Pb0.5 + xMg xZr2 – x(PO4)3(x = 0, 0.5) Phosphates: Structure and Thermodynamic Properties

Article abstract of DOI:10.1134/S0036023620050137

Abstract—Crystalline Pb_{0.5 + x}Mg_xZr_{2 – x}(PO₄)₃ (x = 0, 0.5) phosphates of NaZr₂(PO₄)₃ (NZP) structural type were synthesized. The heat capacity of Pb_0.5Zr₂(PO₄)₃ was measured by adiabatic vacuum and differential scanning calorimetry (DSC) within the temperature range 8–660 K. The studied phosphates were found to experience a reversible phase transition in the region 256–426 K. According to the results of Rietveld structural study, this transition occurred due to an increase in disorder of lead cation positions in cavities of the NZP structure. The measurements of PbMg_0.5Zr_1.5(PO₄)₃ heat capacity in the temperature range 195–660 K showed that it experienced a similar phase transition at 255–315 K. Based on the measured experimental data, the thermodynamic functions of Pb_0.5Zr₂(PO₄)₃, such as C_p ⁰(T) [H⁰(T) – H⁰(0)], S⁰(T), and [G⁰(T) – H⁰(0)] were calculated for the temperature range 0–660 K. The standard formation enthalpy of Pb_0.5Zr₂(PO₄)₃ was determined at 298.15 K.

Full text of DOI:10.1134/S0036023620050137

ISSN 0036-0236, Russian Journal of Inorganic Chemistry, 2020, Vol. 65, No. 5, pp. 711–719. © Pleiades Publishing, Ltd., 2020.
Russian Text © The Author(s), 2020, published in Zhurnal Neorganicheskoi Khimii, 2020, Vol. 65, No. 5, pp. 660–668.
THERMODYNAMICS
OF INORGANIC COMPOUNDS
Pb0.5 + xMgxZr2 – x(PO4)3 (x = 0, 0.5) Phosphates:
Structure and Thermodynamic Properties
P. A . M ayorov a, E. A. Asabinaa, V. I. Pet’kova,
A. V. Markina, *, N. N. Smirnovaa, and A. M. Kovalskyb
aNational Research Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, 603950 Russia
bNational Research University of Science and Technology “Moscow Institute of Steels and Alloys”, Moscow, 119049 Russia
*e-mail: markin@calorimetry-center.ru
Received October 22, 2019; revised November 6, 2019; accepted December 27, 2019
Abstract—Crystalline Pb0.5 + xMgxZr2 – x(PO4)3 (x = 0, 0.5) phosphates of NaZr2(PO4)3 (NZP) structural
type were synthesized. The heat capacity of Pb0.5Zr2(PO4)3 was measured by adiabatic vacuum and differen-
tial scanning calorimetry (DSC) within the temperature range 8–660 K. The studied phosphates were found
to experience a reversible phase transition in the region 256–426 K. According to the results of Rietveld struc-
tural study, this transition occurred due to an increase in disorder of lead cation positions in cavities of the
NZP structure. The measurements of PbMg0.5Zr1.5(PO4)3 heat capacity in the temperature range 195–660 K
showed that it experienced a similar phase transition at 255–315 K. Based on the measured experimental data,
0
the thermodynamic functions of Pb Zr (PO ) , such as
[H0(T) – H0(0)], S0(T), and [G0(T) – H0(0)]
Cp(T ),
2
4 3
were calculated for the temperature0r.5ange 0–660 K. The standard formation enthalpy of Pb0.5Zr2(PO4)3 was
determined at 298.15 K.
Keywords: phosphates, lead, NZP structural type, heat capacity, polymorphic transition, thermodynamic
functions
DOI: 10.1134/S0036023620050137
INTRODUCTION
the types of cationic positions in framework cavities
with the number of occupied positions in each type
(inside the columns of polyhedra (M1) abd between
them (М2)).
