Ronald M. Harris Worcester State College Worcester. Massachusens 01602
I
Standard Entropy of Crystalline Iodine from Vapor Pressure Measurements A physical chemistry experiment
An introduction to statistical thermodynamics has become an integral nart of the undereraduate nhvsical . " chemistrv program at many colleges and universities. Yet, relatively few exneriments illustratine the a~nlicationsof this important theoretical tool have heen deveiiped, and all of theserestrict ( 1 4 .The comnutations to small molecules in the gas .. phase . present communication deals with an application of statistical thermodynamics to the diatomic solid IASJ,and was inspirrd by the experiment drsrrihed hy Stafford ('2). Our ohjertive is to enhance the studrnt's appreciation of the pnwrr of the statistical formulatim of thermodynamics. The simple Einstein model is used to derive an expression (valid atioom temperature and above) giving the vibrational partition function of solid 12 in terms of a single parameter, 8, the characteristic temperature of the crystal (5). In the ideal case, 8 would he evaluated from some mechanical property of the crystal (the frequency distribution, e.g.), and the partition function would then he used to compute all of the thermodynamic properties. However, since this kind of data is not available, 0 must be evaluated using thermodynamic information. In this paper we show that 8 can he obtained from measurements of the vapor pressure of the crystal. In essence, these are measurements of the free energy difference between the crystal and its vapors. The value of Ofound in thismanner is then used to compute another property of the crystal, its absolute entropy. In practice, we have found that remarkably accurate values of the entropy can he obtained by this approach.
- .
Statlstlcal Thermodynamic Considerations For a crystal composed of N, iodine molec~des(ZN, atoms) the Einstein model, which assumes there is one vihrational frequency, leads to the following expression for the vibrational partition function (5) QY
- [ e - B i 2 T / ( 1 - e-fl/T)]31ZN~l = [ e - 0 / 2 T / ( 1 - e-WT)]6N.
Here, G(T) is the free energy per Ip molecule. The equations for E and Cv are an expression of the equipartition theorem (6) and should look familiar to the student. In addition to their vibrational component the energy and free energy functions for 12(s) have a contribution of w e per molecule from the electronic energy of the crystal. More percisely, w,, a negative quantity, is the energy of the crystal (per I2 molecules) with all atoms at rest a t their lattice points relative to a zero in which all atoms are at infinite separation. The total thermodynamic free energy for the solid, therefore is
Equating the free energy of the solid to the free energy of the vapor, which can he computed from statistical formulas using spectroscopic data, one obtains the condition for equilibrium between Iz(s) and its vapor. As shown in the appendix, the vapor pressure of Idg) can then he related to the characteristic temperature, 0, through
Table 1. Parameters lor Computing the Free Energy of Ir(g)' M
8, =
Molecular Weight
h2 Characteristic Rotational Temperature 8rzlk
8 . = hulk Characteristic Vibrational Temperature a
o 0 .
Svmmetrv Number Degeneracy of Ground Electronic State Ground State Electronic Energy
4.214 X IO-*'kg 0.0538'K 308.8'K
2 1
valuer of 8, and 8 . obtained fmm Reference (10)
(1)
where 8 = hulk is the characteristic temperature of the crystal. Although eqn. (1) does not satisfactorily describe the low temperature ( T < 8) quantum effects, it is valid a t higher temperatures, as the classical limit is approached. In this limit ( T > 0) and the partition function becomes
Q" =
(9"
(2)
Because I&) is composed of massive atoms hound by relatively weak forces, we expect it to have a low characteristic temperature and that the "high temperature" approximation is valid even near room temperature. Equation (2) in hand, one can proceed to derive expressions for the vibrational contribution to any desired thermodynamic function. For example
-
Figure 1. Energy diagram. Do D. Is the zero-point energy of an h molecule In the vapor phase. wo w. Is the zero-point energy of the crystal (per I. molecule). A a is the energy of sublimation of iodine at O°K.
