Sublimation of a monatomic element

(1)

SUBLIMATION

_{OF A}

MONATOMIC

_ELEMENT

b

Ronald L. Kerber

and

Din-Yu Hsiel,

Div1sI0

_of

_Engineering

and Applied

_Science

CALIFORNIA

_INSTITUTE _OF

TECHNOLOGY

Pasadena

_California

Office of

_{Naval Researcljecln)jsche}

IIogesclio0ß

Department of the Navy

_DeIft

Contract

_{N00014..67..00940009}

Repòrt No.

_85-45:

Approved: M.

_{S. Plesset}

(2)

SUBLIMATION OF A MONATOMIC ELEMENT

by

Ronald L. Kerber

and Din-Yu Hsieh)C

Reproduction in whole or in part is permitted for any purpose of the United States Government

This docüment has been approved for public

release and sale; its distribution is unlimited.

Division of Engineering and Applied Science

California Institute of Technology Pasadena, California

*

Present Address: Brown University

Providence, Rhode Island

Approved: M. S. Plesset

(3)

niation of monatomic elements. According to this model, the solid and

gas phases are two facets of a single physical system. The nature of the

phase transition is clearly revealed and the relations between the vapor pressure, the latent heat, and the transition temperature are derived. The results are applied to the experimental data of argon, krypton, and

xenon, with good agreement. Extension of the model to the melting

(4)

Introduction

In the past, studies of sublimation have been focused on the deter-rninat:ion of the vapor pressure of the solid. By noting that the two

physical systenis, the gas state and the solid state, are in equi1ibriuin, a relation between the vapor pressure and the critical temperaturè may

be calculated' 2

For a monatomic solid, the vapor pressure is given by.the

.1

expression L

rTdTt (!T

3 C(T")dT" OkT'2 o + £n [( h3 (1)

where L0 is the heat of sublimation per molecule at O°K, is the

heat çàpacity at constant pressure per molecule of the crystal, m is the

atomic mass, k is Boltzmann's constant, and l'i

is Planck's constant

divided by 2*.

Empirically, the vapor pressure may be related to the tempera-ture by a relation of the form

[

frip nT

(ZTr?/2kh/2

(2)

where E0 is the lattice zero-point energy per molecule and (A)g is the

"geòrrietric mean frequency" of the lattice vibrational spectum. Salter has derived Eq. (2) from first principles assuming perfèct crystal

(5)

structure, quasiharmonic lattice vibrations, and a nearly ideal vapor for

T 9. _OD/2 where is the Debye temperature.

The calculations of the vapor pressure of solids as exemplified by (1) and (Z) do not shed much light on the nature of the phase transition

In those analyses, the solid and gaseous states are treated as if they were

different physical systems rather than two phases of a single element. It is the purpose of this paper to develop the theory of sublimation in a new manner by considering sublimation as a bridge between the solid

and the gas states. The solid and gas phases arise as two components of

a single particle system. A critical temperature, T, can be defined

such that at temperatures less than T, the system behaves like a solid;

and for temperatures greater than T, it is a gas. As the temperature

increases across T, we find that a latent heat,

L, will accompany the

phase transition; and L is related to Tc

Moreover, the variation of

the vapor pressure with temperature can be calculated. It is found

that-the vapor pressure is related to that-the environment of that-the particle in that-the gas

state. Also, it is indicated how a similar model might be extended to the melting transition.

Theory

To represent sublimation, we assume that the solid-gas system can be represented by N independent particles, each in its own cell. The potential, which characterizes the cell, is assumed to represent the

aggregate interaction with all other atoms. When the energy of the

particle is lower than a certain energy, E, we assume every particle

lies in a three-dimensional harmonic potential

(6)

where w is the Einstein frequency3 and r is the excursion from the equilibrium position.

At E, the particle suddenly experiences a free particle potential

r<R

i o

E> E

(4)

i

Voc

r)R

Each particle is confined to a cell which is shown schematically

inFig. 1

For energies below E; the particle energy states4 are

E =hw(n+3/2)

with degeneracy

(n±1)(n+Z)

g-

2

For simplicity, we neglect the interaction of the free particle with the harmonic potential when the energy of the particle is greater than E

.,

--12

This is justifiable since R »

_{for all cases under consideration.} o _mw2

We will find that the free particle states are essentially continuous; therefore, we can express the density of, energy states (the number of energy states between E and E + dE) in the form5

I

3V

D(E-E ) Zm c - E

i _2Tr213

(7)

density of energy states is independent of the shape of V. We expect

to decrease with increasing pressure. The idea that dense gas

particles should be considered as confined to a volume much less than the total volume of the container was proposed by Lennard-Jones and Devon-shire6.