Phosphates with mixed {[L2(РO4)3]p–}3∞ octahedral–
tetrahedral frameworks (NaZr2(PO4)3 (NZP, NASI-
CON, kosnarite), Sc2(WO4)3 (SW), K2Mg2(SO4)3 (lang-
beinite), etc.) are intensively studied due to their
chemical, thermal and radiation stability, and control-
lable (including ultrasmall) thermal expansion [1–3].
Such compounds are of interest as matrix materials for
the immobilization of toxic elements, in particular,
heavy metals (Pb, Cd, Sr, etc.) from industrial wastes,
including radioactive ones [4, 5]. In the work [6], we
have studied the possibility of incorporating the ele-
ments in the oxidation state of +2 into the structure of
framework phosphates to demonstrate that the series
containing large М2+ cations (beginning from cad-
mium) crystallize in the NZP structural type. In the
present work, two lead-containing NZP phosphates
Pb0.5 + xMgxZr2 – x(PO4)3 with x = 0 and 0.5 were
selected as the subjects of study.
Depending on the specifics of cation distribution in
the cavities and the negligible displacements of frame-
work ions, it is possible to observe a slight distortion of
a cell and a change of symmetry. The highest possible
symmetry of an NZP structure can be observed for the
family parent NaZr2(PO4)3, which crystallizes in
space group
(Z = 6). However, the ordering of
R3c
alkali-earth metal cations M2+ in the М1 cavities of
some M0.5Zr2(PO4)3 compounds [7] leads to the split-
ting of each extraframework M1 position into two
positions and the alternation of populated and vacant
positions inside the columns. These phosphates are
characterized by space group
R3.
Based on the literature data, it is possible to state
The crystallochemical formula of NZP phosphates that the framework of the Pb0.5Zr2(PO4)3 structure
is (M1)0 → 1(M2)0 → 3{[L2(PO4)3]p–}3∞, where L is the (x = 0) is built of ZrO6 octahedra and PO4 tetrahedra,
and Pb2+ occupy half of the available M1 positions in
the cavities. The partial substitution of zirconium by
magnesium ions in the framework of the
octahedrally coordinated cationic positions of frame-
work-forming cations, and the framework itself is built
of LO6 and PO4 polyhedra, (M1)0 → 1 and (M2)0 → 3 are
711

712
MAYOROV et al.
PbMg0.5Zr1.5(PO4)3 structure (x = 0.5) leads to the veld method [8] using the RIETAN-97 software [9].
complete population of M1 cavities by Pb2+ ions.
The profiles of peaks were approximated by the mod-
ified pseudo-Voigt function (Mod-TCH pV). The
conditions used for the recording and processing of X-
ray diffraction patterns provided the detection of crys-
talline phases at their content in the sample of no less
than 1%
For the Pb0.5 + xMgxZr2 – x(PO4)3 phosphates, the
synthesis and phase formation conditions were stud-
ied, and the regions of solid solutions were revealed
[6]. The thermal expansion of PbMg0.5Zr1.5(PO4)3
within a temperature range of 293–1073 K character-
izes this compound as a low-expansion material resis-
tant to thermal shocks. However, to provide the com-
plete characterization of the thermal behavior of lead-
containing ceramics, it is necessary to have the infor-
mation about their fundamental properties, such as
the heat capacity, the thermodynamic functions, and
the thermodynamic characteristics of phase transi-
tions.
The chemical composition and homogeneity of the
synthesized Pb0.5 + xMgxZr2 – x(PO4)3 samples (x = 0,
0.5) were monitored on a JEOL JSM-7600F scanning
electron microscope (SEM) with a field-emission
electron gun (Schottky cathode). The microscope was
equipped with a microanalysis system, i.e., a Premium
OXFORD X-Max 80 energy dispersive spectrometer
with a semiconductor silicon drift detector. The preci-
sion of determining the elemental composition of
samples was 0.5–2.5 mol %.
In the present work, the heat capacity–tempera-
0
ture dependence
= f(T) and crystal structure of
Cp
Pb0.5 + xMgxZr2 – x(PO4)3 (x = 0, 0.5) NZP phosphates
were studied, and their thermodynamic functions
were determined in the region of 8–660 K.
