Tall column of gas in a gravitational field

The earth s gravitational field is an example of an external force field that acts on a system placed in it. Usually we ignore its effects on the state of the system. If, however, the system s vertical extent is considerable we must take the presence of the field into account to explain, for example, why gas pressure varies with elevation in an equilibrium state. 

A tall column of gas whose intensive properties are a function of elevation may be treated as an infinite number of uniform phases, each of infinitesimal vertical height. We can approximate this system with a vertical stack of many slab-shaped gas phases, each thin enough to be practically uniform in its intensive properties, as depicted in Fig. 8.1. The system can be isolated from the surroundings by confining the gas in a rigid adiabatic container. In order to be able to associate each of the thin slab-shaped phases with a definite constant elevation, we specify that the volume of each phase is constant so that in the rigid container the vertical thickness of a phase cannot change. 

We can use the phase of lowest elevation as the reference phase a, as indicated in the figure. We repeat the derivation of Sec. 8.1.2 with one change for each phase a the volume change dU is set equal to zero. Then the second sum on the right side of Eq. 8.1.6, with terms proportional to dU , drops out and we are left with 

In the equilibrium state of the isolated system, d5 is equal to zero for an infinitesimal change of any of the independent variables. In this state, therefore, the coefficient of each term in the sums on the right side of Eq. 8.1.7 must be zero. We conclude that in an equilibrium state of a tall column of a pure gas, the temperature and chemical potential are uniform throughout. The equation, however, gives us no information about pressme. 

We will use this result to derive an expression for the dependence of the fugacity / on elevation in an equilibrium state. We pick an arbitrary position such as the earth s surface for a reference elevation at which h is zero, and define the standard chemical potential 11° (g) as the chemical potential of the gas under standard state conditions at this reference elevation. At h=0, the chemical potential and fugacity are related by Eq. 7.8.7 which we write in the following form, indicating the elevation in parentheses  

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Suppose we have to deal with a nonuniform region in which intensive properties vary continuously in space along one or more directions—for example, a tall column of gas in a gravitational field whose density decreases with increasing altitude. There are two ways we may treat such a nonuniform, continuous region either as a single nonuniform phase, or else as an infinite number of uniform phases, each of infinitesimal size in one or more dimensions. 

An example of a system influenced by an external field is a tall column of gas in a gravitational field (Sec. 8.1.4). In order for an equilibrium state to be established in this field, the pressure must decrease continuously with increasing elevation. 

In this section we consider a system of a single substance in two or more uniform phases with distinctly different intensive properties. For instance, one phase might be a liquid and another a gas. We assume the phases are not separated by internal partitions, so that there is no constraint preventing the transfer of matter and energy among the phases. (A tall column of gas in a gravitational field is a different kind of system in which intensive properties of an equilibrium state vary continuously with elevation this case will be discussed in Sec. 8.1.4.)... 

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