The low-temperature heat capacity and ideal gas thermodynamic properties of isobutyl tert-butyl ether

Article abstract of DOI:10.1016/j.jct.2005.02.003

The heat capacities of isobutyl tert-butyl ether in crystalline, liquid, supercooled liquid, and glassy states were measured by vacuum adiabatic calorimetry over the temperature range from (7.68 to 353.42) K. The purity of the substance, the glass-transition temperature, the triple point and fusion temperatures, and the enthalpy and entropy of fusion were determined. Based on the experimental data, the thermodynamic functions (absolute entropy and changes of the enthalpy and Gibbs free energy) were calculated for the solid and liquid states over the temperature range studied and for the ideal gas state at T = 298.15 K. The ideal gas heat capacity and other thermodynamic functions in wide temperature range were calculated by statistical thermodynamics method using molecular parameters determined from density-functional theory. Empirical correction for coupling of rotating groups was used to calculate the internal rotational contributions to thermodynamic functions. This correction was found by fitting to the calorimetric entropy values.

Full text of DOI:10.1016/j.jct.2005.02.003

J. Chem. Thermodynamics 38 (2006) 10–19
www.elsevier.com/locate/jct
The low-temperature heat capacity and ideal gas
thermodynamic properties of isobutyl tert-butyl ether
A.I. Druzhinina , O.V. Dorofeeva , R.M. Varushchenko a,*, E.L. Krasnykh c
a
b
a
Department of Chemistry, Moscow State University, 119992 Moscow, Russia
b
Institute for High Energy Densities of Associated Institute for High Temperatures, Russian Academy of Sciences,
Izhorskaya St. 13/19, Moscow 125412, Russia
Samara State Technical University, Galaktionovskaya 141, 443010 Samara, Russia
c
Received 15 November 2004; received in revised form 2 February 2005; accepted 2 February 2005
Available online 7 November 2005
Abstract
The heat capacities of isobutyl tert-butyl ether in crystalline, liquid, supercooled liquid, and glassy states were measured by vac-
uum adiabatic calorimetry over the temperature range from (7.68 to 353.42) K. The purity of the substance, the glass-transition
temperature, the triple point and fusion temperatures, and the enthalpy and entropy of fusion were determined. Based on the exper-
imental data, the thermodynamic functions (absolute entropy and changes of the enthalpy and Gibbs free energy) were calculated
for the solid and liquid states over the temperature range studied and for the ideal gas state at T = 298.15 K. The ideal gas heat
capacity and other thermodynamic functions in wide temperature range were calculated by statistical thermodynamics method using
molecular parameters determined from density-functional theory. Empirical correction for coupling of rotating groups was used to
calculate the internal rotational contributions to thermodynamic functions. This correction was found by fitting to the calorimetric
entropy values.
Ó 2005 Elsevier Ltd. All rights reserved.
Keywords: Adiabatic calorimetry; Heat capacity; Isobutyl tert-butyl ether; Thermodynamic functions
1
. Introduction
tert-butyl ether (MTBE) has been widely used all over
the world for the past 20 years to replace toxic lead com-
pounds as octane enhancers. Despite some desirable
properties, MTBE has high volatility in the summer sea-
son and has been found to contaminate ground and
drinking water. It is likely that heavier tertiary ethers
having lower volatility and solubility in water will soon
mainly replace MTBE as oxygenating fuel additive. Key
thermodynamic properties (enthalpies of combustion
and formation, heat capacity, enthalpy of vaporization,
and saturated vapor pressure) for alternative ethers need
to be considered in the objective selection of them as the
octane enhancing additives to the engine fuels.
Tertiary ethers have a number of desirable physico-
chemical properties that make them perspective for
using as fuel additives [1]. First, they have a high octane
number, therefore adding the ethers to a gasoline in-
creases its octane rating that helps to prevent an engine
from a detonation. Second, an addition of combined
oxygen to the gasoline promotes more complete com-
bustion of the fuel that reduces an emission of such ex-
haust pollutants as carbon monoxide, nitrogen oxides,
volatile organic compounds and solid particles. Methyl
*
Reference [2] reports our determination of the low-
temperature heat capacities of n-propyl tert-butyl ether
(NPTBE) and isopropyl tert-butyl ether (IPTBE) by
Corresponding author. Tel.: +7 095 939 5396; fax: +7 095 939 1240.
E-mail address: varusch@thermo.chem.msu.ru (R.M. Varush-
chenko).
0
021-9614/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jct.2005.02.003

A.I. Druzhinina et al. / J. Chem. Thermodynamics 38 (2006) 10–19
11
vacuum adiabatic calorimetry and calculation of ther-
ꢀ
ture range from (7.68 to 353.42) K. The accuracy of the
Csat,m values are about 2% between the temperatures
from (7.6 to 15) K, (0.5 to 1)% in the temperature range
from (20 to 80) K, and (0.2 to 0.3)% above 80 K. The
difference between Csat,m and Cp,m values that was eval-
uated at T = 298.15 K, was smaller than the uncertain-
ties of Csat,m and not taken into account for the whole
temperature interval studied.
ꢀ
ꢀ
modynamic functions S (T), H (T) ꢁ H (0) and
ꢀ
ꢀ
G (T) ꢁ H (0) in the ideal gas state at T = 298.15 K.
In Ref. [3], the standard enthalpies of formation and
ideal gas thermodynamic functions of NPTBE and
IPTBE were calculated based on DFT (density-func-
tional theory). Both experimental and calculation meth-
ods were combined in [4,5] for studying ethyl tert-butyl
ether (ETBE) and ethyl tert-amyl ether (ETAE) in wide
temperature range from (5 to 1000) K. In this paper, we
report an application of these methods to isobutyl tert-
butyl ether (IBTBE) that is promising fuel additive
and can also be used for extracting pure isoolefins from
the gas hydrocarbons fraction in oil processing.
