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By Kleppner .D And Kolenkow R.

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The energy of phase l(g) is Ul(g) , entropy is Sl(g) and the number of molecules in that phase is Ni,l(g) . The volume of the phase is Vl(g) . The surface also has some energy Us and entropy Ss , and some molecules Ni,s can be on the surface, not in either of the phases l or g. The surface has no volume, only area A. We study reversible processes where the surface moves. The energy change of the surface in such a process is dUs = Ts dSs + σdA + µi,s dNi,s . i The energy chance of phase g is dUg = Tg dSg − Pg dVg + µi,g dNi,g i and the energy change for phase l is 16 2 Phase equilibrium Tg Pg Vg Ni,g Sg Ug Ts s Tl Pl Vl Ni,l Sl Ul A Ni,s Ss Us Fig.

A time-independent force field (gravity) changes the picture for pressure. A column of air in the Earth’s atmosphere, however, is not an equilibrium system. Why? There is a temperature gradient and heat flows. dS = 0 is not enough to guarantee that we have a maximum of S. It could be a minimum or an inflection point without being an extrema at all, or minimum in direction of one variable and maximum in another. For a pure 2 maximum we also have negative second derivative ∂ S2 < 0 for all independent ∂xi variables xi .

The volume of the sphere is Vl = 34 πr3 , and with the radius change dr it changes by dVl = 4πr2 dr. We proceed in the same way as when deriving eq. 1) for the change of total entropy. 1 surface area change dA was zero, now it is not): dStot = + 1 1 − dUg + Tg Tg Pg Pl − dVg + Tg Tl + i 1 1 − dUl Tl Ts µi,g µi,s − dNi,g T Ts g i µi,l µi,s σ − dNi,l − dA Tl Ts Ts = 0 for equilibrium. 5) 20 2 Phase equilibrium A l g Fig. 4. Two phases l and g separated by a spherical surface with area A in an insulated and rigid container.

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An introduction to mechanics Solutions Manual by Kleppner .D And Kolenkow R.


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