LIQUID-SOLID TRANSITION IN CYCLOHEXANOL STUDIED
BY NEUTRON TRANSMISSION
R
Fulfaro
R
PUBLICAgAO lEA
CFN
57
423
L
A
Vinhas
and
Liguori Neto
JUNHO/1976
LIQUID SOLID TRANSITION IN CYCLOHEXANOL STUDIED
BY NEUTRON TRANSMISSION
R
Fulfaro
R
L A
Liguori
COORDENADORIA
DE
Vinhas and
Neto
FÍSICA
NUCLEAR
(CFN)
INSTITUTO
DE ENERGIA
SAO P A U L O -
ATÔMICA
BRASIL
A P R O V A D O PARA P U B L I C A Ç Ã O EM FEVEREIRO/1976
C O N S E L H O SUPERIOR
Eng? Roberto N
Prof
Dr
Jafet
Emilio Mattar
-
Presidente
-
VIce-Prasidehte
Prof Dr Jo«é Augusto Matins
Dr
Ivano Humbart Marchas!
Eng9 Hálelo Modesto da Costa
SUPERINTENDENTE
Prof
Dr
R ó m u l o Ribeiro Pieroni
I N S T I T U T O DE E N E R G I A A T Ô M I C A
Caixa Postal 11 0 4 9 (Pinheiros)
Cidade Universitária
A r m a n d o de Salles Oliveira
SAO P A U L O
-
BRASIL
NOTA Este trabalho foi conferido pelo autor depois de composto e sua redação está conforme o original sem qualquer
correção ou mudança
APPENDIX
FORr«AN
r
LFVÍEL
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
r
C
C
C
C
C
C
r
0001
0002
0003
0004
0005
0006
0307
OOOd
000?
0310
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
Q02?
0023
0024
0025
21
MAIN
DATF
76037
14/16/30
A L C l I L n TEÓRICO OA SECCAO OF CHOOUF r O T A L EM F J N C A J OU TE><PFRATUi< A OE O t S Y F
METIJOn OF K Ü T H A R I - S I N G W I
SaMA = S«-SMIN(JSCU1.0
l O E F I N I O O PELA EO
5 ) EM BARNS
8 K - C n H S T A N r E DE RfILTZMANN ( E V / K )
HC=CnNSTA'^TE DE P L A N C K / 2 P I
(EV*SI
AMN-MASSA Dl) NEUTRÓN (AMU)
A«1A1-MASSA DA AMOSTRA ( A M U I
TLMP = TEMPFi<ATURA DA AMOSTRA I K )
E - E N F R r i A Dll NEUTRÓN I N C I D E N T E
(EV)
N = NU1FR0 OE PiJ'ITOS A SEREM CALCULADOS
T E ^ P D I I l - T b M P F R A T U R A DE DFBYF ( K )
S I C M A I I l = S t C C A I ) DF CHOOUE TOTAL ( B A R N S )
ENTkADA Df DADOS
P R I M E I R O C \ R T A O - SOMA BK HC AMN AMAT E TEMP-NO FORMATO 5 E 1 4 8
S F Í U N O T C A R T A U - N NO FORMATO 13
A SECUIR SAO L I D O S OS VALfJRFS DA TEMPERATURA DE DE8YE NO FORMATO F5 1 ( 1 5
P IR CARTAOl
S A Í D A o n s RESULTADOS
I M P K l M f n VALOR DA TFMPFRATUR4 DA AMOSTRA NO FORMATO F7 7 ( K l E A S E b U r *
A TARi-LA
OM r V A I IW DA TEMPERATURA DE DE^iYE ( K l E A SFCCAO DE CHOqUE E'H
BARNS
3 PROCRAMA U T I L I Z A A SUBR ITI^IA TALEG QUE CALCULA A INTEí-RAL DF DFBVF PFLO
MFTUOO DE t . \ U ' ~ S - L r ENDRF
A SUBROTINA T A L E T U T I L I Z A AS FUNCOFS E F E l E VLEG
HMFN/SION S I C M A d O O l 7 ( 1 0 0 )
TFMPO ( 1 0 0 ) , R ( 16 (
R F A D ( 5 U S I M A BK HC AMN AMAT E . T E M P
R F A 0 ( 5 3|f.|
REAOÍ 5 ? ) ( T F M P U ( r I , 1 = 1 N»
^ R I T F ( 6 I I TEMH
8 F O R M A T I D X l 2H TEMPE RA TUR A= F7 ? )
^ R I T E ( 6 4)
A=0
DO 5
I-1,N
7(11=TFMP3(I)/TEMP
H=Z(II
CALL G A L E G ( A , B H I N T )
F=
?5+HINr/(Z(II)•*?