-
Volume 55. Number 11. November 1978 1 745
Table 2. Entropy of Crystalline lodlne (J mole-' deg-')
9(7Y
rl°K)
94.98 99.91 104.52 108.74 112.72 116.15 116.48 120.04
200 220 240 260 280 298.15 300 320
0
S(nd
S(qE
96.82 101.59 105.90 109.91 113.60 116.73 117.07 120.29
96.7 101.2 105.9 109.6 113.4 116.7 116.7 120.0
~aicviatedhorn e m 181vslw 8 = 78.3'K. obtained horn thestdamdata displayed
in Table 3
Figure 2. Logarithmic plot of lhe vapm preswre data. Circles rapresent &data of (7)and (8). and the triangles represent student data obtained on in our labor a l ~ yThe . slope of the line is -8050.7'K-'.
Table 3. Vapor Pressure of Iodine (mmHg) T(W
P
303.35 308.55 313.15 318.15 322.95 343.15
0.462 0.746 1.07 1.465 2.09 8.23
(0.478)' (0.725) (1.025) (1.498) (2.128) (8.196)
value in psrantheoe are horn References17) and (8). The definitions and values (where appropriate) of the terms M, OR, Our and Do are given in Table 1.According to eqn. (a), a graph of ln[PT5/2(1- c a " l T ) ] against 1IT should be a straight line having an intercept given by 2rrMk 312 k + 6 IU0 intercept = ln (9)
[(
) z]
f n m which a value of 0 can be extracted. The slope of the line is (ru, - IIn,lk and vields an estimate of the energy difference Do ur., which iLthe sum of the zero point energy of the crystal and the sublimation energy of iodine a t T = O°K (refer to Fig. 1). The value of 0 obtained in this manner can then be used to compute other thermodynamic functions, such as the entropy. Before introducing this method of analysis into our laborntorv nroeram a t Worcester State Colleee. ~" .. . i t was tested on the arcurate vapor pressure data reponed by Haxter et al. (7, HI. These measurements. re~resentedbv solid rircles in Fieure fit well t o a straight line tl;at interrepk the ordinate at 44.53, as detrrmined bv the method of least swares. Equation (9) then yields a characteristic temperature of 9 = 78.05"K. Estimates of the entropy of Ids) calculated from eqn. (5) with this value of 0 compare well (Table 2) with the third law entropy values derived by Shirley and Gianque from precise heat capacity measurements (9).The calculated S ( T ) approach the literature values from above. This is the expected result. for equation (5) was derived in the high temderature limit assumine that all vibrational enerev levels are accessible. Even a t 20o0E, well below room tempe&ure, the difference is only about 2%.' ~~
~
~A
.~~
2;
he slope of the line shown in Figure 2 is -8050.7'K-', which corres~ondsto an enerw eao .- w.). = 66.931 Jlmole. If -." . of A E +~ (wo the di&ihution of normal modes were known, one could then compute the zero-point energy of the crystal, wo - w., and in turn estimate AH: AE!),for which the latest reported value is 65,513 Jlmole (9).While it is not possible for us toobtain aprecise estimate of AH!,it is reassuring to note that our "least squares" slope is consistent with Reference (9) to the extent that it yields a positive zero-point energy. (i;
746 / Journal of Chemical Education
Experimental A spectroscopicmethod for obtaining the vapor pressure of iodine isgiven in reference (2).In our laboratory we use a Varian Techtron 635 equipped with thermostated cell holders, and cell temperatures are read to *O.l°C. All absorbance measurements are taken at 520 nm with a wide slit setting (f2 nm). At this wavelength the absorption coefficientof In(g) is 730 1 mole-' cm-'. Accurate pressure measurements over a wide range of temperatures are required. Consequently, our students are asked to take readings at temperatures from about 30°C up to at least 70°C and to allow the system to come to equilibrium at each temperatuie. In