u, z) == 1( u) + - log z

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(7.69)

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CANONICAL ENSEMBLE AND GRAND CANONICAL ENSEMBLE

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Using (7.65) we obtain

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</>(u, z)

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1 1 -logz + u u

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fV du'f3P (u')

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(7.70)

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By (7.67), we have a 2 </>ja(lju)2 ~ 0, or a 2> </ 2 a</> -+--<0 u au au 2

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(7.71)

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We now calculate the grand partition function. For a fixed volume V the partition function QN(V) vanishes whenever

N > No(V)

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where No(V) is the maximum number of particles that can be accommodated in the volume V, such that no two particles are separated by a distance less than the diameter of the hard sphere in the interparticle potential. Therefore fl(z, V) is a polynomial of degree No(V). For large V it is clear that

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No(V)

(7.72)

where a is a constant. Let the largest value among the terms in this polynomial be exp [V</>o (z )1, where

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(N=O,1,2, ... )

(7.73)

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Then the following inequality holds:

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eV</>o(z)

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fl(z, u)

No(V)

eV</>o(z)

Using (7.72) we obtain or 1 log (aV) </>o(z) ~ V log fl(z, V) ~ </>o(z) + - - V Therefore

1 lim - log fl ( z, V) = </>0 ( z ) v ..... oo V

(7.74)

(7.75)

Let iJ be a value of u at which </>( u, z) assumes its largest possible value. Since </>( u, z) is differentiable, iJ is determined by the conditions (7.76)

(7.77)

By virtue of (7.71) the first condition implies the second. Therefore iJ is de-

P(u)

STATISTICAL MECHANICS

L-_..L-

L - _ - L_ _~ U

Fig. 7.2 Typical isotherm of a substance in the transi-

tion region of a first-order phase transition.

termined by (7.76) alone. By (7.69) and (7.65) we may rewrite it in the form

du' P(u') - oP(o)

-kTlogz

[~~ du' P(u') -

(0 - uo)P(o)] - uoP(o)

-kTlogz

(7.78)

A geometrical representation of this condition is shown in Fig. 7.2. The value of o is such that the difference between the area of the region A and that of the region B is numerically equal to - kT log z. The result is shown in Fig. 7.3. It is seen that to every value of 0 greater than the close-packing volume there corresponds a value of z. This answers question (b) in the affirmative.

L-.._ _- L

...J.U2

Fig. 7.3 z as a function of

CANONICAL ENSEMBLE AND GRAND CANONICAL ENSEMBLE

There is a value of z that corresponds to all the values of interval VI sus v2 . This value, denoted by zo, is given by log Zo

v lying

in the

(7.79)

/3v 1P( VI)

dv' /3P( v')

7.7 BEHAVIOR OF W(N)

In (7.44) we introduced the quantity W(N), which is the (unnormalized) probability that a system in the grand canonical ensemble has N particles. Comparing (7.44) to (7.68) we see that

W(N) = exp

[V$(:' z)]

(7.80)

Hence it is of some interest to examine the function $( v, z) in more detail. Suppose P( v) has the form shown in the P - v diagram of Fig. 7.2. For values of v lying in the range VI S V S v2 , P has the constant value Po. For this range of v we have

$( v, z)

which is the same as

~ [log z + [1 dv' /3p( v') V

/3Pov1 ] + /3Po

$(v, z)

~ log (~) v Zo

+ /3Po

v2 )

(7.81)

where Zo is defined by (7.79). Hence we can immediately make a qualitative sketch of a family of curves, one for each z, for the function $( v, z) in the interval VI S v S v2. The result is shown in Fig. 7.4.

cP(v,Z)

(3Po

z =zo

O':------P---------'V2------* V

Fig. 7.4 Qualitative form of <j>(v, z) for a

physical substance.

STATISTICAL MECHANICS

To deduce the behavior of the following facts:

</>( v,

z) outside the interval just discussed we use

(a) J</>/Jv is everywhere continuous. This is implied by (7.70). (b) J</>/Jv = 0 implies J 2</>/Jv 2 s O. That is, as a function of v, </> cannot have a minimum. This follows from (7.71). (c) For z"* Zo, </> has one and only one maximum. This follows from (b).

Guided by these facts we obtain the curves shown in Fig. 7.4. The behavior of W(N) can be immediately obtained from that of </>(v, z). It is summarized by the series of graphs in Fig. 7.5. For z "* zo, W(N) has a single sharp peak at some value of N. This peak becomes infinitely sharp as V ~ 00. For z = zo, all values of N in the interval

(7.82)

are equally probable. The number of N values corresponding to (7.82) is (7.83) This situation corresponds to the large fluctuation of density in the transition

W(N)