The solid phase of IBTBE was formed by cooling the
liquid sample from the room temperature at a rate of (1–
ꢁ
3
ꢁ1
2) Æ 10 K Æ s followed by quenching it at T = 77.4 K
during 24 h. The heat capacity curve (figure 1) reveals a
glass, supercooled liquid, and crystalline phase of the
ether. During the transitions of the glass into the super-
cooled liquid, spontaneous liberation of the heat took
place at the temperatures from (30 to 34) K higher than
the glass-transition temperature. This phenomenon was
caused by crystallization of the sample. To obtain the
stable crystalline phase, the solid specimen was annealed
at the temperatures by (10 to 15) K below the triple
point for 24 h and then quenched at T = 77.4 K for
the same period.
2
. Experimental
2
.1. Methods and results
The IBTBE was prepared in a reaction between iso-
butanol and isobutylene at T = 313 K in the presence
of a helium cationite Levatit S-100 as a selective catalyst.
The reaction product was purified from residual of alco-
hol and isobutylene by washing with distilled water and
blowing through with nitrogen under vacuum. Then the
ether was dried over Na SO and rectified in the vac-
Table 1 contains the Cp,m/R values of the IBTBE in
the crystalline, liquid, supercooled liquid, and glassy
states. The heat capacity values could not be measured
over the whole range of the supercooled liquid. They
were obtained only for the temperature intervals by 23
2
4
uum. According to a gas–liquid chromatography, sam-
ple purity was 0.9992 mass fraction at analysis
sensitivity 0.0001 mass fraction. The total amount of
impurities was determined by calorimetric fractional
melting as described in [6] and found to be 8 Æ 10 mole
fraction. The major impurities were isobutanol and
isobutylene.
K above glass-transition temperature, T and 7.6 K be-
gl
low the triple point temperature, Ttp.
The purity and the triple point temperature of the
ether were determined from fractional melting study
the dependence of the equilibrium fusion temperatures,
ꢁ
4
T , on the reciprocal fraction of the sample melted, 1/
i
Fi. Three experiments were performed to obtain the
T = f(1/F ) dependence. An example is given in table 2.
The heat capacity of IBTBE was measured in a fully
automated setup [7] that consisted of a vacuum adia-
batic calorimeter, data acquisition and control system,
and a personal computer. The sample of the ether was
put in a stainless steel cylindrical container (volume
i
i
350
3
300
ꢂ1 cm ) which was supplied with a brass lid and sealed
by means of an indium gasket. The temperature of the
calorimeter was measured using (rhodium + iron) resis-
250
tance thermometer (R ꢂ 100 X) calibrated on ITS-90.
200
0. 5
. 4
0. 3
. 2
O
0
An accuracy of the temperature measurement is
±
ꢁ
3
150
5 Æ 10 K. The temperature difference between the cal-
0
T
orimeter and the adiabatic shield was measured by a
four-junction (copper + 0.001 mass of iron) – Chromel
thermocouple. The data acquisition system controlled
the adiabatic condition of experiment with a stability
of (1–3) mK throughout the whole temperature interval.
A small cryostat with the calorimeter was placed directly
in transport Dewar flasks that made it possible to cut
down cooling agents (liquid helium and nitrogen). The
cryostat was evacuated by a cryosorption using a carbon
adsorber. The heat capacity of IBTBE, Csat,m, was mea-
sured at the saturated vapor pressure over the tempera-
C
100
0. 1
0
C
0
50
100
2
150
350
50
2
T
, K
Tgl
Ttr
0
0
50
100
150
200
250
300
400
T/K
FIGURE 1. Experimental molar heat capacity, Cp,m, of isobutyl tert-
butyl ether as a function of temperature, T, where Tgl and Ttp denote
the temperatures of the glass transition and triple point. In the insert, a
2
dependence of Cp,m/T on T is given for the helium range of the
temperatures.

12
A.I. Druzhinina et al. / J. Chem. Thermodynamics 38 (2006) 10–19
TABLE 1
Experimental molar heat capacity Cp,m of isobutyl tert-butyl ether (M = 130.231 g Æ mol , mass of the sample used = 0.738214 g, R = 8.314472
ꢁ1
ꢁ1
ꢁ1
)
J Æ K Æ mol
T/K
C
p,m/R
T/K
Cp,m/R
T/K
C
p,m/R
T/K
Cp,m/R
Crystal
7
8
8
9
.68
.33
.97
.61
0.1610
0.2096
0.2666
0.3342
0.3993
0.4978
0.6213
0.7564
0.8960
1.048
1.205
1.374
1.542
1.716
1.887
2.069
2.258
2.460
2.658
2.863
3.070
3.258
3.435
3.623
3.817
4.013
4.175
4.346
4.521
4.677
4.841
5.007
5.151
5.300
5.449
5.596
42.60
5.753
5.883
6.044
6.155
6.301
6.429
6.574
6.683
6.804
6.953
7.098
7.249
7.404
7.596
7.759
7.765
7.853
8.445
8.509
8.679
8.793
8.905
8.953
9.076
9.220
9.325
9.603
9.780
9.890
9.993
10.30
10.41
10.38
10.50
10.65
10.73
83.57
10.71
11.13
11.22
11.30
11.44
11.55
11.68
11.82
11.92
12.05
12.20
12.32
