The Specific Heat of Ag 2 S in a-phase
Hideo Okazaki
General Education Department, Niigata University, Niigata, Japan
Akio T a k a n o
Department of Physics, Faculty of Science, Niigata University, Niigata, Japan
Z. Naturforsch. 40a, 9 8 6 - 9 8 8 (1985); received May 14, 1985
The specific heat of Ag 2 S has been measured from room temperature up to the melting point
(838 °C). The temperature dependence of the specific heat of its a-phase in the temperature
range from the ß-y. transition point at 179 ° C to 586 °C, at which temperature there occurs
another crystallographic transition, is well reproduced by supposing a Schottky-type specific heat
(excitation of mobile ions) in addition to the normal lattice heat capacity. The latent heat of the
latter transition, where the structure of the lattice changes from bcc to fee, is obtained to be
774 J/mol.
T h e specific heat of solid superionic conductors
gives i m p o r t a n t i n f o r m a t i o n a b o u t the b e h a v i o u r of
the m o b i l e ions. M e a s u r e m e n t s of the specific heat
of Ag 2 S have yet been insufficient, although m a n y
m e a s u r e m e n t s have been d o n e on Agl, whose structure in its superionic phase is similar to that of
Ag 2 S. Moreover, there exist discrepancies concerning the t e m p e r a t u r e d e p e n d e n c e of the specific h e a t
of Ag 2 S [1, 2], and the m e a s u r e m e n t s have been
restricted to below 400 °C. This p r o m p t e d us to
r e e x a m i n e the specific heat of Ag 2 S and to e x p a n d
the range of t e m p e r a t u r e s up to the melting p o i n t
(838 °C).
T h e m e a s u r e m e n t s at high t e m p e r a t u r e were
carried out on specimens enclosed in silica tubes in
v a c u u m in order to avoid dissociation. An a d i a b a t i c
type e q u i p m e n t for measuring specific heats (produced by Sinku R i k o Co., J a p a n as the type SH3000) was used. Details of the e q u i p m e n t and
m e t h o d are p u b l i s h e d elsewhere [3].
Ag 2 S has three phases in the solid state. T h e
crystallographic transitions take place at 179° and
685 ° C , the crystal structure changing f r o m m o n o clinic to bcc and f r o m bcc to fee, respectively. W e
call these phases as ß-, a B - and a F - p h a s e in the o r d e r
f r o m lower to higher t e m p e r a t u r e . At the First
transition, the superionic conduction by A g + ions
begins. In this p a p e r we pay attention to the a B p h a s e because its crystal structure, necessary to
analyse the experimental results, is well known.
Reprint requests to Prof. H. Okazaki, Physics Laboratory. General Education Department, Niigata University,
Niigata 950-21, Japan.
O u r results are shown in Fig. 1, together with
those o b t a i n e d by Jost and Kubaschewski [1], and
by Perrott and Fletcher [2]. T h e results obtained
were estimated to b e within 3% in the accuracy. T h e
question of the rapid increase in the specific heat
a f t e r melting is still open. Below 400 ° C our results
coincide with those of Jost and Kubaschewski and
d i f f e r f r o m those of Perrott and Fletcher. According
to Perrott and Fletcher, stoichiometric Ag 2 S shows a
/.-type a n o m a l y a f t e r the ß -*• a B transition. So, we
have exactly controled the stoichiometry of the
s p e c i m e n within 10 _2 % by C o u l o m e t r i c titration
t h r o u g h a cell Pt s p e c i m e n Agl Ag before m e a -
1 AO
Ag2S
Present work
Experiments
Calculation
Jost and
Perrott
Kubaschewski
and Fletcher
1 mol"/. excess S
•
60
Ag
Stoichiometry
J
0
200
400
600
L
800
T(°C)
Fig. 1. The specific heat of Ag 2 S. Small open circles and
black ones show the specific heat obtained experimentally
and calculated from (1), respectively. Arrows show the
experimental accuracy.
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H. Okazaki and A. Takano • The Specific Heat of Ag2S in a-phase
987
Table 1. Calculated values of 9 a? Vm T/yT and A Cp in units of J / m o l K.
T(°C)
200
250
300
350
400
450
500
550
585
9tfVmT/XT
2.72
2.85
3.01
3.26
3.43
3.64
3.89
4.39
4.66
A Cp
9.12
8.29
7.49
6.78
6.11
5.48
4.98
4.52
4.81
suring the specific heat. We did not find this
a n o m a l y , so t h a t we support the results of Jost and
Kubaschewski. T h e extraordinary increase in the
specific heat n e a r the phase transition ß -* a B is
explained by t h e excess heat necessary to creat the
Frenkel type cationic defects [1]. O u r interpretation
will be limited to the b e h a v i o u r of the specific heat
in the a B -phase, since the crystal structure of the
ap-phase is not sufficiently known.