The heat capacity of Pb0.5Zr2(PO4)3 in the tem-
perature range 8–300 K was measured on a BKT-3.07
fully automated adiabatic vacuum calorimeter (Ter-
mis) [10] as described in [11]. Liquid helium and
nitrogen were used as cooling agents. An ampoule
with the substance was filled with dry helium as a heat
transfer gas to a pressure of 40 kPa at room tempera-
ture. The calorimeter was tested for performance reli-
EXPERIMENTAL
Phosphates Pb0.5Zr2(PO4)3 (x
=
0) and
PbMg0.5Zr1.5(PO4)3 (x = 0.5) were synthesized by the
sol–gel method with subsequent heat treatment. The
used initial reagents of specialty grade were Pb(NO3)2,
MgO, ZrOCl2 ∙ 8H2O, and NH4H2PO4. Before syn-
thesis, magnesium oxide was dissolved in a calculated
amount of nitric acid and the other reagents were in
distilled water.
ability by measuring the 0 of a copper reference sam-
Cp
ple of specialty pure grade, synthetic corundum, and
benzoic acid of K-3 grade. The measurement instru-
ments and method provided the possibility to estimate
the heat capacities of compounds with an error of 2%
at temperatures below 15 K, 0.5% within a range of
15–40 K, and 0.2% in the region of 40–300 K and
determine the temperatures and enthalpies of phase
transitions with an accuracy of 0.01 K and 0.2%,
respectively. The calorimetric ampoule of the adia-
batic calorimeter was filled with 1.47266 g of the stud-
ied compound. The heat capacity of the measured
sample was 40–50% of the total heat capacity of the
calorimetric ampoule with the substance.
The phosphates were synthesized by pouring
together aqueous solutions of metal salts in stoichio-
metric amounts under continuous stirring at room
temperature with further addition of an ammonium
dihydrophosphate solution also in compliance with
the stoichiometry of these phosphates under stirring.
The reaction mixture were dried at 363 K, dispersed,
and subjected to heat treatment under free air access
conditions at 873 (100 h) and 1073 K (50 h). To iden-
tify the samples and determine the completeness of
The heat capacities of phosphates in the region of
interaction between the reagents and the absence of 195–660 K were measured on a Netzsch Gerätebau
foreign phases in the samples, the phosphates were DSC204F1 Phoenix differential scanning calorimeter.
studied by X-ray diffraction, electron microscopy, and The design of the DSC204F1 calorimeter and its
microprobe analysis.
operation procedure were described in [12, 13]. The
performance reliability tests of this calorimeter were
performed by standard calibration experiments mea-
suring the thermodynamic melting characteristics of
n-heptane, mercury, indium, tin, lead, bismuth, and
zinc. The measurement instruments and method pro-
vided the possibility to estimate the temperatures and
enthalpies of phase transitions with an error of 0.5 K
and 1%, respectively. The heat capacity of the stud-
ied phosphates was determined by the ratio method
[13]. Corundum was used as a standard reference. The
ampoule with the substance was heated at an average
0
The X-ray diffraction patterns of the
Pb0.5Zr2(PO4)3 and PbMg0.5Zr1.5(PO4)3 samples were
recorded on a Shimadzu XRD-6000 diffractometer
(CuKα radiation, λ = 1.54178 Å) within a range of
angles 2θ = 10°–80° (scan step, 0.02°) with exposure
for 6 s per point. The X-ray diffraction patterns of
Pb0.5Zr2(PO4)3 for structural studies were taken at 173
and 473 K, and the X-ray diffraction pattern of
PbMg0.5Zr1.5(PO4)3 for the calculation of crystallo-
graphic characteristics was recorded at 298.15 K. The
processing of X-ray diffraction patterns and the
refinement of structures were performed by the Riet-
rate of 5 K/min in an argon atmosphere. The
esti-
Cp
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Pb0.5 + xMgxZr2 – x(PO4)3 (x = 0, 0.5) PHOSPHATES: STRUCTURE
(b)
713
I, counts
(a)
1
×103
(а)
1
3
2
2
20
10
0
5
4
3
4
7
6
6
10 μm
7
10 μm
5
1, 2
3
4
Fig. 2. Electron microscopy results for (a) Pb Zr P O
2 3 12
and (b) PbMg Zr P O .
