Equilibrium, hydrostatic

In order for solvent and solution to be in equilibrium in an apparatus such as that shown in Figure 3.2, the solution side must be at a higher pressure than the solvent side. This excess pressure is what is known as the osmotic pressure of the solution. If no external pressure difference is imposed, solvent will diffuse across the membrane until an equilibrium hydrostatic pressure head has developed on the solution side. In practice, to prevent too much dilution of the solution as a result of the solvent flow into it, the column in which the pressure head develops is generally of a very narrow diameter. We return to the details of osmotic pressure experiments in the next section. First, however, the theoretical connection between this pressure and the concentration of the solution must be established. 

Osmometers consist basically of a solvent compartment separated from a solution compartment by a semipermeablc membrane and a method for measuring the equilibrium hydrostatic pressures on the two compartments. In static osmometers... 

The foregoing equations are those obtained by Enskog using somewhat different methods. Let us examine the results of Longuet-Higgins and Pople. From Eq. 61, the equilibrium hydrostatic pressure is given by... 

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

Relationships from thennodynamics provide other views of pressure as a macroscopic state variable. Pressure, temperature, volume and/or composition often are the controllable independent variables used to constrain equilibrium states of chemical or physical systems. For fluids that do not support shears, the pressure, P, at any point in the system is the same in all directions and, when gravity or other accelerations can be neglected, is constant tliroughout the system. That is, the equilibrium state of the system is subject to a hydrostatic pressure. The fiindamental differential equations of thennodynamics ... 

The situation is more complex for rigid media (solids and glasses) and more complex fluids that is, for most materials. These materials have finite yield strengths, support shears and may be anisotropic. As samples, they usually do not relax to hydrostatic equilibrium during an experiment, even when surrounded by a hydrostatic pressure medium. For these materials, P should be replaced by a stress tensor, <3-j, and the appropriate thermodynamic equations are more complex. 

Another property of importance is the pore volume. It can be measured indirectly from the adsorption and/or desorption isotherms of equilibrium quantities of gas absorbed or desorbed over a range of relative pressures. Pore volume can also be measured by mercury intrusion techniques, whereby a hydrostatic pressure is used to force mercury into the pores to generate a plot of penetration volume versus pres- sure. Since the size of the pore openings is related to the pressure, mercury intrusion techniques provide information on the pore size distribution and the total pore volume. 

A capillary system is said to be in a steady-state equilibrium position when the capillary forces are equal to the hydrostatic pressure force (Levich 1962). The heating of the capillary walls leads to a disturbance of the equilibrium and to a displacement of the meniscus, causing the liquid-vapor interface location to change as compared to an unheated wall. This process causes pressure differences due to capillarity and the hydrostatic pressures exiting the flow, which in turn causes the meniscus to return to the initial position. In order to realize the above-mentioned process in a continuous manner it is necessary to carry out continual heat transfer from the capillary walls to the liquid. In this case the position of the interface surface is invariable and the fluid flow is stationary. From the thermodynamical point of view the process in a heated capillary is similar to a process in a heat engine, which transforms heat into mechanical energy. 

The atmosphere is very dose to being in hydrostatic equilibrium in the vertical dimension. This can be described by the hydrostatic equation ... 

Fluids on the Earth s surface that are in hydrostatic equilibrium may be stable or unstable depending on their thermal structure. In the case of freshwater (an incompressible fluid), density decreases with temperature above ca. 4°C. Warm water lying over cold water is said to be stable. If warm water underlies cold, it is buoyant it rises and is unstable. The buoyant force, F, on the parcel of fluid of unit volume and density p is ... 

As is well known, the earth is mainly a fluid the upper crust is an exception, but it is extremely thin layer with respect to the earth s radius. For this reason it is natural to expect that rotation around its axis makes the shape of the earth practically the same as if it was a fluid, and we will follow this conventional point of view. Suppose that during this motion the mutual position of all elementary volumes of the earth remains the same, and correspondingly each of them is involved only in rotation with angular velocity m. This means that the effect of different types of currents inside the earth is neglected and we deal with hydrostatic equilibrium. 

We see that the gradient of the density and that of the gravitational field are parallel to each other. This means that at each point the field g has a direction along which the maximal rate of a change of density occurs. The same result can be formulated differently. Inasmuch as the gradient of the density is normal to the surfaces where 5 is constant, we conclude that the level surfaces U = constant and 5 — constant have the same shape. For instance, if the density remains constant on the spheroidal surfaces, then the level surfaces of the potential of the gravitational field are also spheroidal. It is obvious that the surface of the fluid Earth is equip-otential otherwise there will be tangential component of the field g, which has to cause a motion of the fluid. But this contradicts the condition of the hydrostatic equilibrium. 

As follows from Chapter 1, we have formulated an external Dirichlet s boundary value problem, which uniquely defines the attraction field. In this light it is proper to notice the following. In accordance with the theorem of uniqueness its conditions do not require any assumptions about the distribution of density inside of the earth or the mechanism of surface forces between the elementary volumes. In particular, these forces may not satisfy the condition of hydrostatic equilibrium. 

We have derived formulas for the gravitational field outside and at the surface of the rotating spheroid with an arbitrary value of flattening /, provided that this surface is equipotential. Such a distribution of the potential U(p) takes place only for a certain behavior of the density of masses. For instance, as follows from the condition of the hydrostatic equilibrium this may happen if the spheroid is represented as a system of confocal ellipsoidal shells with a constant density inside each of them. 

The effect of external pressure on the rates of liquid phase reactions is normally quite small and, unless one goes to pressures of several hundred atmospheres, the effect is difficult to observe. In terms of the transition state approach to reactions in solution, the equilibrium existing between reactants and activated complexes may be analyzed in terms of Le Chatelier s principle or other theorems of moderation. The concentration of activated complex species (and hence the reaction rate) will be increased by an increase in hydrostatic pressure if the volume of the activated complex is less than the sum of the volumes of the reactant molecules. The rate of reaction will be decreased by an increase in external pressure if the volume of the activated complex molecules is greater than the sum of the volumes of the reactant molecules. For a decrease in external pressure, the opposite would be true. In most cases the rates of liquid phase reactions are enhanced by increased pressure, but there are also many cases where the converse situation prevails. 

The existence of dark matter (either baryonic or non-baryonic) is inferred from its gravitational effects on galactic rotation curves, the velocity dispersions and hydrostatic equilibrium of hot (X-ray) gas in clusters and groups of galaxies, gravitational lensing and departures from the smooth Hubble flow described by Eq. (4.1). This dark matter resides at least partly in the halos of galaxies such as our... 

From radiative equilibrium, Eq. (5.23), and hydrostatic equilibrium with the ideal-gas equation of state Eq. (5.15),... 

---