DONALD G. MILLER
536
added to repeat the process a second and third time. After the third hydrolysis the olymer was placed in a buret and washed a t the rate 0?1 ml./minute with a m u m u m of 3 1. of 1 N hydrochloric acid to remove the nitrite and then washed with distilled water until free of acid. The polymer was air-dried and vacuum-dried a t 55’ for a minimum of 18 hours to constant weight prior to titration. Polymer samples containing high ercentage of sulfo group8 absorbed more water and required yonger drying periods. Swelling Characteristics.-The procedure follows that previously described.’ A weighed sample (0.22-0.27 g.) of dried polymer was placed in a graduated cylinder and its apparent volume noted. Two ml. of hexane was added and the volume in excess of 2 ml. was taken as the true volume of the resin. Water was added to displace the hexane and permit swelling. The apparent volume of the swollen resin was noted and corrected by the same factor observed in the ratio of apparent to true value for the unswollen resin. The degree of swelling was taken as the ratio of the corrected volume swollcn to the true value unswollen.
Vol. 60
Capacity Measurements .-Total capacity determinations8 were made by adding 100 ml. of standard (m. 0.1 N ) sodium hydroxide to ca. 2.0 g. of dried olymer in its hydrogen form. The flask was stoppered and alfowed to stand one week with frequent shaking. Two 10.0-ml. aliquot portions were titrated with standard (ca. 0.1 N ) hydrochloric acid to the phenolphthalein end-point. Values are given in the table as meq. of base per gram of dry resin in the hydrogen form. Titrations were conducted by the procedure described previously) The resins were titrated with 0.1 N potassium hydroxide usin 12 Sam les of resin of approximate1 equivalent weiggt to whicf was added increments of 1.0 potassium chloride and 0.1 N potassium hydroxide in 1.0 N potassium chloride to give a total volume of 25 ml. The samples were sealed and allowed to stand for one week with occasional shaking on a machine. The pH of each solution was determined and plotted against meq. of base added per gram of dry resin. A series of such data for the polymers prepared a t 110’ is given in Fig. 1. The equivalence point is given in Table I in terms of meq. of base per gram of dry resin in the hydrogen form required to neutralizc.
k
AN ANALYTICAL PROOF THAT THE EXTREMUM OF THE THERMODYNAMIC PROBABILITY IS A MAXIMUM’ BY DONALD G. MILLER^ University of Louisuilb, Louisuille, Kentucky Received December $0, 1964
An analytical proof is given that the distribution found by extremizing the thermodynamic probability actually is the most probable one. The cases considered are classical statistics, Bose-Einstein and Fermi-Dirac statistics, radiation and Wall’s theory of rubber elasticity. I n these examples, the proof depends only on the form of W.
Introduction
distribution found actually is the most probable one. The argument will be given for classical statistics; and the Fermi-Dirac and Bose-Einstein system and use the most probable distribution so statistics, radiation and rubber elasticity will be obtained to determine the system’s macroscopic discussed briefly.6 SuRcient Condition for a Maximum.-As noted properties. 8 This procedure is permissible since the above, inflection points and saddle points number of particles is very large in most systems. will alsominima satisfy the necessary conditions obtained In this event it can be shown that the most probable by the LM technique. Therefore it is desirable distribution completely dominates all the others. In general a system is subject to various con- to have sufficient conditions which will enable one straints, such as a constant energy or a constant to decide which type of extreme is present. Fornumber of particles. Consequently the location tunately these conditions have been worked out, the following theorem on constrained extremes of the maximum of W is most conveniently handled and be by means of Lagrangian Multipliers. However may Theorem.-Let F ( q , . . ., zn) be subject to the the Lagrangian Multiplier (LM) method only leads to necessary conditions for an extreme, and it is m constraintsfi(z,, . . ., x,) = a i , 1 I imI m,where ordinarily assumed without proof that a maximum the ai are constants. Let 4 = F Xf, where is obtained. Although it seems more or less clear i=l that a maximum does result, nevertheless without X i are the Lagrangian Multipliers and are to be proof it is conceivable that a minimum or a saddle treated as constants. Let xIo,. . . xnobe the values point could be obtained. It is the purpose of this of XI, . . ., Zn at an extreme of 4, as found by the paper to provide a rigorous analytical proof that the usual LM technique. Now form the sequence of determinants (1) Presented before the Physical and Inorganic Division a t the
A standard technique in statistical mechanics is to maximize the thermodynamic probability W of a
(v6
124th ACS meeting, September, 1953. (2) Chemistry Department, Brookhaven National Laboratory, Upton, Long Island, New York. (8) See, for example, Mayer and Mayer, “Statistioal Mechanics.” John Wiley and Sons, Inc., New York, N. Y., 1940. (4) Referenoe 3, appendix VI. (5) I. 8.and E. 8. Sokolnikoff, “Higher Mathematics for Engineera and Physicists,” MoGraw-Hill Book Co., New York, N. Y.. 1941. p. 163.