12.42
12.52
12.63
12.73
12.86
13.00
13.07
13.17
13.29
13.41
13.55
13.64
13.74
13.85
13.95
14.07
14.17
14.27
14.39
14.49
14.59
14.69
14.81
14.94
121.09
123.02
123.99
124.96
125.92
126.91
127.10
127.90
128.89
129.13
130.13
132.12
133.10
134.08
136.02
136.98
137.94
138.89
139.84
140.78
141.71
142.63
143.56
144.49
145.41
146.33
147.24
149.06
150.86
151.76
152.66
153.55
156.22
157.08
157.95
15.01
15.23
15.37
15.45
15.57
15.66
15.70
15.74
15.79
15.86
15.94
16.22
16.28
16.35
16.55
16.67
16.71
16.84
16.85
16.94
17.08
17.14
17.25
17.26
17.34
17.40
17.47
17.61
17.73
17.85
17.92
18.03
(18.43)
(18.48)
(18.74)
43.66
44.72
45.78
46.84
47.90
48.96
50.01
51.07
52.12
53.17
54.22
55.27
56.31
57.36
58.41
59.46
63.63
64.68
65.72
66.76
67.80
68.84
69.88
70.92
71.95
74.01
75.04
76.07
77.10
79.16
80.19
80.56
81.22
82.25
83.28
86.51
87.51
88.51
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
0.26
1.02
1.92
2.83
3.74
4.65
5.57
6.48
7.41
8.34
9.27
0.20
1.22
2.29
3.35
4.42
5.49
6.55
7.63
8.70
9.77
0.84
1.91
2.98
4.05
5.12
6.19
7.26
8.34
9.41
0.47
1.54
89.50
90.50
91.51
92.51
93.50
94.49
95.48
96.48
97.47
98.47
99.46
100.45
101.44
102.44
103.44
104.44
105.43
106.41
107.39
108.37
109.35
110.32
111.31
112.29
113.28
114.26
115.24
116.22
117.20
118.17
119.15
120.12
a
a
a
Liquid
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
63.93
65.92
67.90
69.88
71.84
73.80
75.75
77.69
79.63
81.56
83.49
85.40
87.31
89.21
91.11
93.00
34.20
35.65
36.37
37.10
37.82
38.55
39.27
25.91
26.00
26.08
26.16
26.25
26.38
26.47
26.50
26.59
26.69
26.78
26.84
26.95
27.03
27.11
27.18
28.97
29.10
29.19
29.14
29.13
29.30
29.28
194.89
196.77
198.65
200.52
201.82
202.53
203.25
203.96
204.67
205.41
206.12
206.84
207.55
208.27
208.98
209.69
255.46
256.20
256.95
257.70
258.45
259.20
259.95
27.28
27.35
27.40
27.48
27.61
27.56
27.57
27.62
27.71
27.68
27.76
27.83
27.80
27.80
27.92
27.90
30.08
30.19
30.23
30.15
30.21
30.26
30.37
210.40
211.12
211.83
212.55
213.26
213.97
214.69
215.40
216.12
216.87
217.59
218.30
219.02
219.73
220.45
221.17
279.86
280.69
282.66
283.22
284.18
284.95
285.74
27.94
27.98
28.02
28.03
28.09
28.11
28.09
28.15
28.15
28.12
28.17
28.22
28.37
28.39
28.41
28.39
31.44
31.39
31.55
31.55
31.61
31.71
31.68
221.89
222.60
223.33
224.04
224.76
225.48
226.20
226.92
227.65
228.42
229.14
229.86
230.59
231.31
232.04
233.48
316.58
318.24
319.91
321.57
323.26
324.94
326.61
28.50
28.50
28.47
28.58
28.60
28.60
28.63
28.70
28.70
28.82
28.78
28.82
28.81
28.94
28.91
28.91
33.34
33.49
33.54
33.59
33.77
33.87
33.96

A.I. Druzhinina et al. / J. Chem. Thermodynamics 38 (2006) 10–19
13
TABLE 1 (continued)
T/K
40.05
C
p,m/R
T/K
C
p,m/R
T/K
C
p,m/R
T/K
Cp,m/R
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
29.26
29.32
29.34
29.42
29.49
29.45
29.45
29.57
29.71
29.56
29.71
29.81
29.82
29.86
29.83
29.87
29.95
30.04
30.13
260.71
261.46
262.22
263.85
264.62
265.39
266.16
266.93
267.70
268.48
269.26
270.82
273.20
274.00
274.80
275.61
276.58
277.39
279.03
30.34
30.41
30.50
30.51
30.60
30.60
30.70
30.71
30.78
30.82
30.84
30.89
31.03
31.05
31.06
31.14
31.23
31.20
31.40
287.30
287.56
288.43
288.86
290.42
290.43
291.99
295.17
298.38
299.98
301.59
303.21
304.86
306.49
308.30
309.95
311.60
313.27
314.92
31.75
31.80
31.83
31.78
31.86
31.95
31.93
32.10
32.29
32.36
32.45
32.54
32.67
32.74
32.85
32.95
33.00
33.16
33.25
328.29
329.99
331.66
333.33
335.19
336.85
338.51
340.16
341.81
343.47
345.13
346.78
348.44
350.08
351.76
353.42
34.05
34.13
34.24
34.34
34.39
34.50
34.63
34.76
34.88
35.02
35.07
35.16
35.24
35.43
35.56
35.71
40.77
41.50
42.22
42.94
43.67
44.39
45.12
46.57
47.30
48.03
48.76
49.49
50.22
51.77
52.51
53.24
53.98
54.72
Glass, supercooled liquid
13.43
80.42
81.39
82.32
83.29
84.28
85.22
86.13
87.04
87.98
88.93
89.88
90.83
91.79
92.74
93.68
94.62
95.55
96.51
a
11.01
11.15
11.24
11.45
11.54
11.71
11.69
11.89
12.09
12.20
12.38
12.49
12.57
12.73
12.91
12.98
12.98
13.21
97.47
98.41
99.31
114.06
114.91
115.82
116.80
117.80
118.80
119.80
120.79
121.77
122.76
123.76
124.76
125.75
126.75
127.74
128.73
129.70
130.64
19.53
23.32
23.82
24.01
24.05
24.10
24.14
24.17
24.23
24.30
24.31
24.33
24.40
24.40
24.47
24.56
24.62
24.64
131.57
132.51
133.45
134.39
135.33
136.26
137.20
149.71
151.77
153.82
155.87
157.90
159.92
161.93
163.93
165.92
167.90
169.88
24.66
24.72
24.76
24.80
24.82
24.88
25.00
25.26
25.34
25.48
25.56
25.66
25.74
25.82
25.91
26.00
26.08
26.16
13.56
13.77
13.69
13.86
14.11
14.17
14.31
14.46
14.61
14.80
15.00
15.20
15.50
15.63
16.12
16.71
17.57
100.20
101.10
101.97
102.84
103.72
104.61
105.51
106.42
107.34
108.29
109.25
110.23
111.21
112.18
113.14
Given in parentheses are the heat capacity Cp,m values in the region just before the fusion.
