Equilibrium Condition Under the Influence of Gravity

The work term in Eq. (1.4) of Chapter 1 includes only the expansion or compression contribution d W —PdV represents the work of expansion (or compression) done by the system on the surroundings. Now consider a mass m undergoing both expansion (or compression), and change in position in the vertical direction (see Fig. 2.1). In order to raise mass m to some height dz, a certain amount of work must be done d W — mgdz, provided there is no change in the volume of mass m. When there are both change in the volume of mass m and displacement in the vertical direction by the distance, dz, the work term is simply 

Now let us apply the Gibbs criterion of equilibrium dG must vanish at equilibrium. As a result, the independent terms on the right side of Eq. (2.5) must vanish. Pressure, P, and vertical position, e, are not independent. Therefore at equilibrium. 

Equation (2.6) states that the temperature T must be the same everywhere in the system. Since p = mjV, then Eq. (2.7) becomes 

Note that for a single-component system, the last term on the right side of Eq. (2.10) is absent (see Eq. (2.2)). 

At equilibrium, dG must vanish. Since z and P are dependent, then 

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