At low temperatures, we expect the energies of most particles to

be less than E, thus we have essentially the Einstein

model3 which is

in good agreement with experiment for temperatures above ?i/k for a

solid.

At high temperatures, we have an ideal gas.

The partition function, Z, for the system is

Z =

Zn/N! (8)

where Z is the partition function for a single particle.

exp(-E /kT)g (9)

allE_n

where E_n denotes a single particle energy state. For our model we have

z=

(n+l)(n+2) exP(-(n+1(L)/kT)+ _{D(-E)exp(-E/kT)ck} _. ₍₁₀₎

where M is the largest integer satisfying the inequality

(M + E

(8)

and

V=aV

_c _s

j;3

V5.)3 A= 4.rr3IZ ( 3/Z (1.4) C=kT/i'io

_...

(lì)

where V5 is the average atomic Volume in thesolid, which is

essential-ly constant. . .

Using (7) and (11) --(17), the,n Eq. (10) becomes

Z Xth[cru1

+ZXIt+I+AeiP(-Lo/kT)C]

, (.18)

qu3ntum effects f the harmonic oscillators, are present near the triple

point of rare gas solids. This has been noted by Moelwyn-Hughes7.

the ratio of kT and h

is large, we can use a continupus approximation

for the density of states of the solid. This approximation is treated in

Appendix I.

Let u.s denote

=E --1

(12)

which-can be interpreted as the. zero-point latent heat. Also, it is

con-venient to define the new variables:

X = exp(-/kT)

, (1.3)

where I' and I"

re respèctirely thé first and'the second derivatives of

(9)

The free energy and the internal energy are given by

F = - kT logZ , (19)

8 (F / T

E=-T2

Using (8),(18), (19), and (20), we find the energy

3]

x3 ₁₁₁₁₊ _x21"+6X1'+ _{- I+aAexp(-L0/kT} _°+(C+1)

E2

N'ïo

-I1t+2XV+I+aAexp(L0/kT)C3/2

where P" is the third derivative of I with respect to X. We note that

I ₌

ixM

(22)

It is clear that the only undetermined parameter of the model is

a, the number of volumes V available to the free particle. We also

noted that VC decreases with increasing pressure. Let us then take

a = a(p) (23)

where a(p) decreases with increasing pressure.

It is instructive to examine Eq. (21) in detail; we note

E = E(T,w, L, Aa(p) (24)

For low temperatures, the energy is that of a collection of harmonic

oscillators and for high temperatures, we have the energy of an ideal gas. In fact, for either temperature extreme, the energy is independent of

a(p). In the application of this model to àrgon, krypton, and xenon, we and

(10)

8G

8T

_T=Ç

(25)

where C is the specific heat at constant volume. For this reason, we

define the pressure of Eq. (23) to be the equilibrium vapor pressure.

From (24) and (25), we have

T

= T(w,L0,A(p)

(26)

For a particular solid, (26) becomes ,.

T

T(a(p) )

. (27)

Equation (27) yields the equilibrium vapor pressure curve, once a(p)

is known. '

For most practical applications, T is essentially independent

of o; therefore, Eq. (27) can be written more simply as

Tc = T(L, Aa(p) )

. (28)

This is exactly true fOr the cias sicai.solid described in Appendix I

if A

if replaced by A.

Therefore, for solids with similar molecular structure, such as the rare gas solids, oné might expect from the law of corresponding

states8 that for some characteristic pressure

T =T(L)

_c _o

In Table I, we illustrate the experimental relationship of T

argon, krypton, and xenon at their, triple point témperatures, T.

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(11)

TABLE I

Rat-10 of zero-point latent héàt and

*Calculated _{from values given by G. L. Pollack, Rev. Mod. Phys}

3,

748 (1964).

Application. to Argon, Krypton, and Xenon

To apply the theory., we need to calculate E(T) given by (21) and

its derivatives. To calculate the.constants of Eq. (21), we need the

fol-lowing physical constants: the zero-point latent heat, L0, the zero

pÒint energy, E.,

the atomic mass, m, and the average atomic volume

in the solid state, V. We will alsö need the triple point préssure,

_Pt,

and the triple point temperature, _Tt for future reférence. These

constants are given in Table II.

