- Email: [email protected]
The isothermal Joule-Thomson coefficient and equation of state for methanol and ethanol vapours
M-983 1. Chem. Thermodynamics 1979,11,741-756
The isothermal Joule-Thomson coefficient and equation of state for methanol and ethanol vapours P. G. FRANCIS” and R. C. PHUTELA Department of Chemistry, The University, Hull HU6 7RX, U.K. (Received 2 November 1978) Measurements have been made of the isothermal Joule-Thomson coefficient of methanol and of ethanol vapours at temperatures between 323 and 430K and at pressures up to 80 kPa. The results are analysed in terms of association in the vapour and are compared with other thermodynamic data.
1. Introduction The apparatus used was the same as that described previously.(l) Methanol and ethanol are particularly suitable substances for Joule-Thomson measurements becausethis technique is independent of adsorption errors. In an isothermal Joule-Thomson apparatus a stream of vapour is passedat known amount-of-substance flow rate f through a calorimeter containing a throttle and a heater. Current I and potential difference V at the heater maintain constant temperature when the gas passesbetween inlet pressurep1 and outlet pressurepz. The energy input per amount of substanceof gas is then the enthalpy change between the pressure limits, and a quantity 4 is defined by 4 = ~~/f(P, -Pl) = omz~ T) - WPl, T>l/(P* -PI), This is related to the virial equation of state: pV = RT+Bp+C’pZ+D’p3+
(1)
...,
by the equations 4 = ~,+3(C’-TdC’/dT)(p,+pz)+g(o’-TdD’/dT)(p:+p,pz+p,2)+ &, = B- TdBldT.
(2)
...,
(3) (4)
2. Materials The materials used were obtained at high purity and only drying with anhydrous calcium sulphate was necessary.Analysis by g.1.c.using a flame ionization detector gave purities of 99.96and 99.9 massper cent for the methanol and ethanol respectively. The water contents were determined by Karl Fisher titration to be 0.02 and 0.07 mole per cent respectively. a To whom correspondence should be addressed. 0021-9614/79/080747+10 $01.00/O
0 1979 Academic Press Inc. (London) Ltd.
748
P. G. FRANCIS
AND
R. C. PHUTELA
3. Results Measurements were made near to 7 temperatures for methanol and 6 temperatures for ethanol in the range 320 to 430 K. The maximum inlet pressure used was near saturation or 100 kPa and the pressure difference was about 25 kPa. The results are shown in tables 1 and 2 and in figures 1 and 2. TABLE 1. Methanol. Mean experimental values of the calorimeter temperature 7’, power P, flow ratef, inlet pressurepl, outlet pressurepz, the value of the quantity b(T), defined as d(T) = P/f(pz-PI), corrected for change in kinetic energy, and the deviation S) from the fitting equations defmed in the text. &, is the value of 45extrapolated to zero pressure T E
P mW
f mm01 s-1
PI k%
P2 kx
d(T) cm3 mol-l
54 cm3 mol - 1
323.09 323.10 323.09 323.11 323.11
39.488 62.380 81.680 96.551 97.926
0.22225 0.29109 0.30294 0.31844 0.30642
28.410 33.390 37.931 41.617 44.860
6.302 10.502 14.753 19.915 25.678
-8034 -9362 -11632 -13971 -16660
-187 +193 -68 -101 -71
328.17 328.26 328.18 328.26 328.18 328.26
32.110 46.428 61.113 80.576 103.706 130.128
0.23123 0.27516 0.31177 0.34429 0.37164 0.39214
338.44 338.44 338.44 338.44 338.45 338.44 338.44
17.041 34.048 57.618 59.666 106.950 102.880 95.369
0.15861 0.21576 0.26380 0.27186 0.35566 0.33044 0.32777
348.49 348.50 348.51 348.41 348.46 348.52
12.092 14.852 24.673 37.519 43.115 50.038
0.13788 0.15830 0.22055 0.27988 0.28903 0.30034
363.35 363.62 363.50 363.43 363.36 363.53 363.42 363.60 363.43
10.125 14.334 23.682 24.711 23.395 28.880 33.047 41.359 31.921
0.13308 0.18539 0.24514 0.25932 0.25466 0.28292 0.31404 0.36085 0.31692 do = -3206
do = -4735
qSo= -4525
do = -4124
q& = -3743
cm3 mol-’ 28.379 34.266 39.520 44.359 49.223 54.380
at 323.15 K 6.423 11.231 16.885 21.260 26.189 31.055
-6322 -7324 -8660 -10131 -12114 - 14266
-!-235 +225 t-155 +135
-4892 -6281 -8041 -8877 -10128 -11624 -12754
-166 -355 -395 -89 -98 -139 $75
-3963 -4013 -4701 -5251 -5879 -7316
-65 -13
-58
+66
cm3 mold1 at 328.15 K 29.685 44,456 57.861 63.328 71.906 77.097 80.717 cm3 mol-1 29.139 34.004 52.296 62.297 72.070 87.730
7.726 19.332 30.698 38.603 42.212 50.312 57.904 at 338.15 K 7.016 10.629 28.499 36.766 46.784 64.957
cm3 mol-l 31.662 43.544 54.434 54.287 63.158 69.400 73.573 83.507 94.192
at 348.15 K 7.800 19.488 24.856 25.207 37.314 40.910 44.782 53.899 70.467
cm3 mol-l
at 363.15 K
-3188 -3214 -3266 -3277 -3555 -3579 -3655 -3871 - 4248
42 +94 +169
+23 +53 -t88
+a -58 -15 -12 -57 -93
ISOTHERMAL
JOULE-THOMSON
COEFFICIENTS
74Y
TABLE l-continued T it
P mW
mm01 s-l
383.14 383.17 383.17 383.13 383.15 383.15
7.825 14.663 22.23 1 24.721 28.289 21.638
0.13656 0.23562 0.30079 0.31949 0.36711 0.35303
427.74 427.96 427.71 427.89 427.71 427.84 427.67
3.894 7.597 10.128 8.940 16.334 12.056 11.265
0.16159 0.27083 0.33943 0.35695 0.49248 0.45590 0.46724
f
PX
PZ
kx
kE
31.505 45.002 55.698 65.738 74.680 96.919
#o = -2538
cmarn01-~
cm3 mol-’
-2388 -2310 -2331 -2389 -2451 -2643
7.521 18.069 23.988 33.347 43.245 73.731
w cm3 mol -X +126 +201 +186 +142
$96 -13
at 383.15 K
28.413 43.238 53.658 62.079 75.039 81.407 92.035
40 = -1133
4(T) cm3 mol-l
6.467 18.075 27.128 39.679 46.353 58.378 71.088
-1096 -1114 -1124 -1118 -1156 -1148 -1151
-I-42 +11
+6 t3 -33 -31 -32
at 428.15 K
~ 0
I I -o-o-*-o-o-o------o-
-270
0
0
0
4
I
O-0
8 (pl 2 +p,
I
I
428.15 K 383.15 KO-
12
p2 +/+)/lo 2
16 20 3 kPa 2
24
FIGURE 1. Measured values of &T*> for methanol, corrected to standard temperature T* and plotted against (~2 + plpz + ~2) at seven standard temperatures between 323 and 428 K. Curves according to equations given in the text, assuming a pressure difference of 25 kPa.