(6) At the time this work waa originally carried out. no proofs had ever been published. It han oome to the author’s attention that L. Page (“Theoretioal Phyaics,” 3rd edition, D. Van Nostrand, New York, N. Y., 1952) has independently given a, somewhat different one for classioal statistics. ( 7 ) C. G. Phipps. A m . Math. Monthly, 59, 230 (1952). (8) R. P. Gillespie, “Partial Differentiation,” Interscience Publishera, Inc., New York, N. Y., 1951. (9) T. F. Chaundy, “Differential Calculus,” Oxford, 1935.
PROOF THAT EXTREMUM OF THERMODYNAMIC PROBABILITY Is A MAXIMUM
May, 196G
....0 . . . . 0. ....0 0.
f,' .....ftl
.
,t =m,m+l,
f,'.
in (l), the quantities b N / h i , b E / h i , and &,, = b2tpc/h,hr are required. It will also be convenient for reasons of notation to denote & by -ac. Then by means of the Stirling approximation and suitable differentiations, one finds that
. . .
fp . . . . . f , m
*fP411. * * * * & l t
..,
537
~t (1)
bN - = 1,-bE bni bnr
di
and & ~ * - ' ? = - , , # j k E1 . O
c
(7)
Therefore the determinants of (1) become" Each entry of these determinants is to .........1 00 1 I... be evaluated at no,.... xno. Then (1) if At is positive for all t = m, .... aI . . ".......e. 00 E1 n, F(zlo, .... xno)is a relative minimum of F; (2) if At has the sign (-1); for r 1 el -cry o........... 0 all t = m, .... n, ~(zlo,.... zn0) is a 1 a2 0 -uf 0. ......0 relative maximum of F. . . . 0. . 0 . . . . Since this theorem is proved by in.. .. .. duction, it cannot be applied rigorously .. .. .. . o . . to those classical cases where there are .. .. .. .. - 0 an infinite number of variables. Such cases arise when there is a continuous ....... .1 et 0 0 UC distribution of enerdes. It will be assumed however thac the theorem holds 1 = 2,3, ..- To prove that the In W is a maximum, it is for an infinite number of variables also. Quantum cases present no such difficulties though, since necessary to show that the A: alternate in sign as solutions of the Schrodinger equation lead to a finite (-1);. This is most readily accomplished by an but indefinitely large number of energies. induction on t. For t = 2 The Proof.-Before proceeding with the proof, it should be pointed out that physically the num100 1 1 I ber of particles ni must actually be discrete. HowO O ever it is customary both to treat the ni as con4 = ( 1 a1 4 0en 3 (e2 - a1)S (9) tinuous and to use Stirling's approximation for factorials. Since the result of applying these approximations agrees with the ones obtained by more elaborate methods, they will be adopted here which has the proper sign. la also. Suppose A: has the proper sign for t = IC. ConIt will be more convenient in all cases to maxisider A$+1. It is given by mize the logarithm of W instead of W itself. Classical Statistics.-Quantities associated with classical statistics will be denoted by a superscript C. In this case the thermodynamic probability is given by'O
*:
. . .
. . . e
I
FVC = N!ngin{/nni!