Element

L/kT*

o t

Argon 11.08

Krypton 11.58

(12)

Zero-point latent heat Element

L(cal/mol)e

Zero -point energy E(cal/mo].e )e TABLE II

Physical Constants Atomic

Average mass atomic ¿roiume m(:arnu.) V5(A3) Triple Point P (mm) T(° K)

Argon Krypton Xenon

1846a 2666a 3828 a 184 123a 39 948 83. 8O 131. 39. 51 6 86a 548. 7a 612. a 83. 810a 115. 78a 161. 37a aG

L. Pollack, Rev. Mod. Phys. 36, 748 (:1964)

bPo1lack

gives 187 cal/mole and I. J. Zucker and G.

G. Cheli

J. Phys. C(Proc. P1ys. Soc. ),,

Ser. 2., 1, 35(1968), give 182.3 cal/mole. We

use 184.6 cal/nole for numerical convenience.

CR.

C. Weast, Handbook of Chemistry and Physics (The

Chemical Rubber Co., Cleveland, Ohio,

1967 - 1968) 48th ed.

p. B3.

dCaicui!ated

from nearest neighbor distances given by Pollack.

eWe

(13)

The zero-point energy is given by

E_o _Z

-Using the physical constants of Table II and Eqs. (11) and (16), we list

M and A for each element in Table Ill.

TABLE III

Model Constants

The energy may now be calculated from Eq. (21) as a function of

temperature with the parameter a. The internal energies of argon,

krypton, and xenon are shown graphically in Fig. 2, 3, and 4 respectively,

for selected values of a. The phase transition is obvious. We have noted

the critical temperature of each curve.

The specific heat,

aE

- 8T

_y

may also be calculated from Eq. (21) by differentiation. The shape of the

specific curve is shown in Figs. 5, 6, and 7 for argon, krypton, and

xenon, respectively. The critical temperature of each curve is about 75

percent the triple point temperature. We note that krypton and xenon are behaving as nearly classical solids before the transition temperature is

(30) (31) Element M A Argon 15 921.3 Krypton 27 2212 Xenon 46 4472

(14)

Argon 1.00

Krypton 1. 08

Xenon 1.42

By numerically differentiating Eq. (21), we may caiculäte the critical

curve by using (32) in (25). These results are compared with

experi-9 10

mental data in Fig. 8 obtamed by Flubacher, et al , Freeman and Halsey

(32) reached, but that argon is not.

To this point, we have been using a as a free parameter As we

vary a, we change the critical temperature, this is shown in Figs. 2,

3, and 4. This behavior was predicted by Eq. (27). As we noted in the

previous section, if we know the functional form of a(p) we can

deter-mine the vapor pressure curve from (27). With the knowledge that a(p)

is a decreasing function with pressure and that a(p) is proportional to the volume seen by the free particle, we assume

G

a = -p.

where G is a constant and p is the equilibrium vapor pressure.

Since

our model has no provision for a third phase, the liquid state, we shall

adjust G to the triple point data. We list in Table IV the values of G

found for the elements studied.

TABLE IV

Product of ap = G

(15)

and Clark, et aiU for argon. In Fig. 9, we cOmpare the vapor pressure

of kryptonmeasured by Beaumont, et al12, Fisher and McMillian13, and Freeman and Halsey10 with our theoreticäl curve. Finally, we compare

our results with the vapor pressüre of xenon measured by Fréeman and

.10

:,:.,

..l4.

..l5

Halsey , Podgurski and Davis , and Peters and Weil in Fig. 10.

We. nòte that Freëthan and Hàlsey' °gave an experimental curve; the

figures show selected points of these curves. - We conclúde that our as

sum-ed relation (32) is valid.

Although a theoretical interpretation of G has not been found,,

we mention that G is the same order of magnitude as kTt/V.

Analytically setting the second derivative of E with respect to

T equal to zero is very tedious. In Appendix II we calculate an

approxi-mate expressin for the vapor pressure analytically. We find

L

lnp= -

-

-nC +n[CZ(1_e1?] +n[1+f(

.,T)j+.2n[GA]

(33)

\ hCA.

where f

-- .T) -

O as _t--

- 0.