750
P. G. FRANCIS
AND
R. C. PHUTELA
TABLE 2. Ethanol. Mean experimental values of the calorimeter temperature T, power P, flow rate J inlet pressure pl, outlet pressure p2, the value of the quantity d(T), defined as /(T) = P/f (pa -pl), corrected for change in kinetic energy, and the deviation Se from the fitting equations defined in the text. & is the value of 4 extrapolated to zero pressure T x
P mW
333.15 333.15 333.15 332.96 333.15 333.05 332.96 333.15 333.05 333.15 333.03
43.182 64.436 108.687 89.196 48.092 100.682 90.189 72.600 74.886 46.368 60.747
0.17160 0.20708 0.24480 0.22071 0.18204 0.23602 0.22762 0.20697 0.20345 0.16948 0.18546
347.79 347.67 347.72 347.65 347.73 347.79
15.931 43.839 67.814 54.613 59.438 63.487
0.12784 0.21700 0.28633 0.23909 0.27527 0.26954
363.65 363.47 363.66 363.64 363.47 363.57 363.64 363.48
12.342 17.935 31.518 43.718 40.400 37.107 33.054 41.641
0.13345 0.18619 0.25316 0.30960 0.30690 0.29079 0.28725 0.31341
382.81 382.81 382.75 382.83 382.81 382.74 382.83 382.72
7.689 15.009 17.924 17.267 18.517 26.922 18.013 20.630
0.12193 0.18573 0.22083 0.22904 0.25192 0.31200 0.27747 0.30349
402.86 402.82 402.90 402.86 402.87 402.84 402.88 402.84
4.824 8.174 9.608 12.774 11.938 13.188 16.472 15.318
0.11016 0.16056 0.19333 0.23497 0.23991 0.26832 0.30759 0.30799
PZ kPa
& = -7315
& = -5595
do = -4326
do = -3128
& = -2317
33.078 36.450 43.942 43.080 38.291 44.212 43.187 41.778 42.398 40.263 43.874 cm3 mol-l
9.783 9.030 10.485 13.342 18.233 12.498 14.655 16.105 15.899 21.155 23.291
i(T) -___ cm3 mol - 1 -10802 -11348 -13270 -13589 -13165 -13450 -13887 -13663 -13890 -14319 -15914
w cm3 mol- 1 -9 *0 -54
+46 +20 +246 +7 -28 -51 -94 -155
at 333.15 K 7.483 12.341 21.987 35.506 38.739 50.367
-5988 -6697 -7681 -8973 -9233 -10904
-25 i-78 +93 +220
30.011 42.851 52.928 64.524 72.115 72.172 80.417 91.167
at 348.15 K 7.365 20.995 25.816 35.859 47.269 48.064 60.223 69.913
-4083 -4407 -4592 -4926 -5298 -5293 -5698 -6252
+226 +71 +14 -43 -84 -82 -101 -132
cm3 mol-1
at 363.15 K -2902 -2967 -3032 -3100 -3174 -3262 -3410 -3523
+204 1-135
-2182 -2182 -2210 -2258 -2282 -2321 -2366 -2399
+113 +1m -t63 +g -20 -63 -110 -140
28.289 42.507 52.820 60.962 62.074 71.968 cm3 mol-1
29.018 39.064 46.163 52.454 60.440 69.860 82.241 90.705 cm3 mol-1
7.296 11.836 19.399 28.133 37.279 43.412 63.202 71.413
28.279 37.973 46.793 56.442 63.192 73.426 83.109 91.180
at 383.15 K 8.215 14.647 24.304 32.362 41.380 52.253 60.479 70.434
cm3 mol-l
at 403.15 K
$28 $12
$76 +14 -39 -95 -168 -217
ISOTHERMAL
JOULE-THOMSON
2-continued
TABLE
T E
P mW
mm01 s-l
430.55 430.28 430.48 430.28 430.15 430.28 430.19
3.217 7.282 9.010 9.233 9.745 12.850 10.499
0.10784 0.18897 0.23207 0.25162 0.27180 0.32320 0.31040
f
0r
Pl
PZ
N’)
ks
kPa
cm3 mol-’
27.547 43.300 53.108 63.090 71.488 82.074 91.118
do = -1615
6.992 17.526 27.600 39.639 48.748 57.203 70.284
cm3 mol-l
I
751
COEFFICIENTS
I
-1450 -1495 -1522 -1565 -1577 -1598 -1623
64 cm3 mol-’ +140 4-89 $50
+2 -14 -46 -76
at 430.15 K
I
I
I
430.15 K
o-o-o-o-o-o-o -2
-4
t
I -6 7 3 E 2 “0 T! s-
-8
-10
L
t
10
b
348.15 K
-12 I-
\ -\
-14
L
.
t “\
\
-16
L
0
O333.15K \
I
i
I
I
I
I
4
8
12
16
20
24
1 (~1+plpz +P$W 3 kPa* FIGURE 2. Measured values of qS(T*) for ethanol, corrected to standard temperatures T* and plotted against (p,” -f- plpz + p.$) at six standard temperatures between 333 and 430 K. Curves according to equations given in the text, assuming a pressure difference of 25 kPa.
