Free volume of liquid

In a later work Cohen and Turnbull [90] defined the free volume of liquid as a part of the thermal expansion being freely redistributed throughout the volume without any change in energy. Macedo and Litovitz [91] pointed out that Cohen-TurnbulTs equation (74) ensues from an already known equation obtained independently by Fulcher [92] and Tamman [93] ... 

The relationship between viscous flow and the free volume of liquids capable of undergoing glass transition was established in the so-called valence configurational theory of Nemilov [94], who obtained this relationship in the general form ... 

Amplitude a can he related to the temperature coefficient of linear expansion of liquid Oi a = 2a T. At the same time, the amplitude can be related to the free volume of liquid Vf... 

In turn, the free volume of liquid can be estimated, e.g., by the heat of evapora-... 

Neither the volume occupied by a mole of gas at the boiling point nor the free volume of a liquid vary too widely from substance to substance. Taking the former to be about 30,000 ml and the latter to be about 3 ml gives... 

The permachor method is an empirical method for predicting the permeabiUties of oxygen, nitrogen, and carbon dioxide in polymers (29). In this method a numerical value is assigned to each constituent part of the polymer. An average number is derived for the polymer, and a simple equation converts the value into a permeabiUty. This method has been shown to be related to the cohesive energy density and the free volume of the polymer (2). The model has been modified to liquid permeation with some success. 

For a very dilute suspension, i.e., e = 1 and (e) = 1, the settling veloeity will be equal to the free-fall veloeity. As no valid theoretieal expression for the funetion (e) is available, eommon praetiee is to rely on experimental data. Note that a unit volume of thiekened sludge eontains e volume of liquid and (1 - e) volume of solid phase, i.e., a unit volume of partieles of sludge eontains e/(l - e) volume of liquid. Denoting a as the ratio of partiele surfaee area to volume, we obtain the hydraulie radius as the ratio of this volume, e/(l - e), to the surfaee, a, when both values are related to the same volume of partieles ... 

L. L. Blyler and T. K. Kwei [39] proposed the direct opposite (to 4). In their reasoning, they proceeded from the known and generally acceptable Doolittle equation, which puts liquid viscosity in exponential dependence on the inverse value of the free volume of the latter. According to [39], gas has a volume of its own, the value of which it contributes to the free volume of the polymer when it dissolves therein as a result, viscosity falls. The theoretical formula obtained by the authors was experimentally confirmed in the same work. The authors measured pressure values at the entrance of cylindrical capillaries, through which melts of both pure polyethylene, and polyethylene with gas dissolved in it, extruded at a constant rate. 

As discussed, the intuitive notion that there should be a connection between the statistics of the free volumes of a fluid and its measurable macroscopic properties has a long history in studies of the liquid state. In fact, it turns out that this connection is precise in the case of the thermodynamics of the single-component hard-sphere fluid. Specifically, Hoover, Ashurst, and Grover77 and Speedy82 have provided independent derivations that predict the relationship between the hard-sphere compressibility factor Z = P/pksT and the geometric properties of its free volumes, as follows ... 

The mathematical treatment of surfaces was developed much earlier than the direct examination of superficial films. It is based primarily upon the observed tendency of a volume of liquid to adopt when unconstrained a shape with a minimum surface. The property of having a minimum surface for a given volume is possessed by a sphere, and this shape is assumed by a free body of liquid. A falling raindrop is spherical upon this fact indeed is based the... 

We begin by formulating the free energy of liquid-crystalline polymer solutions using the wormlike hard spherocylinder model, a cylinder with hemispheres at both ends. This model allows the intermolecular excluded volume to be expressed more simply than a hard cylinder. It is characterized by the length of the cylinder part Lc( 3 L - d), the Kuhn segment number N, and the hard-core diameter d. We assume that the interaction potential between them is given by... 

Liquid holdup is critical in the downflow operation of fixed beds, in contrast to the upflow operation where the liquid occupies practically the whole external free void volume of the bed. Total liquid holdup ht consists of two parts static h, and dynamic holdup liA. Static holdup is related to the volume of liquid that is adherent to the particles surface, whereas dynamic holdup is related to the flowing pari of the liquid. 

Covacs6 proposed another definition of the fractional free-volume. As the extrapolated volume of supercooled liquids reaches the crystalline state, vc, at temperature rc > 0 K, it is possible to compare this critical temperature with rg , the limit glass temperature at infinitely slow cooling. Free-volume in the crystalline state is assumed to be zero. In this case the value /g according to Doolittle may be compared with /g, c, which characterizes the excess free-volume of glass as compared with the crystalline state ... 

Mason77 developed ideas about the distribution of the free-volume to explain the existence of the broad transition region from glassy to rubberlike state. He believed that there is some localization of that part of the free-volume that distinguishes the rubberlike state from the true liquid state in which the free-volume is not localized. In the non-crosslinked state of some rubbers there may be an arbitrary distribution of the free-volume Vf connected with the free-volume of each monomeric... 

FREE VOLUME. A liquid differs from a solid having the same type of packing of the molecule in having a certain additional volume, the free volume, which provides the necessary looseness in the slructure lo permil free movement of the molecules. The concept of free volume is used in several theories of the liquid state. 

The free amine rapidly absorbs carbon dioxide. It is therefore essential to protect the distillation apparatus from the atmosphere with a guard-tube filled with soda-lime. As the product tends to foam excessively during distillation, the apparatus used should be larger than is customary for the volume of liquid to be distilled. 

It is, therefore, not surprising that there exists a definite relationship between Aand the enthalpy of vaporization, Av H, the former constituting a fraction between 0.2 and 0.3 of the latter, as is readily obtained from the data in Tables 3.1 and 3.9. The pressure dependence of the viscosity is also closely related to the free volume of the solvent. The fluidity (O = l/r ) is proportional to the ratio between the free and the occupied volume, the former, as mentioned above, being the difference between the actual molar volume and the intrinsic molar volume (Tables 3.1 and 3. 4) (Hildebrand 1978). In fact, the logarithm of the viscosity of liquids was found (Marcus 1998) to be described well for some 300 liquids by the empirical relationship ... 

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