In deriving Eq. (33), we have assumed

aAC3IZ

_{» X(1-XíM}

₍₃₄₎ and 65 . 9

1. XC8

L IT=LL2

C8 C

l-X 2 + (1-xy L (35) We can not neglect f T in Eq. (33) since it is as large as

0. 6. For, argon, we find

L L.Z_o

(16)

This approximate solution for the equilibrium vapor pressure of

argon is shown as the dashed line in Fig. 8. Since Eq. (33) is already

more cumbersome than Eq. (2) no further approximations were

attempted.

In concluding our study of the equilibrium vapor pressure, we demonstrate one of the advantages of this model. As we noted, previous studies have considered sublimation as the equilibrium of two physical

systems. We can do this by considering the solid and gas phases to be in

equilibrium. For the partition function of the solid, we have from Eq. (18),

z

L 2

and for the gas,

-L /kT

Z = aAe C3/2X31Z

g

By equating the chemical potentials of these separate systems and using Eq. (32) we find the vapor pressure is described exactly by Eq. (33) with

f(-, T) equal to

zero.

This leads to a large error in p.

With this

approach we have lost the inherent corrections due to vacancy formations of our cell model. These corrections must be handled separately as

Salter demonstrates.

Finally, we want to calculate the latent heat. Since the energy is a continuous function of T, the beginning and end of the transition is not precisely defined. Therefore, the latent heat is somewhat arbitrary.

The latent heat per molecule may be calculated from

(17)

We have found that (for a » 1),

pv=paV

(40)

Using Eq. (32), we have

pAv=GV5 (41)

Figures 2 3 and 4 indicate that

E is the order of L for all T

o c

We choose to define AhE by extending the Ttnaturallt tangents of the energy

curve above and below Tc as shown in Fig. 11. With this definition of

we find by similar triangles

31 __C

J-E

((c)

'kT

o o I o

where Ni«c) is the average specific heat at the point _{of inflection of the}

specific heat curve of the solid as sublimation begins and

_{J is the}

inter-cept of this tangent line.

Therefore, as an approximate expression for the latentheat,we have

L E

/ \

3kTGV

+

J

-E L

o o o o

For argon, krypton, and xenon, we find <C) to be 2.50, 2. 75, and

2.95, respectively, and J tobe .046, .018, and .005, respectively

We may take (Cv> to be constant over a wide range of temperatures

since the corrections resulting from its dependence on T are small

and these corrections are less than the inherent uncertainty of (43). We

note that an error in the estimate of <C> is compensated for when J

is calculated. For a variation of

.

02 in <C), we find L varies less

than _%.

(18)

values with the experimental data at the triple point in Table V

TABLE V

Latent heats at the triple point

*

G. L1 Pollack, Rev. Mod. Phys. 36, 748 (1964).

Discussion of Results and Extention to Melting

Sublimation

The agreement between the theoretical and experimental results

as shown in Figs. 8, 9, and 10, is indeed very remarkable. At this

point, it is especially noteworthy that for each element, the only unknown

parameter is G,

since L0 and

may be obtained from theories of

the solid

state'6.

One point on the vapor pressure curve, e.g. the triple

point, then determines unequivocally. In this aspect, we may remark

that Eq. (2), as derived by Salter, consists also of _{one parameter which}

has to be determined by experiment in practice. But the range of validity

of Eq. (2) is somewhat less than the present theory. For example., Eq.

(2.) begins to deviate from experimental data for krypton _{at about 75° K;}

.but this model'is consistent with all experimental data available.

Element Theoretical

(L/L)

Experim,ntal(L/L0)

Argon 1.006 1.008

Krypton 0. 963 . 967

(19)

The sublimation process according to the present model does not

representa singularly sharp phase transition. Although, whether the

transition is in fact a sharp transition is still not a settled question, we do not intend to raise this issue here. We only point out that the energy and specific heat curves as shown in Figs. 2, 3, 4, 5, 6, and 7 clearly reveal a phase transition across the temperature region in the

neighbor-hood of T .

In fact, the transition becomes sharper as T

becomes

c c

smaller.

The mechanism of the sublimation process can also be seen from this physical model. The tendency for the system to stay at a lower

energy in the harmonic potential is constantly competing with the tendency

to be in the free particle cell at higher energy. The Boltzmann factor

will favor the lower energy states. On the other hand, the free particle cell provides a much larger number of available states. The sudden predominance of the large density of states for the free particle cell over the Boltzmann factor across a narrow temperature range results in the

sublimation transition. This competition between the Boltzmann factor and the density of states, we believe, underlies all the phenomena of

phase transitions. The system will change from one phase to another

when the latter has a much larger density of states even at the expense

of a finite jump in energy. This jump in energy gives rise to the latent

heat.

