Open Science Journal of Modern Physics
2015; 2(4): 50-54
Published online July 30, 2015 (http://www.openscienceonline.com/journal/osjmp)
Specific Heat and Thermal Expansion of Refractory
Nonmetal: CaO
Vladimir Yu. Bodryakov*
Institute of Mathematics, Informatics and Information Technologies, Ural State Pedagogical University, Yekaterinburg, Russia
Email address
[email protected]
To cite this article
Vladimir Yu. Bodryakov. Specific Heat and Thermal Expansion of Refractory Nonmetal: CaO. Open Science Journal of Modern Physics.
Vol. 2, No. 4, 2015, pp. 50-54.
Abstract
In this paper in the development and dissemination ideas of previously published works on a new class of objects (refractory
nonmetals), a correlation study of molar specific heat C(T), volumetric coefficient of thermal expansion ο(T) and molar volume
V(T) of solid calcium oxide has been made. As for earlier investigated solids, for CaO clear correlation ο(C) takes place not only
at low temperatures, but also to a much wider temperature range. A significant deviation from the linear low-temperature
behaviorο(C) occurs on reaching heat capacity the classical limit 6R by Dulong and Petit law. Temperature dependence is
estimated for calcium oxide of the differential Grüneisen parameter.
Keywords
Calcium Oxide (CaO), Coefficient of Thermal Expansion, Correlation, Differential Grüneisen Parameter, Heat Capacity,
Molar Volume
1. Introduction
Alkaline earth metal oxides (AEMO) have numerous
technological applications in view of their peculiar physical
and chemical properties and high melting points; their
behavior at high temperatures and pressures is important from
the Geophysics point of view of [1], [2]. Alkaline earth metal
oxides, as well as alkali halides, are also considered as a
convenient model objects, because of their simplest, as is
believed, ionic type of chemical bonding, permitting the
creating and testing a variety of relatively uncomplicated
models of their properties. However, a number of important
physical properties of AEMO insufficiently studied yet; data
of different authors do not agree well with each other. In
particular, as further seen for CaO, it relates to basic
thermodynamic properties, such as specific heat C(T),
volumetric coefficient of thermal expansion (VCTE) ο(T),
molar volume V(T), and others.
Previously published author’s papers [3]–[10] actually
establish a new standard for statistical processing and mutual
(correlation) analysis of the fundamental thermodynamic
quantities of solids, so here there is no need for a detailed
description of the calculations. The aim of this research is to
spread the ideas [3]–[10] for a new class of objects (refractory
nonmetals). Namely, the results are presented for solid
calcium oxide (also called lime, quicklime, calcia) of the
correlation study of heat capacity and thermal expansion in the
whole range of solid state, far beyond the known limits of
applicability of low-temperature Grüneisen law. Note an
essential uncertainty in the melting point Tm of the oxide (K):
Tm = 3108 [1]; 3200 [2]; 3200 [11]; 2900–50+300 [12]; > 3086
[13]; 2903 [14]; 3200 [15]. In this paper accepted value is Tm =
3100 K (±100 K). As in [3]–[10], the correlation dependence
ο(C) is called Grüneisen plot (GP), and according correlation
analysis – GP-analysis. Cited original sources, without
claiming to be exhaustive list, give quite a clear idea of the
temperature behavior of the properties in question.
2. Results and Discussion
Temperature dependence of the molar heat capacity C(T) of
solid calcium oxide is shown in Fig. 1. The data [1], [11]–[22]
on C(T) (total 13 sets of data, more than 290 points), in general,
reasonably consistent with each other, except for clear
overestimated data [16]. Data [16] were excluded from the
calculation and are just for comparison. It should be
emphasized the urgency for additional modern calorimetric
investigations in the entire temperature range of solid state CaO,
especially at low and high temperatures. These measurements
51
Vladimir Yu. Bodryakov:
Specific Heat and Thermal Expansion of Refractory Nonmetal: CaO
should be made on fused and well certified the material. Solid
(trend) line in Fig. 1 is the result of statistical averaging and
smoothing the data C(T) by different authors; the trend
reasonably presents the empirical data on the heat capacity of
calcium oxide. For convenience, the smoothed data C(T), along
with VCTE ο(T) for CaO are given in Table 1.
Table 1. Recommended heat capacity C (JK–1mol–1) and VCTEο (10–6K–1) of solid CaO.