0.5
0.5 1.5
3 12
10
×103
20
20
30
40
50
60
composition, according to microprobe analysis data,
corresponds to the theoretical values within method
error (Tables 3 and 4).
(b)
Heat Capacity–Temperature Dependence
and Polymorphic Transitions
The heat capacity of Pb0.5Zr2(PO4)3 was studied by
adiabatic vacuum calorimetry (8–300 K, 110 experi-
10
mental values of 0) and differential scanning (195–
Cp
0
660 K) calorimetry (Fig. 3). Experimental values of
Cp
1, 2
3
4
were smoothed with the use of power and semi-loga-
0
rithmic polynomials such that the mean-square devi-
0
ation of experimental points from the smoothed
=
Cp
10
20
30
40
2θ, deg
50
60
f(T) curves was less than the error of heat capacity
measurements. For example, the heat capacity of
Pb0.5Zr2(PO4)3 can be described by the following
0
Fig. 1. Experimental (1), calculated (2), difference (3),
and stroke (4) X-ray diffraction patterns of phosphate
Pb Zr (PO ) at temperatures of (a) 173 and (b) 473 K.
equations ( J/(mol K)):
Cp
0.5
2
4 3
0
p
ln C = 5.518527 + 24.74717ln T /30
(
)
2
3
+ 92.14430 ln T /30 + 184.9730 ln T /30
(
(
)
(
(
)
]
[
[
]
[
mation error by the described method was less than
2%.
4
5
+ 196.7086 ln T /30 + 106.4614 ln T /30
)
]
[
)
]
6
+ 230.3970 ln T /30
[
(
)
]
RESULTS AND DISCUSSION
in the temperature range 8–20 K,
Characterization of the Samples
0
p
C = 26.42048 + 45.29058ln T /30
(
)
The synthesized Pb0.5 + xMgxZr2 – x(PO4)3 (x = 0,
0.5) samples represent colorless polycrystalline pow-
ders. According to powder X-ray diffraction data, the
synthesized phosphates are single phases, crystallize in
the NZP structural type, and are identical to the
phases described in [6]. Their X-ray diffraction pat-
2
3
+ 31.49051 ln T /30 – 4.823553 ln T /30
(
(
)
)
(
(
)
]
)
]
[
[
]
]
[
[
4
5
− 2.324907 ln T /30 + 1.513020 ln T /30
6
+ 3.097185 ln T /30
[
(
)
]
in the range 20–50 K,
terns were indexed within space group
(Fig. 1,
R3
0
p
C = 375.9106 – 3267.399ln T /30
(
)
Tables 1 and 2). The content of impurities in the stud-
ied samples does not exceed the detection limit under
the given conditions used for the recording and Riet-
veld refinement of X-ray diffraction patterns (1%).
The homogeneity of the Pb0.5 + xMgxZr2 – x(PO4)3
phosphate samples (x = 0, 0.5) was confirmed by the
electron microscopy data (Fig. 2). Their chemical
2
3
+ 12817.11 ln T /30 – 25702.30 ln T /30
(
(
)
)
(
(
)
)
[
[
]
]
[
[
]
4
5
]
+ 28313.25 ln T /30 –16203.82 ln T /30
6
+ 3773.261 ln T /30
[
(
)
]
at 50–80 K,
RUSSIAN JOURNAL OF INORGANIC CHEMISTRY Vol. 65 No. 5 2020

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MAYOROV et al.
Table 1. Recording conditions and refinement results for the crystal structure of Pb0.5Zr2(PO4)3
T, K
173
473
Space group
R
(No. 148)
6
3
Z
Range of 2θ angles, deg
Unit cell parameters
10−80
a, Å
c, Å
8.7009(4)
23.4822(11)
1539.57(13)
295
8.6996(16)
23.479(3)
1538.9(4)
301
V, Å3
Number of reflections
Number of refined parameters*
23 + 28
23 + 28
Reliability factors, %:
Rwp; Rp
S
3.47; 2.75
1.41
3.48; 2.75
1.39
* The first number is the background and profile parameters, scale factor, and unit cell parameters; and the second number is the posi-
tion and thermal parameters of atoms and their populations.
