Enthalpy and Heat-Capacity Standard Reference Material: Synthetic Sapphire (α-Al₂O₃) from 10 to 2250 K

Abstract

Heretofore unpublished enthalpy data which were used in the derivation of smooth enthalpy and heat-capacity data for NBS SRM 720 (α-Al₂O₃, heal eapacily and enthalpy standard) are presented along with some details of the high-temperature experiments. Recent NBS low-temperature measurements on SRM 720 are smoothed by a least-squares spline technique and a revised table of certified values for enthalpy and heal capacity of α-Al₂O₃ from 10 K to near the melting point (2250 K) is presented.

1. Introduction

Standard Reference Material 720 (α-Al₂O₃) has been offered by the NBS Office of Standard Reference Materials since 1970 [1]¹ as a heat-capacity and enthalpy standard certified in the temperature range 273.15 K to 2250 K. The relative enthalpy data, whose smoothed representation appears in [1], were obtained in two different types of drop calorimeter: a Bunsen ice calorimeter was used from 273.15 K to 1173.15 K. [2] and an adiabatic receiving calorimeter from 1173.15 K to 2250 K. The smoothed relative enthalpy values of [1] rely as well for the absolute ice-point enthalpy (H_273.15 K — H_{O K}) upon a much earlier low-temperature heat-capacity study [3] on the “Calorimetry Conference Sample” of pure α-Al₂O₃.

A detailed presentation of the original ice-calorimeter enthalpy data along with a description of the smoothing procedure used to obtain the enthalpy and heat-capacity values appearing in [1] has been given in [2]. Unfortunately, the original receiving-calorimeter enthalpy data were never published due to the death of one of the principal experimenters and subsequent personnel changes.

The present work presents this enthalpy data and describes certain aspects of the experiments in the receiving calorimeter. In addition, it documents a re-smoothing of the NBS low-temperature heat-capacity data [4] on SRM 720 obtained since 1970. New smooth heat-capacity data for SRM 720 in the temperature interval 10 K to 2250 K are presented. The present smooth enthalpy data above 273.15 K differ by less than 0.01% from the corresponding valnes given in [1].

2. High-Temperature Enthalpy Data, 1173 K to 2250 K

Enthalpy data above 1173.15 K were obtained by S. Ishihara in a high-temperature adiabatic receiving calorimeter. Some physical and operational details of this apparatus have been given in [5] and [6]. These experiments were carried out with the calorimeter and furnace containing purified argon gas at about 0.1 atm. pressure. The single-crystal segments of α-Al₂O₃ were contained in a molybdenum capsule (mass: 9.26727 g) with a close-fitting, though not hermetically-sealed, lid. The capsule was suspended from a doubled and twisted loop of 8-mil tungsten wire by a small tantalum hook (hook mass: 0.56097 g). Thirteen evenly-spaced temperatures from 1173 K to 2250 K were chosen, and one day’s experiments consisted of four enthalpy measurements at a single one of these temperatures. The temperature for any particular day was chosen randomly with the aid of a table of random numbers. The first and last enthalpy measurements of a day were made on the same system (either the empty capsule or the same capsule filled with α-Al₂O₃). Experience has shown that this method of scheduling one day’s experiments makes it possible to take into account small changes in the pyrometer characteristics or sample capsule mass changes due either to interaction with the sample or with the carbon atmosphere created by the induction furnace susceptor.

Results of the high-temperature measurements are given in table 1. The initial sample capsule temperatures are given in column 1. These were measured with an automatic optical pyrometer which was focused on the bottom of the sample capsule through a small aperture in the furnace susceptor. A separate output signal from the pyrometer was used to control the furnace temperature. Column 3 gives the measured heat to the calorimeter at 298.15 K. The aetual terminal temperature of the ealorimeter and capsule in the equilibrating period after an experiment was usually less than 320 K. In order to reference all heat data to 298.15 K, it was necessary to add to each measured heat a correction equal to the enthalpy of the capsule (plus sample and carbon contamination, if present) at the terminal temperature of the calorimeter relative to 298. ? 5 K. These corrections ranged from one to two percent of the measured heat. The enthalpy data necessary to calculate the corrections for carbon, tantalum, aluminum oxide, and molybdenum, were taken from references [7], [8], [9] and [2], respectively. The heat content of the aluminum oxide constituted about 55 percent of the total measured heat at all temperatures. The differing values for specimen mass (column 4) indicate that in some experiments different amounts of α-Al₂O₃ were used, though the difference can correspond to at most one or two small pieces of specimen. In the correction of specimen mass data for atmospheric buoyancy, a density of 3.97 g cm⁻³ for α-Al₂O₃ was used. The molar enthalpy values in column 5 were obtained from net heat values (F-C differences from column 3) by division by the applieable specimen mass (column 4).

