- Email: [email protected]
Specific heat study of solid chlorine and bromine
ThermochimicaActa, 92 (1985) 529-534 Elsevier Science PublishersB.V., Amsterdam SPECIFIC Carlo6 A. Mart&+,
HEAT
STUDY
Facultad de
dad National de C&doba,
OF
SOLID
Matem&ica,
CHLORINE
AND
BROMINE*
Astronomfa y Ffsica,
Laprida 854, 5000 C&doba,
Universi-
Argentina.
ABSTRACT The specific heat at constant pressure as a function of temperature (c vs T) in solid Cl and Br is analyzed by recourse to a method developPin a companion pap&. Valugs for the lattice frequencies, their temper ature dependence6 and the Debye temperature (T ) have been determined for both halogens. Activation energies correspondi!?gto lattice vacancy forms tion were determined in both cases. INTRODUCTION Although Cp
vs
T has been measured in solid Cl2 and Br2 (1,2), a dy-
namical analysis like that proposed by Martin (3)
has not been carried
out in any of these halogens. They are isomorphous, crystsllizing in a face centred orthorombic lattice, spatial group Cmca (D$), ecules per primitive unit cell (4,5),
with two mol-
therefore there are twelve normal
modes of vibration: nine optic and three acoustic. The optic modes may be divided into four librational, three translational and two internal stretching. The zone center eigenvectors for the optic modes are shown in Fig. 1 along with their symmetry species. Mode Au is both Raman end InfraRed inactive. Collected data on the frequencies of the librational modes are shown in Fig. 2 (6-12). As may be seen a linear temperature dependence seems reasonable, and the parameters corresponding to the straight lines, written in the form Wx=W,x(l-CxT), are given in Table 1 (the notation used throughout this paper is that used in Ref. 3). Mode B2g in Cl2 has not been detected by Raman and Infra-Red spectroscopies, and the result shown in Fig. 2 has been deduced by analyzing the temperature dependence of the chlorine Nuclear Quadrupole Resonance transition frequency (13). This analysis also made possible the assignment of the various librational frequencies to their symmetry species, in agreement with Ref. 9. In order to evaluate Cp vs T as indicated in Ref. 3 it is necessary to know the temperature dependence of the frequencies for all the modes. On the basis of the results shown in Fig. 2 it seems reasonable to adopt a linear tempcratura dependence for all the remaining made frequencies in eluding the acoustic modes since these are essentially translatory. The next question is what particular linear dependence is to be chosen for the modes not included in Fig. 2. Since experimental information is lacking, Proceedingsof ICTA 85, Bratislava
- 530 -
Fig. 1: Zone center eigenvectors for optic modes along with their symmetry species. Arrows indicate in-plane motion while + and indicate out-of-plane motions.
and in order to introduce as few parameters to be adjusted as possible, we make a choice based on the following observation on the temporature dependence of the librational mode frequencies in Cl2 and Br2: from Fig. 2 and the values on Table 1 we find reasonable to write Wx=Wox(l-F Wax T), where P has different values for the librational and translational modes and for each of the halogens under investigation in this work. The internal stretching mode frequeliciesare considered temperature independent. The frequencies of the modes B2u and Blu have been given the above form forcing the straight line to go through the experimental values at liquid nitrogen temperatures (6)j the acoustic and Au modes were given the above form. Therefore, the paraneters to be fitted are F, WoAu
and TD, where TD
is the Debye temperature. At high temperatures an additional contribution was found necessary to be taken into account, which is attributed to vacancy formation. The term to be added to Eq. 4 in Ref. 3 is (14): Acp
= R exp(Sf/k) (EfpdT12 exp(-Ef/kT)
(1)
,whereS f and E f are the entropy and energy associated to the creation of
- 531 -
k..
B2g
65
55
Pig. 2: Full
circles
:orrespond to frequez
I
85-
. _I___jj B1g
)us librational modes
;o C12, while those
I\ .
.
110
. .
?ar temperature depe_n lence provides a rea:onable description
w.
120
\
Ls may be seen, a lip
A,
.
90
Q
If figures correspond it the right to Br2.
1oc
819
l
.
1-14.The left column
‘5 75 * 3
.
obtained from Refs.
.
c
.
:y data of the vari-
.
)f data. The parame.ers associated with .he various straight ines are given in
B3g
1x
A,
1
'able 1.