TABLE 2
Equilibrium melting temperatures T
The Mastrangelo and Dornte method [9] did not indi-
cate the presence of any solution in the solid state
of the sample. Table 3 lists mean values of T , mole
a
i
, reciprocal of the sample fraction
a
melted 1/F
i
, the Ti(calc) values calculated from the linear dependence of
tp
T
i
on 1/F
i
for isobutyl tert-butyl ether.
b
T
i
/K
i
q /J
1/F
i
Ti(calc)/K
a
a
a
a
a
a
a
a
a
a
a
a
a
a
TABLE 3
1
1
1
1
1
1
1
1
1
62.142
62.239
62.266
62.280
62.288
62.297
62.307
62.314
62.321
5.399
5.399
5.396
5.355
5.316
5.343
5.366
5.375
5.366
8.95
4.47
2.98
2.24
1.80
1.50
1.28
1.12
1.0
162.144
162.235
162.266
162.281
162.290
162.296
162.301
162.304
162.306
162.327
The values of triple point temperature, Ttp, enthalpy, Dfus
entropy, Dfus , of fusion, total amount of impurities, N
cryoscopic constants Acr and Bcr, glass transition temperature, Tgl
H
m
, and
S
m
2
, and
,
and temperature interval of the vitrification, DTgl, of isobutyl tert-
butyl ether
T
D
D
tp/K
162.33 ± 0.01
8.65 ± 0.02
53.29 ± 0.15
ꢁ
m
H /(kJ Æ mol )
1
1
fus
fus
ꢁ
ꢁ1
)
S
m
/(J Æ K Æ mol
0
.00
values were used for calculation of N
from 1.2 to 610 according to [8].
denotes a quantity of energy used for melting of the F
substanceꢀs fraction.
N
2
/mole fractions
ꢁ
0.0008 ± 0.0001
0.03948 ± 0.00012
0.00280 ± 0.00001
114.1
1
a
A
cr/K
The T
i
and 1/F
i
2
, Ttp, Acr and
ꢁ1
B
T
cr/K
gl/K
B
cr in the range of 1/F
i
b
The q
i
i
DTgl/K
110.2–116.8

14
A.I. Druzhinina et al. / J. Chem. Thermodynamics 38 (2006) 10–19
TABLE 4
a
The molar enthalpy of fusion, Dfus
H
m
, of isobutyl tert-butyl ether calculated by equation (1)
ꢁ1
ꢁ1
ꢁ1
ꢁ1
ꢁ1
)
T
1
/K
T
2
/K
DHtot/(J Æ mol
)
DHemp/(J Æ mol
)
DH
3
/(J Æ mol
)
DH
4
/(J Æ mol
)
D
fus
m
H /(J Æ mol
1
1
1
56.874
53.095
53.505
168.902
170.580
169.384
13350
15427
14771
2460
3564
3233
843
1420
1785
1525
mean
8627
8664
8661
1414
1352
8651 ± 24
a
The meanings of enthalpy increments DHtot, DHemp, DH
3
and DH
4
are given in text under equation (1).
X
i
fractions of impurities, N , and cryoscopic constants A
2
Cp;m
¼
A fðT ꢁ A Þ=B g ;
ð3Þ
cr
i
k
k
and Bcr calculated from three independent experiments.
Melting point depressions amount from (0.017 to 0.020)
where (T ꢁ A )/B is the normalizing term. The RMS
k
k
deviation between the calculated and experimental
Cp,m values is equal to 0.24% that corresponds to
the experimental errors at T > 80 K. Smoothed values
K. The N values obtained by chromatographic and
2
calorimetric methods are in good agreement. The
glass-transition temperature Tgl was found graphically
ꢀ
of C /R and thermodynamic functions S ðTÞ=R;
p,m
as an inflexion point of the C -curve for the transition
p
m
ꢀ
ꢀ
ꢀ
fH ðTÞ ꢁ H ð0Þg=RT ; and U ðTÞ=R are listed in table
of glass to the supercooled liquid. A correlation between
the glass-transition and triple point temperatures of
IBTBE, T ꢃ 2/3T , is typical for glasses [10]. The en-
m
for the whole temperature interval studied. Table 6
m
m
5
lists the standard ideal gas values of the thermody-
namic functions and entropy and free Gibbs energy
of formation of IBTBE at T = 298.15 K. The errors
of the functions were estimated by the law of the ran-
dom errors accumulation [11] based upon the uncer-
tainties of the heat capacity measurements. In the
gl
tp
thalpy of fusion, D H , of IBTBE was determined
fus
m
calorimetrically based on the total enthalpy absorbed
during the fusion with a correction for the normal heat
capacities of the crystal and liquid that the substance
had in the fusion region. The DfusH value was obtained
m
calculation of the ideal gas thermodynamic functions,
ꢀ
from the equation:
the enthalpy and entropy of vaporization, Dvap
H
m
D
fus
H
m
¼ DtotH ꢁ DH
1
ꢁ DH
2
ꢁ DHemp
;
ð1Þ
ꢀ
ðT Þ; and DvapS ðTÞ, and entropy of compression of
m
the ideal gas, R Æ ln{p(298.15)/101.325 kPa}, have been
used. The values of these entropies were calculated
where D H is the total enthalpy absorbed in heating the
tot
calorimeter from initial temperature T < T to final
temperature T > T , DH and DH are the heating
1
tp
ꢀ
ꢁ1
using the DvapH ðT Þ ¼ ð39:12 ꢄ 0:20Þ kJ ꢅ mol value
2
tp
1
2
m
measured calorimetrically [12] and saturated vapor
pressure of IBTBE (our unpublished data), respec-
tively. The thermodynamic consistency of the results
of the vaporization calorimetry and vapor pressure
enthalpies calculated from the normal heat capacities
of the crystal and liquid in the temperature intervals
from T to T and Ttp to T , respectively, DH is
emp
1
tp
2
the enthalpy increment needed for heating the empty
calorimeter from T to T . The enthalpy of fusion of
IBTBE is listed in table 4.