Table 1.

Heat-Content Data for α-Al₂O₃(s), SRM 720.

Furnace Temperature T68	Date (1969)	Heat to Calorimeter at 298.15 K	Speeimen Mass	HT – H298.15	Deviation from eq. (3) of [2]
K		J	molb	J mol−1	J mol−1	%
		F 5386.39a	0.0295095	99871.6	+ 24.06	+0.024
1173.18	24 Mar.	C 2439.23
		C 2439.35
		F 5384.10	0.0295094	99794.3	− 53.24	−0.053
		F 5681.19	0.0303910	103184.8	− 0.57	−0.001
1199.25	7 Apr.	C 2545.30
		C 2545.25
		F 5680.87	0.0303914	103174.6	− 10.77	−0.010
		C 2855.86
1299.16	4 Apr.	F 6474.38	0.0311633	116114.8	+ 41.71	+0.036
		F 6474.87	0.0311633	116156.8	+ 83.71	+0.072
		C 2855.04
		C 3060.22
1401.65	10 Mar.	F 6474.38	0.0317601	129563.7	+123.18	+0.095
		F 7175.51	0.0317601	129553.7	+113.38	+0.088
		C 3060.87
		C 3467.73
1501.15	19 Mar.	F 7914.37	0.0311937	142549.3	+ 3.60	+0.003
		F 7913.75	0.0311937	142612.1	+ 66.40	+0.047
		C 3565.15
		F 8441.51	0.0303913	156264.1	− 66.65	−0.043
1604.90	6 Mar.	C 3692.44
		C 3691.51
		F 8353.12	0.0298344	156254.7	− 76.05	−0.049
		C 4002.05
1702.22	4 Mar.	F 9003.15	0.0295487	169249.4	−110.34	−0.065
		F 9003.65	0.0295487	169251.4	−108.34	−0.064
		C 4002.49
		C 4461.57
1799.86	27 Mar.	F10147.78	0.0311923	182307.8	−210.67	−0.115
		F10147.78	0.0311923	182307.8	−207.47	−0.114
		F 4461.18
		F10946.89	0.0311712	196456.3	+ 12.42	+0.006
1902.65	17 Mar.	C 4823.11
		C 4823.88
		F10947.56	0.0311671	196479.3	+ 35.42	+0.018
		F11264.44	0.0295326	210507.7	+242.33	+0.115
2004.13	12 Mar.	C 5047.60
		C 5047.53
		F11264.44	0.0295218	210509.9	+244.53	+0.116
		C 5125.11
2101.61	21 Apr.	Fl 1697.26	0.0293640	223816.6	+214.15	+0.096
		F11692.47	0.0293223	223971.5	+369.05	+0.165
		C 5125.11
		C
2203.28	16 Apr.	Fc		237748.5	+173.77	+0.073
		F		237625.0	+ 50.27	+0.021
		C
		F13886.37	0.0315029	244676.5	−322.90	−0.132
2257.11	14 Mar.	C 6178.35
		C 6177.04
		F13855.68	0.0313787	244708.7	−290.70	−0.119

^a“C” prefixes data for empty capsule.

“F” prefixes data for capsule and sample.

^bMolecular Weight = 101.9613.

^cOriginal heat and mass data are available but it was not possible to trace the corrections applied to yield the molar values (col. 5), which were used in evaluation of the smoothing function (eq. (3) of [2]).