752
P. G. FRANCIS
AND R. C. PHUTELA
Whereasfor non-polar vapours, Q,is Iinear in mean pressure(pr +p2)/2, the pressure dependence of $ for these alcohols is dominated by the fourth virial coefficient. Figures 1 and 2 show (P(T*), corrected to standard temperatures T* by the fitting equations described later, plotted against the quantity (p:+pIp2+pi) given in equation (3). Although r$(T*) is not a unique function of this quantity, the approximate linearity shows the dominance of this term. The very strong dependence on pressure at low temperature is remarkable. The points at the lowest pressuresare the least reliable becausethe calorimeter response is then poorest.
4. Theory In the lower half of the temperature range studied the properties of the vapours are dominated by hydrogen-bond interactions. The approximate linearity in the pressure function (p:+p2p2 +p$, or dependence upon the fourth virial coefficient, can be regarded as evidence for association into tetramers. This makes the extrapolation of the results to zero pressure, where only the second virial coefficient is significant, very uncertain at these low temperatures. Literature values of the second virial coefficient, shown in reduced form in figure 3, are discordant due to the effects of adsorption and to contributions from higher terms in the virial equation. The strong association in these vapours has previously been detected by observation of the pressure dependenceof the heat capacity, from which enthalpies and entropies
-4.0 I
I 0.7
I
I 0.8
I
I 0.9
I
I 1.0
I
I 1.1
T/Tc FIGURE 3. The reduced second virial coefficient Bp,IRT, plotted against reduced temperature T/T,. MethanoI: 0, Kretschrner and Wiebe;oa) A, Kudchadker and Eubank;tx3) Cl, Kell and McLaurin;t’4) X, Bottomley and Spu.rhng;(lb) +, Foz, Morcillo, and Mendez;u6) A, Lambert et a1.(17)Ethanol: 0, Kretschmer and Wiebe;oa) @, Hanks and Lambert;@) Dash-dot curve: corresponding-states curve for argon. (I) Dashed curve : calculated for methanol. Full curve : calculated for ethanol.
ISOTHERMAL
JOULE-THOMSON
COEFFICIENTS
753
of association have been deduced.c2-+ The measurements reported here contain sufficient detail to show inadequacy in the simple model used to interpret heatcapacity measurements. The equations relating to a simple association model of a vapour have been given by Woolley. c6)Positive values of the second virial coefficient then imply a negative association constant, the significance of which was discussedby Kilpatrick’7) and by Winston.@) The parameters used have been termed “sociation constants” by Guggenheimtg,lo) to emphasizetheir form as an excessnumber of molecular clusters over the random number. Heat-capacity measurements have been interpreted’2-5’ using this association model with the assumption that the second virial coefficient can be represented by a dimerization constant. For alcohols this is inadequate and modified equations are now needed in terms of the second virial coefficient together with an association model for higher terms. This may conveniently be obtained by following the treatment of Guggenheim.(“) The Helmholtz free energy is written in terms of an activity t as A/kT = N In c- V(5 + b2t2 + b&f3 + b&f4 -I- . . .). (5) The second virial coefficient B is then - b2. The formal definitions N,/Y = 5, NJV = c2, NJV = c3, and N4/V = t4 lead to sociation constants b, = N,V2/N:
= (kT)2n3/n:,
(6)
b4 = N,V3/N:
= (kT)3n,/n:.
(7)
N, and Z, can be regarded as numbers and partial pressures respectively of y-mers. Following Woolley,@) we write KS = n&r: and K4 = 71,/n:. The equation for the total pressure is p/kT = <+b2t2+b3t3+b4t4+
...,
(8)
which may be inverted to give 5 in the form t =p/kT-b2(p/kT)2+(2b~-b3)(p/kT)3+(-5b~+5b,b3-b,)(p/kT)4+
....
(9)
For minimum Helmholtz free energy N/V = t+2b2g2+3b3t3+4b4C4+
....
WV
Combination of equations (8), (9), and (lo), with division of one series by another, gives V = RT/p+B+RT(3(B/RT)2-2K3}p
+RT{10(B/RT)3-12(B/RT)K3-3K4)p2+ .... (11) This is analogous to the equation obtained by Woolley except for retention of the second virial coefficient B rather than an association constant for dimeezation. The temperature dependence of V is required for interpretation of Jo&-Thomson measurements. This can be written in terms of an empirical equation for B(T), together with crude expressions for the temperature dependence of K3 and ~~ in the form: K,(T) = K,(T+)exp{AH,(TTt)/RTTt) (12) K,(T) = K,(T+)exp(AH,(TTt)/RTTt} (13)
754
P. G. FRANCIS AND R. C. PHUTELA
-2
-10
-12 0.6
0.7
0.8
1.0 0.9 T/T,
1.1
1.2
1.3
FIGURE 4. The reduced zero-pressureintercept &p,/RT, plotted against reduced temperature 0, Methanol; 0, ethanol. Dashed curve: corresponding-statescurve for argon.“) Full curves according to equations given in the text. T/T,.