TFMP 1 - T F M P 0 ( I )
ALEA=3 * H C * » 2 » F / ( B K * T E M P O |
S I G M A ! I ) - - 4 «SQMA*AMN*ALFA*E/(HC**2 * A M A T I * 3 *SnMA*AMN*(BK*TEMPO 1 *
* * S / I A M A T Í F * * 5 1 * ( ( 4 / 7 ( I ) - l / 7 * Z ( I I / 5 4 I «-E/( B < * T E M P T 1 * ( 5 / ( 3
*7(
• 1 ) 1 - 5 + 5 * Z ( I ) / B 4 ) + ( E / ( S K * T E M P f l | ) « * 2 » ( T / ( • »7 1 I ) ) - 7 / 2 4 » 7
• Z d l
*/240 )KE/(BK*TEMP1)|**3 *(3 / ( 3 »Z(Il)+3 /16 - Z ( I ) / 0 6
))+SOMA*-46
*5
»(RITE(6 6 I S I G M A ( I )
TtMPD(I)
5 CONTINUE
1 FQRMAT(5F14 BI
2 FORMATI 1 6 F 5
II
3 F0RM4Tn3)
4 F O K M A T d X / / / , 3 0 X , 16HSFCCA0 DF CHOQUE 1 0 X , 5 H T E M P 0 I
6 Fi)RMAT( 33X EB 2 1 5 X , F 5 1 )
STOP
FNO
FO(fTK*N
IV
0001
0002
0003
0004
0005
0006
0007
0008
0009
0010
=
76037
14/16/30
EFEl
DATE
=
75037
14/16/30
VLEG
DATE
=
75037
14/16/30
FJY»=nB-AI*YtA+B)/2
R( 1 ) = 9 8 9 4 0 0 9 3
R{2)=
94457502
R{3)=
86563120
R(4)=
75540441
R{5)=
61787624
«(6)=
45901678
R{7)=
28160355
RI8»=
09501250
R(9)=-R(1)
R(10I=-R(2)
RJ i n = - R « 3 )
R(121=-R(4)
R{13)=-R(5I
R(14)=-R(6)
R(15I=-R(71
R(16)=-Rt8l
M=16
Nl=M*l
fcNEl=Nl
SIGMA=0
DO 2 J = 1 , M
E N U H = E F E 1 ( F ( R ( J ) ) - R ( J ) * * 2 »
DEN=ENE1*FNE1*VLF5(NIfR<J))**2
2 SIGHA=FNUM/OEN+SIGHA
HINT=(B-A»»SIGMA
RETURN
ENO
IV
G LEVEL
21
FUNCTION E F E l l X )
EFF1=X/(EXP{X|-1
RETURN
ENO
0001
0002
0003
0004
FORTRAN
DATE
GALEG
21
SUBROUTINE G A L E S ( A , 8 , H I N T I
DIMENSION R(16»
0001
0002
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
FORTTRAN
G LEVEL
IV
G LEVEL
21
1
FUNCTION V L E G I M . X I
VLEG1=7 8 7 5 * X * * 5 - 8 7 5 * X * * 3 + 1
875*X
VLFG2=14 4 3 7 5 * X » * 6 - 1 9 5 8 7 5 « X * * 4 + 5 5 6 2 5 * X * X 3125
DO 1 1 = 7 , M
FLI = I
VLEG=I(2 * F L I - 1 )»X*VLEG2-(FLI-1
l*VLEGll/FLt
VLEGI=VLEG2
VLEr2=VLEG
RETURN
END
LIQUID SOLID TRANSITION IN CYCLOHEXANOL STUDIED
BY NEUTRON TRANSMISSION*
R
Fulfaro
R
L A
Vinhas and
Liguori Neto
ABSTRACT
The total cross section o f the yclohexcnol was measured for neutrons w i t h wavelength of X = 6 13 A, varying
the sample temperature in a range that includes the temperature of melting From these experimental results and by
comparison w i t h theoretical calculations based in the Placzek model i t was possible t o obtain the Debye temperatures for
b o t h states These temperatures were used m the calculation of the disorder entropy variation near the fusion p o i n t
INTRODUCTION
Slow neutron transmission methods can be used t o investigate physical properties such as freedom
o f m o t i o n o f molecules and molecular groups in condensed states of matter*^ 6 1 0 1 1 )
this regard the
plastic solids f o r m e d b y globular molecules are of particular interest since their physical properties are
intermediate between liquids and solids These solids called organic globular compounds usually presents a
very small e n t r o p y o f melting'^ i> « 5 cal/mole K) the term globular* is applied t o t h e m because the
molecules involved possess a spherical s y m m e t r y about the center o f mass as a result either of the
distribution o f the atoms or of the rotation o f the molecule about its center The solid f o r m e d by such a
molecule when it freezes f r o m the l i q u i d state displays rather unusual physical properties that have been
explained as resulting f r o m practically unhindered r o t a t i o n of the molecules about their lattice positions