Table
2. Crystallographic
characteristics
of
0
p
C = –890.1582 + 1588.630 T /30
(
)
PbMg0.5Zr1.5(PO4)3 at 298.15 K
2
3
− 1102.190 T /30 + 409.6157 T /30
V, Å3
(
(
)
)
(
(
)
a, Å
c, Å
4
5
− 83.89098 T /30 + 8.984215 T /30
)
8.6997(4)
23.4591(8)
1537.63(15)
6
− 0.3939832 T /30
(
)
in the range 80–150 K, and
Table 3. Microprobe analysis results for Pb0.5Zr2P3O12
0
p
Point no.
O
P
Zr
Pb
C = –9971.374 + 9698.560 T /30
(
)
2
3
1
2
3
4
5
6
7
12
12
12
12
12
12
12
12
3.02
2.94
2.97
2.96
2.99
3.04
3.03
2.06
2.05
1.95
2.06
2.08
2.03
1.98
0.52
0.48
0.54
0.53
0.5
− 3877.457 T /30 + 825.8579 T /30
(
)
(
(
)
4
5
− 98.45857 T /30 + 6.225879 T /30
(
)
)
6
− 0.1631033 T /30
(
)
at 150–300 K.
0.54
0.52
All 0 experimental points and the smoothed curve
Cp
for Pb0.5Zr2(PO4)3 are shown in Fig. 3. According to
DSC data, the studied phosphate experiences a revers-
ible phase transition in the range 256–426 K. In com-
pliance with the McKallaf thermodynamic classifica-
tion of phase transitions, it may be classified with G-type
transitions. This transition does not appear up to 300 K
when the heat capacity of the same Pb0.5Zr2(PO4)3 sam-
ple is measured in the adiabatic calorimeter, probably,
due to its nonequilibrium character (it does not appear
under near-equilibrium recording conditions).
Average composition
3.03(4) 2.03(4) 0.52(2)
Table 4. Microprobe analysis results for PbMg0.5Zr1.5P3O12
Point no.
O
P
Mg
0.48
Zr
Pb
1
2
3
4
5
6
7
12 3.06
12 2.95
12 3.03
12 2.98
12 3.05
12 3.04
12 2.99
1.54
1.53
1.49
1.53
1.48
1.54
1.51
0.97
1.03
1.03
0.97
1.01
0.97
0.98
0.49
0.47
0.53
0.5
To clarify the nature of the considered phase tran-
sition, the structural study of Pb0.5Zr2(PO4)3 was per-
formed at 173 and 473 K by the Rietveld method using
the powder X-ray diffraction data (Fig. 1, Table 2).
The initial data used for structural refinement at the
mentioned temperatures were the coordinates of atoms
in the structure of phosphate Cu0.5Mn0.25Zr2(PO4)3
0.47
0.51
(NZP, space group R [14]). The coordinates of
3
Average composition 12 3.02(4) 0.49(2) 1.52(2) 0.99(3)
atoms in the refined models are given in Table 5.
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Pb0.5 + xMgxZr2 – x(PO4)3 (x = 0, 0.5) PHOSPHATES: STRUCTURE
Table 5. Coordinates of atoms in the structure of Pb0.5Zr2(PO4)3
715
x
y
z
Atom/position
173 K
473 K
173 K
473 K
173 K
473 K
Pb*/18f
0.02(6)
0
0
0.2939(13)
0.1701(21)
0.073(3)
0.193(3)
0.8293(28)
–0.033 (26)
0
0
0.2996(13)
0.1655(23)
0.1513(25)
0.2022(25)
0.7999(26)
0.00(12)
0
0
0.0201(16)
0.9654(22)
0.859(3)
0.178(2)
0.7724(23)
–0.034 (22)
0
0
1.0237(15)
0.9585(26)
0.9306(23)
0.1516(22)
0.7733(27)
0.0044(11)
0.15061(18)
0.64726(17)
0.2524(6)
0.1901(11)
0.7036(7)
0.0840(9)
0.6114(8)
0.0035(13)
0.15073(20)
0.64701(19)
0.2455(6)
0.1927(10)
0.7024(8)
0.0980(8)
0.5936(8)
Zr(1)/6с
Zr(2)/6с
P/18e
O(1)/18f
O(2)/18f
O(3)/18f
O(4)/18f
*The occupancy of this position is 0.167.