T h e m e a s u r e d molar specific heat at constant
pressure, Cp, can b e expressed by three terms:
w h e r e Cv is t h e m o l a r specific heat at constant
volume. T h e second term is due to the a n h a r m o n i c
lattice v i b r a t i o n of the constituent ions; cc\ means
the linear e x p a n s i o n coefficient, Vm the m o l a r
volume, XT the isothermal compressibility and T
the absolute t e m p e r a t u r e . ACp is a Schottky-type
excess specific heat caused by j u m p s of mobile
ions f r o m stable e q u i l i b r i u m positions to activated
ones. If we a s s u m e that each ion vibrates independently w i t h i n a h a r m o n i c well, which may be a
good a p p r o x i m a t i o n at higher t e m p e r a t u r e , then Cv
b e c o m e s 3 nR (Dulong-Petit's law), w h e r e R is the
gas constant a n d n = 3. Therefore, C, of a B - A g 2 S
is equal to 74.9 J / m o l K.
F o r the second term on the r.h.s. of (1) we need
the k n o w l e d g e of ai, Vm and xi a s a function of
temperature. F o r the compressibility of Ag 2 S we use
the value at r o o m t e m p e r a t u r e (2.9 x 10 - 1 2 c m 2 /
d y n e [4]). If t h e second term on the r.h.s. of (1) is
written as LCjT
with L = 9a? F m / y r C j , then L
turns out to be almost constant for a wide range of
t e m p e r a t u r e s ( N e r n s t - L i n d e m a n n relation). Using
the /?-phase value of x, = 2.0 x 1 0 " 5 / K [5] and
Vm = 34 cm 3 (density 7.3 g/cm 3 ), we have L = 0.69 x
10~6 m o l / J . W e a s s u m e L to be constant t h r o u g h o u t
in the ß- and a B - p h a s e , then we obtain the values of
9 a 2 Vm T/xt s h o w n in T a b l e 1*.
T h e last t e r m in (1) may be derived f r o m the
following considerations. In a B -Ag 2 S, sulphur ions
f o r m a bcc lattice and silver ions are distributed
r a n d o m l y over t e t r a h e d r a l 12 (h) sites. If a silver ion
occupies a 1 2 ( h ) site, however, o t h e r silver ions
m a y not occupy n e i g h b o u r i n g sites at the s a m e t i m e
because the d i a m e t e r of the silver ion (2.32 A) is
larger t h a n the distance of the n e i g h b o u r i n g 12 (h)
sites (1.72 A). According to the recent study on the
crystal structure of x B -Ag 2 S by n e u t r o n d i f f r a c t i o n
by Cava et al. [6], w h o h a v e used a single crystal,
the silver ions are delocalized along the ( 1 0 0 ) directions. This fact suggests that silver ions prefer to
m o v e in the ( 1 0 0 ) direction, i.e. a silver ion moves
f r o m a t e t r a h e d r a l site (referred to as A-site hereafter) to one of its next nearest A-sites via the octahedral 6 (e) site (referred to as B-site h e r e a f t e r ) as
illustrated in F i g u r e 2. T h i s situation is r e m a r k a b l y
d i f f e r e n t to that in a-Agl, w h e r e m o b i l e ions prefer
to move along the ( 1 0 0 ) direction f r o m a 12 (h) site
to one of its n e i g h b o u r i n g 12 (h) sites via the 24 (j)
site [7]. T h e ionic conductivity of a B -Ag 2 S increases
with increasing t e m p e r a t u r e , which m e a n s that the
p o p u l a t i o n of the c o n d u c t i n g A g + ion on B-sites
increases with increasing t e m p e r a t u r e . If Ah is the
enthalpy d i f f e r e n c e b e t w e e n A- and B-sites, the
G i b b s free energy associated with the p r o m o t i o n of
iVB ions f r o m A- into B-site, AG, is written as
AG = JVb Ah —
TAs,
where As is the entropy change,
the configurational entropy Asc
change associated with the change
f r e q u e n c y of ions on B-sites, Asf.
o r d e r e d system, such as a B -Ag 2 S,
a n d m a y t h e r e f o r e b e neglected.
culated by the e q u a t i o n
Asc=
which consists of
and the entropy
of the vibrational
F o r a highly disAs{ will be small
Asc m a y be cal-
kB\n(WAWB),
where WK and
to distribute NA
(3)
are the n u m b e r s of possibilities
ions on jAN A-sites (j\> 1) and
* The values of 9 a, Vm 7 7 / r estimated by Perrott and
Fletcher are about four times larger than our values. Probably they have used the value of
in the a B -phase, which
is about two times larger than that in the /?-phase.