(2)
where gi are the degeneracies, ni the number of particles with energy ei, and N the total number of particles. The ni correspond to the xi of (1). The constraining relations corresponding to the f of (1) are
the large ~ e r o sindicating that all terms in the dotted box are zero except the principal diagonal. Now expand the right-hand column of this deter(4) minant by minors. One of these determinants is where E is the total energy. The quantities N, Expand the other two in minors by their E and the e i are constants. For this case 4 is - &+ bottom rows. One obtains defined as (11) The valuea of u(" at the extreme are found by the usual LM +C = In W - XN - p E (5)
-
Zni = N Znrci E
where X and
p
(3)
are the LM. For the determinants
(10) Mayer and Mayer, "Statistioal Mechanics," John Wiley and
Sons,N. Y., 1940, pp. 111-112.
-
method and are n(" (I/oi) exp (A 4- per). These valueaneed not be put into the determinant aince they are all positive. For our purposes, only the signs of the determinant8 are of interest. oan never be aero beaause the possibility that a e* ia (12) removed by the degeneracy faotor.
9
-
538
DONALD G. MILLER 0
,......... ....e3
0 el.
-uf 0.........0
(-1)2kt1
0
*
.
. .
.
0
1
.
. 0. . -.
o . . 0.......0 '-O; 10 1 ............1
€k
*
1: -7
1
0
0
(-l)*k+*a+,
ek
0
.
*
.......0 s-u;o 0 1
...........1 -upo .........0 1
.
0 . . o. . . . O f . . . - 0 1 0 ........0 -Ukc
.
. . o . .
I.
0
.
I
............0
+
..........eL
- u y o ......... 0
1
.
.
VOl. 60
* OC 0 ........ 0-un
If the first four determinants of (11) are evaluated by means of the theorem in the appendix and the various terms collected together, it is found that = (-l)"'[(ekti
-~I)*u~***u~ +(€&I
E#
uiua
U:
+
.**
+(ek+i - C~)'U?***U;-~] d- (-1b;+lA; (12) By the induction hypothesis A; has the sign (- l)R, and since all the terms inside the bracket are positive, A:+1 has the sign ( - l ) k + l . Therefore by the principle of mathematical induction, A; has the sign (- 1)t for all t . It has thus been proved that the extreme of In W is a true maximum in classical statistics. It is easy to show by a similar induction argument that the explicit form for A? is given by
Bose-Einstein Statistics-A11 Bose-Einstein quantities will be denoted by a superscript B. In this case, the thermodynamic probability is given by lo
and the system is again subject to the constraints (3) and (4). Applying Stirling's approximation and carrying out the required differentiations, one obtains equations 6 and
&
-u:
- ( S i - I)/w (gi
+;
=0
+ ni - 1) (15)
4; -o; = -gi/ni(gi - n;), $$ 0 (17) are obtained. The substitution of (6) and (17) in (1) leads to (13) as before, except that U: is replaced by u?. It remains to consider the Clearly g r and nr are positive, and in sign of.:u general g r > ni.13 Consequently A: has the sign (-l)t, and Fermi-Dirac statistics also leads to a most probable distribution. Radiation.-In radiation, photons are assumed to obey the Bose-Einstein statistics. However, the number of photons is not conserved so that WB is subject only to the constraint of constant energy. The evaluation of the A: for this case is much easier as the general expression can be written down immediately using the theorem in the appendix. Thus E
The A: are clearly of exactly the same form as A? except that uy is replaced by uy. Hence the (13) with the same final expression for A?- is just 0 e l . . ..... . . E t substitution. The sign of A: thus depends on the -u?O .......0 sign of nu:. Considerations of the terms of (15) * o yield the following. The nf are positive as can be . . 0 . = . . . determined from the expression found by the LM . - 0 . . method, and the g, are positive for physical reasons. . . Moreover the ncand g c are ordinarily greater than .o .-u; o €1 0s.. 1. Consequently the u? are positive for all i, and as a result A; has the sign (- l)t. It has thus been proved that In W is maximum for BoseEinstein statistics also. Fermi-Dirac Statistics.-All Fermi-Dirac quantities will be denoted by a superscript F. Here the This expression also has the sign (- l)t,and again it has been proved that the maximum is obtained. thermodynamic probability is given bylo Rubber Elasticity.-A less ordinary case comes W F= Hgi/(gi - nr)!ni! (16) from Wall's theory of rubber elasticity.l 4 Wall (13) If a case where gi = ni arises, its contribution to W ia 01 = 1. and as before, W" is subject to the constraints (3) terms are removed before taking logarithms and differentiating. and (4). Again applying Stirling's approximation Such F. T. Wall, J . Chem. Phzrs., 11, 527 (1943). See this paper and carryin.g out the desired differentiations, for(14) the physical meaning of the symbols. To keep a uniform notation equations 6 and Wall'a N i have been replaced by ni.
