Alan J. Markworth
Battelle Memorial Institute Columbus, Ohio 43201
I I
Liquid Rise in a Capillary Tube A treatment on the basis of energy
A
now-classic prohlem in capillary phenomena is the calculation of the height h to which liquid rises in a vertically positioned tuhe. One elementary approach to treating this prohlem consists of equating the weight of the liquid column to the vertical component of "force" resulting from the surface tension of the liquid. Another involves equating the pressure change across the meniscus to the hydrostatic pressure differenceacross the liquid column. I n either case, it is shown, assuming the tuhe has circular cross section of radius R, that where y r is the surface tension of the liquid, 0 the contact angle between the liquid and the surface of the tube, p the density of the liquid, and g the acceleration due to gravity. Let us examine an alternate, and physically illuminating, approach which consists of viewing this prohlem from the standpoint of the total energy change, AE, associated with formation of the liquid column. We begin with the relation AE = AE.
+ AE.
(2)
where AE, is the work done against gravity to construct the liquid column and AE, is the corresponding change of interfacial energy. Of course, the latter contribution to AE arises from the fact that when the liquid column is formed within the tuhe, an interface characterized by the surface tension of the tuhe material, r., is changed to one charact,erized by that of the solid-liquid interface, y,, (or vice-versa, depending upon the magnitude of y. relative to that of y,,).
528 / Journol of Chemicol Educofion
It is easy to show that AE.
=
'/mpgRW
(3)
and AE. = (re, - r,)2=Rh
(4
From eqn. ( 4 ) ,we infer that h must he positive if 7 , > rsland negative if y. < y , ~ ,the former situation corresponding to an elevated liquid column, the latter to a depressed column. We may now use the familiar equation of Young and DuprB, i.e. together with eqns. (2)-(4), to obtain The equilibrium height of the liquid column corresponds to that value of h for which AE has its minimum value, whence, we again obtain eqn. (1). It is also interesting to note that eqns. (1) and (5) can he used to show that the minimum value of AE can he expressed as with AE, being equal to -2AE, at this particular value of AE. As expected, (AE),;, is negative, regardless of whether the liquid column is elevated or depressed, and, in addition, is independent of R. Although this approach is somewhat more involved than the others mentioned above, it does lend considerable insight into the details of the physical changes which occur as the liquid column is formed.