. 9C
~
103 0
I 60
I 120
-II
1
T (K)
80
165
?L
T(K)
a vacancy. Sf was given a value of 1.5 k (15) (k and R are Boltzmann and gas constant, respectively). This additional contribution renders one more parameter to be adjusted in the fitting to C vs T, i.e. Ef. P RESULTS
AND
DISCUSSION
Fitting, as indicated above, by means of a non-linear least-squares procedure (16) the values for the parameters F, Woiu, TU and Ef shown in Table 1 are obtained. Fig. 3 shows the results achieved, and as may be seen the fits are excellent. Curve 1 corresponds to the specific heat
as-
suming the mode frequencies are temperature independent, their values taken as those at 0 K; curve 2 is obtained adding to curve 1 the contribution
- 532 -
Fig. 3:
The
circles in the UJ per and lower fig urefi represent the Cp vs T data in solid Cl2 and Br2 respectively, up to the melting point. The dotted line corresponds to the specific heat produced by
90
130 T(K)
temperature independent lattice frequencies. The dashed line is 02 tained by adding :o the dotted the :ontributions due to the anharmonicities. Adding to this line the contribution due to vacancy formation, the full line is obtained. As may be seen, the agreement with Cp vs T data is, in both cases, excellent.
-
533
-
TABLE 1 Values
due to the temperature curve
ior
change of
2 the contribution
goes it
Deduced for
through
ergies
as expected
are somewhat smaller
these
discrepancies
grounds. tained
However, for
support
values.
the anharmonic
with temperature,
Chlorine
form&ion
(17).
The obtained
the various
to the present
quantities analysis
the fits
vacancy
to
which
formation
of specific
1 bring
behav-
(la),
and the reasonable
shown in Table
3,
a linear
reported
choices
curve
of approximation
exhibits
previously
may be due to different of
and by adding
we obtain
To a good degree
contribution
than those
the excellence
and Bromine
the mode frequencies,
from vacancy
the experimental
may be seen that
Solid
es
and heat
back
values
confidence
ok
and
(19).
PEFEFLENCES *
+
This research was partially supported by Consejo National de Investi gaciones Cientificas y Te’cnicas and by Consejo de Investigaciones Cientificas y Tecnoldgicas de la Provincia de Cbrdoba, Argentina. Fellow nicas,
of the Consejo Argentina.
National
de Investigaciones
Clentfficas
y TIC-
- 534 1
k Glnuu'ueand T. ':.Po:vell,J. Am. Chcn. Sot. 6l, 197~;(1939). L;:5: ?ildebr,lnd,'1.R. Krx!sr, R. A. !!cDonaldan1 D. I?.Stull, J. A3. Chn,?.Sot. 2, 412? (1153). C. A. 'ortin, cozpsnlon :)-i!,‘r in thlc journal. 3. L. Collin, Acta Crystnliot:r.21 431 (1952). 3. I,.Collin, Acta CryctalloSr. 2, 537 (1956). !:.Suzuki, T. 'fokoynmaand t!.Ito, J. Chem. Phys. z, 3392 (1969). ,I.r. Cahill and G. E. Leroi, J. Chem. Phys. 1 4514 (1969). A. Anderson an.1T. S. Sun, Chem. Phys. Lett. %I _, 611 (,970). 3. B. iialacand E. Bonadeo, J. Chen. Phys. D, 643 (1983). E. B. halac, Pd. D. thesis work, Universidad National de &rdoba, 1983. P'.Suzuki, T. Yokoyana and :*I. Ito, J. Chem. Phys. 2, 1929 (1969). G. G. 3unst-, F. Voville and J. P, Viennot, Eel. Phys. 2!3,1345 (1974). C. A. Martl'n,Accepted for publication in J. Kagn. Rescnance. R. H. Beaumont, t!.Chihara and J. A. Morrison, Proc. Phys. Sot. a 1462 (1961:. A. S.': I:',?. Schumacher, 1. Schilling and J. Diehl, eds., WacanNorth Holland, Aasterdar, 1970. ties and Intcrsticials in I\!etalsu, '!!. Z. SeminE, "Statistical Adjustnint of Datan, Dover, New York, 1364. of Crystals", Wiley, 9ew York, 1972. D. C. Xallacc, ~*Thcrmodynomics I:. ?:a.kamura and I!.Chlhara, J. Phys. Sot. Japan 22, 201 (1967). for his help in me preparation of I thank Lit. Nnriano J. Zuriaga this manuscript.