Table 6. Selected interatomic distances in the polyhedra
forming the structure of Pb0.5Zr2(PO4)3
Both Pb0.5Zr2(PO4)3 polymorphs belong to the
NZP structural type, and structural refinement was
Т = 173 K
bond
Pb–O(3)
Т = 473 K
bond
Pb–O(3)
2.43 (6) Pb–O(3')
2.53(4) Pb–O(3'')
3
performed in both cases within space group R char-
acterized by the splitting of M1 crystallographic posi-
tions (6a) in the cavities inside the polyhedra columns
into two types of non-equivalent positions (3a and 3b).
d, Å
2.41(3)
d, Å
2.62(2)
2.70(5)
2.72(8)
2.90(7)
2.92(5)
2.99(6)
Pb–O(3')
Pb–O(3'')
Pb–O(3''')
Pb–O(3'''')
Pb–O(3''''')
In the structures of most NZP compounds
M0.5Zr2(PO4)3 [7], the positions of one type (3a or 3b)
2.59 (5) Pb–O(3''')
are occupied by М2+ cations, and the other type
remains vacant. In this case, М2+ cations are in an
2.68(6)
Pb–O(3'''')
Pb–O(3''''')
2.70(3)
octahedral environment with six metal–oxygen bonds Zr(1)–O(1) (×3) 1.894(21) Zr(1)–O(1) (×3) 1.922(23)
of identical length. However, in the case of Zr(1)–O(3) (×3) 2.250 (19) Zr(1)–O(3) (×3) 2.012(17)
Pb0.5Zr2(PO4)3, the Pb2+ cations occupying the struc-
Zr(1)–O(4) (×3) 1.974 (23) Zr(1)–O(4) (×3) 2.141(19)
ture cavities have a stereoactive 6s2 electron pair,
Zr(1)–O(2) (×3) 2.107 (27) Zr(1)–O(2) (×3) 2.252(19)
which usually results in the distortion of lead–oxygen
polyhedra [15, 16]. The degree of their distortion in
the known lead-containing compounds depends on
the temperature and their structural features. As a
consequence, the Pb2+ ions are shifted from the cen-
ters of octahedrally coordination cavities (disordered),
and the symmetry of their positions decreases to 18f.
P–O(3)
P–O(2)
P–O(4)
P–O(1)
1.413(21) P–O(3)
1.443(29) P–O(2)
1.535 (26) P–O(4)
1.636(27) P–O(1)
1.426(17)
1.436(12)
1.599(25)
1.699(27)
A
fragment of the crystal structure of
Сp0, J/(mol K)
Pb0.5Zr2(PO4)3 at 173 K is shown in Fig. 4a. The NZP
framework of this compound is built of ZrO6 octahe-
dra and РО4 tetrahedra. Fragments of two octahedra
and three tetrahedra form the columns along crystal-
lographic axis с. A half of the cavities inside these col-
umns are occupied by Pb2+ ions shifted from their cen-
ter due to slight disordering. The interatomic distances
(Table 6) in the coordination polyhedra composing
the crystal structure of the studied compound at both
temperatures agree with the literature data for the
other phosphates with an NZP structure [6, 14].
400
200
The results of structure refinement for
Pb0.5Zr2(PO4)3 at 173 and 473 K showed that, as the
temperature grows, the degree of ordering for the lead
cations sustains an increase accompanied by stronger
distortion in the PbO6 octahedra (Fig. 4b, Table 6).
Hence, the polymorphic transitions on the heat
capacity curves of the studied phosphates occur due to
Tt0rs = 299 K
0
200
400
600
T, K
Fig. 3. Heat capacity of Pb Zr (PO ) versus temperature.