I
?
CRYSTAL STRUCTURE OF PHOSPHORUS DIIODIDE
May, 1956
539
has derived the expression
symmetry of At and the fact that $jk = 0. Therefore the sign of At depends only on the sign of the WE = N ! IIpinilIIni! (19) product of diagonal terms. These terms in turn which is subject to the constraints (3) and arise solely from the form of W , and consequently 12 = CZnixiZ (20) the type of extreme is determined solely by the ~ ~ all The. form of WE is analogous to that of classical form of W . In our examples, the c $ were negative. Since each term in the expansion of statistics, and it is easily shown that At has the product of t - 2 of the &i as a factor, the sign of At must alternate as t increases. Consequently if the first At has the proper sign for a maximum, all the others will also. These observations may be reduced to a simple Again A: has the sign (- 1) I, and a most probable rule. If = In W - ZhrCrwhere hr are Lagrangian distribution is proved to result. Comrnent.--In the above examples, it is seen Multipliers and the C, are constraining relations, that the squared terms in equations 13, 18 and 21 if 4 j k = 0, and if &i are negative for all i, then the arise from the constraints. This is due to the extreme of W will be a true maximum. Appendix The following theorem is very useful 'in evaluating the determinants which arose in the previous ana I ysis. It is
+
0 a l . . ....... .ak bl x 1 0....... .O
I: : ....... . bk 0..
0 Ox k I
It is very easily proved as follows. First expand the determinant in minors with respect to t,he first column. Expand each ith of the resulting determihants by its ith row. One obtains
o....o
x2 0
ll
=
(-1)albl
f
.
Ix1:
0 .
o'''''o
. + ...
+.
. o . .
*o
0 ......0 Xk
* o
0.......0 xk
THE CRYSTAL STRUCTURE OF PHOSPHORUS DIIODIDE, PJ, BY YUENCHU LEUNG AND JURGWASER Contribution from the Chemistry Department, The Rice Znstitute, Houston, Texas Received A m 7 491966
X-Ray examination reveals crystals of P2I4 to be triclinic, space group P-T, with one molecule per unit cell. The two P atoms are linked together at a distance of 2.21 A. with a standard deviation of 0.06 A. while each P atom is linked to two I-atoms at an average distance of 2.475 A. with a s.d. of 0.028 A. The molecular symmetry is 2 / m .
Of the two known iodides of phosphorus, PI, and PJ4, the former has been studied by electron diffra~tionl-~ while little work has been reported concerning the structure of the latter. The molec(1) A. H. Gregg, G. c. Hampson, G. I. Jenkins, P. L. Jones and L. E. Sutton, Trans. Faraday Soc., 83, 852 (1937). (2) 0. Hassel and A. Sandbo. 2. pfiysik. Cfiem., B11,75 (1988). (8) S. M. Swingle, reported by P. W. Allen and L. E. Sutton, Acta C T ~ E8~, .46 , (1950).
ular unit in the gas phase is reported to be P214,4 and we found this to hold in the crystalline phase also. The substance thus provides a good object to study the P-P bond under conditions of little or no Strain. In addition,, the bonding angles Of phosphorus, the P-I distances, as well as the molecular symmetryof p2r4are of interest. (4) L. Troost. Compl. rend., 96, 293 (1882).
