Scientific journal
European Journal of Natural History
ISSN 2073-4972

Simple Kelvin Equation applicable in the Critical Point Vicinity

Valeev A.A. 1
1 Kamyshin Technological Institute of Volgograd State Technical University
New simple form of the Kelvin equation is offered that applies near the gas-liquid transition critical point.
1. Thomson W. On the Equilibrium of Vapour at a Curved Surface of Liquid, Proc. Roy. Soc. Edinburgh, 7 (1870) Р. 63–68.
2. Thomson W. On the Equilibrium of Vapour at a Curved Surface of Liquid, Philosophical Magazine Series 4, 42 (282) (1871) 448–452.
3. Gregg S.J., Sing K.S.W. Adsorption, Surface Science and Porosity, Academic Press, New York, 1982.
4. Parry A.O., Evans R. Universal fluctuation-induced corrections to the Kelvin equation for capillary condensation, J. Phys. A: Math. Gen. 25 (1992). – Р. 275–284.
Well known form of the Kelvin Equation [1, 2]

val01.wmf (1)

has been proved to work excellent for the phase equilibrium of rare gas and liquid systems, the ones it was originally designed for. Here val02.wmf,
val03.wmf are the saturated vapor pressures against a plain surface and a meniscus with effective curvature radius r, respectively, the latter accounting for such effects as the adsorption at the pore wall. Number “2” stays for spherical geometry of the meniscus. It should be replaced by “1” in the case of the cylindrical geometry. μ is the molar mass of the substance in use, ρL is the liquid phase density, R is the universal gas constant. Quite often, this equation is misused when applied to high-presser gases and to liquids at near-critical temperatures.

To account for the gas’s nonideality, another equation [3] was developed:

val05.wmf (2)

where f is the replacement of the pressure by the fugacity. In the next paragraph, it will be shown that this equation, as well as those obtained in [4], does not work quite well near the critical point and the replacement will be presented.

Kelvin equation from scratch

We consider a liquid-gas equilibrium system divided by a plain surface. Now we can put down the change in the Gibbs energy for each phase:

val06.wmf (3)

where i denotes the phase that can be either G or L. μi, ni, Si, Ti, Vi, pi are chemical potential, amount of substance, entropy, temperature, volume and pressure of the phase i, respectively. Taking into account that both phases are in the state of thermodynamical equilibrium,
we can put:

val07.wmf (4)

Generally speaking, pG ≠pL, due to additional pressure by the curved surface tension. The eq. (3) will take the following form:

val08.wmf (5)

Now we can put down the change in the chemical potential:

val09.wmf (6)

where si, vi, are the molar entropy and the molar volume, respectively. Now we
can see that

val10.wmf (7)

Taking into account that val11.wmf, we can put

val12.wmf (8)

Omitting vLdpG term is what has been done to obtain the eq’s (1), (2) and the analogous equations in [4], because the gas phase there has been considered dilute. This is not our case, because we operate in the critical point vicinity, i.e. vL is comparable with vG. Integrating the eq. (8)

val13.wmf (9)

we obtain approximate equation

val14.wmf, (10)

which can be used if vG – vL does not change significantly within interval val15.wmf.

Finally, we have

val16.wmf (11)

where ρG is the gas phase density.

Now we can rewrite eq. (8) in the following form

val17.wmf (8*)

Integrating it we obtain

val18.wmf, (12)

where val19.wmf is the pressure of the liquid phase in equilibrium with the saturated vapor (both divided by the curved meniscus).

val20.wmf, (13)

Now we can write

val21.wmf. (14)

Conclusions

The eq. (11) (along with the eq. (14) and the eq’s (9), (12) in the integral form) is the new form of the Kelvin equation that applies near the gas-liquid transition critical point. It can be helpful in researches of nano-structured materials. It should be applied carefully, due to the geometry dependence of the surface tension.