0.5
2
4 3
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MAYOROV et al.
(a)
(b)
Pb
ZrO6
PO4
O
O
O
O
O
O
Pb
Pb
O
O
O
O
O
O
c
173 К
473 К
a
b
Fig. 4. Structure of Pb Zr (PO ) : (a) crystal structure fragment at 173 K and (b) distortion of PbO octahedra at 173 and 473 K.
0.5
2
4 3
6
an increase in the disordering of lead cation positions the M1 cavities inside the columns of framework-
in the NZP structure cavities.
forming polyhedra and, therefore, the possibilities of
structural distortions due to the displacement of these
ions are more limited.
The heat capacity–temperature dependence for
PbMg0.5Zr1.5(PO4)3 in the range 195–660 K (Fig. 5) is
similar to the above-considered dependence for
Pb0.5Zr2(PO4)3. This compound also experiences a
polymorphic phase transition, but at lower tempera-
tures (255–315 K), and a heat capacity jump is less
pronounced in this region. This thermal behavior of
PbMg0.5Zr1.5(PO4)3 can be explained by the fact that
Heat Capacity and Thermodynamic Functions
Except the regions of transitions, the heat capacity
of phosphates has no singularities and smoothly rises
throughout the range of temperatures studied. To calcu-
late the thermodynamic functions of Pb0.5Zr2(PO4)3, the
temperature dependence of its heat capacity was
the Pb2+ cations in its structure completely populate
Cp0
extrapolated to 0 K by the Debay function
=
Сp0, J/(mol К)
nD(θD/T), where D is the Debay function symbol and
n = 3, θD = 88.4 K are fitted parameters. Together with
the mentioned parameters, this equation describes the
experimental values of 0 in the range 8–12 K with an
Cp
400
error of 2.4%. It was assumed that Eq. (2) reproduced
0
the values of
at Т < 8 K with the same error.
Cp
The low-temperature heat capacity data (30–50 K)
were processed with consideration for the multifrac-
tality of the vibrational states of atoms [17, 18]. The
fractal dimension D [19, 20] in the multifractal model
serves as an indirect characteristic of the structural
topology of solids and provides the possibility to judge
the geometric character of their structure: D = 1 for
solids with a chain structure, D = 2 for a layered struc-
ture, and D = 3 for a spatial structure. Assuming with-
0
200
Tt0rs = 286 K
0
200
400
600
T, K
out appreciable accuracy loss that
=
0 at T < 50 K
C
Cp
and using the experimental heat capacity vdata, we find
with an error of 1.0% that the studied phosphate
Fig. 5. Heat capacity of PbMg Zr (PO ) versus tem-
perature.
0.5 1.5
4 3
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Pb0.5 + xMgxZr2 – x(PO4)3 (x = 0, 0.5) PHOSPHATES: STRUCTURE
717
Table 7. Standard thermodynamic functions of crystalline Pb0.5Zr2(PO4)3 (FW = 570.957 g/mol, p0= 0.1 MPa)
0
0
0
0
0, J/(mol K)
Cp
S0(T), J/(mol K)
− G T − H 0
( ) ( )







Т, K
, kJ/mol
, kJ/mol

H T − H 0
( ) ( )
Crystal I
0
5
0
0
0
0
0.352
2.60
0.000400
0.00680
0.08060
0.2743
0.6154
1.747
0.117
0.910
5.611
13.28
22.99
45.53
70.41
96.29
122.2
147.7
172.8
197.3
221.1
244.2
266.7
288.4
309.5
328.2
329.1
0.000146
0.00232
0.03165
0.1239
0.3041
0.9847
2.140
10
20
30
40
12.87
26.42
41.83
71.15
103.5
129.5
154.7
177.4
198.2
217.2
234.7
250.8
265.9
278.3
291.4
303.4
303.9
60
80
3.492
5.822
100
120
140
160
180
200
220
240
260
280
298.15
299
3.807
8.668
5.992
11.99
15.75
8.692
11.90
19.91
15.60
24.43
29.29
19.78
24.44
29.55
34.46
39.89
45.59
35.10
41.08
50.99
46.87
51.24
47.15
Crystal II
299
300
320
340
360
380
400
420
440
460
480
500
520
540
560
580
600
620
640
660
367.7
368
368
369
369
370
370
371
372
372
374
375
377
379
380
382
383
385
386
388
51.24
51.6
59.0
66.3
73.7
81.1
329.1
330
354
376
397
417
436
455
472
488
504
520
534
549
562
576
589
601
613
47.15
47.5
54.3
61.6
69.4
77.5
86.1
95.0
88.5
95.9
103
104
111
118
126
133
141
148
156
114
124
134
145
155
166
178
190
201
214
226
164
171
179
187
625
RUSSIAN JOURNAL OF INORGANIC CHEMISTRY Vol. 65 No. 5 2020

718
MAYOROV et al.