The name plastic crystals for the solids arises because among these properties one f i n d the ease of plastic
f l o w under o n l y moderate pressures In every case it is observed a transition at some definite temperature
below the freezing p o i n t f r o m the plastic solid f o r m t o a crystalline f o r m t h a t behaves like an ordinary
s o l i d * * ' In the rotational f o r m stable at temperatures just below the freezing p o i n t the crystal structure is
usually of the cubic f a m i l y
In the present w o r k the cyclohexanol was selected f o r the studies because its solids state
crystallizes in a cubic system and f o r its properties as a globular substance The liquid-solid transition in
cyclohexanol was investigated t o obtain informations about the transition itself and also t o obtain
informations a b o u t the cyclohexanol in the liquid state A t o m i c or molecular vibrational m o t i o n is one of
the physical parameters whose changes play an i m p o r t a n t part in this transition It can be successfully
investigated by means o f slow neutron transmission methods*
The t o t a l cross section measurements as a
f u n c t i o n o f temperature of the sample in a range near the melting p o i n t allows a clear observation on
changes in the molecular dynamics at state transition In this paper f r o m the slow neutron transmission
measurements carried o u t on solid and liquid cyclohexanol near the melting p o i n t (25 5 ° C ) , it was possible
t o determine the Debye temperature f o r the l i q u i d cyclohexanol and evaluate a disorder entropy at the
melting p o i n t transition
I 1 This research v^as partially supported b y the Comissão Nacional de Energia Nuclear
EXPERIMENTAL
Neutron transmission measurements in the temperature interval 3°C t o 35°C were performed w i t h
neutrons o f 6 13 Ã wavelength selected using a single crystal spectrometer*^' at the l E A 2 MW light water
reactor A magnetite crystal monochromator and an appropriate choice of polycrystalline filters o f Be and
Pb were used t o select only neutrons f r o m first order Bragg s reflections the elimination o f higher order
contamination was checked by t o t a l cross sections measurements of gold and water chosen as standard A t
6 13 A the intensity was enough to have in one day a series of measurements covering the w h o l e
temperature interval o f interest
The sample was a commercial cyclohexanol (J T Baker Chemical) w i t h p u r i t y better than 9 9 % I t
was in a alumnium holder w i t h 0 253 cm internal spacing The sample cooling was performed b y circulation
o f cold air w i t h variable f l o w m a isolated volume surrounding the sample the temperatures were controlled
w i t h i n 2 ° C b y thermocouples attached t o the a l u m i n i u m holder The t o t a l neutron cross section is given b y
(ij = In (T~M/n where T is the measured neutron transmission and n is the number o f molecules/cm^
obtained f r o m the sample thickness and b u l k density The melting p o i n t for cyclohexanol is 25 S^C and the
curve of density v s temperature is shown in figure 1 The density for solid cyclohexanol was calculated
using the f o r m u l a obtained f r o m ultrasonic measurements'*' normalized f o r the experimental value
p = 9 6 2 4 g / c m ' at 2 0 ° C ' ' ' 2 ' f o r the liquid cyclohexanol the density was obtain through the f o r m u l a
