Internal Pressure
A. F. M. Barton
Victoria University of Wellington Wellington, New Zealand
A fundamental liquid property
Although many equations have been developed to express pressure-volume-temperature h V T ) data for liquids, notably the isothermal equations of state (1-4) the resulting parameters or constants do not in general provide an immediate and visual comparison of liquid properties. I t is possible to express graphically the results of pVT measurements in such a way that both the balance between repulsive and attractive intermolecular effects and also a direct comparison between different liquids are apparent. This approach was described by Hildehrand and Scott (5, 6) but has since been rather neglected. It deals with the fundamental cohesive force which is the resultant of forces of attraction and forces of repulsion between liquid molecules. This interpretation of pVT data for liquids is made in terms of eqn. (3) which is often called a "thermodynamic equation of state" (7) because it provides a relationship between pressure p, molar volume V , temperature T , and molar internal energy U which is valid for all substances and which is closely related to ordinary pVT data. From the basic thermodynamic relations dU
=
TdS - p d V
one obtains (aW/bV),
=
T(aS/bV)r - p
and hence ( b U I b V h = T(bp/aT)v - p
(3)
The isothermal internal energy-volume coefficient (bU/dV)Tis often called the internal pressure, rr. For an ideal gas, the internal pressure is zero, since intermolecular forces are absent, but for imperfect gases and liquids it becomes appreciable, and is frequently much greater than the external pressure p. The internal pressure may he measured experimentally by the difference between the thermal pressure term T(bp/bT)" and the external pressure p. The isochoric or constant volume thermal pressure ccefficient 0 = (bp/bT), is directly accessible from the determination of isochoric p versus T plots (8-13).
and The Constant Volume Thermometer
li Figure 1. (left) Pierometer for tion of irochoric pT doto (12).
Re
determino-
Figure 2. (above) Pressure verse1 designed (15) for gas pressure operotion to 2 kbor and
500°C.
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Journol of Chemical Educotion
A typical "constant volume thermometer" or "piezometer" (14) is shown in Figure 1. This was designed for use with an electrically-conducting liquid, a molten salt (12, 13); and the liquid level is determined by an electrical indicator circuit consisting of platinum wire contacts sealed into capillary tubing and connected externally by a relatively high resistance. When the liquid rises due to increasing temperature or decreasing pressure, successive probes are shorted by the low resistance melt, and the electrical resistance shows a sharp decrease. The spiral of capillary tube is designed to minimize diffusion of the pressurizing gas into the sample liquid. Pressure and temperature must be closely controlled; a suitable pressure vessel for the range 0-2 kbar (16) is shown in Figure 2 (1 bar = lo5N m-2). After the filled piezometer has been inserted in the pressure vessel, which is maintained at a constant temperature in some form of thermostat, the pressure is increased to reduce the liquid level to its lowest required position. The pressure is then allowed to decrease slowly, and the pressure corresponding to the liquid level reaching each probe is recorded. The procedure is repeated at different temperatures, and for different quantities of melt in the piezometer. Typical temperature versus pressure isochores for fused salts are shown in Figure 3.
0
Figure 3.
P bor
500
1000
1500
McLeod gauge: the suspended column of liquid is causing a negative pressure in the liquid at the top of the capillary (20). I n principle it is possible to produce a negative pressure by heating liquid in a sealed glass tube until the liquid fills the whole space, then cooling the tube gently so that the liquid remains adhering to the walls; but in practice the formation of this metastable condition is both difficult and rather dangerous (21). Briggs (22) has measured the experimental limiting negative pressures of water, mercury, and organic liquids by a centrifugal method using filled capillaries open at both ends and rotating in a horizontal plane. Observations were made along the rotation axis to determine when the liquid column was broken by the force acting outwards in the liquid a t each end of the capillary. With the assumption that the intermolecular forces in the liquid are weaker than the forces between the liquid and the glass walls, the tension at which the column breaks provides an estimate of the greatest negative pressure attainable: At room temperature these were found to range from -431 bar for Hg to -132 bar for benzene. A maximum value of -159 bar has been observed for benzene (23). A pressure vessel method has also been described (24)~.
Temperature venur pressure irochorer for fused KNOOl122.
Internal Energy and Molar Volume Internal Pressure, Entropy, and Internal Energy
These are also often close to straight lines, for nonpolar liquids, even at the critical point (14). When 0 is a function of V only in this way, it follows from eqn. (3) that ?r is also necessarily a pure volume function within experimental error (16-18). The relation between internal pressure and liquid structure may also be considered from the point of view of entropy. For a closed system undergoing a reversible isothermal volume change, eqn. (1) holds, and thus when the external pressure p is zero, the internal pressure is proportional to the volume derivative of entropy. The molar entropy S and molar energy U of the liquid may he evaluated by integration of eqns. (1) and (2) if the necessary pVT data are available, as in the recent equation of state measurements for liquid argon (19) S
U
=
= 0 0
So
+
+
s,
s,
V
(~p/M')vdV
v i T ( b ~ / a T ) v- p1dV
The reference points were S = 0 at T = OoI
110) Grnaolr, R. E.. nso LOEPFLEA, 0.H., J . Amer. Chem. Soc.. 61, 2515 ,>,,?O>
Iinnnc~a,A. W.. Ind. Bng. Cham.. 49, 1779 (1957). RnEToN, A. r. M., H ~ L L SG, . J., FRAY.D, J., A N D TOYLINSON, J, lK, H i g h Temp.. Ili0b Plaaaurer to be published. I l n ~ l m n n . J. E.. P1r.D. Thesis. University of Soatllsmpton, 1969. Reference (7).n. 30: Pnnwnerow, J. R.. "An Advanced Treatise on I'lrysieal Cliernistry," vol. 2 , "The Pro~ertierof Lirrairln:' Lonemans. London. 1951, P. 58. D n n r o ~A, . F. M.,C m ~ v E nH, . , nno HILLS.G. J.. Trana. Paraday Soc., 64, 208 (1968).
Volume 48, Number 3, March 1971
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