a3 and b2, and acentric factor ro. The magnitudes and signs of a3 and b2 strongly depend on molecular size, shape, symmetry, departures from near-spherical form. Fig. 20 demonstrate acentric factor dependence of the present derived coexistence curve diameter asymmetry coefficients, together with previous reported values for other fluids [13, 14]. As this figure shows the present results for a3 and b2 of methanol, ethanol, propanol, and DEE are in good consistence with the previous reported data for other fluids [13, 14]. The same correlations were found also between asymptotic critical amplitudes (A^, To, B0) and acentric factor ro (by Perkins et al. [51]). The present derived asymptotic critical amplitudes values (A+, B0) together with reported data [51] are presented in Fig. 21. As one can see from Fig. 20 and 21, the values of the asymptotic critical amplitude (A+, B0) and asymmetric coexistence curve parameters (a3, b2) can be readily predicted from the acentric factor data for the fluids.
4. Conclusions. Using calorimetric (quasi-static thermo- and barograms) technique, the saturated liquid and vapor densities of pure fluids (methanol, ethanol, propanol, DEE) and binary mixtures (H2O+methanol, H2O+ethanol, H2O+ammonia, H2O+D2O, and
-0.10 0.05 0.20 0.35 0.50 0.65
ra
0.01 -
-0.01 -
-0.03 -
-0.05 -
-0.07
-0.09
-0.10 0.05 0.20 0.35 0.50 0.65
ra
Fig. 20. Asymmetric parameters a3 and b2 of the coexistence curve singular diameter ("complete scaling" theory parameters) as a function of acentric factor m
Fig. 21. Asymptotic critical amplitude of the isochoric heat capacity (A+) and liquid—gas coexistence curve (B0) as a function of acentric factor w:
1 — from [51]; 2 — calculated from the correlation [51]; 3 — n-propanol; 4 — ethanol; 5 — methanol;
6 — DEE (this work); 7 — n-butanol [53]; solid line is the prediction from the correlation by Gerasimov [52]; dashed line is the correlation by Perkins et al. [51]; dot-dashed — calculated with the correlation by Alekhin et al. [54]
CO2+n-decane) has been accurately measured near the critical point. Simultaneously measured values of saturated liquid and vapor density and isochoric heat capacity near the critical point has been used to calculate the asymmetric parameters a3 and b2 of the coexistence curve singular diameter ("complete" scaling theory parameters) near the critical point. The derived values of a3 and b2 were used to calculate the Yang—Yang anomaly strength parameter. The contributions of a "complete" scaling term, t2 on the coexistence-curve diameter behavior and second temperature derivatives d2PS/dT2 and d2^/dT2 on the divergence of two-phase heat capacity Cv2 near the critical point were studied. The divergence of the coexistence curve diameter, dpd/dt, for methanol, ethanol, n-propanol, and DEE is shared between the terms, B2t1-a and t2^. The Yang—Yang anomaly strength parameter R^ for DEE is about 0.45, which means that Cv2 anomaly almost equivalently shared between the vapor-pressure and chemical potential terms. For ethanol the value of R^ is very small (0.024), which means that the contribution of the chemical potential d2^/dT2 to Cv2 divergence is almost zero i. e. singularity of Cv2 caused by vapor pressure term, d2PS/dT2. For methanol and n-propanol also R^ is relative small (0.245 and 0.171, respectively), therefore the singularity of the Cv2 basically caused by vapor-pressure term. The values of R^ derived from Cv2 measurements in good agreement with the values derived from the coexistence curve density or singular diameter data. The amplitude of the "complete" scaling term a3 for methanol, ethanol, n-propanol, and DEE is positive, i. e., B4 = a3/(1 + a3)B^ > 0, while the "incomplete" scaling asymmetric parameter b2 is negative for ethanol, i. e., B2 = -b2A-/(1 — a) > 0. Therefore, the singular diameter for ethanol
asymptotically curves towards the liquid phase. Both non-analytical contributions (t1-a and t2IJ) for methanol, n-propanol, and DEE are compensate each other, producing an imitation of a rectilinear diameter.
The critical parameters for binary mixtures (critical curve data) (H2 O+methanol, H2O+ethanol, H2O+ammonia, H2O + D2O, CO2+n-decane) were determined using the saturated density data near the critical points. The correlations between the asymptotic critical amplitudes and the asymmetric coexistence curve diameter (a3, b2) and the acentric factor were found. The B0—m dependence is almost linear in the wide range of m, while A+—m dependence deviate from the linearity at high acentric factor values (for alcohols).
One of us, I. M. Abdulagatov, thanks the Thermophysical Properties Division of the National Institute of Standards and Technology for the opportunity to work as a Guest Researcher during the course of this research. The authors also thank Prof. Sengers for useful discussion acentric factor dependence of the asymptotic critical amplitudes and his suggestions.
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Статья поступила в редакцию 19 сентября 2012 г.