given m the International Critical Tables <8' Also the experimental values given in the literature'^ ^ 8) ,5
shown in figure 1
T w e n t y independent series of independent measurements of neutron transmission were taken b y
cooling and/or heating the sample Results obtained f o r n o y = I n T " * in f u n c t i o n of temperature are
displayed in figure 2 as average over series o f cooling and heating The indicated errors are obtained b y
propagation o f the statistical errors o f the several points measured at each temperature Results indicate on
cooling an step at 2 2 5°C the temperature difference f r o m the tabulated melting p o i n t (25 5°C) is
probably due t o a small water contamination On heating the variation occurred near 15°C T h e
temperature difference f r o m heating and cooling cannot be ascribed t o an enror in the temperature
measurement and may be due t o a premelting phenomenon and partially due t o further water
contamination in the cooling process''"
The values of a-^ as a f u n c t i o n of temperature in the interval 3 ° C t o 3 5 * 0 are shown m figure 3 f o r
cooling and heating series The densities were chosen f r o m figure 1 considering the sample as being solid
only below 22 5 ° C f o r the cooling and o n l y below 15°C f o r the heating case Amean value o f o-p = 1103
barns can be ascribed f o r the cyclohexanol in the solid state and other mean value of ay = 1135 barns f o r
the l i q u i d state
C A L C U L A T I O N OF T O T A L CROSS SECTION V S D E B Y E T E M P E R A T U R E
The Debye a p p r o x i m a t i o n is also valid f o r molecular crystals w h e n in c o m p u t i n g t h e r m o d y n a m i c
functions only the lattice vibrations are considered The cyclohexanol has a f c c l a t t i c e ' * ' and belongs t o
the globular molecule class then i t can be likened t o a m o n o a t o m i c crystal whose atoms are changed in
more complex units l e m o l e c u l e s ' ' " "
For cubic monoatomic polycrystals it is possible t o c o m p u t e the total scattering cross sections
using the Debye temperature as a parameter'^' | n the Placzeck expansion the total scattering cross section
IS the sum of individual cross sections for 0 1 2
phonon process 1 e
0 3 = 1
ag
£=0
(11
10
c
«
\
E
o
90
95
100
30
("O
40
O Experimental points
ref
8 and 13
temperature
Temperature
20
vs
Figure 1 — Density of ciclohexanol as a function of temperaturt;
10
density
CYCLOHEXANOL
50
CJI
6
where a j is the total incoherent cross section in t h e Debye approximation generalized t o the case o f £
phonons involved in the scaterring process In the case of heavy nuclei the Debye Waller factor occuring m
Ofi IS expanded in powers of 1/M Collecting terms o f same power in M " '
one can w r i t e
(jM
a.-
S
0
~ -
(2)
= 0
With
a<">=
2
ap<">
(3)
£= 0
Alternating terms in equation (3) are of opposite sign cancel the c o n t r i b u t i o n o f each others
Because of this the series of equation (2) converges very rapidly For sufficiently large M i t is n o t necessary
t o go b e y m d the first inverse power o f M
Considering the complete expression*^) f o r o g * " ' and using the equations (2) and (3) one can
write
„,= „ , O U ^ i i -
,4,
a<0' = ffo*°* = S + s
(5)
with
t h a t IS the sum o f coherent and incoherent scaterring cross sections of hydrogen atoms in the cyclohexanol