The present, high-temperature enthalpy results (table 1) and those from [2] in the range 273.15 K to 1173.15 K were represented by a single smoothing funetion derived by the method of least squares (eq. (3) of [2]). The last two columns of table 1 give the absolute and percent deviation of the present high-temperature data from this equation.

3. Low-Temperature Heat Capacity Data, 8.6 K to 273.15 K

Chang [4] has measured in an automated adiabatic calorimeter [10] the low-temperature heat capacity of α-Al₂O₃ chosen from the same NBS SRM 720 lot as was the material for the high-temperature measurements presented above. A piecewise representation of this low-temperature heat-capacity data, smooth in derivatives to order two, has been obtained using a least-squares spline-fitting technique. The value of this function, as well as its first and second derivatives, match precisely at 273.15 K corresponding values derived from the function [2] representing the enthalpy above 273.15 K.

The piecewise representation is illustrated in figure 1., where P1, P2, and P3 are polynomials of the form

$$\sum _{i=0}^6\frac{A_i}{i!}{\left(T-T_0\right)}^i$$ (1)

T₀ is a reference temperature, different for each temperature interval.

$$45.0\mathrm{K}>T⩾8.61\mathrm{K};C_p=\exp \left(\mathrm{P}1\right)\mathrm{J}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-1};T_{\mathrm{o}}=8.61\mathrm{K}$$

$$\begin{gathered} A_0=-0.5147E+01 \\ {A}_1=+0.34127E+00 \\ {A}_2=-0.333446E-01 \\ {A}_3=+0.450764E-02 \\ {A}_4=-0.51464E-03 \\ {A}_5=+0.397864E-04 \\ {A}_6=-0.152136E-05 \end{gathered}$$ (1a)

$$125\mathrm{K}>T⩾45.0\mathrm{K};C_p=\mathrm{P}2\mathrm{J}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-1};T_0=40.0\mathrm{K}$$

$$\begin{gathered} A_0=+0.6966E+00 \\ {A}_1=+0.59387E-01 \\ {A}_2=+0.40357E-02 \\ {A}_3=+0.95173E-04 \\ {A}_4=-0.35910E-05 \\ {A}_5=-0.6498E-07 \\ {A}_6=+0.4089E-08 \end{gathered}$$ (1b)

$$273.15\mathrm{K}>T⩾125\mathrm{K};C_p=\mathrm{P}3\mathrm{J}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-1};T_0=125\mathrm{K}$$

$$\begin{gathered} A_0=+0.21993E+02 \\ {A}_1=+0.38853E+00 \\ {A}_2=+0.13955E-02 \\ {A}_3=-0.83967E-04 \\ {A}_4=+0.19133E-05 \\ {A}_5=-0.31778E-07 \\ {A}_6=+0.29562E-09 \end{gathered}$$ (1c)

Figure 1.

Temperature ranges for spline fit of low-temperature heat-capacity data for Al₂O₃.

The low-temperature heat-capacity data [4] are represented by these smoothing functions with computed percent standard deviations (S_c) for each of the three fitting intervals as follows:

$$\begin{gathered} 45.0\text{K}>T⩾8.61\text{K};{\text{S}}_c=0.97 \\ 125.0\text{K}>T⩾45.0\text{K};{\text{S}}_c=0.13 \\ 273.15\text{K}>T⩾125.0\text{K};{\text{S}}_c=0.05 \end{gathered}$$

According to Chang [4], “It is estimated that the accuracy of the smoothed (low-temperature) values are better than 0.1% at temperatures between 100 and 350 K … Below 100 K, the inaccuracy is estimated to become progressively larger, reaching perhaps 1% around 50 K and 10% around 10 K.”