our lab we find 30 to60 min is required for bringing the system to a new temperature and establishing equilibrium. We therefore allow two laboratory periods (-6 hr) for the completion of the experiment. Results and Discussion Pressure data taken from a student reDort are c o m ~ a r e d in Table 3 with Baxter's accurate values, and the two sets of values agree quite well. This particular set of data is represented in Figure 2 by triangles and yields a "best" straight line (least squares) that is virtually indistinguishable from the one obtained from the Baxter data. The line intercepts the ordinate a t 44.5s corresponding to a characteristic temperature of 0 = 78.3"K. Entropy values computed on the basis of this temperature are in close agreement with literature values. In view of the scatter in the data, this close agreement is somewhat fortuitous. It is, nevertheless, striking. In conclusion, standard entropy values are ordinarily derived from extensive heat capacity measurements a t temperatures ranging upward from about 10°K and requiring specialized equipment found in only a relatively small number of research laboratories. The methods described in this DaDer. .. . on the other hand, enable a student to perform accurate entropy calculations for crystalline iodine on the basis of vapor pressure data readily obtained in two laboratory periods using eauinment available in most undermaduate ~hvsical chem. . " ish;laboratories. ~
~~
Appendix The condition for equilibrium between crystalline iodine and its vapors is (10) G,(T) = Gg(T) where the subscripts have their usual meaning. In the high temperature limit G. is given by eq. ( I ) ,and an expression for the free energy of a diatomic gas, such as Idg), is developed in Chapter 8 of Reference (5).There it is shown (p. 158),in the rigid rotor-harmonic scillator approximation, that G,(T) = G,(trans)
+ Gg(rot)+ Gg(vih)+ G,(elec)
where G,(trans)
= -kT
In
[(2rz73'2k~] + P -
kTln
(11)
G,(rot)
= -kT
In
(Ty
8rr21k
G,(vib) = kT In (1- e-h"/kT)
=
-kT in
(-)T
sor
k8 + h" = kT In (1- ec8JT) + 2 2 2
G,(elec) = D.
- kT In w
The meanings and values of the terms appearing in eq. (11) are summarized in Table 1. Substitution of this equation and (7) into equation (10) yields (with w = 1) we-6kTln(;)
= k T l n ~ - k T l n[ ( 2*y)a/2k~]
- kT In (2) + kT In (1 - e-#JT) +Do car where -Do = -(D,
+
(12)
k0./2) is the dissociation energy of a gas phase IS molecule measured from the vibrational zero-point level (see Fig. 1).Solving eqn. (12) for 1nP then gives eqn. (8) above.
Acknowledgment The author wishes to thank Professor Leonard Nash for his encouragement and advice and Mr. Frederick Peterson for his assistance during the developmental stages of this experiment. Literature Cited (1) Stafford. F.E.. Holt, C. W., and PauBon,G. L., J. CHEM. EDUC.,40,246 (19831. (2) Stafford, F.E., J. CHEM. EDUC., 40,249 (1963). (3) LittkR., J. CHEM.EDUC.,32,2 (1966). (4) Ruark, J. E., Iven, J. E.,snd Roberta,J. L., J. CHEM. EDUC., 49,758 (19741. ( 5 ) Hill. Tenell L.,"An lntrodudion to Statistid Thermodyomica," Addlaon-Wcalry Publishing Co., Ine.. Reading. 1960, pp. 86-93. (6) For a dlsnuvian of ~yuipartitionsee Moore, W.,"Pbyaid CbmYhy).)3rd Ed., Pmntiti H&, h.. E D ~ I ~cliffs, W ~1963,pp. 239-243. (7) ~ a r t e r ,0. P.,~ i c k e y C. , H., and ~ o l m e s W. , c., J. ~ m e rchem. sor, 19. 127 (19(J7). (8) h t e r , G. P. aod Gmse,M. R., J. A m r . Chom. Soc. 31,1061 (19151. (9) Shirley, D.A. and Gisuquc. W. F..J Amen Chem. Sac., 8l.4778 (19591. (10)Blinder, S. M., '"Advanced P h y s i d Chemistry." Maemillan, New York, 1969, p. 450.
Volume 55, Number 11. November 1978 / 747