studies was proved by satisfactory agreement between
ꢀ
1
2
DvapH ðTÞ values measured calorimetrically and those
m
calculated from the (p, T) data (39.24 ± 0.56)
ꢁ
1
kJ Æ mol . The standard Gibbs free energy of forma-
tion was calculated from
2
.2. Treatment of the experimental data
The heat capacity of IBTBE was extrapolated to
ꢀ
ꢀ
ꢀ
D
f
G ðgÞ ¼ D
f
H ðgÞ ꢁ T ꢅ D
f
S ðgÞ
ð4Þ
m
m
m
T ! 0 with the equation:
3
using the experimental value of absolute entropy (ta-
ble 6), and calorimetric value of the enthalpy of for-
C
p;m ¼ A
D
T :
ð2Þ
ꢁ
3
ꢁ4
The coefficient A = (3.042 ± 0.001) Æ 10
J Æ K
was calculated by the least-squares method
Æ
ꢀ
D
mation in liquid state, D H ðlÞ, [13] and enthalpy of
f
m
vaporization of IBTBE [12], which led to
ꢁ
1
mol
(
(
ꢀ
ꢁ
1
LSM) from Cp,m data in the temperature interval from
7.68 to 12.82) K. The root-mean-square (RMS) devia-
D H ð298:15; gÞ ¼ ðꢁ369:9 ꢄ 1:6Þ kJ ꢅ mol . The func-
f
m
ꢀ
tion D
f
S ðgÞ in equation (4) is the difference between
m
tion between the calculated and experimental Cp,m val-
ues (±1.9%) lies within the limits of experimental
errors. Thus, the temperature dependence Cp,m(T) corre-
the values of absolute entropies of the final and initial
products in reaction
sponds to the Debye equation (2) and C = 0 and
p,m
16 ꢅ Cðcr; graphiteÞ þ 18 ꢅ H
2
ðgÞ þ O ðgÞ ¼
2
ꢀ
S ¼ 0 at T ! 0.
m
2 ꢅ ðCH
3
Þ CHCH
2
OCðCH
3
Þ ðgÞ:
ð5Þ
2
3
The experimental heat capacities were divided into
several intervals and fitted with polynomials of the fol-
lowing form:
The absolute entropies of the initial products were taken
from [14].

A.I. Druzhinina et al. / J. Chem. Thermodynamics 38 (2006) 10–19
15
TABLE 5
Smoothed molar thermodynamic properties of isobutyl tert-butyl ether
TABLE 6
Calorimetrically derived thermodynamic functions in ideal gas state of
ꢁ1
ꢁ1
ꢀ
(
R = 8.314472 J Æ K Æ mol
)
isobutyl tert-butyl ether, the standard entropy, D
f
S ðTÞðgÞ,and free
m
ꢀ
o
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
a
energy of formation, D
f
G
m
ðTÞðgÞ, where Dvap
H
m
ðTÞ and Dvap
S
m
ðTÞ are
T/K
C
p,m/R
fH ðTÞ ꢁ H ð0Þg=RT S ðTÞ=R
Crystal
U ðTÞ=R
m
m
m
m
the enthalpy and entropy of vaporization and R Æ ln {p(T)/101.325
kPa} is ideal gas entropy of compression at T = 298.15 K
5
0.04573
0.3659
1.109
2.027
2.971
3.857
4.662
5.388
6.060
6.700
7.326
7.942
8.549
9.152
9.748
10.34
10.93
11.52
12.10
12.68
13.82
14.91
15.96
16.90
17.72
18.35
0.0115
0.0914
0.2999
0.6151
0.9925
1.397
1.812
2.210
2.601
2.978
3.346
3.702
4.052
4.395
4.732
5.063
5.392
5.716
6.036
6.354
6.981
7.597
8.201
8.788
9.357
9.900
10.02
0.0152
0.1220
0.4027
0.8453
1.400
2.021
2.677
3.347
4.022
4.693
5.362
6.026
6.685
7.340
7.992
8.640
9.285
9.926
10.57
11.20
12.46
13.71
14.95
16.16
17.36
18.52
18.79
0.0037
0.0306
0.1028
0.2302
0.4075
0.6240
0.8650
1.137
1.421
1.715
2.016
2.324
2.633
2.945
3.260
3.577
3.893
4.210
4.534
4.846
5.479
6.113
6.749
7.372
8.003
8.620
8.770
ꢀ
ꢁ1
DvapH ðTÞ=ðkJ ꢅ mol
Þ
ꢁ1
39.12 ± 0.20
131.21 ± 0.67
ꢁ28.45
m
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
0
5
0
5
0
5
0
5
0
5
0
5
0
5
0
5
0
5
ꢀ
ꢁ1
DvapS ðTÞ=ðJ ꢅ K ꢅ mol
Þ
m
ꢁ
1
ꢁ1
)
R Æ ln{p(T)/101.325 kPa}/(J Æ K Æ mol
ꢀ
ꢁ1
ꢁ1
S ðTÞðgÞ=ðJ ꢅ K ꢅ mol
Þ
456.96 ± 0.89
93.98 ± 0.23
ꢁ42.26 ± 0.35
ꢁ866.5 ± 0.9
ꢁ111.6 ± 1.6
m
ꢀ
ꢀ
m
ꢁ1
fH ðTÞ ꢁ H ð0ÞgðgÞ=ðkJ ꢅ mol
Þ
m
fG ðTÞ ꢁ H ð0ÞgðgÞ=ðkJ ꢅ molꢁ1Þ
ꢀ
ꢀ
m
m
ꢀ
ꢁ1
ꢁ1
Df S ðTÞðgÞ=ðJ ꢅ K ꢅ mol
Þ
m
ꢀ
ꢁ1
Df G ðTÞðgÞ=ðkJ ꢅ mol
Þ
m
3
1G(d,p) level. Vibrational frequencies, zero-point ener-
gies, and thermal corrections were calculated at the same
level. The scaling factors of 0.958 and 0.975 were applied
to C–H stretchings and to all other frequencies, respec-
tively. These values were obtained from fitting the exper-
imental vibrational fundamentals of some ethers.
