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(7.96)

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There is a value of z, denoted by zo, at which (7.96) has two roots VI and for which <I>(v 1 , z) = <I>(v 2 , z). The conditions for this to be so are that

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(7.97)

The second condition is equivalent to Pcan ( VI) Combining these conditions, we obtain

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Pcan( v2 ), by virtue of (7.94).

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(7.98)

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which means that VI and v2 are the end points of a Maxwell construction on Pcan' as shown in Fig. 7.6. In general we can find z as a function of v by solving (7.96) graphically, in a manner similar to that used in the last section for (7.78). The result is qualitatively sketched in Fig. 7.7. As explained before, the interval a < v < b must be excluded. By definition of the 'Maxwell construction, the portions of the curves outside the interval VI ~ V ~ v 2 , shown in solid lines in Fig. 7.7, coincide with the corresponding portions in Fig. 7.3. We need to discuss further only the dashed portions of the curves. Consider the points A and B in Fig. 7.7. Let their volumes be, respectively, VA and VB and let their common z value be z'. The fact that they are both solutions of (7.96) means that the function <I>(v, z') has two maxima, located respectively at V = VA and V = VB' These maxima cannot be of the same height, because that would mean that VA and VB are, respectively, V 2 and VI' which they are not. To determine which maximum is higher we note that by (7.85), (7.94),

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Fig. 7.7 z as a function of

and the fact that z' is common to both,

fA dv' Pcan(v')

vAPcan(VA) - vBPcan(V B)

(7.99)

Suppose Pcan(v B) < Pcan(vA), Consider the point C indicated in Fig. 7.6. By inspection of Fig. 7.6 we see that

[A dv' Pcan(V') < (VA Vc

VC)pcan(V

Subtracting (7.99) from this inequality, we obtain

dv' Pcan(v') < VBPcan(VB) - VCPcan(v A )

which, by the original assumption, implies

[8 dv' Pcan ( V') < (VB Vc

VC)pcan(V B)

By inspection of Fig. 7.6 we see that this is impossible. Therefore we must have Pcan ( VB) > Pcan( VA)' By (7.94), this means that

<1>( VB' z,) > <1>( VA' z,)

In a similar fashion we can prove that, for the points A' and B' in Fig. 7.7,

<1>( VA"

Zll) > <1>( VB" Zll)

Therefore the dashed portions of the curves in Fig. 7.7 must be discarded. In Fig. 7.8, Pgr(ij) is shown as the solid curve. It is the same as Pcan(ij) except that the portion between VI and V2 is missing because there is no z that will give a v lying in that interval. In other words, in the grand canonical ensemble the system cannot have a volume in that interval. We can, however, fill in a horizontal line at Po by the usual arguments, namely, that since the systems

CANONICAL ENSEMBLE AND GRAND CANONICAL ENSEMBLE

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Fig. 7.8 The pressure in the grand canonical ensemble (solid lines).

at Vi and v2 have the same temperature, pressure, and chemical potential, a system at Vi can coexist with a system at V2 with any relative amount of each present. It is an experimental fact that a / aV ~ p It could not be otherwise, for then the system would be in the highly unstable situation in which releasing the pressure on it leads to a shrinkage. The quantity Pcan is the result of a (generally approximate) calculation, and mayor may not have this desirable property. However, the corresponding pressure in the grand canonical ensemble always satisfies the stability condition because the ensemble explicitly includes all possible density fluctuations of the system.