An arbitrary reference temperature, Tt = 298.15 K, has been adopted. The factors AH, and AH, may be regarded as enthalpies of association of monomer units into trimers and tetramers respectively. The quantity 4 measured in this work is then given by 4 = P(P,, T)-Hh = 40 + 13WW90
T)h--PI) +&AH&1 +PZ) + WUWW240 - 4&$, +4(BIRT)K,AH,+K,AH,}(p~ +pIp2+p3+
...
(14)
5. Discussion The second virial coefficient measurements shown in figure 3 have been plotted in reduced form BpJRT,, using critical data from the compilation by Ambrose and Townsend. The values given for the critical volumes are thought to be wrong. The Joule-Thomson
measurements reported here cannot be fitted using values of
&, consistent with the second virial coefficient measurementsat low temperature. At high temperatures the reduced second virial coefficients converge into the corresponding-states curve previously obtained for argon. (r) As the temperature is decreased, deviations from this curve set in so rapidly that a simple smooth function of temperature cannot be fitted. Accordingly, a reference temperature TR was adopted below which a deviation occurs of the form (TJT- T,/TdB; this applies only at temperatures
ISOTHERMAL
JOULETHOMSON
755
COEFFICIENTS
below TP The equations for B and +e are then the argon corresponding-states curves together with terms containing three additional parameters: Bp,/RTc = 0.139-0.336(T,/T)-0.1055(T,/T)2-0.0313(T,/T)3-0.00038(T,/T)8 0" G Td, -a(T/T,)(W-'T,/TJBfl,
(15)
$J,P,/RT, = 0.139-0.672(T,/T)-0.3165(T,/T)2-0.1252(T,/T)3 -0.00342(Tc/T)8 -a@+ l)(T,/T- Tc/TR)@,(T < T,J.
(16)
The measurementsof 4 were fitted by least squaresto the combination of equations (12), (13), (14), and (16). A reference temperature Tt = 298.15 K was chosen for the sociation constants, the search parameters being a, p, and TRof equation (16) together with K3(Tt), K,(T'), AH,, and AH4 of equations (12) and (13). This, in common with other forms of equation also tried, shows that these results do not allow meaningful values of K3(Tt) and AH, to be obtained, the contribution from K,(T) being zero within experimental error. The pressure dependence of 4 is dominated by the K,(T) terms. Allowance for temperature dependenceof AH,, equivalent to a non-zero value of AC, for the formation of tetramers,(I 4, did not significantly improve the fit. The results obtained for the fitting parameters are shown in table 3. The values obtained from heat capacities by Counsel1 et a1.(4’5)are shown in brackets, and values of their parameter AS, have also been included. TABLE 3. Values of the fitting parameters. Constants are for association, monomer = tetramer
; T,/TB &(298.15 K)/mPab3 A&/kJ mol-l A&(101.3 kPa)/J K-l mol-l
Methanol
Ethanol
0.839 7.415
3.532 4.690
1.163 664.5 (631.9) -96.27 (-98.94) -325.96 (-335.34)
0.777 1906.7 (1734.4) -102.40(-101.85) -337.76 (-336.70)
6. Conclusions These measurementsare free from adsorption errors. Direct compressibility measurements of second virial coefficient are shown to be too negative, probably for this reason. The pair interactions in the two alcohols are in close conformity with each other at corresponding states at reduced temperatures above 0.65, and with the inert gases at reduced temperatures above 0.9. At lower temperatures the expected negative deviation for the larger molecule is found. The equation of state is dominated by association into tetramers at reduced temperatures below 0.6, this interaction being specific and not following corresponding states.The dominant association contribution to the Joule-Thomson coefficient makes extrapolation to zero pressure very uncertain at low temperatures.
756
P. G. FRANCIS
AND R.. C. PHUTELA
REFERENCES 1. Clarke, P. H.; Francis, P. G.; George, M.; Phutela, R. C.; Roberts, G. K. St. C. J. Chem. Thermodynamics 1979, 11, 125. 2. Weltner, W.; Pitzer, K. S. J. Am. Chem. Sot. 1951, 73, 2606. 3. Barrow, G. M. J. Chem. Phys. 1952,2O, 1739. 4. Counsell, J. F.; Fenwick, J. 0.; Lees, E. B. J. Chem. Thermodynamics 1970,2, 367. 5. Counsell, J. F.; Lee, D. A. J. Chem. Thermodynamics 1973, 5, 583. 6. Woolley, H. W. J. Chem. Phys. 1953, 21, 236. 7. Kilpatrick, J. E. J. Chem. Phys. 1953, 21, 1366. 8. Winston, H. J. Chem. Phys. 1953,21,2245. 9. Guggenheim, E. A. Trans. Faraday Sot. 1%0,56, 1159. 10. Guggenheim, E. A. Applications of Statistical Mechanics. Oxford University Press. 1966. 11. Ambrose, D.; Townsend, R. Vapour-liquid critical properties. National Physical Laboratory Report 1977. 12. Kretschmer, C. B.; Wiebe, R. J. Am. Chem. Sot. 1954,76,2579. 13. Kudchadker, A. P.; Eubank, P. T. J. Chem. Eng. Data. 1970,15,7. 14. Kell, G. S.; McLaurin, G. E. J. Chem. Phys. 1969, 51, 4315. 15. Bottomley, G. A.; Spurling, T. H. Aust. J. Chem. 1%7,20, 1789. 16. Fez, 0. R.; Morcillo, J.; Mendez, A. An. R. Sot. Esp. Fis. Quim. 1954, 17B, 23. 17. Lambert, J. D.; Roberts, G. A. H.; Rowlinson, G. S. ; Wilkinson, V. J. Proc. R. Sot. London 1949, 196A, 113. 18. Hanks, P. A.; Lambert, J. D. 1971. Taken from Dymond, J. H.; Smith, E. B. The Virial Coefficients of Gases. Oxford University Press. 1969.