molecule^where S = 4jra^ and s = 47r(a^ - a^) being a the scattering length
For the l i m i t i n g case E < <
and T >>
M > 10 amu it is n o t necessary t o consider o * ^ '
some approximations can be performed For
since its c o n t r i b u t i o n t o total cross section w o u l d n o t
amount more than one per cent
For the case T > >
0 [ j the a * ^ ' is given b y * ^ '
„ „ ,v,,,2
,n.= - 2 ( S . s ) 2 ^ « E 3(S+s)mo
.--^^W
^^[(f
iTin
3(S+s)mo
V
2
T
1
1
*b
^
(6)
<o^D
•^
3
do
3
1
16
96
2
84
T
kle^
4
6^
24
240
T
k¿0¿
8
0
^D_
T
*1
The theoretical scattering cross section
is calculated f r o m equations (4)
(5) and (6)
Here M is the
molecular mass m^, is the neutron mass and E is the neutron energy
where
a^^^^withF =—+
) 0<2)= - ~
f^-~—
dx
(7)
IS the Debye temperature, T is the absolute temperature h = h/ltr h and
are the Planck and
B o l t z m a n n constants respectively
In the present w o r k for the fixed X = 6 1 3 A corresponds
E = 0 0 2 1 7 6 e V The temperature interval for solid cyclohexanol was 2 7 3 ° K - 2 9 0 ° K IV = 100 16
amu f o r the cyclohexanol ( C e H i l O H ) and having twelve hydrogen atoms result S + s = 9 7 8 barns The
total cross section Oj =
+
where
is computed according t o formula(4) and cTg is t h e
absorption cross section In
are included the c o n t r i b u t i o n t o Oj of all atoms different of hydrogen
presents in the cyclohexanol molecule for the fixed neutron energy E this total c o n t r i b u t i o n is equal t o
46 5 barns
The theoretical ay(T) was calculated using the mean tabulated value
~ 6 0 ° K as a parameter
for all the temperature interval f o r solid cyclohexanol the m a x i m u m variation of the calculated values o f
OjiJ] was 0 2% Therefore the OjiOQ) was calculated as a f u n c t i o n of Debye t e m p e r a t u r e i n the
intervallO°K - 100°K using the f i x e d value T = 2 8 3 ° K as a parameter The range of variation f o r 6 ^
was chosen according t o reference**', the resulting p l o t of Oj versusfl^ is displayed in figure 4 T o t h e
experimental mean value 0 j = 1 103 barns obtained f o r the solid cyclohexanol (figure 3) corresponds
t w o values of Debye temperatures in the figure 4 0^ = 2 3 ° K and 6^ = 6 5 ° K this last value agree w i t h
the values usually t a b u l a t e d ' * ' F r o m the same figure 4 the experimental mean value a-j- = 1 135 barns
obtained f o r the liquid cyclohexanol (figure 3) near the break leads t o a Debye temperature value
~ 2 0 ° K f o r the liquid cyclohexanol
C A L C U L A T I O N OF A D I S O R D E R ENTROPY A T T H E POELTING POINT
The melting of a crystal is accompanied by an abrupt change in internal energy molar volume
and entropy When the crystal is t h a t of a molecular c om pound contributions t o the increase of internal
energy and entropy arise f r o m change in the character o f the vibrations among the u n i t molecules
disorder o f the structure change in the internal vibrations w i t h i n each molecule and acquisition o f
rotational degrees o f freedom A simpler case t o treat is relative t o crystals made up of atoms t h a t
possess an entropy of melting w i t h only the t w o first mentioned contributions since the atoms do n o t
have internal degrees of freedom Another interesting case where there is no change in the internal