For the present physical model, sublimation is not a discontinuous

process. So there is some ambiguity in defining the latent heat. We

have defined L as shown in Fig. 11. The inferences of such a definition

(20)

defined varies slightly with T. From the physical model, we can see

that it tends to

L0' at O°K. For the nearly classical'soli4s in the

temperature range where this theory is valid, L decreases as Tc

in-creases.

We have represented the solid phase by the Einstein model3 mainly

because we are prim interested in the problem of sublimation. In

the temperature range we have been interested, the Einstein model gives

nearly 'as good a representation as the Debye model'7. The use of the Einstein model yields a simple picture of a particle confined in a cell.,

and enable.s us to visualize graphically the. process of sublimation. At low temperatures', we need to revise our representation of the solid state

to accommodate the inadequacy of the Einstein model. Also, we may need to incorporat.e the anharmonic effects in our model to deal with. the situation in the immediate vicinity of the triple point.

In this analysis, we have presented a. very simple cell model. Although the model is crude, and may be improved on various points, it describes very well qualitatively and quantitively many aspects of the sublimation process.

Melting

It is our ultimate purpose to deal with the much more, complex problem of melting. The study of the sublimation process serves as an

initial step. So far, the study of melting has been concentrated in two

18

areas.

The general study of phase transitions by Yang and Lee and

the studies of melting by Kirkwood and Monroe'9 and Brout2° deal main-ly with the establishment of the existence of the transition. The second

(21)

area is concerned with correlating the physical quantities in connection

21,22

with the melting process. Lennard-Jones and Devonshire have

considered melting as an order-disorder transition. Tsuzuki23 has

approximated the liquid state with rough estimates of the pair correlation function and the free volume; the melting transition was then

demonstrat-ed. These studies illustrate the main difficulty found in. studying the

melting transition, i. e., the lack of a good theoretical representation of

the liquid state.

At the present, the authors have found encouraging qualitative

agreement from preliminary studies by representing the liquid as a

two-component substance. For the transition, we consider N of the particles

behaving as though they are in the sublimation cell presented in this paper,

and N of the particles in harmonic oscillator states at all times. We

find a phase transition accompanied with a latent heat. Details of our findings on the problem of melting are to be presented in a subsequent paper.

(22)

REFERENCES

1. R0H. Fowler, Statistical Mechanics (Cambridge University Press,

Cambridge, England, 1936), 2nd ed., p. 173.

2. L. S. Salter, Trans. Faraday Soc.. 59, 657 (1963).

3. A. Einstein, Ann. Physik 22, 180 (1907).

4. G. S. Rushbrooke, Introduction to Statistical Mechanics (Oxford

University Press, London, 1949), P. 32.

5. C. KitteL1 Introduction to Solid State Physics (John Wiley and Sons,

Inc. , New York, 1966), 3rd ed. , p. 206.

6. J. E. Lennard-Jones and A. F. Devonshire,

(London) 163, 53(1937).

7. E. A. Moelwyn-Hughes, Z. Physik. Chem.

8. I. Z. Fisher, Statistical Theory of Liquids

Chicago, 1964), p. 19.

9.

P. Flubacher, A. J. Leadbetter, and J. A.

Soc. (London) 78, 1449 (1961).

10. M. P. Freeman and G. D. Halsey, Jr., J.

(1956).

11. A. M. Clark, F. Din, J. Robb, A. Michels, T. Wassenaar, and

Th. Zwietering, Physical7, 876 (1951).

12. R. H. Beaumont, H. Chihara, and J. A. Morrison, Proc. Phys.

Soc. (London) 78, 1462 (1961).

13. B. B. Fisher and W. G. McMillan, J. Phys. Chem. 62, 494 (1958). 14. H. H. Podgurski and F. N. Davis, J. Phys. Chem. 65, 1343 (1961). 15. K. Peters and K. Weil, Z. Physik. Chem. (Leipzig) 148A, 27(1930). 16. N. Bernardes, Phys. Rev. 112, 1534 (1958).

17. .P. Debye, Ann. Physik, 789 (1912).

18. C. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (1952).

19. J. G. Kirkwood and E. Monroe, J. Chem. Phys. 9, 514 (1941).

20. R. Brout, Physica 29

21.

