Experimental and Regression VLE Data for Isobutanol + 1-Butanol, Isobutanol + 2-Ethyl-1-hexanol, and 1-Butanol + 2-Ethyl-1-hexanol Binary Systems

Abstract

Vapor–liquid equilibrium (VLE) experimental data are reported for three binary systems: isobutanol + 1-butanol, isobutanol + 2-ethyl-1-hexanol, and 1-butanol + 2-ethyl-1-hexanol. Due to the limited and incomplete data available in the literature, we determined the p-T-x experimental VLE data for these binaries using an equilibrium apparatus, designed and built in our laboratory, which had been used extensively in various determinations. The temperature and pressure ranges for determining the VLE data are as follows: (305.15–388.15) K and (2.284–99.779) kPa for the isobutanol + 1-butanol system, (305.15–455.15) K and (2.284–99.779) kPa for the isobutanol + 2-ethyl-1-hexanol, and (320.15–455.15) K and (3.635–98.039) kPa for the 1-butanol + 2-ethyl-1-hexanol. The experimental VLE data for these binary systems were regressed using the nonrandom two-liquid (NRTL) model. The results indicate a reasonably good agreement between the model and the experimental data, with maximum deviations of 7% in the liquid-phase composition of the most volatile component from the binary and 4.5% in pressure.

1. Introduction

The absence of sufficient equilibrium data over a wide range of temperatures and pressures for the binary systems isobutanol + 1-butanol, isobutanol + 2-ethyl-1-hexanol, and 1-butanol + 2-ethyl-1-hexanol was noted during earlier research on a technological study of the 2-ethyl-1-hexanol recovery from a dioctyl phthalate production plant. This technological study involved simulating different technological solution options. The recovery of 2-ethyl-1-hexanol was achieved through distillation in the existing octanols separation plant [1]. In the recovered 2-ethyl-1-hexanol mixture, the most problematic components, i.e., those that induce deterioration of the quality of dioctyl phthalate as the recovered fractions are returned to manufacturing, are those found under the generic name of “light products”. The isobutanol and 1-butanol are light components present in a typical proportion of 3 mass % in the 2-ethyl-1-hexanol recovery mixture; these are responsible for increasing the color index of the synthesized dioctyl phthalate from the 2-ethyl-1-hexanol. Motivated by our previous observations, this study focuses on the vapor–liquid equilibrium (VLE) characteristics of the three selected binary mixtures. The two inferior alcohols presented in the studied binary systems (isobutanol and 1-butanol) have important uses that have been widely studied in recent years, namely as fuel additives and as alternatives to fossil fuels [2,3,4,5,6,7]. Isobutanol and 1-butanol are not only used in the field of fuels, but also serve as excellent organic solvents, extracting agents, and reactants in the production of chemicals such as butylated amino resins, butyl acrylate, and high-grade epoxy resins [3,8].

2-Ethyl-1-hexanol is an essential and vital chemical reactant and solvent used in the production of plasticizers, coatings, adhesives, and in the manufacture of acrylate and methacrylate esters, as well as a volatile solvent for resins, disinfectants, and detergents [9,10].

According to the specialized literature, the binary system isobutanol + 1-butanol VLE data have been studied since 1955, but the sources in the literature are ancient. Still, we managed to find and study the temperature and pressure range of the equilibrium data reported for the isobutanol + 1-butanol system (see

Table 1. VLE experimental data available in the specialized literature for the isobutanol + 1-butanol binary system.

Pressure, kPa	Temperature, K	Source
2757.9–4136.9	518.92–558.58	[12]
101.32	381.52–388.72	[13]

). The experimental data found in the literature do not present uncertainty values and were not obtained over a wide temperature and pressure range, like those we performed using our equilibrium apparatus. Therefore, we cannot use them in a feasible comparison. Furthermore, for the binary system isobutanol + 1-butanol, there are binary parameters for the NRTL thermodynamic model in the simulation program we use, namely PRO/II software, version 2024 (AVEVA Group plc, Cambridge, UK) [11]. We decided to perform VLE experimental data for the isobutanol + 1-butanol mixture to validate it by comparison with the VLE data present in the simulation program. This approach is validated once again, as we have used this VLE experimental procedure in the study of many binary mixtures.

The VLE data for isobutanol + 2-ethyl-1-hexanol is reported in only one article [14] at a constant pressure of 26.67 kPa and for the temperature range of 349.29 K–415.18 K. Ghanadzadehet al. [15] report VLE data for 1-butanol + 2-ethyl-1-hexanol at atmospheric pressure in the temperature range of 392 K–458.15 K. Furthermore, using this equilibrium data, the authors develop a neural network model to estimate VLE data for this binary system [16,17]. The available VLE data found in the literature are often reported at a limited number of constant pressures, and several studies do not provide uncertainty estimates.

In the databank of the PRO/II software, version 2024 (AVEVA Group plc, Cambridge, UK), no VLE data are available for the binary interaction parameters of systems containing isobutanol, 1-butanol, and 2-ethyl-1-hexanol.