T, K
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
110
120
130
140
C
0.0025
0.020
0.068
0.162
0.316
0.547
0.868
1.29
1.84
2.51
3.30
4.22
5.26
6.39
7.61
8.89
10.21
11.58
12.94
14.30
16.99
19.56
21.98
24.21
ο
0.002
0.015
0.051
0.125
0.243
0.42
0.67
0.99
1.42
1.92
2.53
3.24
4.03
4.91
5.84
6.83
7.84
8.89
9.94
10.99
13.05
15.02
16.88
18.59
T, K
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
350
400
450
500
550
600
650
700
C
26.26
28.13
29.83
31.37
32.76
34.03
35.17
36.20
37.14
38.03
38.84
39.58
40.27
40.92
41.52
42.15
44.87
46.72
47.97
48.94
49.63
50.22
50.71
51.12
ο
20.17
21.60
22.91
24.09
25.16
26.13
27.01
27.81
28.53
29.21
29.83
30.39
30.92
31.42
31.89
32.37
34.48
35.93
36.98
37.84
38.49
39.10
39.74
40.29
T, K
750
800
850
900
950
1000
1050
1100
1150
1200
1250
1300
1350
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
C
51.47
51.81
52.10
52.37
52.63
52.89
53.12
53.33
53.53
53.73
53.92
54.11
54.30
54.49
54.68
54.87
55.06
55.25
55.44
55.63
55.82
56.01
56.20
56.39
ο
40.79
41.24
41.68
42.06
42.49
42.87
43.22
43.54
43.86
44.15
44.44
44.73
45.01
45.30
45.59
45.88
46.16
46.45
46.74
47.03
47.32
47.61
47.89
48.19
T, K
1950
2000
2050
2100
2150
2200
2250
2300
2350
2400
2450
2500
2550
2600
2650
2700
2750
2800
2850
2900
2950
3000
3050
Tm
C
56.58
56.77
56.96
57.15
57.34
57.53
57.73
57.94
58.16
58.39
58.63
58.89
59.17
59.47
59.79
60.13
60.49
60.87
61.28
61.73
62.23
62.79
63.42
64.15
ο
48.47
48.76
49.05
49.34
49.62
49.91
50.21
50.53
50.87
51.22
51.59
51.98
52.41
52.86
53.35
53.86
54.41
54.98
55.61
56.29
57.05
57.90
58.86
59.97
handbook [24]). There aren’t clear reasons for preferring one set
of data to another, so all of them, including theoretical
estimations, were involved in the computation. Much greater
inconsistency in VCTE compared with heat capacity is
probably due to methodological difficulties in carrying
high-temperature measurements of thermal expansion, and
essentially higher quality requirements to the dilatometric, than
calorimetric, specimens. It is necessary to carry out additional
modern dilatometric studies of well certified fused solid
calcium oxide over the entire temperature range.
Fig. 1. Temperature dependence of molar heat capacity C(T) for solid CaO.
Symbols – tabular data: 1 – [16], 2 – [17], 3 – [18], 4 – [19], 5 – [11], 6 – [12],
7 – [20], 8 – [13], 9 – [14], 10 – [21], 11 – [1], 12 – [15], 13 – [22], 14 (solid
line) – trend.
Temperature dependence of the volumetric coefficient of
thermal expansion of calcium oxide ο(T) is shown in Fig. 2.
Data from [1], [14], [21]–[30] for thermal expansion (a total of
12 sets of data, more than 230 points) are only in
semi-quantitative agreement; dispersion increases coming to
the melting temperature of CaO. Original data of direct
measurements of thermal expansivity of the oxide are very
scarce and contradictory (see e.g. survey in well-known
Fig. 2. Temperature dependence of volumetric coefficient of thermal expansion
ο(T) for solid CaO. Symbols – tabular data: 1 – [23], 2 – [24], 3 – [25], 4 – [14],
5 – [21], 6 – [26], 7 – [1], 8 – [27], 9 – [28], 10 – [22], 11 – [29], 12 – [30], 13
(solid line) – trend.
Open Science Journal of Modern Physics 2015; 2(4): 50-54
52
change in the character of ο(C) dependence. Above 600 K,
where C> 6R, the slope ο(C) increases and remains
approximately constant up to the melting point Tm= 3100 K.
The temperature of the kink in the dependence ο(C)
approximately accords to Debye temperature of calcium oxide
at this condition: θ(600 K) ∼ 650 K [1], [21]. Estimates [1],
[21] of Debye temperature θ(T) from the elastic constants
show a monotonic decrease θ(T) from ∼ 670 K at T = 300 K
down to ∼ 620 K at T = 1200 K. Calorimetric estimates [18]
indicate more complex temperature behavior and somewhat
smaller values θ(T):θ(300 K) ∼ 570 K.
Fig. 3. GP-diagram – correlation dependence ο(C) for CaO. Symbols –
averaged and smoothed data of VCTE and molar heat capacity; straight line
– linear regression οlin(C) for low-temperature values ο(C). Arrow indicates
classical limit 6R by Dulong and Petit for heat capacity.
Fig. 5. Temperature dependence of molar volume V(T) for solid CaO: 1 –
[31], 2 – [23], 3 – [32], 4 – [33], 5 – [34], 6 – [35], 7 – [36], 8 – [24], 9 – [21],
10 – [1], 11 – [29], 12 – [37], 13 – [30], 14 (solid line) – trend.
Fig. 4. The same as Fig. 3 for difference ∆ο(C)for CaO.
Accuracy of smoothed values C(T) and ο(T) in Fig. 1, 2 can
be estimated visually – by an extent to which the trend lines
accord to empirical data [1, 2, 11-30], and quantitatively – by
standard deviation (SD) of empirical points from the trends
C(T), ο(T) at each temperature point.