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988
H. Okazaki and A. Takano • The Specific Heat of Ag2S in a-phase
shared by NB ions is then expressed as AQ = NB Ah,
and the excess specific heat AC P is calculated by
d i f f e r e n t i a t i n g AQ with respect to t e m p e r a t u r e . T h e
following relation is accordingly o b t a i n e d for A C p :
ACp =
2 N0(Ah)2
1
knT2
F
(7)
w h e r e 2 N0, kB and F are the n u m b e r of m o b i l e ions
in 1 mol Ag 2 S (N0 m e a n s the A v o g a d r o n u m b e r ) ,
the Boltzmann constant and a numerical constant
d e p e n d i n g on the configurational d i s t r i b u t i o n of N
ions on A- and B-sites as expressed by
F=
Fig. 2. Three kinds of positions on the (100) plane of
a B -Ag 2 S. Big circles show sulphur ions. Small circles,
hexagons, and squares mean 12 (h). 6 (e). and 24 (j) sites,
respectively. A cation (black circle) on a 12 (h) site (an
A site) prefers to move to one of its second neighbouring A
sites in the ( 1 0 0 ) direction via the 6 (e) site (B site) like
shown by arrows.
7Vb ions on / B iV B-sites (j B > 1), respectively. Here,
TV = yVA + iVB is the total n u m b e r of cations a n d / x
the mole ratio of X-sites in the unit cell {jA = 12/4
= 3 and / B = 6 / 4 = 3/2). Since two A g + ions m a y
not occupy n e i g h b o u r i n g A-sites at the s a m e time,
W A is given by
UAN-NA+
'NA\(jAN-2NA+
1)!
1)!
(4)
WB is similarly given by
W» =
OB AO!
(5)
NB\(jBN-NBy.
T h e e q u i l i b r i u m value of NB at a t e m p e r a t u r e is
o b t a i n e d as
f ( \ + 2 f y
= e x p ( - Ah/kBT)
(6)
/
+
!-•
4
2
+
1+2/
1
2+f
1+/
+
1 - /
9
2 + 3/'
(8)
respectively. The evaluated values of the excess
specific heat using (7) are also shown in T a b l e 1 for
the case of /? = 0.11 eV, which corresponds to the
activation enthalpy evaluated f r o m the ionic conductivity [8]. The calculated values of the specific
heat are also shown in Figure 1. T h e a g r e e m e n t
with the experiments is satisfactory in view of the
fact t h a t we have no knowledge on XT of t h e a B p h a s e and have assumed L to be constant in spite of
the p h a s e transition.
Finally, we have o b t a i n e d a latent heat 774 J / m o l
for the transition taking place in the s u p e r i o n i c
phase w h e r e the anion lattice changes f r o m a bcc to
a fee arrangement. T h e entropy associated with the
p h a s e transition is estimated to be 0.92 J / m o l K,
which is fairly small. It is an interesting p r o b l e m to
understand the change of the anion lattice f r o m bcc
with a lower packing density to fee with the highest
one, retaining the superionic property. T h i s problem is now being studied.
f r o m the condition of (dAG/dNB)T
= 0. Here, / is
the ratio Nß/N.
T h e excess heat, AQ, which is
A u t h o r s express their thanks to Professor T a m a k i
for valuable discussion. One of the authors (A. T.) is
grateful to Drs. S. T a k e d a and S. H a r a d a for their
h e l p f u l discussions.
[1] W. Jost and P. Kubaschewski, Z. Physik. Chemie N. F.
60,69 (1968).
[2] C. H. Perrott and N. H. Fletcher, J. Chem. Phys. 50,
2344 (1969).
[3] S. Takada. H. Okazaki. and S. Tamaki. J. Phys. Soc.
Japan 54, in press (1985).
[4] Landolt-Börnstein, Zahlenwerte und Funktionen aus
Physik. Chemie. Astronomie. Geophysik und Technik,
6. Auflage, II. Band, 1. Teil, Springer-Verlag, Berlin
1971.
Unpublished. The method of lamp and scale was used.
B. J. Cava, F. Reidinger, and B. J. Wuensch, J. Solid
State Chem. 31,69 (1980).
P. Vashishta and A. Rahman, Phys. Rev. Lett. 40, 1337
(1978).
H. Okazaki. J. Phys. Soc. Japan 23,355 (1967).
(l-/)(2+/) a - f )
[5]
[6]
[7]
[8]
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