The isothermal Joule-Thomson coefficient and equation of state for methanol and ethanol vapours P. G. FRANCIS” and R. C. PHUTELA Department of Chemistry, The University, Hull HU6 7RX, U.K. (Received 2 November 1978) Measurements have been made of the isothermal Joule-Thomson coefficient of methanol and of ethanol vapours at temperatures between 323 and 430K and at pressures up to 80 kPa. The results are analysed in terms of association in the vapour and are compared with other thermodynamic data.
1. Introduction The apparatus used was the same as that described previously.(l) Methanol and ethanol are particularly suitable substances for Joule-Thomson measurements becausethis technique is independent of adsorption errors. In an isothermal Joule-Thomson apparatus a stream of vapour is passedat known amount-of-substance flow rate f through a calorimeter containing a throttle and a heater. Current I and potential difference V at the heater maintain constant temperature when the gas passesbetween inlet pressurep1 and outlet pressurepz. The energy input per amount of substanceof gas is then the enthalpy change between the pressure limits, and a quantity 4 is defined by 4 = ~~/f(P, -Pl) = omz~ T) - WPl, T>l/(P* -PI), This is related to the virial equation of state: pV = RT+Bp+C’pZ+D’p3+
(1)
...,
by the equations 4 = ~,+3(C’-TdC’/dT)(p,+pz)+g(o’-TdD’/dT)(p:+p,pz+p,2)+ &, = B- TdBldT.
(2)
...,
(3) (4)
2. Materials The materials used were obtained at high purity and only drying with anhydrous calcium sulphate was necessary.Analysis by g.1.c.using a flame ionization detector gave purities of 99.96and 99.9 massper cent for the methanol and ethanol respectively. The water contents were determined by Karl Fisher titration to be 0.02 and 0.07 mole per cent respectively. a To whom correspondence should be addressed. 0021-9614/79/080747+10 $01.00/O
0 1979 Academic Press Inc. (London) Ltd.
748
P. G. FRANCIS
AND
R. C. PHUTELA
3. Results Measurements were made near to 7 temperatures for methanol and 6 temperatures for ethanol in the range 320 to 430 K. The maximum inlet pressure used was near saturation or 100 kPa and the pressure difference was about 25 kPa. The results are shown in tables 1 and 2 and in figures 1 and 2. TABLE 1. Methanol. Mean experimental values of the calorimeter temperature 7’, power P, flow ratef, inlet pressurepl, outlet pressurepz, the value of the quantity b(T), defined as d(T) = P/f(pz-PI), corrected for change in kinetic energy, and the deviation S) from the fitting equations defmed in the text. &, is the value of 45extrapolated to zero pressure T E
P mW
f mm01 s-1
PI k%
P2 kx
d(T) cm3 mol-l
54 cm3 mol - 1
323.09 323.10 323.09 323.11 323.11
39.488 62.380 81.680 96.551 97.926
0.22225 0.29109 0.30294 0.31844 0.30642
28.410 33.390 37.931 41.617 44.860
6.302 10.502 14.753 19.915 25.678
-8034 -9362 -11632 -13971 -16660
-187 +193 -68 -101 -71
328.17 328.26 328.18 328.26 328.18 328.26
32.110 46.428 61.113 80.576 103.706 130.128
0.23123 0.27516 0.31177 0.34429 0.37164 0.39214
338.44 338.44 338.44 338.44 338.45 338.44 338.44
17.041 34.048 57.618 59.666 106.950 102.880 95.369
0.15861 0.21576 0.26380 0.27186 0.35566 0.33044 0.32777
348.49 348.50 348.51 348.41 348.46 348.52
12.092 14.852 24.673 37.519 43.115 50.038
0.13788 0.15830 0.22055 0.27988 0.28903 0.30034
363.35 363.62 363.50 363.43 363.36 363.53 363.42 363.60 363.43
10.125 14.334 23.682 24.711 23.395 28.880 33.047 41.359 31.921
0.13308 0.18539 0.24514 0.25932 0.25466 0.28292 0.31404 0.36085 0.31692 do = -3206
do = -4735
qSo= -4525
do = -4124
q& = -3743
cm3 mol-’ 28.379 34.266 39.520 44.359 49.223 54.380
at 323.15 K 6.423 11.231 16.885 21.260 26.189 31.055
-6322 -7324 -8660 -10131 -12114 - 14266
-!-235 +225 t-155 +135
-4892 -6281 -8041 -8877 -10128 -11624 -12754
-166 -355 -395 -89 -98 -139 $75
-3963 -4013 -4701 -5251 -5879 -7316
-65 -13
-58
+66
cm3 mold1 at 328.15 K 29.685 44,456 57.861 63.328 71.906 77.097 80.717 cm3 mol-1 29.139 34.004 52.296 62.297 72.070 87.730
7.726 19.332 30.698 38.603 42.212 50.312 57.904 at 338.15 K 7.016 10.629 28.499 36.766 46.784 64.957
cm3 mol-l 31.662 43.544 54.434 54.287 63.158 69.400 73.573 83.507 94.192
at 348.15 K 7.800 19.488 24.856 25.207 37.314 40.910 44.782 53.899 70.467
cm3 mol-l
at 363.15 K
-3188 -3214 -3266 -3277 -3555 -3579 -3655 -3871 - 4248
42 +94 +169
+23 +53 -t88
+a -58 -15 -12 -57 -93
ISOTHERMAL
JOULE-THOMSON
COEFFICIENTS
74Y
TABLE l-continued T it
P mW
mm01 s-l
383.14 383.17 383.17 383.13 383.15 383.15
7.825 14.663 22.23 1 24.721 28.289 21.638
0.13656 0.23562 0.30079 0.31949 0.36711 0.35303
427.74 427.96 427.71 427.89 427.71 427.84 427.67
3.894 7.597 10.128 8.940 16.334 12.056 11.265
0.16159 0.27083 0.33943 0.35695 0.49248 0.45590 0.46724
f
PX
PZ
kx
kE
31.505 45.002 55.698 65.738 74.680 96.919
#o = -2538
cmarn01-~
cm3 mol-’
-2388 -2310 -2331 -2389 -2451 -2643
7.521 18.069 23.988 33.347 43.245 73.731
w cm3 mol -X +126 +201 +186 +142
$96 -13
at 383.15 K
28.413 43.238 53.658 62.079 75.039 81.407 92.035
40 = -1133
4(T) cm3 mol-l
6.467 18.075 27.128 39.679 46.353 58.378 71.088
-1096 -1114 -1124 -1118 -1156 -1148 -1151
-I-42 +11
+6 t3 -33 -31 -32
at 428.15 K
~ 0
I I -o-o-*-o-o-o------o-
-270
0
0
0
4
I
O-0
8 (pl 2 +p,
I
I
428.15 K 383.15 KO-
12
p2 +/+)/lo 2
16 20 3 kPa 2
24
FIGURE 1. Measured values of &T*> for methanol, corrected to standard temperature T* and plotted against (~2 + plpz + ~2) at seven standard temperatures between 323 and 428 K. Curves according to equations given in the text, assuming a pressure difference of 25 kPa.