4. Enthalpy of SRM 720 (α-Al₂O₃), 10 K to 2250 K

Smooth heat-capacity values were computed from eqs (1), above. Below 8.6 K, a T³ dependence of heat capacity was assumed. Using these values, the heat-capacity functions were integrated to obtain the absolute enthalpy of α-Al₂O₃ in the temperature range 10 K to 273.15 K. Above 273.15 K, enthalpy and heat-capacity values were derived from the following equation (reproduced for convenience from [2]):

$$\begin{gathered} H_T-H_{273.15}=A{T}^{-2}+B{T}^{-1}+C\ln T+K+DT \\ +E{T}^2+F{T}^3+G{T}^1+H{T}^5\text{J}{\text{mol}}^{-1} \end{gathered}$$ (2)

$$\begin{array}{ll}A=+6.6253E+07& E=-8.57516E-02\\ B=-4.54238E+06& F=+4.299063E-05\\ C=-5.475599E+04& G=-1.15192E-08\\ K=+2.5819702E+05& H=+1.26351E-12\\ D=+2.574076E+02& \end{array}$$

Table 2 presents these smooth heat-capacity and enthalpy data for α-Al₂O₃.

Table 2.

Enthalpy and heat capacity of standard reference material 720.

Temp	HT − HOK	Cp	Temp.	HT − HOK	Cp
K	J mol−1b	J mol−1 K−1	K	J mol−1	J mol−1 K−1
			440	22953	100.69
10	0.023	0.0091	450	23965	101.71
15	0.115	0.0307	460	24987	102.68
20	0.364	0.0732	470	26018	103.60
25	0.898	0.146	480	27059	104.48
30	1.905	0.265	490	28108	105.33
35	3.646	0.443	500	29165	106.13
40	6.460	0.697	510	30230	106.90
45	10.77	1.046	520	31303	107.64
50	17.11	1.506	530	32383	108.35
60	38.18	2.793	540	33470	109.02
70	74.68	4.592	550	34563	109.67
80	131.7	6.901	560	35663	110.29
90	214.2	9.678	570	36769	110.89
100	326.6	12.855	580	37881	111.46
110	472.4	16.347	590	38998	112.02
120	654.3	20.07	600	40121	112.55
130	874.3	23.95	610	41249	113.06
140	1133.7	27.93	620	42382	113.55
150	1433.1	31.95	630	43520	114.02
160	1772.7	35.95	640	44663	114.48
170	2152.0	39.90	650	45810	114.92
180	2570.3	43.75	660	46961	115.35
190	3026.7	47.50	670	48117	115.76
200	3519.9	51.12	680	49276	116.16
210	4048.7	54.61	690	50440	116.55
220	4611.6	57.95	700	51607	116.92
230	5207.1	61.14	720	53953	117.64
240	5833.9	64.18	740	56313	118.32
250	6490.3	67.00	760	58685	118.96
260	7175.0	69.82	780	61071	119.56
270	7886.3	72.42	800	63468	120.14
273.15	8115.6	73.21	820	65876	120.69
280	8622.8	74.87	840	68295	121.21
290	9383.2	77.20	860	70724	121.71
298.15	10020	79.01	880	73163	122.20
300	10166	79.41	900	75612	122.66
310	10971	81.51	920	78070	123.11
320	11796	83.49	940	80536	123.55
330	12641	85.37	960	83011	123.97
340	13503	87.16	980	85495	124.37
350	14383	88.84	1000	87986	124.77
360	15280	90.45	1020	90486	125.16
370	16192	91.97	1040	92992	125.53
380	17119	93.41	1060	95507	125.90
390	18060	94.7a	1080	98028	126.26
400	19014	96.08	1100	100560	126.61
410	19982	97.32	1120	103090	126.95
420	20961	98.50	1140	105640	127.29
430	21951	99.62	1160	108180	127.61
1180	110740	127.94	1700	179080	134.31
1200	113300	128.25	1750	185810	134.73
1250	119730	129.01	1800	192550	135.13
1300	126200	129.74	1850	199320	135.50
1350	132710	130.43	1900	206100	135.89
1400	139240	131.08	1950	212900	136.18
1450	145810	131.70	2000	219720	136.50
1500	152410	132.29	2050	226550	136.80
1550	159040	132.84	2100	233400	137.10
1600	165700	133.36	2150	240260	137.41
1650	172380	133.85	2200	247140	137.73
			2250	254030	138.06

^aTemperatures expressed on IPTS-68 seale.

^bMolecular Weight = 101.9613.