The barriers for internal rotation of CH3 and
C(CH ) rotors were determined from optimization of
corresponding transition states. Potential functions for
3
3
1
1
1
1
1
1
1
1
00
10
20
30
40
50
60
internal rotation of i-C H and i-C H rotors were
9
3
7
4
determined by scanning the torsional angles from 0° to
60° at 15° increments and allowing all other structural
parameters to be optimized at the B3LYP/6-31G(d,p) le-
vel. The calculated energy values were fitted to the tor-
sional potential, which is a cosine-based Fourier
3
62.33 18.47
Liquid
function:
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
62.33 25.83
16.43
16.87
17.39
17.89
18.36
18.81
19.23
19.64
20.03
20.41
20.78
21.15
21.50
21.85
25.20
26.40
27.92
29.36
30.77
32.11
33.42
34.69
35.93
37.14
38.32
39.47
40.60
41.71
8.770
9.530
10.53
11.47
12.41
13.30
14.19
15.05
15.90
16.73
17.54
18.32
19.10
19.86
X
70
80
90
00
10
20
30
40
50
60
70
80
90
26.18
26.63
27.06
27.48
27.92
28.36
28.83
29.31
29.82
30.33
30.86
31.38
31.90
V ðuÞ ¼ V þ 1=2
V ð1 ꢁ cos nuÞ:
ð6Þ
0
n
n
ꢀ
ꢀ
The entropies, S ðTÞ, heat capacities, C ðTÞ, and
m
p;m
ꢀ
ꢀ
m
enthalpies, H ðT Þ ꢁ H ð0Þ, (100 K 6 T P 1500 K) were
m
calculated by standard statistical thermodynamics for-
mulae using the rigid-rotor harmonic-oscillator approx-
imation with correction for internal rotation. The
torsional frequencies were omitted in the calculation of
thermodynamic functions. Internal rotational contribu-
tion for each rotor was calculated by direct summation
over the energy levels obtained by diagonalizating the
one-dimensional Hamiltonian matrix associated with
98.15 32.32 ± 0.06 22.13 ± 0.04
42.59 ± 0.08 20.46 ± 0.2
00
10
20
30
40
50
a
32.43
32.95
33.52
34.12
34.76
35.42
22.19
22.53
22.87
23.20
23.53
23.86
ꢀ
42.81
43.88
44.93
45.97
47.00
48.01
20.62
21.35
22.06
22.77
23.47
24.15
potential function from equation (6).
ꢀ
Enthalpy of formation, D H
, was calculated at
the B3LYP/6-311+G(3df,2p)//B3LYP/6-31G(d,p) level
f
m;298
using isodesmic reactions and available recommended
ꢀ
D
f
H
values for ethers [16].
m;298
ꢀ
ꢀ
ꢀ
m
U ðTÞ ¼ S ðTÞ ꢁ fH ðTÞ ꢁ H ð0Þg=T.
m
m
m
3.2. Enthalpy of formation
3
. Theoretical calculations
.1. Methods of calculations
The DFT calculations were performed using the
The enthalpy of formation of IBTBE, (ꢁ369.9 ± 1.6)
ꢁ1
3
kJ Æ mol , was obtained from available experimental
data (see Section 2.2). To check the reliability of this va-
lue, in this work, the enthalpy of formation of IBTBE
was calculated using the isodesmic reactions with group
balance. From our previous studies [3,5], it was shown
Gaussian 98 software package [15]. The structural
parameters were fully optimized at the B3LYP/6-

16
A.I. Druzhinina et al. / J. Chem. Thermodynamics 38 (2006) 10–19
TABLE 7
TABLE 8
ꢀ
Enthalpies of reactions ðD
r
H
of isobutyl tert-butyl ether calculated at B3LYP/6-
Þ and enthalpies sof formation
Moments of inertia, vibrational frequencies and internal rotational
molecular constants of isobutyl tert-butyl ether calculated at B3LYP/6-
m;298
ꢀ
ðD
f
H
11+G(3df,2p)//B3LYP/6-31G(d,p) level from isodesmic reactions
Þ
m;298
3
31G(d,p) level
ꢀ
ꢀ
Reaction
DrH
Df H
TG conformer
m;298
m;298
ꢁ1
ꢁ1
Point group: C
Symmetry number: r = 1
Ground electronic state: X A
Molar mass: 130.2296
1
Moments of inertia:
kJÆmol
ꢁ4.8
kJÆmol
ꢁ369.3
ꢁ39
2
I
A
I
B
I
C
I
A
= 29.6826 Æ 10
= 104.1552 Æ 10
= 112.0862 Æ 10
g Æ cm
1
ꢁ39
2
2
i-Bu–O–t-Bu + Me–O–Et =
Pr–O–t-Bu + Me–O–i-Pr
i-Bu–O–t-Bu + Me–O–Pr =
Bu–O–t-Bu + Me–O–i-Pr
i-Bu–O–t-Bu + Me–O–i-Pr =
Pr–O–t-Bu + Me–O–t-Bu
i-Bu–O–t-Bu + Me–O–Et =
i-Pr–O–t-Bu + Me–O–Pr
i-Bu–O–t-Bu + Et–O–Et =
i-Pr–O–t-Bu + Et–O–Pr
i-Bu–O–t-Bu + Et–O–Pr =
i-Pr–O–t-Bu + Pr–O–Pr
~
g Æ cm
ꢁ39
g Æ cm
117 3
ꢁ
6
ꢁ3.2
1.2
ꢁ371.7
ꢁ371.4
ꢁ371.1
ꢁ369.3
ꢁ370.6
I I
B C
= 346525 Æ 10
g Æ cm
a
Vibrational frequencies: 3007, 3001, 3000, 2994, 2993, 2990, 2985,
2
1
1
1
5
1
981, 2972, 2967, 2927, 2920, 2918, 2911, 2906, 2904, 2886, 2854, 1501,
491, 1489, 1479, 1477, 1476, 1467, 1466, 1464, 1461, 1450, 1405, 1399,
387, 1378, 1372, 1371, 1347, 1304, 1252, 1251, 1228, 1203, 1174, 1162,
130, 1087, 1022, 1017, 955, 944, 936, 915, 913, 902, 889, 877, 814, 735,
0.1
0.1
b
b
b
b
b
01, 480, 446, 417, 382, 356, 331, 288, 270 ,259, 245, 238, 216, 206,
b
41, 123, 65, 16
b
b
0.3
a
Scaling factor of 0.958 was used for C–H stretchings and 0.975 for
other modes.