750
P. G. FRANCIS
AND
R. C. PHUTELA
TABLE 2. Ethanol. Mean experimental values of the calorimeter temperature T, power P, flow rate J inlet pressure pl, outlet pressure p2, the value of the quantity d(T), defined as /(T) = P/f (pa -pl), corrected for change in kinetic energy, and the deviation Se from the fitting equations defined in the text. & is the value of 4 extrapolated to zero pressure T x
P mW
333.15 333.15 333.15 332.96 333.15 333.05 332.96 333.15 333.05 333.15 333.03
43.182 64.436 108.687 89.196 48.092 100.682 90.189 72.600 74.886 46.368 60.747
0.17160 0.20708 0.24480 0.22071 0.18204 0.23602 0.22762 0.20697 0.20345 0.16948 0.18546
347.79 347.67 347.72 347.65 347.73 347.79
15.931 43.839 67.814 54.613 59.438 63.487
0.12784 0.21700 0.28633 0.23909 0.27527 0.26954
363.65 363.47 363.66 363.64 363.47 363.57 363.64 363.48
12.342 17.935 31.518 43.718 40.400 37.107 33.054 41.641
0.13345 0.18619 0.25316 0.30960 0.30690 0.29079 0.28725 0.31341
382.81 382.81 382.75 382.83 382.81 382.74 382.83 382.72
7.689 15.009 17.924 17.267 18.517 26.922 18.013 20.630
0.12193 0.18573 0.22083 0.22904 0.25192 0.31200 0.27747 0.30349
402.86 402.82 402.90 402.86 402.87 402.84 402.88 402.84
4.824 8.174 9.608 12.774 11.938 13.188 16.472 15.318
0.11016 0.16056 0.19333 0.23497 0.23991 0.26832 0.30759 0.30799
PZ kPa
& = -7315
& = -5595
do = -4326
do = -3128
& = -2317
33.078 36.450 43.942 43.080 38.291 44.212 43.187 41.778 42.398 40.263 43.874 cm3 mol-l
9.783 9.030 10.485 13.342 18.233 12.498 14.655 16.105 15.899 21.155 23.291
i(T) -___ cm3 mol - 1 -10802 -11348 -13270 -13589 -13165 -13450 -13887 -13663 -13890 -14319 -15914
w cm3 mol- 1 -9 *0 -54
+46 +20 +246 +7 -28 -51 -94 -155
at 333.15 K 7.483 12.341 21.987 35.506 38.739 50.367
-5988 -6697 -7681 -8973 -9233 -10904
-25 i-78 +93 +220
30.011 42.851 52.928 64.524 72.115 72.172 80.417 91.167
at 348.15 K 7.365 20.995 25.816 35.859 47.269 48.064 60.223 69.913
-4083 -4407 -4592 -4926 -5298 -5293 -5698 -6252
+226 +71 +14 -43 -84 -82 -101 -132
cm3 mol-1
at 363.15 K -2902 -2967 -3032 -3100 -3174 -3262 -3410 -3523
+204 1-135
-2182 -2182 -2210 -2258 -2282 -2321 -2366 -2399
+113 +1m -t63 +g -20 -63 -110 -140
28.289 42.507 52.820 60.962 62.074 71.968 cm3 mol-1
29.018 39.064 46.163 52.454 60.440 69.860 82.241 90.705 cm3 mol-1
7.296 11.836 19.399 28.133 37.279 43.412 63.202 71.413
28.279 37.973 46.793 56.442 63.192 73.426 83.109 91.180
at 383.15 K 8.215 14.647 24.304 32.362 41.380 52.253 60.479 70.434
cm3 mol-l
at 403.15 K
$28 $12
$76 +14 -39 -95 -168 -217
ISOTHERMAL
JOULE-THOMSON
2-continued
TABLE
T E
P mW
mm01 s-l
430.55 430.28 430.48 430.28 430.15 430.28 430.19
3.217 7.282 9.010 9.233 9.745 12.850 10.499
0.10784 0.18897 0.23207 0.25162 0.27180 0.32320 0.31040
f
0r
Pl
PZ
N’)
ks
kPa
cm3 mol-’
27.547 43.300 53.108 63.090 71.488 82.074 91.118
do = -1615
6.992 17.526 27.600 39.639 48.748 57.203 70.284
cm3 mol-l
I
751
COEFFICIENTS
I
-1450 -1495 -1522 -1565 -1577 -1598 -1623
64 cm3 mol-’ +140 4-89 $50
+2 -14 -46 -76
at 430.15 K
I
I
I
430.15 K
o-o-o-o-o-o-o -2
-4
t
I -6 7 3 E 2 “0 T! s-
-8
-10
L
t
10
b
348.15 K
-12 I-
\ -\
-14
L
.
t “\
\
-16
L
0
O333.15K \
I
i
I
I
I
I
4
8
12
16
20
24
1 (~1+plpz +P$W 3 kPa* FIGURE 2. Measured values of qS(T*) for ethanol, corrected to standard temperatures T* and plotted against (p,” -f- plpz + p.$) at six standard temperatures between 333 and 430 K. Curves according to equations given in the text, assuming a pressure difference of 25 kPa.