65
HEAT
STUDY
Facultad de
dad National de C&doba,
OF
SOLID
Matem&ica,
CHLORINE
AND
BROMINE*
Astronomfa y Ffsica,
Laprida 854, 5000 C&doba,
Universi-
Argentina.
ABSTRACT The specific heat at constant pressure as a function of temperature (c vs T) in solid Cl and Br is analyzed by recourse to a method developPin a companion pap&. Valugs for the lattice frequencies, their temper ature dependence6 and the Debye temperature (T ) have been determined for both halogens. Activation energies correspondi!?gto lattice vacancy forms tion were determined in both cases. INTRODUCTION Although Cp
vs
T has been measured in solid Cl2 and Br2 (1,2), a dy-
namical analysis like that proposed by Martin (3)
has not been carried
out in any of these halogens. They are isomorphous, crystsllizing in a face centred orthorombic lattice, spatial group Cmca (D$), ecules per primitive unit cell (4,5),
with two mol-
therefore there are twelve normal
modes of vibration: nine optic and three acoustic. The optic modes may be divided into four librational, three translational and two internal stretching. The zone center eigenvectors for the optic modes are shown in Fig. 1 along with their symmetry species. Mode Au is both Raman end InfraRed inactive. Collected data on the frequencies of the librational modes are shown in Fig. 2 (6-12). As may be seen a linear temperature dependence seems reasonable, and the parameters corresponding to the straight lines, written in the form Wx=W,x(l-CxT), are given in Table 1 (the notation used throughout this paper is that used in Ref. 3). Mode B2g in Cl2 has not been detected by Raman and Infra-Red spectroscopies, and the result shown in Fig. 2 has been deduced by analyzing the temperature dependence of the chlorine Nuclear Quadrupole Resonance transition frequency (13). This analysis also made possible the assignment of the various librational frequencies to their symmetry species, in agreement with Ref. 9. In order to evaluate Cp vs T as indicated in Ref. 3 it is necessary to know the temperature dependence of the frequencies for all the modes. On the basis of the results shown in Fig. 2 it seems reasonable to adopt a linear tempcratura dependence for all the remaining made frequencies in eluding the acoustic modes since these are essentially translatory. The next question is what particular linear dependence is to be chosen for the modes not included in Fig. 2. Since experimental information is lacking, Proceedingsof ICTA 85, Bratislava
- 530 -
Fig. 1: Zone center eigenvectors for optic modes along with their symmetry species. Arrows indicate in-plane motion while + and indicate out-of-plane motions.
and in order to introduce as few parameters to be adjusted as possible, we make a choice based on the following observation on the temporature dependence of the librational mode frequencies in Cl2 and Br2: from Fig. 2 and the values on Table 1 we find reasonable to write Wx=Wox(l-F Wax T), where P has different values for the librational and translational modes and for each of the halogens under investigation in this work. The internal stretching mode frequeliciesare considered temperature independent. The frequencies of the modes B2u and Blu have been given the above form forcing the straight line to go through the experimental values at liquid nitrogen temperatures (6)j the acoustic and Au modes were given the above form. Therefore, the paraneters to be fitted are F, WoAu
and TD, where TD
is the Debye temperature. At high temperatures an additional contribution was found necessary to be taken into account, which is attributed to vacancy formation. The term to be added to Eq. 4 in Ref. 3 is (14): Acp
= R exp(Sf/k) (EfpdT12 exp(-Ef/kT)
(1)
,whereS f and E f are the entropy and energy associated to the creation of
- 531 -
k..
B2g
65
55
Pig. 2: Full
circles
:orrespond to frequez
I
85-
. _I___jj B1g
)us librational modes
;o C12, while those
I\ .
.
110
. .
?ar temperature depe_n lence provides a rea:onable description
w.
120
\
Ls may be seen, a lip
A,
.
90
Q
If figures correspond it the right to Br2.
1oc
819
l
.
1-14.The left column
‘5 75 * 3
.
obtained from Refs.
.
c
.
:y data of the vari-
.
)f data. The parame.ers associated with .he various straight ines are given in
B3g
1x
A,
1
'able 1.