Average
Average value corrected for
mixture of TG and TT conformers
ꢁ370.6
ꢁ370.3
b
Instead of these torsional modes, the contributions due to the
internal rotation were calculated using the following coefficients V in
equation (6), reduced moment of inertia I and symmetry number of
n
a
r
Experimental
ꢁ369.9 ± 1.6
Calculated using experimental data from Refs. [12,13] (see Section
m
rotor r :
a
ꢁ39
ꢁ39
2
ꢁ1
,
CH
CH
CH
CH
CH
3
3
3
3
3
group: I
group: I
group: I
group: I
group: I
r
r
r
r
r
= 0.5230 Æ 10
= 0.5222 Æ 10
= 0.5217 Æ 10
= 0.5230 Æ 10
= 0.5174 Æ 10
g Æ cm , r
m
m
m
m
m
= 3, V
= 3, V
= 3, V
= 3, V
= 3, V
3
3
3
3
3
= 1169 cm
= 1103 cm
= 1122 cm
= 1121 cm
= 1042 cm
2.2).
2
ꢁ1
ꢁ1
ꢁ1
ꢁ1
g Æ cm , r
,
,
,
,
ꢁ39
2
g Æ cm , r
ꢁ39
2
g Æ cm , r
that such calculations led to reasonable accuracy and
even allowed to adjust recommended values derived
from calorimetric measurements.
ꢁ39
ꢁ
2
g Æ cm , r
39
2
ꢁ1
t-C H group: I = 7.6485 Æ 10
4
9
r
g Æ cm , r = 3, V = 980 cm
m
3
,
ꢁ
39
2
i-C
V
4
H
9
group:
I
r
= 7.3771 Æ 10
g Æ cm ,
r
m
= 1,
= 120.0,
V
1
= 3973.1,
= 7.2 (in
2
= ꢁ1564.5,
V
3
= 726.1,
V
4
= ꢁ173.0,
V
5
V
6
ꢁ1
cm ),
i-C
ꢁ
39
g Æ cm
= 74.4,
ꢁ1
,
3
H
7
group:
I
r
= 7.4922 Æ 10
r
m
= 1,
V
1
= ꢁ438.1,
5000
4000
3000
2000
1000
0
V
cm ).
2
= 230.8,
ꢁ1
V
3
= 1680.2,
V
4
V
5
= 25.2,
V
6
= ꢁ79.7 (in
(
3 2 2 3 3
CH ) CHCH –OC(CH )
Fifty five isodesmic reactions were used to estimate
ꢀ
the D H
value. The calculated values varied from
f
m;298
ꢁ1
(
ꢁ347 to ꢁ389) kJ Æ mol with the average value of
ꢁ
1
ꢁ369 kJ Æ mol . Since among these 53 reactions, there
were not well-balanced reactions with rather large values
TG
ꢀ
of enthalpies of reactions, D
with small values of D
r
H
, finally six reactions
were chosen (Table 7).
m;298
0
60
120
180
240
300
360
ꢀ
r
H
m;298
2000
1800
1600
1400
1200
1000
As is seen from table 7, all reactions lead to close values
and the average value agrees with experimental one
(
3
CH )
2 2 3 3
CH–CH OC(CH )
TABLE 9
Comparison between experimental (calorimetric), Sexp and calculated,
8
6
4
2
00
00
00
00
0
Scalc values of the absolute entropy of isobutyl tert-butyl ether in ideal
gas state
TT
TT
T/K
S
exp
/
S
calc
/
S
exp ꢁ Scalc
/
TG
TG
ꢁ1
ꢁ1
ꢁ1
ꢁ1
ꢁ1
ꢁ1
)
(
J Æ K Æ mol
)
(J Æ K Æ mol
)
(J Æ K Æ mol
-
200
0
60
120
180
240
300
360
2
3
3
3
3
3
98.15
456.96 ± 0.89
466.88 ± 1.65
473.37 ± 1.47
479.92 ± 1.35
486.53 ± 1.31
493.21 ± 1.28
456.92
466.82
473.44
480.03
486.57
493.08
0.04
0.06
Torsional angle ϕ, degree
13
23
ꢁ0.07
ꢁ0.11
ꢁ0.04
0.13
FIGURE 2. Torsional potential functions of isobutyl tert-butyl ether
plotted against angle of rotation about C–O and C–C bonds. Points
are calculated values at the B3LYP/6-31G(d,p) level of theory. Line is
Fourier expansion with the coefficients listed in table 8.
33
43
53

A.I. Druzhinina et al. / J. Chem. Thermodynamics 38 (2006) 10–19
17
within its uncertainty. A more positive value was ob-
tained with correction for conformational energy differ-
ence based on Boltzman averaging.
ꢀ
It is worth stressing that the estimated D
f
H
value,
m;298
ꢁ
1
ꢁ
349.3 kJ Æ mol , for IPTBE was used in last three
reactions in table 7. This value was calculated from
isodesmic and homodesmotic reactions [3] and was quite
different from the recommended experimental value of
ꢁ358.1 ± 3.0) kJ Æ molꢁ [16]. When used in isodesmic
1
(
and different relative positions of bulky groups around
the C–C bond [(CH ) CH–CH OC(CH ) ]:
reactions, the estimated value leads to good agreement
with experiment for NPTBE [3], ETBE ethyl [5], and
IBTBE (Table 7). Thus it may be concluded that well-
3
2
2
3 3
balanced isodesmic reactions for ethers yield the
ꢀ
D
f
H
values of calorimetric accuracy. For IBTBE,
m;298
theoretical calculations support the experimental value;
this latter was adopted in further calculations of
ꢀ
ꢀ
D H ðT Þ and D G ðTÞ.