752
P. G. FRANCIS
AND R. C. PHUTELA
Whereasfor non-polar vapours, Q,is Iinear in mean pressure(pr +p2)/2, the pressure dependence of $ for these alcohols is dominated by the fourth virial coefficient. Figures 1 and 2 show (P(T*), corrected to standard temperatures T* by the fitting equations described later, plotted against the quantity (p:+pIp2+pi) given in equation (3). Although r$(T*) is not a unique function of this quantity, the approximate linearity shows the dominance of this term. The very strong dependence on pressure at low temperature is remarkable. The points at the lowest pressuresare the least reliable becausethe calorimeter response is then poorest.
4. Theory In the lower half of the temperature range studied the properties of the vapours are dominated by hydrogen-bond interactions. The approximate linearity in the pressure function (p:+p2p2 +p$, or dependence upon the fourth virial coefficient, can be regarded as evidence for association into tetramers. This makes the extrapolation of the results to zero pressure, where only the second virial coefficient is significant, very uncertain at these low temperatures. Literature values of the second virial coefficient, shown in reduced form in figure 3, are discordant due to the effects of adsorption and to contributions from higher terms in the virial equation. The strong association in these vapours has previously been detected by observation of the pressure dependenceof the heat capacity, from which enthalpies and entropies
-4.0 I
I 0.7
I
I 0.8
I
I 0.9
I
I 1.0
I
I 1.1
T/Tc FIGURE 3. The reduced second virial coefficient Bp,IRT, plotted against reduced temperature T/T,. MethanoI: 0, Kretschrner and Wiebe;oa) A, Kudchadker and Eubank;tx3) Cl, Kell and McLaurin;t’4) X, Bottomley and Spu.rhng;(lb) +, Foz, Morcillo, and Mendez;u6) A, Lambert et a1.(17)Ethanol: 0, Kretschmer and Wiebe;oa) @, Hanks and Lambert;@) Dash-dot curve: corresponding-states curve for argon. (I) Dashed curve : calculated for methanol. Full curve : calculated for ethanol.
ISOTHERMAL
JOULE-THOMSON
COEFFICIENTS
753
of association have been deduced.c2-+ The measurements reported here contain sufficient detail to show inadequacy in the simple model used to interpret heatcapacity measurements. The equations relating to a simple association model of a vapour have been given by Woolley. c6)Positive values of the second virial coefficient then imply a negative association constant, the significance of which was discussedby Kilpatrick’7) and by Winston.@) The parameters used have been termed “sociation constants” by Guggenheimtg,lo) to emphasizetheir form as an excessnumber of molecular clusters over the random number. Heat-capacity measurements have been interpreted’2-5’ using this association model with the assumption that the second virial coefficient can be represented by a dimerization constant. For alcohols this is inadequate and modified equations are now needed in terms of the second virial coefficient together with an association model for higher terms. This may conveniently be obtained by following the treatment of Guggenheim.(“) The Helmholtz free energy is written in terms of an activity t as A/kT = N In c- V(5 + b2t2 + b&f3 + b&f4 -I- . . .). (5) The second virial coefficient B is then - b2. The formal definitions N,/Y = 5, NJV = c2, NJV = c3, and N4/V = t4 lead to sociation constants b, = N,V2/N:
= (kT)2n3/n:,
(6)
b4 = N,V3/N:
= (kT)3n,/n:.
(7)
N, and Z, can be regarded as numbers and partial pressures respectively of y-mers. Following Woolley,@) we write KS = n&r: and K4 = 71,/n:. The equation for the total pressure is p/kT = <+b2t2+b3t3+b4t4+
...,
(8)
which may be inverted to give 5 in the form t =p/kT-b2(p/kT)2+(2b~-b3)(p/kT)3+(-5b~+5b,b3-b,)(p/kT)4+
....
(9)
For minimum Helmholtz free energy N/V = t+2b2g2+3b3t3+4b4C4+
....
WV
Combination of equations (8), (9), and (lo), with division of one series by another, gives V = RT/p+B+RT(3(B/RT)2-2K3}p
+RT{10(B/RT)3-12(B/RT)K3-3K4)p2+ .... (11) This is analogous to the equation obtained by Woolley except for retention of the second virial coefficient B rather than an association constant for dimeezation. The temperature dependence of V is required for interpretation of Jo&-Thomson measurements. This can be written in terms of an empirical equation for B(T), together with crude expressions for the temperature dependence of K3 and ~~ in the form: K,(T) = K,(T+)exp{AH,(TTt)/RTTt) (12) K,(T) = K,(T+)exp(AH,(TTt)/RTTt} (13)
754
P. G. FRANCIS AND R. C. PHUTELA
-2
-10
-12 0.6
0.7
0.8
1.0 0.9 T/T,
1.1
1.2
1.3
FIGURE 4. The reduced zero-pressureintercept &p,/RT, plotted against reduced temperature 0, Methanol; 0, ethanol. Dashed curve: corresponding-statescurve for argon.“) Full curves according to equations given in the text. T/T,.