Table 8. Heat capacities smoothed with polynomials for
crystalline PbMg0.5Zr1.5(PO4)3 (FW = 641.098 g/mol, p0 =
0.1 MPa)
cr, 298.15 K) = –1136.4 1.3 J/(mol K). The found
value corresponds to the process
0.5Pb(cr) + 2Zr(cr) + 3P(cr, white) + 6О2(gas)
0, J/(mol K)
Cp
0, J/(mol K)
Cp
= Pb0.5Zr2(PO4)3 (cr).
Т, K
Т, K
Crystal I
214
380
400
420
440
460
480
500
520
540
560
580
600
620
640
660
347
353
357
361
365
369
371
374
377
379
381
383
384
385
386
CONCLUSIONS
195
200
220
240
260
280
286
Crystalline phosphates Pb0.5Zr2(PO4)3 and
PbMg0.5Zr1.5(PO4)3 of NZP structural type were syn-
thesized by the sol–gel method. The isobaric heat
capacities of Pb0.5Zr2(PO4)3 in the temperature region
of 8–660 K and PbMg0.5Zr1.5(PO4)3 in the range 195–
660 K were studied for the first time. Phase transitions
of G type were observed in the heat capacity–tempera-
ture dependences of both phosphates. Based on the
results of the structural study of Pb0.5Zr2(PO4)3 at 173
and 473 K, these polymorphic transitions were eluci-
dated to occur due to an increase in the position disor-
dering of lead cations in NZP structure cavities. The
thermodynamic functions of Pb0.5Zr2(PO4)3 were cal-
culated in the region from Т → 0 to 660 K, and the
standard enthalpy of its formation from simple com-
pounds at 298.15 K was determined. Based on the low-
temperature heat capacity, the fractal dimension of
Pb0.5Zr2(PO4)3 was calculated, and the conclusion
about the applicability of this model to framework
NZP compounds was made.
222
245
261
272
280
282
Crystal II
286
298.15
300
320
340
307
312
312
321
330
340
360
Pb0.5Zr2(PO4)3 has D = 3, which corresponds to a spa-
tial (framework) structure of this compound.
The enthalpies [H0(T) – H0(0)] and entropies
S0(T) of Pb0.5Zr2(PO4)3 were calculated via the
FUNDING
0
This work was supported by the Russian Foundation for
Basic Research (project nos. 19-33-90075 and 18-29-
12063) and by the Ministry of Science and Higher Educa-
tion of the Russian Federation (state assignment
no. 4.8337.2017/BCh).
numerical integration of the corresponding
= f(T)
Cp
0
and
= f(lnT) dependences, and the Gibbs func-
Cp
tions [G0(T) – H0(0)] were estimated from the enthal-
pies and entropies of compounds at the corresponding
temperatures. The smoothed heat capacities and ther-
modynamic functions of Pb0.5Zr2(PO4)3 are given in
Table 7. The PbMg0.5Zr1.5(PO4)3 heat capacities
smoothed with polynomials are listed in Table 8.
CONFLICT OF INTERESTS
The authors declare that they have no conflict of inter-
ests.
The heat capacity of the studied phosphates at high
temperatures approaches the values estimated by the
0
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RUSSIAN JOURNAL OF INORGANIC CHEMISTRY Vol. 65 No. 5 2020