f
m
f
m
3.3. Geometry, vibrational frequencies, and torsional
potentials
Calculated potential energy curves as a function of
torsion angles u(C–O) and u(C–C) are shown in figure
2
. For rotation about the C–C bond, the minima on
The DFT calculations predict the existence of two
the potential curve at 0° and 120° correspond to the
TT and TG conformers, respectively. Moments of iner-
tia for the optimized geometry, vibrational frequencies,
stable low energy conformers of IBTBE: the trans–
gauche (TG) and trans–trans (TT) conformers with C1
and C symmetry, respectively. (TG) conformer is 1.1
and coefficients V in the expansion for the torsional
n
s
ꢁ
1
ꢁ1
kJ Æ mol (95 cm ) more stable than (TT). Both con-
formers have trans location of bulky groups relative to
the C–O bond [(CH ) CHCH –OC(CH ) ]:
potentials (equation (6)) are given in table 8. The ther-
modynamic functions of IBTBE (Section 3.4) were cal-
culated for an equilibrium mixture of TG and TT
3
2
2
3 3
TABLE 10
Ideal gas thermodynamic properties of isobutyl tert-butyl ether (R = 8.314472, p° = 101.325 kPa)
ꢀ
a
ꢀ
ꢀ
b
ꢀ
ꢀ
ꢀ
ꢀ
T
C
ðTÞ=R
S ðTÞ=R
U ðTÞ=R
fH ðTÞ ꢁ H ð0Þg=RT
7.18
9.08
Df H ðTÞ=RT
Df G ðTÞ=RT
p;m
m
m
m
m
m
m
1
00
50
00
50
98.15
00
50
00
50
00
00
00
00
00
11.01
14.62
17.77
20.91
24.02
24.14
27.38
30.50
33.43
36.13
40.91
44.95
48.42
51.42
54.02
56.28
58.23
59.93
61.41
62.70
36.89
42.06
46.71
51.01
54.96
55.10
59.07
62.93
66.69
70.36
77.38
84.00
90.23
96.11
101.67
106.93
111.91
116.64
121.14
125.42
29.72
32.99
35.85
38.45
40.80
40.89
43.20
45.43
47.58
49.68
53.72
57.57
61.27
64.82
68.23
71.51
74.67
77.72
80.66
83.50
ꢁ407.70
ꢁ278.59
ꢁ213.61
ꢁ174.55
ꢁ149.22
ꢁ148.40
ꢁ129.55
ꢁ115.23
ꢁ103.94
ꢁ94.76
ꢁ80.66
ꢁ70.26
ꢁ62.19
ꢁ55.71
ꢁ50.37
ꢁ45.87
ꢁ42.04
ꢁ38.72
ꢁ35.83
ꢁ33.29
ꢁ324.23
ꢁ186.81
ꢁ116.46
ꢁ73.31
ꢁ44.86
ꢁ43.94
ꢁ22.55
ꢁ6.23
6.67
1
2
2
2
3
3
4
4
5
6
7
8
9
10.86
12.55
14.15
14.22
15.87
17.50
19.11
20.68
23.66
26.42
28.96
31.30
33.44
35.42
37.24
38.92
40.48
41.92
17.13
33.08
44.70
53.53
60.47
66.06
1000
1100
1200
1300
1400
1500
a
70.64
74.46
77.69
80.46
82.84
ꢀ
ꢁ2
2
3
At temperatures from 150 K to 500 K thermodynamic functions can be calculated using equation C ðTÞ=R ¼ a1 þ a2T þ a3T þ a4T þ a5T
p;m
4
ꢁ2
ꢁ4
ꢁ7
.
with a
b
1
= 10.890, a
ꢀ
2
= ꢁ2.680 Æ 10 , a
3
= 1.361 Æ 10 , a
4
= 1.516 Æ 10 , a
5
= ꢁ1.550 Æ 10
ꢀ
ꢀ
ꢀ
m
U ðTÞ ¼ S ðTÞ ꢁ fH ðTÞ ꢁ H ð0Þg=T.
m
m
m

18
A.I. Druzhinina et al. / J. Chem. Thermodynamics 38 (2006) 10–19
conformers. We did not take into account the difference
between geometry and vibrational frequencies of TG
and TT conformers of IBTBE because this results in
energy) were calculated for the solid and liquid states
over the temperature range studied and for the ideal
gas state at T = 298.15 K.
ꢀ
ꢀ
insignificant changes of S ðT Þ and C ðT Þ values (0.05
The ideal gas thermodynamic properties of IBTBE
were calculated over the temperature range (0 to 1500)
K by combining the DFT results with absolute entropies
determined from calorimetric investigation. The B3LYP
method provides a good way to calculate the structure
and vibrational frequencies of the molecules, and tor-
sional potential of independent rotors. A multi-dimen-
sional model for the energy levels of internal rotation
has not yet been solved and so an empirical correction
for rotor–rotor coupling was used. Since the calculated
entropy values were fitted to the experimental ones,
the accuracy of calculated thermodynamic functions
was believed to be rather high. As to enthalpy of forma-
tion values, this work and our previous calculations [3,5]
demonstrate that DFT results applied to isodesmic reac-
tions lead to values whose accuracy is comparable with
experimental.
m
p;m
ꢁ
1
ꢁ1
and 0.15 J Æ K Æ mol , respectively, at T = 298.15 K).
3
.4. Thermodynamic functions
ꢀ
ꢀ
ꢀ
Thermodynamic functions, S ðT Þ; C ðT Þ; and H
m
p;m
m
ꢀ
ðTÞ ꢁ H ð0Þ, were calculated within the framework of
m
the rigid-rotor harmonic-oscillator approximation for
all rotation and vibration modes, except for internal
rotation modes for which the independent-rotor model
was employed. This model results in overestimated val-
ues of thermodynamic functions as compared with
ꢀ
ꢀ
S ðTÞ and C ðT Þ values determined from calorimetric
m
p;m
measurements for alkyl ethers [3]. The greater is the
number of rotors, the greater are the discrepancies be-
tween calculated and experimental values. Especially
large discrepancies are found for tert-alkyl ethers. It is
most likely that the increased error is associated with
the independent-rotor assumption.
As in our previous works [3,5], in this work the con-
tribution to the thermodynamic functions due to cou-
pling of rotor potentials was taken into account by
multiplying the partition function for uncoupling inter-
nal rotations by the empirical factor K The K value
Acknowledgment
This work was supported by the Russian Foundation
for Basic Research under Grant No. 02-02-17009.
rꢁr
rꢁr
of 0.843 was found by fitting to the calorimetric entro-
pies of IBTBE. The difference between experimental
and thus calculated entropies is shown in table 9. It
should be noted that the employed procedure for correc-
tion of the thermodynamic functions does not violate
well-known thermodynamic relations between calcu-
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ꢀ
ꢀ
ꢀ
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p;m
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JCT 04-242