coordinates during melting is that o f the organic and inorganic crystals whose molecules possess free
rotation in three dimensions in the crystalline state'^*
A t the melting there is a discontinuous change in properties accompanying the discontinuous
change i i volume A volume expansion in a crystal and in a liquid is accompanied by an increase in
entropy chiefly due t o the change in the vibrational degrees of freedom of the assembly while the
configuration remains essentially unchanged On t h e other hand a t the crystal liquid transition there is
in addition an entropy change associated w i t h the change in the structure since the liquid lacks the long
range regularity o f the crystal*^' It is possible t o compute the change in entropy associated w i t h the
volume change together w i t h the change in vibrational character AS^ by subtracting this entropy change
f r o m the experimental entropy of fusion ASf one can obtain the entropy change associated w i t h the
disordering melting A S ^ * ^ '
As a consequence of the foregoing arguments one can write
ASD
= ASf -
AS^
(8)
T H E AS„ C A L C U L A T I O N
From the statistical definition o f entropy and using some approximations one can write the
expression o n entropy f o r an Einstein s o l i d ' ^ '
S = k [ (q + 3N) £n (q + 3N) - q£nq -
3N £n3N ]
(9)
SERIES
•
HEATING
SERIES
15
18'C
0
Temperature
Temperature
O COOLING
In T " ' v s .
CYCLOHEXANOL
0
( ' O
20
28 8 »0
0
25
o
30
Figure 2 — Results obtained for no = £nT"* for cyclohexanol in f u n c t i o n of temperature averages over several series of cooling and heating
1600
t 610
1620-
630-
640
650 -
660
00
O
O
i—
O
if)
Ü
<u
o
o
—
Figure 3 -
1060
1080
MOO
COOLING
HEATING
•
L-y
SERIES
10
SECTION
SERIES
CROSS
O
TOTAL
vs
,
.^
Te mpera
15
¿L
liquid
1
5!,
JL
0
rip
t u r e (°C)
20
il35 barns
TEMPERATURE
CYCLOHEXANOL
^
25
^solid
CJl
^
5 ^
i—X—v
^
J30
1103 barns
Values of Oj as a function of temperature The mean values f o r cyclohexanol in the solid and liquid state are indicated
1120|-
11401-
1160
^
10
4.000 h
3,500 \ -
o
^
3,000
c
o
o
to
2,500
-
o
o
o
o
2,000 h
1,500 h-
I.OOOh
50
Debye
Figure 4
—
75
Temperature
100
125
GoCK)
T o t a l cross section as a f u n c t i o n of Debye temperature f o r T = 2 8 3 K calculated using t h e
computer program m A p p e n d i x
11
where N is the number of atoms (or molecule)
q is the number of energy quanta or phonons of
magnitude hv and k is the Boltzmann constant
The thermodynamic f u n c t i o n free energy F is given b y F = U - TS where U = qhi* is t h e
amount o f energy raised above the zero p o i n t energy after the absorption of q vibrational quanta by the
solid So,F can be w r i t t e n as
F = qhi' -
k T [ ( q + 3N) Cn (q + 3N) - qCnq - 3Nfin3N]
(10)
The number o f quanta q is determined regarding t h a t no quanta at all w o u l d be absorbed i f the
solid were able t o submit completely t o its striving towards a m i n i m u m value of energy O n the other hand
if It were governed completely b y t h e striving towards m a x i m u m entropy the number of quanta absorbed
w o u l d continue t o increase'^) T h e equilibrium state lies at t h e p o i n t where the free energy is a m i n i m u m
thus where
-lL(5L = o
(11)
dq
Combining equations (10) and (11) follows
q + 3N
hi; - kTen^^
= 0
or
3N
q =-
Rearranging the eq (f))
S = k[qen(^^)
q
^.^Cn(54-^)]
(12)
3N
or
S = 3 N k [ - < ^ l ^
+ M 1 -
e-f^^^Tplj
(1.)
