Presentation of the Microcanonical Assembly in the Gibbs Triangle R. Lacmann lnstitut fur Physikalische und Theoretische Chemie der Technischen Universitit Braunschweig. Hans-Sommer-Str. 10, D-3300 Braunschweig, West Germany The purpose of this paper is t o illustrate the equilihrium distrihution for systems with three energy levels and N particles a t a given temperature by means of a Gihbs triangle. This device assists understanding of the derivation of the distrihution laws. In the derivation of the Boltzmann distribution law, the Boltzmann statistics and the Boltzmann method are used. For the case of a constant number of particles ( Z N j = N = const.) and a constant total enerev Y N,i . c; = E = const.). ... ( themacrostate that issought isthe one to which thegreatest number of microstates (W(N. N,)) corres~onds.For nondegenerate energy levels and di&inguish&le particles, the number of the microstates for a mamostate is given by1 h'! W(N, Nj) = (1) Na!N,!N2! , .. Introducing the restrictive conditions corresponding to the Lagrange method, one obtains N, with gets
u
and
= exp(-n).
exp(-8.
(2)
as Lagrange multipliers. With X N j = N o n e
-~microcanon
Fiours 2. of - The number ~~
~
~
microstatesassociated with thedifferent macrostates ~
~
ca assemoly (Bohmann nal st csl Each po nt ol me I gure represents one m8crostate g ven oy a aeterm neo tripe N,. 4 4 ~ 8 t (hN T N, k = 101. Tne aashed lhnes 1- - -1 cannecl tne macrostates with canslant of a
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Here x e x p (-0 cj) = Z is the molecular partion function. Considering two microcanonical assemblies in thermal equilibrium, one can show that p = l l k T is only dependent on temperature. The Gibhs triangle is often used for representation of phase diagrams in ternary systems (cf, footnote 2). I t can also he used to develop the Boltzmann distrihution law for a system with only three energy levels. For each point in the Gibhs (equilateral) triangle the sum of the distances to the three sides (hA h s hc) is constant and equals the height (h) of the triangle (Fig. I). Because of the relationship hetween the heights h ~he, , h c and h, each point in the triangle
+ +
+
energy E
corresponds to a concentration in the ternary system (A, B, C). For the mole fractions XA
= h ~ l hx,, = h,lh and zc = hclh
is valid. The pure components are represented hy the triangle corners and the binary iystems by the triangle sides. The Preseniatlon of the Macrostates and Number of Microstates with the Glbbs Triangle In a system with N particles and three energy levels we identify NOIN= h,lh, N,NV
= h,lh
and N,IN = h,lh
That means, for all points, No + N I + N2 = N i s valid. Each corner of a Gibhs triangle is equivalent to one N j = N. The macrostates with one Ni = 0 = 0,1,or 2) are represented by points on the triangle s~des.Figure 2 shows this for N = 10. Each intersection point for the straight lines N j = const. is -
-
--
--
' Findenegg. G. H. Staiistische Thermwnamik; Stelnkopff: Darm-
Figure 1. The Gibbs niangle
stadt, 1985. Smith-D'Ans. Einfiihrung in die allgemeine und anorganische Chemie. XI; Braun: Karlsruhe, 1947. Landolt-Barnstein. Zahlenwerle und Funktionen aus Physik, Chemie, Asronomie, Gwphysik und Technik, 6: Auflage, II. Band. 3. Teilband: Springer: Berlin. 1956. Mersmann, A. Thermische Verfahrenstechnik, Grundlagen und Meth oden; Springer: Berlin, 1980. Volume 67
Number 11 November 1990
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Maxlmum Values 01 w,. and Population Numbers (N.) tor Different Total Energies (0
Figure 4. The number of microstates associatedwith mttdifferent macmstates of the Bose-Einstein statistics (cf. Flg. 2).
-
of No > Nl > N2 and N2/N1 NdN0 are the E values physically reasonable (see eq 3). This outcome occurs only forE = 9 to 12 hu. The higher E valuesrequire further energy levels. For E > 12 hv, the description with only three energy Ievels is not more adequate. One has to take into account also the energy levels with r, = 7 huI2, 9hv/2. . . and the N,'s. Setting E = 11hv the maximum is located between No, NI, N2 = 6, 2, 2 and 5, 4, 1. For a more exact determination naturally the procedure above can he performedfor N = 100. Figure 3 shows the Wlw values for the straight line E = 120 hv in the range of Nl = 20 t o 40 in dependence on NI. The maximum is located between N1 = 28 and 30 corresponding to the macrostates 100 (56,28,16)and 100 (55,30,15).With increasing N the maximum becomes more marked. The maximum for N m is the equilihrium distribution. Finally Figure 4 shows an example for the Bose-Einstein statistics with N = 10, gl = 1,g, = 3, and g3 = 5. g, is the degeneracy of the energy level e,, that is, the number of different states with the same energy. W is given' by
-
Figure 3. W,aoas function of N, far E = 120 hv
equivalent to a macrostate N(No, NI, N2) with x N j = 10. Based on eq 1, the WIo values for the macrostates are shown a t each intersection point. The total number of microstates is 31° = 59 043. Because of the commutativity all Wlo values for No z N1 z N2appear six times. The maximum with WIO = 4 200 is located a t the equidistribution (No N1 o Nz). The second restrictive condition ZN; c; = const. is taken into account by connecting all points with constant total energy. They are arranged on a line. A straight line connects the macrostates with the total energy E = const. if the ej's have equally spaced values, that means €2 - €1 = €1 - €0 is valid. The W;? values are to be searched for on these straight lines. For example in the case of a harmonic oscillator r , is given by (n 1/2)hv,hence co = hv/2, €1 = 3hvl2 and e3 = 5hv/2. Figure 2 shows the straight lines for E = 9,10,. . ., 15and 20 hv. The table contains the distributions lO(No,NI, N2)corresponding to the maximum W1ovalues. Only in case
-
+
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Journal of Chemical Education
F;.
The straight lines are drawn in for E = 25, 30, and 35 a.u. with rI = 2, E~ = 3, c3 = 4 a.u. Conclusions The maximum of the number of microstates per macrostate can be shown with the Gihbs triangle when restricted to three energy levels. This presentation is useful for understanding the derivation of the distribution law by means of the Boltzmann method.
Acknowledgment The author thanks the "Fonds der Chemischen Industrie" for financial support of this work, S. Ascheberg, K. Talk, H. Kempa, and Brennecke for their help.