An arbitrary reference temperature, Tt = 298.15 K, has been adopted. The factors AH, and AH, may be regarded as enthalpies of association of monomer units into trimers and tetramers respectively. The quantity 4 measured in this work is then given by 4 = P(P,, T)-Hh = 40 + 13WW90
T)h--PI) +&AH&1 +PZ) + WUWW240 - 4&$, +4(BIRT)K,AH,+K,AH,}(p~ +pIp2+p3+
...
(14)
5. Discussion The second virial coefficient measurements shown in figure 3 have been plotted in reduced form BpJRT,, using critical data from the compilation by Ambrose and Townsend. The values given for the critical volumes are thought to be wrong. The Joule-Thomson
measurements reported here cannot be fitted using values of
&, consistent with the second virial coefficient measurementsat low temperature. At high temperatures the reduced second virial coefficients converge into the corresponding-states curve previously obtained for argon. (r) As the temperature is decreased, deviations from this curve set in so rapidly that a simple smooth function of temperature cannot be fitted. Accordingly, a reference temperature TR was adopted below which a deviation occurs of the form (TJT- T,/TdB; this applies only at temperatures
ISOTHERMAL
JOULETHOMSON
755
COEFFICIENTS
below TP The equations for B and +e are then the argon corresponding-states curves together with terms containing three additional parameters: Bp,/RTc = 0.139-0.336(T,/T)-0.1055(T,/T)2-0.0313(T,/T)3-0.00038(T,/T)8 0" G Td, -a(T/T,)(W-'T,/TJBfl,
(15)
$J,P,/RT, = 0.139-0.672(T,/T)-0.3165(T,/T)2-0.1252(T,/T)3 -0.00342(Tc/T)8 -a@+ l)(T,/T- Tc/TR)@,(T < T,J.
(16)
The measurementsof 4 were fitted by least squaresto the combination of equations (12), (13), (14), and (16). A reference temperature Tt = 298.15 K was chosen for the sociation constants, the search parameters being a, p, and TRof equation (16) together with K3(Tt), K,(T'), AH,, and AH4 of equations (12) and (13). This, in common with other forms of equation also tried, shows that these results do not allow meaningful values of K3(Tt) and AH, to be obtained, the contribution from K,(T) being zero within experimental error. The pressure dependence of 4 is dominated by the K,(T) terms. Allowance for temperature dependenceof AH,, equivalent to a non-zero value of AC, for the formation of tetramers,(I 4, did not significantly improve the fit. The results obtained for the fitting parameters are shown in table 3. The values obtained from heat capacities by Counsel1 et a1.(4’5)are shown in brackets, and values of their parameter AS, have also been included. TABLE 3. Values of the fitting parameters. Constants are for association, monomer = tetramer
; T,/TB &(298.15 K)/mPab3 A&/kJ mol-l A&(101.3 kPa)/J K-l mol-l
Methanol
Ethanol
0.839 7.415
3.532 4.690
1.163 664.5 (631.9) -96.27 (-98.94) -325.96 (-335.34)
0.777 1906.7 (1734.4) -102.40(-101.85) -337.76 (-336.70)
6. Conclusions These measurementsare free from adsorption errors. Direct compressibility measurements of second virial coefficient are shown to be too negative, probably for this reason. The pair interactions in the two alcohols are in close conformity with each other at corresponding states at reduced temperatures above 0.65, and with the inert gases at reduced temperatures above 0.9. At lower temperatures the expected negative deviation for the larger molecule is found. The equation of state is dominated by association into tetramers at reduced temperatures below 0.6, this interaction being specific and not following corresponding states.The dominant association contribution to the Joule-Thomson coefficient makes extrapolation to zero pressure very uncertain at low temperatures.
756
P. G. FRANCIS
AND R.. C. PHUTELA
REFERENCES 1. Clarke, P. H.; Francis, P. G.; George, M.; Phutela, R. C.; Roberts, G. K. St. C. J. Chem. Thermodynamics 1979, 11, 125. 2. Weltner, W.; Pitzer, K. S. J. Am. Chem. Sot. 1951, 73, 2606. 3. Barrow, G. M. J. Chem. Phys. 1952,2O, 1739. 4. Counsell, J. F.; Fenwick, J. 0.; Lees, E. B. J. Chem. Thermodynamics 1970,2, 367. 5. Counsell, J. F.; Lee, D. A. J. Chem. Thermodynamics 1973, 5, 583. 6. Woolley, H. W. J. Chem. Phys. 1953, 21, 236. 7. Kilpatrick, J. E. J. Chem. Phys. 1953, 21, 1366. 8. Winston, H. J. Chem. Phys. 1953,21,2245. 9. Guggenheim, E. A. Trans. Faraday Sot. 1%0,56, 1159. 10. Guggenheim, E. A. Applications of Statistical Mechanics. Oxford University Press. 1966. 11. Ambrose, D.; Townsend, R. Vapour-liquid critical properties. National Physical Laboratory Report 1977. 12. Kretschmer, C. B.; Wiebe, R. J. Am. Chem. Sot. 1954,76,2579. 13. Kudchadker, A. P.; Eubank, P. T. J. Chem. Eng. Data. 1970,15,7. 14. Kell, G. S.; McLaurin, G. E. J. Chem. Phys. 1969, 51, 4315. 15. Bottomley, G. A.; Spurling, T. H. Aust. J. Chem. 1%7,20, 1789. 16. Fez, 0. R.; Morcillo, J.; Mendez, A. An. R. Sot. Esp. Fis. Quim. 1954, 17B, 23. 17. Lambert, J. D.; Roberts, G. A. H.; Rowlinson, G. S. ; Wilkinson, V. J. Proc. R. Sot. London 1949, 196A, 113. 18. Hanks, P. A.; Lambert, J. D. 1971. Taken from Dymond, J. H.; Smith, E. B. The Virial Coefficients of Gases. Oxford University Press. 1969.