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Fig. 7.5 The function W(N) for three different fugacities (hence three different densities). For curves a and c the system is in a single pure phase. For curve b the system is undergoing a first-order phase transition.
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CANONICAL ENSEMBLE AND GRAND CANONICAL ENSEMBLE
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region and may be stated in more physical terms as follows: The pressure is unchanged if we take any number of particles from one phase and deliver them to the other. Each time we do this, however, the total number of particles in a given volume changes, because the densities of the two phases are generally different. Let us start with the system in one pure phase and then transfer the particles one by one to the other phase, until the system exists purely in the other phase. The number of transfers we can make is proportional to V. Each transfer corresponds to a term in the grand partition function, and all these terms have the same value.
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7.8 THE MEANING OF THE MAXWELL CONSTRUCTION
It has been shown that if the pressure P calculated in the canonical ensemble satisfies the condition aplav ::;; 0, the pressure calculated in the grand canonical ensemble is also P. We show that the converse is also true. We then have the statement (a) The pressure P calculated in the canonical ensemble agrees with that
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calculated in the grand canonical ensemble if and only if aplav::;; O. It will further be shown that (b) If aPlav > 0 for some v, the pressure in the grand canonical ensemble is obtainable from P by making the Maxwell construction. Suppose the pressure calculated in the canonical ensemble is given and is denoted by Pcan ( v). At a certain temperature we assume Pcan( v) to have the qualitative form shown in the P - v diagram of Fig. 7.6. The partition function of the system under consideration is
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(7.84)
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F( v)
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It is easily seen that
fV dv' f3Pcan ( v')
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(7.85)
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(7.86) Let
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<1>( v,
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z) == F( v) + -log z
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(7.87)
It is easily verified that
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a <1> + ~ a<l> av 2 v av
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v av ::;; 0
(a<v<b) (otherwise)
(7.88)
STATISTICAL MECHANICS
L...-_.L-_ _----'-_ _.l..-
--J._ _.L-
Fig. 7.6 Isotherm with apcanlav> 0 for v lying in the range a <v < b.
(7.75) is independent of the sign of
To calculate the grand partition function we recall that the derivation of aplau. Hence, in analogy with (7.75), we have in the present case lim - log 2 ( z, V) v ..... OO V where
= <I> (
v, z)
(7.89)
<I>(v,z)
max[<I>(u,z)]
(7.90)
This determines u m terms of z, or vice versa. The pressure m the grand canonical ensemble, denoted by Pgr ( v), is given by
/3Pgr (v)
<I>(v, z)
(7.91)
From (7.87) and (7.85) we see that both <I> and a<I> Iau are continuous functions of u. Hence (7.90) is equivalent to the conditions a<l =0 ( au v~v (7.92) a2 <1 -<0 2
( au
v=v-
with the following additional rule: If (7.92) determines more than one value of we must take only the value that gives the largest <1>( v, z). The first condition of (7.92) is the same as
U2 (
aF )
= log z
(7.93)
CANONICAL ENSEMBLE AND GRAND CANONICAL ENSEMBLE
Substituting this into (7.86) we obtain
/3Pcan (ij) = F( v) + -:-log z = <1>( v, z) v
Comparing this with (7.91) we obtain
(7.94)
Pcan ( v) = Pgr( v)
(7.95)
That is, if there is a value v that satisfies (7.92), then at this value of the specific volume the pressure is the same in the canonical and grand canonical ensemble. Therefore it only remains to investigate the possible values of V. It is obvious that v can never lie between the values a and b shown in Fig. 7.6, because, as we can see from (7.88), in that region J <1>/ J v = implies J 2 <I> / J v 2 > 0, in contradiction to (7.92). On the other hand, outside this region, J<I>/Jv = implies J 2 <1>/Jv 2 ~ O. Hence the first condition of (7.92) alone determines V. Using (7.85) we can write this condition in the form
dv' Pcan(v') - vPcan(v)