_{J. E. Lennard-Jones}

169, 317 (1939).

22.

_{J. E. Lennard-Jones}

170, 464 (1939).

23.

T. Tsuzuki, J. Phys. Soc. Japan 21, 25

(1966).

Proc. Roy. Soc.

(Frankfurt) 15, 270 (1958).

(Univ. of Chicago Press,

Morrison, Proc. Phys.

Phys. Chem. 60, 1119

1041 (1963).

and A. F. Devonshire, Proc. Roy. Soc. (London) and A. F. Devonshire, Proc. Roy. Soc. (London)

(23)

The partition function becomes

E

and

z=

APPENDIX I

For the quasi-classical solid, kT »15c, one may consider the

energy states as a continuous spectrum. That is (5) and (6) become

E

n

i

Therefore, the number of energy states between e and E + dE is

n2

dn = - dE

EN

2y2_e[ 1+2+2y2-Aa]

For the density of states in this approximation, we have Z D(E) E _(I-4) 2 (hw)3 exp(-E/kT)dE +Ç cV o 2(?x _E 2,r ti3 Je-E exp(-e/kT)d

Using (8), (18), and (19), we have

E 6y3 e -i [i+3'y+6y2+6y3 _A1cr( - +i)]

(I-2) (I-3) (I-5) (I-6) where = kT (I-7) i and n2

=

(24)

-.-A b (I-8)

I _3/2 )3

From Eq. (16) we have,

A

= (I-9)

The energy relation (21) is compared with the quasi-classical

energy for argon in Fig. 12. The critical temperature of the classical

model is slightly higher than the critical temperature for the Einstein mQdel.

(25)

where

APPENDIX II

Using (22), the energy in E4. (21) can be written

EQ

N?Iw - W - _- _{M-6) +xM+z(- M3 +} M2 + 9).+ xM+l (

M

-L/kT

L M2 _- _M - )+ - (X+1)](l-XY+aAe ° _{3"2(2 4(C+1))} (II-la) and M2 3M _{i) +xM+Z(M2+4M3).+xM+} i

-L/kT

+lj(1_X13+aAe ° CI

-M-3)

Near the fra.nsition region that we have studied, we find that C

varies from 1.05 to 1.35, 1.23 to ¿.38, and 1.95 to 3.91 for argon,

krypton, an xenon respectively. In the region where

(fl-i)

Condition (II-Z) is true for argon over the entire temperature range of interest; however, it is only valid for krypton and xenon at temperatures

well below the triple point.

Setting the second derivative of Eq (II-3) to zero, We find the » X(1-X)4M

we may approximate the energy equation by

-L/kT

rL

E

- (l+X)(1-X+aAe

°

_{C'._2}

, + - (c+1)j

i

(II-2) (II-3) -L (1-X)3+aAe

o/kT/z

(26)

critical curve in the form

-3L /kT

-2L /kT

-L /kT

c3A3e O +a2A2e O B +aAe O

B +B = O i Z 3 4 where - 6x(1-x) (5+X)i +

c

_-

(lX)13(9

+ .-

(l-X)'

2 _1/2r J

L()

L2

5L2 2L +X ° + ° 2 (hc)2 t1() ( (1+2X)+1) + 30x2(1-x)10(1+x) (II-4) = - .- (1-X)1

c5/2:[

.

(1X)13('0

+ (l-X)12(-49 +

1i5))j

i-

c"[

l.X)l3(l:C)Lo + 15 + (1

_X)(

'- +

7x(o

+ BZ

(lx)'0c2

+ 1

¡L

+ )+ (l-X)9(l+x)] 2L 2L _6L 18L

.9(1TX)10(1

_{+3)+(1-x)9(.._2 x(}

o 3)) - _{X(1 -X)8 (29 + 25X) + C}

_{[.3X(i X)9(._2}

₊ ₂₎ + 3(1-X)8 f-°+ _{5 + 3X(} _{? +} ₂₎₎ +

4X2(lX)7(2x)1

(I6).

