System in a Gravitational Field

Let us now analyze some specific systems. First, we examine a column of pure fluid perpendicular to the surface of the earth [Fig. 21.1 (c)] and at equilibrium. In this case, it can be shown by the following argument that the pressure within the fluid varies with position in the gravitational field. 

The column of pure fluid [Fig. 21.1(c)] is at a constant temperature, and the external pressure on it is constant. Thus, for the column of fluid 

Let us now analyze the contributions to dG if we take one mole of the pure fluid in the column at position x and move it to the position x + dx. At each level within the column, the pressure is different (as the weight of fluid above it is different), although it remains fixed at each level. Hence, as the temperature and the composition remain fixed, when a unit of pure fluid is being moved from one position to another. Equation (21.8) can be written as 

If the pure fluid were in the shaft extending below the surface [Fig. 21.1 (/ )], our analysis would correspond in every detail to that in the column above the surface except that the negative sign in Equation (21.11) would be replaced by a positive sign, because x is positive in the direction of the gravitational field. Thus, the pressure within the fluid would increase as the depth x down the shaft increases. 

From an equation analogous to Equation (9.25), it follows (see Exercise 4 of Chapter 9) that 

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Wliat is probably the simplest fonn of work to understand occurs when a force is used to raise the system in a gravitational field ... 

The thermodynamic properties of a system in a gravitational field are emphasized in the following sections. However, because of the similarity of the two fields, the concepts and equations for the centrifugal field are the same. 

When external fields are considered, mechanical equilibrium no longer corresponds to a state of uniform pressure. This is most important for systems in a gravitational field. Equating the upward and downward forces on the volume element of fluid shown in Fig. 13 gives... 

E.A. Boucher, M.J.B. Evans and T.G.J. Jones, The Computation of Interface Shapes for Capillary Systems in a Gravitational Field. Adv. Colloid Interface Set 27 (1987) 43. (Anailysis of equations for interfacial shapes and numerical analysis of the involved differential equations contains computer program in BASIC.)... 

When the pressure of the system is not uniform, as in a gravitational field, the system may be divided into parts in which the pressure is uniform. Then H = X (E + PV) = E + (PI7), the sum being taken over the individual parts. If the volumes of the regions are infinitesimal, then X (PI7) may be substituted by J P d V. The integral is taken over the entire volume of the system. 

A work reservoir is similarly defined as any body or combination of bodies, used as part of the surroundings, whose only interaction with the system is one that may be described in terms of work. We may have a different type of reservoir for each mode of interaction other than thermal interaction. A work reservoir then is used to perform work across the boundary separating the reservoir and the thermodynamic system and to measure these quantities of work. In the following we are, in order to simplify the discussion, primarily concerned with mechanical work, but this limitation does not alter or limit the basic concepts. A reservoir for mechanical work may be a set of weights and pulleys in a gravitational field, an idealized spring, or a compressible fluid in a piston-and-cylinder arrangement. In any case the reservoir must... 

Let us consider a closed, single-phase system containing C components in a gravitational field. The state of the system is fixed and the system is in equilibrium. The condition of equilibrium is that for each component the quantity (pt + Af ) must have the same value in each homogenous region of the phase. In general, (pt + Af, ) is a function of the temperature, pressure, (C — 1) mole fractions, and the potential, so the differential of (pt + M,0) may be written as... 

Until now the L/V surface in the vicinity of TL assumed to have a negligible curvature. Hereafter, this assumption will be removed by minimizing the free energy of the whole S/L/V system in the gravitational field, rather than the free energy of the region of radius r around TL. In this case, the curvature at each point Q of the L/V surface has to satisfy the equation of Laplace (1805) (see Appendix A) ... 

Potential energy Energy due to the position of the system in a potential field (such as a gravitational or electromagnetic field). In this text, we will deal only with gravitational potential energy. 

Figure 7-1 sketches the steps of the induction process. The b-SoC is represented as blocks of various shapes and sizes (representing particular psychological structures) forming a system/construction (the state of consciousness) in a gravitational field... 

Dependence of Vapour Pressure Upon External Pressure If we have a system consisting of liquid and saturated vapour in a cylinder, we have stated that the system will be in equilibrium if the pressure put on the piston at the top of the cylinder is /0> where p0 is the pressure of the saturated vapour This is how the equihbnum is usually regarded It must not be forgotten, however, that all experiments are earned out in a gravitational field of force, and hence a column of vapour exerts a hydrostatic pressure downwards just as a column of liquid would do In fact, the pressure exerted by the vapour at the... 

Figure 8.2 Kinds of states that are possible in a mechanical system, such as a roller coaster in a gravitational field...

Emulsions are thermodynamically unstable systems and will, as a function of time, separate to minimize the interfacial area between the oil phase and the water phase. If a density difference exists between the dispersed and continuous phases, dispersed droplets experience a vertical force in a gravitational field. The gravitational force is opposed by the fractional drag force and the buoyancy force. The resulting creaming rate vq of a single droplet is given by Stokes law ... 

A classical mechanical system is characterized by a set of mechanical state variables velocity, elevation in a gravitational field, electrical charges, and so forth. If these variables are given and the external fields are known, the system is fully specified and its behavior at any instant of time, past or future, can be calculated. Thermodynamic systems are characterized by an additional state variable temperature. 

An effort or a flow, which is a gate for communicating with the exterior, can be imposed (or supplied) by an external system (another dipole or dipole assembly, for instance). An example of external effort is when a force is imposed on a mass placed in a gravitational field. An external flow may correspond to the convection phenomenon, when a fluid transporting an object imposes its own velocity. 

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