Since t h e Debye temperatures d o n o t d i f f e r greatly f r o m t h e Einstein temperatures*^) t h e f o r m e r
defined f o r a solid as 6^ =
^
one can p u t the equation (13) f o r the entropy of an Einstein solid in the
K
form
Ssoi
0 /T
= 3Nk[—^
e^s^-1
+ Cn(1 - e ~ ^ s ^ ) " ' ' ]
(14)
12
The mam interest is t o determine AS,, in the melting temperature T ^
In a solid the atoms vibrate
about mean positions which are f i x e d b u t in a liquid at temperatures near the melting p o i n t i t is generally
recognized t h a t the atoms vibrate about mean positions which though n o t fixed move slowly compared
with
the
velocity
with
which
the
atoms vibrate
quasi crystalline w i t h weaker binding forces i e
Then
one can assume the
w i t h a Debye temperature
liquid phase t o be
f o r I'qiiid lower Ifian in the
the solid state 6 5 " ' The entropy f o r the liquid phase S|,p can be calculated using the same equation (14)
b u t n o w introducing the
T h e A S ^ = S|,q -
Ssoi IS given b y
[
(15)
+ £n
e^'-Z'^m - 1
e^s'Tm _ ,
1 -
e~^L/Tm
T H E ASp C A L C U L A T I O N
According w i t h equation (8) t o obtain A S Q it is necessary t o have the experimental e n t r o p y of
fusion defined a s ' ^ ^ l
AHc
ASF
=
-
V here A H p is the gram atomic heat of fusion
For t h e cyclohexanol
cal/mole°K
where T ^
= 2 9 8 5 ° K and A H p = L M = 4 1 9 67 cal/mole ASp = 1 406
Here L is the latent heat of fusion equal t o 4 19 cal/gram and M is the cyclohexanol molecular
mass
The equation (8) f o r the disorder e n t r o p y at the melting p o i n t can be rewritten m the f o r m
ASD=
-3Mk
+ 8n
e^L'T^-l
where Nk
=
e V T m - l
—
]
(17)
l-e-^/'Tm
R = 1 9 8 7 cal/mole ° K is the Gas constant
The experimental values
and 8^
obtained f r o m the total cross sections measurements m
cyclohexanol were used t o p e r f o r m the <ialcutatiortof the eq (17)
The f i r s t set o f experimental Debye temperatures
cyclohexanol
respectively
= 6 5 ° K and
gives a c o m p u t e d AS^ = 7 015
experimental entropy o f fusion ASp
cal/mole°K
= 2 0 ° K f o r solid and l i q u i d
this value
is larger
than
A l t h o u g h the value 9^ = 6 5 ° K is w i t h i n t h e range of values usually
f o u n d f o r the solid c y c l o h e x a n o l ' * ' the e m p l o y m e n t of this 9^ in the calculations lead t o a negative value
o f A S Q Since t h e c o n d i t i o n AS^ <
Debye temperatures
ASD =
9^ = 2 3 ° K
5 7 3 cal/mole ° K
ASf must be s a t i s f i e d ' ^ ' the AS^ was calculated f o r the second set o f
and
= 20°K
obtaining AS^ =
8 3 3 cal/mole ° K and t h e value
ACKNOWLEDGMENTS
We w o u l d like t o thank A M Figueiredo Neto and C Furhmann for the collaboration in t h i
computacional w o r k and I M Yamazaki f o r the drawing of the figures
RESUMO
F O I medida a secçao de choque total do ciclohexanol para neutrons com comprimento de onda f i x o de
X = 6 13 A , variando se a temperatura da amostra no intervalo que incluí a temperatura da mudança de estado A partir
destes resultados experimentais e comparando com cálculos teóricos baseados no modelo de Placzek foram obtidos valores
das temperaturas de Debye d o ciclohexanol para ambos os estados Estas temperaturas foram utilizadas no cálculo da
variação de entropía de desordem próximo ao ponto de fusão
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