. 9C
~
103 0
I 60
I 120
-II
1
T (K)
80
165
?L
T(K)
a vacancy. Sf was given a value of 1.5 k (15) (k and R are Boltzmann and gas constant, respectively). This additional contribution renders one more parameter to be adjusted in the fitting to C vs T, i.e. Ef. P RESULTS
AND
DISCUSSION
Fitting, as indicated above, by means of a non-linear least-squares procedure (16) the values for the parameters F, Woiu, TU and Ef shown in Table 1 are obtained. Fig. 3 shows the results achieved, and as may be seen the fits are excellent. Curve 1 corresponds to the specific heat
as-
suming the mode frequencies are temperature independent, their values taken as those at 0 K; curve 2 is obtained adding to curve 1 the contribution
- 532 -
Fig. 3:
The
circles in the UJ per and lower fig urefi represent the Cp vs T data in solid Cl2 and Br2 respectively, up to the melting point. The dotted line corresponds to the specific heat produced by
90
130 T(K)
temperature independent lattice frequencies. The dashed line is 02 tained by adding :o the dotted the :ontributions due to the anharmonicities. Adding to this line the contribution due to vacancy formation, the full line is obtained. As may be seen, the agreement with Cp vs T data is, in both cases, excellent.
-
533
-
TABLE 1 Values
due to the temperature curve
ior
change of
2 the contribution
goes it
Deduced for
through
ergies
as expected
are somewhat smaller
these
discrepancies
grounds. tained
However, for
support
values.
the anharmonic
with temperature,
Chlorine
form&ion
(17).
The obtained
the various
to the present
quantities analysis
the fits
vacancy
to
which
formation
of specific
1 bring
behav-
(la),
and the reasonable
shown in Table
3,
a linear
reported
choices
curve
of approximation
exhibits
previously
may be due to different of
and by adding
we obtain
To a good degree
contribution
than those
the excellence
and Bromine
the mode frequencies,
from vacancy
the experimental
may be seen that
Solid
es
and heat
back
values
confidence
ok
and
(19).
PEFEFLENCES *
+
This research was partially supported by Consejo National de Investi gaciones Cientificas y Te’cnicas and by Consejo de Investigaciones Cientificas y Tecnoldgicas de la Provincia de Cbrdoba, Argentina. Fellow nicas,
of the Consejo Argentina.
National
de Investigaciones
Clentfficas
y TIC-
- 534 1
k Glnuu'ueand T. ':.Po:vell,J. Am. Chcn. Sot. 6l, 197~;(1939). L;:5: ?ildebr,lnd,'1.R. Krx!sr, R. A. !!cDonaldan1 D. I?.Stull, J. A3. Chn,?.Sot. 2, 412? (1153). C. A. 'ortin, cozpsnlon :)-i!,‘r in thlc journal. 3. L. Collin, Acta Crystnliot:r.21 431 (1952). 3. I,.Collin, Acta CryctalloSr. 2, 537 (1956). !:.Suzuki, T. 'fokoynmaand t!.Ito, J. Chem. Phys. z, 3392 (1969). ,I.r. Cahill and G. E. Leroi, J. Chem. Phys. 1 4514 (1969). A. Anderson an.1T. S. Sun, Chem. Phys. Lett. %I _, 611 (,970). 3. B. iialacand E. Bonadeo, J. Chen. Phys. D, 643 (1983). E. B. halac, Pd. D. thesis work, Universidad National de &rdoba, 1983. P'.Suzuki, T. Yokoyana and :*I. Ito, J. Chem. Phys. 2, 1929 (1969). G. G. 3unst-, F. Voville and J. P, Viennot, Eel. Phys. 2!3,1345 (1974). C. A. Martl'n,Accepted for publication in J. Kagn. Rescnance. R. H. Beaumont, t!.Chihara and J. A. Morrison, Proc. Phys. Sot. a 1462 (1961:. A. S.': I:',?. Schumacher, 1. Schilling and J. Diehl, eds., WacanNorth Holland, Aasterdar, 1970. ties and Intcrsticials in I\!etalsu, '!!. Z. SeminE, "Statistical Adjustnint of Datan, Dover, New York, 1364. of Crystals", Wiley, 9ew York, 1972. D. C. Xallacc, ~*Thcrmodynomics I:. ?:a.kamura and I!.Chlhara, J. Phys. Sot. Japan 22, 201 (1967). for his help in me preparation of I thank Lit. Nnriano J. Zuriaga this manuscript.
65