+ 6X(1-x 1

(27)

'7L

-X)7C1"2

+cu/2[-

(l-X)71_°

+ 4

= -- (1

- - (lX)6(1+l3X)]+ 3/2r1 (l-X)7

k

I o + 3) '9L_° + 8))+ - X(1-X)5(-13 + 7X)J - _{- (l-X)}

5,rL

IL (lX)6(_2_2 - _{(l-X)7t_-2+}_t1) / 5L2 8L L ° ₊ °

_{+2 +3X(1_X)5-2+4}

2 tw (i'ic) ¡ 5L_° +6 + 6x2(l-x)4(4-X)

],

and B4 3XC4(l-X2) -

6xc3(1-x)2

We now assume that

L

(II-9)

L

Actually for argon

_ç

is 15 and increases to 46. 6 for xenon. Taking

only the smallest order of t/L0 the critical curve (II-4) reduces to

-L/kT

_r

aAe °

_C(1X?L1

+f(, T)1= i

(11-10)

where f_, T)- O as

_t-

- O.

o o

We must limit any higher order approximation to argon; since Eq. (11-3) would not be valid for a higher order approximation in krypton

and xenon.

(28)

f Noting that we have:

-C

C19

.

çç.

sixz

2 8 1-X-

Z Ï-X

2 _(1X)2 G

P= a

-L/kT

r p =:cAe _.° _{C31z(i_X?L} i

After taking the logarithth of Eq. (II-13) and rearranging, ters, we find

- _nC .1n[C2(1_e_1'.Cfl+n[ _i _{+ f(} _, _T)] _{+ n[GAJ}

We note that Eq. (II-14) has roughly the same form as Eq. () since

) ] is nearly constant over the terpperatu.re.range of argon.

(29)

Fig. 1 - Energy Cell

L. + E.

CELL RADIUS

(30)

o

J

z

LU 2.O o 0.1 0.2 0.3 0.4 r O5 0.6 0.7 TEMPERATURE, kT/Lo

(31)

0.i 0.2 I I I 0.3 0.4 0.5 TEMPERATURE, kT/L0

Fig. 3 - Internal Energy of Krypton

a 500 107.1 2O0OO 82.4°)< 0.6 0.7

(32)

o _-J

z

_w I I 0.1 0.2 0.3 0.4 0.5 TEMPERTURE,

kT/L0

Fig. 4 - Internai Energy of Xenon

Tc 2,000 136.0 °K 2,000,000 90.4 °K 0.6 I î I I I 0.7

(33)

.1 I

o.os ' 0.10 0.15 0.20

TEMPERATURE, kT/L0

Fig. 5--Specific Heat of Argon

a 8,000

I

62.9°K 50 . 40 L) I-w

i

o

30 L) w '0-(1) 20 Io

(34)

t I

Q.05 0.10 0.15

TEMPERATURE, kT/L0

Fig. 6 - Specific Heat of Krypton

0.2Ö a 8,000 Tc' 87.4 °K 50 .

z

o

40

J

w :1:

o

30

u-o

w Q-. ci) 20 I0

(35)

0.05 0.10

TEMPERATURE , kT/L0

Fig. 7 - Specific Heat of Xenon

0.15 0.20 a = 10,000 Tc 121.7 °K 50 .

z

40 L) I-Lu

i

30 L) IL

o

w G-(J) 20 I0

(36)

I 0'

iO2 .50 D FLUBACHER et al FREEMAN 8 HALSEY D . CLARK et al THEORETICAL ÇURVE ANALYTICAL APPROXIMATION

60. ...70

TEMPERATURE , °K

Fig. 8 - Vapor Pressure of Argon

(37)

i02 E E

Io-w a: u) u, w a: a-a: o o->

I 0'

o _BEAUMONT _et _al

o _{FISHER & McMILLAN}

FREEMAN & HALSEY

THEORETICAL CURVE

IO2 I. I I i t

59 .60 70 .80 90 lOO 110 ₁₂₀

TEMPERATURE., °K

(38)

E E o Q. > Io-I THEORETICAL CURVE TEMPERATURE ,

Fig. lo -

Vapor Pressure of Xenon

(39)

o -J

z

w >-E0 (

I+t_

Lt w

z

w -J

z

Lt w

I

z

E. / L, J k T / L0 TEMPERATURE, kT/L0

Fig. il - Method of defining the change of internal energy for sublimation.