GP – correlation dependence ο(C), where averaged and
smoothed values of molar heat capacity and volumetric
thermal expansivity of CaO are taken in the according
temperature points, – is shown in Fig. 3. In the temperature
range 0 <T≤ 500 K, for which the specific heat 0 <C≤ 48.94 J
K-1 mol-1, and VCTE 0 <ο≤ 37.84×10-6 K-1, the dependence
ο(C) is linear with the high level of correlation (R2 = 0.999997,
n = 47 points):
οlin(C)=(0.7684±0.0002)×С,
(1)
where in VCTE is in units 10-6 K-1, and the heat capacity is in
the J K-1 mol-1. SD of ο(C) points out of the regression straight
line (1) is of small value σ≈0.040×10-6 K-1. In achieving heat
capacity the classical limit by Dulong and Petit CDP = 6R≈
49.88 J K-1 mol-1 for CaO, there is a drastic, close to the kink,
Fig. 6. Temperature dependence of differential Grüneisen parameter γ′(T) for
solid CaO. Dashed line – a piecewise linear approximation.
As clearly seen from the difference GP-diagram (Fig. 4)
∆ο(C) = ο(C) – οlin(C),
(2)
points ο(C) well above CDP fit well with smooth growing
dependence close to the straight line. Shown in Fig. 3, 4, the
results can be interpreted as almost strict constancy below ∼
600 K of differential Grüneisen parameter:
γ′ = V0K0(∂ο/∂C),
(3)
where’s accepted: V0 = 16.71 ± 0.02 cm3 mol-1 and K0 = 112 ±
2 GPa – low-temperature limits of the molar volume and bulk
53
Vladimir Yu. Bodryakov:
Specific Heat and Thermal Expansion of Refractory Nonmetal: CaO
modulus for CaO. The value K0 is estimated based on the
values of the elastic constants [21]. The value V0 is obtained
from dilatometric data [1], [21], [23], [24], [29]–[37] (13 sets
of data, about 230 points). Additional checking molar volume
of CaO was necessary because of the valuable dispersion of
thermal expansion data by various authors. For this, the trend
dependence ο(T) was numerically integrated and temperature
dependence of molar volume was get:
V(T) = V0exp
ο
.
(4)
The comparison of molar volume V(T) computed using Eq.
(4) with the dilatometric data [1], [21], [23], [24], [29]–[37]
(Fig. 5) confirms its quantitative adequacy and therefore
established trend ο(T). For convenience, the smoothed
temperature dependences of molar volume V(T) and density ρ
= µ/V, where µis the molar mass of CaO, are given in Table 2.
Table 2. Recommended molar volume V (cm3mol–1) and densityρ (kgm–3) of solid CaO.
T, K
5
50
100
150
200
250
300
350
400
V
16.71 ± 0.02
16.71
16.72
16.73
16.75
16.77
16.80
16.83
16.86
ρ
3356 ± 4
3356
3355
3352
3348
3344
3338
3333
3327
T, K
500
600
700
800
900
1000
1100
1200
1300
V
16.92
16.98
17.05
17.12
17.19
17.27
17.34
17.42
17.49
ρ
3315
3302
3289
3275
3262
3248
3234
3220
3206
Above ∼ 800 K (beyond the transition area) differential
Grüneisen parameter γ′ is close to a greater also constant value
(Fig. 6). To find the parameter γ′ with use of Eq. (3) digital
differentiation of correlation GP-dependence ο(C) was
applied (without further smoothing). Piecewise-linear
approximation is shown in Fig. 6 by dashed line. Increased
dispersion of the points γ′(T) is due to a procedure of
numerical differentiation.
Unusual behavior of differential Grüneisen parameter γ′(T)
of calcium oxide is contrary to traditional views about the
behavior of Grüneisen parameter γ(T) in solids, including
AEMO (see, e.g., [1, 21, 25]). Clearly observed kink in the
dependence γ′(T) at ∼ 600 K for CaO, as well as for other
studied solids [3]–[10], cannot be explained by the
contribution of thermal vacancies; the latter could have a
significant effect higher than ∼ 0,7Tm∼ 2100 K. Therefore, the
observed above 600 K feature in the behavior of correlation
dependence ο(C) allows to assume “turning on” new, needed
additional study, mechanisms of formation of thermodynamic
properties of solid calcium oxide.
T, K
1400
1500
1600
1700
1800
1900
2000
2100
2200
ρ
3191
3177
3162
3147
3132
3117
3102
3087
3072
T, K
2300
2400
2500
2600
2700
2800
2900
3000
Tm
V
18.35
18.44
18.54
18.63
18.73
18.84
18.94
19.05
19.16
ρ
3057
3041
3025
3010
2994
2977
2961
2944
2927
Acknowledgements
The author is grateful to Mrs. Irina V. Ivanova (USPU,
Yekaterinburg) for assistance in providing sources of original
thermodynamic data.
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The GP-analysis formed in the example of solid calcium
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