With this definition of

(40)

o 0.6 0.7 0.1 0.2 0.3 0.4 0.5 TEMPERATURE, kT/L0

(41)

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D D 1 JAN 64FORM

1473

0101 -807-6800 Unclassified

Security Classification

(Security cleaeificacii ei title, body of abstract and indexing ennoIe Clon must be entered when the overall report le classified)

1. ORIGINATIN G ACTIVITY (Cozporate author)

California Institute of Technology

Division of Engineering and Applied Science

2e. REPORT SECURITY C LASSIFICATION

Unclassified

2b. GNT

applicable

3. REPORT TITLE

SUBLIMATION OF A MONATOMIC ELEMENT

4. DESCRIPTIVE NOTES (Type of report and inclusiv, dates)

Technical Report

5. AUTHOR(S) (Last name. first name, initial)

Kerber, Ronald L. and Hsieh, Din-Yu

6. REPORT DATE

January 1969

7e. TOTAL NO. OF PAGES

37 .

7b. NO. OF REFS

23

8e. CONTRACT OR GRANT NO.

N00014-67-0094-0009

b. PROJECT NO. c.

d.

9e. ORIGINATOR'S REPORT NUMBER(S)

Report No. 85-45

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Chie report)

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This document has been approved for public release and sale; its distribution is unlimited.

II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Office of Naval Research

13. ABSTRACT

A simple physical model is constructed to represent. the sublimation of

monatomic elements. According to this model, the solid and gas phases are two

facets of a single physical system. The nature of the phase transition is clearly revealed and the relations between the vapor pressure, the latent heat, and the

transition temperature are derived. The results are applied to the experimental

data of argon, krypton, and xenon, with good agreement. Extension of the model

(51)

Phase transition

Sublimation

Rare gases

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capital letters. Titles in all cases should be unclassified.

If a meaningful title cannot be selected without classifica-tion, show title classification in all capitals in parenthesis immediately following the title.

DESCRIPTIVE NOTES: If appropriate, enter the type of report, e.g., interim, progress, summary, annual, or final. Give the inclusive dates when a specific reporting period is

covered.

AUTHOR(S) Enter the name(s) of author(s) as shown on or in the report. Enter last name, first name, middle initial. If military, show rank and branch of service. The name of the principal author is an absolute minimum requirement.

REPORT DATL Enter the date of the report as day, month, year; or month, year. If more than one date appears

on the report, use date of publication.

TOTAL NUMBER OF PAGES: The total page cOunt should follow normal pagination procedures, i.e., enter the

number of pages containing information.

NUMBER OF REFERENCES: Enter the total number of references cited in the report.

8e. CONTRACT OR GRANT NUMBER: If appropriate, enter the applicable number of the contract or grant under which the report was written.

8b, Sc, & 3d. PROJECT NUMBER: Enter the appropriate military department identification, such as project number,

subproject number, system numbers, task number, etc. 9e. ORIGINATOR'S REPORT NUMBER(S): Enter the

offi-cisl report number by which the document will be identified and controlled by the originating activity. This number must be unique to this report.

9b. OTHER REPORT NUMBER(S): If the report has been assigned any other report numbers (either by the originator or by the sponsor), also enter this number(s).

10. AVAILABILITY/LIMITATION NOTICES Enter any lim-itations on further dissemination of the report, other than those

INSTRUCTIONS

imposed by security classification, using standard statements such as:

"Qualified requesters may obtain copies of this report from DDC."

"Foreign announcement and dissemination of this report by DDC is not authorized."

"U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC users shall request through

"U. S. military agencies may obtain copies of this report directly from DDC. Other qualified users shall request through

'All distribution of this report is controlled. Qual-ified DDC users shall request through

If the report has been furnished to the Office of Technical Services, Department of Commerce, for sale to the public, indi-cate this fact and enter the price, if known.

SUPPLEMENTARY NOTES: Use for additional explana-tory notes.

SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (pay-ing for) the research and development. Include address.

ABSTRACT: Enter an abstract giving a brief and factual summary of the document indicative of the report, even though it may also appear elsewhere in the body of the technical

re-port. If additional space is required, a continuation sheet shall be attached.

It is highly desirable that the abstract of classified reports be unclassified. Each paragraph of the abstract shall end with an indication of the military security classification of the in-formation in the paragraph, represented as (TS), (S), (C),àr ((J).

There is no limitation on the length of the abstract

How-ever, the suggested length is from 150 to 225 words.

KEY WORDS: Key words are technically meaningful terms or short phrases that characterize a report and may be used as index entries for cataloging the report. Key words must be selected so that no security classification is required. Identi-fiers, such as equipment model designation, trade name, military project code name, geographic location, may be used as key words but will be followed by an indication of technical con-text. The assignment of links, r1es, and weights is optional.

Uncias sified