Therefore, it is necessary to measure the VLE data of alcohol–alcohol binary systems across a wide range of temperatures and pressures, as these data play a crucial role in the design and optimization of distillation processes [18] and are essential for evaluating liquid-phase models that are intended to indicate deviations from ideal solution behavior [19]. The synthesis and simulation of any process requires good thermodynamic models based on experimental data. Knowledge of VLE behavior enables process engineers to design more energy-efficient and environmentally friendly processes. The VLE data obtained are used in simulation software for process design and optimization.

This study aims to obtain VLE data for three binary systems, including alcohol + alcohol. The VLE data have been measured within the temperature range of 305.15 to 455.15 K and pressures up to 99.779 kPa for the following binary systems: isobutanol + 1-butanol, isobutanol + 2-ethyl-1-hexanol, and 1-butanol + 2-ethyl-1-hexanol.

Vapor–liquid equilibrium data for these binaries were determined using a static equilibrium apparatus built in our laboratory, as described in our previous articles [20,21]. The p-T-x experimental data obtained were correlated using the NRTL (nonrandom two-liquid) model. The VLE data obtained with NRTL were compared with data calculated with the UNIQUAC Functional-Group Activity Coefficients (UNIFAC) model and the IDEAL model.

The resulting binary parameters of the NRTL model for vapor–liquid equilibrium can be used in various studies of chemical process simulation involving the studied binary systems, such as multi-fractionations to result in a single alcohol from C1–C8 mixed alcohols produced by using coal [18], a separation process based on the characteristics and composition of butanol and octanol raffinates discharged from butanol and octanol plants [19], and the extractive distillation process for n-butanol/isobutanol/water separation, together with the LLE data [3,22], especially when the solvent used is 2-ethyl-1-hexanol [23,24].

2. Materials and Methods

The specifications of the three alcohols used in the experiments are presented in

Table 2. Specifications of the chemicals used.

Chemical Name	CAS Registry Number	Supplier	Mass Fraction Purity (Provided by Supplier)
1-butanol	71-36-3	Sigma Aldrich (St. Louis, MO, USA)	≥0.998
isobutanol	78-83-1	Sigma Aldrich (St. Louis, MO, USA)	≥0.995
2-ethyl-1-hexanol	104-76-7	Sigma Aldrich (St. Louis, MO, USA)	≥0.996

. These substances did not require advanced purification; the purity of the product provided by the supplier was high and sufficient for the VLE experiments.

The equilibrium apparatus used to determine the VLE data for the three binaries is a static one built in our laboratory. The construction of the apparatus was first described in 2013 in one of our previous articles [20]. Since then, the description of the experimental procedure for determining the VLE data for a binary system has been presented in many of our published articles [21]. The equilibrium apparatus and experimental procedure have been validated through the measurement and publication of VLE experimental data for twenty-three binary systems that include propylene glycols, water, and aromatic hydrocarbons.

The p-T-x experimental equilibrium data were determined by measuring the vapor pressure at a maintained temperature value and fixed compositions for the binary system and the pure components. The five mixture compositions for each binary system were prepared by mass using a Mettler Toledo AB204-S (Mettler-Toledo, LLC, Columbus, OH, USA) electronic balance accurate to 0.0001 g. The mixture sample was introduced into the equilibrium cell, subsequently cooled, and degassed. The equilibrium cell was then connected to a U-shaped manometer and placed in a thermostatically controlled oil bath, where the temperature was carefully selected based on the mixture composition and the expected system pressure, and maintained with an accuracy of ±0.05 K. The temperature was maintained until the level of the manometric liquid in the two branches of the U-shaped tube remained constant. Once the system’s pressure remained unchanged for at least 30 min, indicated by a stable liquid level in both branches of the manometer, the pressure was measured.

In the experiments, the vapor pressure was measured using a DPI 705 sensor with a measuring range up to 100 kPa and an accuracy (combined non-linearity, hysteresis, and repeatability) of ±0.1% full-scale, and the temperature was measured with a VWR International, LLC, NIST traceable digital thermometer, with an accuracy of ±0.05% and a resolution of 0.001 K.

The standard uncertainty for the temperature was calculated for each VLE data binary set, taking into account the digital thermometer uncertainty and pressure fluctuations during the experiment. The standard concentration uncertainty, expressed as the mole fraction of the most volatile alcohol for each binary system, is due to the mixture preparation. The standard temperature uncertainty and the standard concentration uncertainty are ±0.01 K and ±0.002, respectively. The vapor pressure was measured three times for each constant temperature; the average value and the standard uncertainty for each average value are presented in

Table 3. Experimental VLE data for temperature T, pressure p with standard uncertainty u(p), and mole fraction for the binary system isobutanol (1) + 1-butanol (2) *.

T/K	p/kPa	u(p)/kPa	T/K	p/kPa	u(p)/kPa	T/K	p/kPa	u(p)/kPa
$x_1=0.0000$
325.15	5.174	0.06	347.65	16.823	0.53	370.15	45.674	1.22
329.65	6.658	0.08	352.15	20.816	1.05	374.65	54.749	1.21
334.15	8.495	0.10	356.65	25.580	1.16	379.15	65.259	1.18
338.65	10.750	0.34	361.15	31.228	1.15	383.65	77.369	1.26
343.15	13.499	0.41	365.65	37.881	1.12	388.15	91.252	1.23
$x_1=0.1023$
326.15	5.823	0.07	348.65	18.421	0.81	371.15	49.525	1.25
330.65	7.453	0.11	353.15	22.864	0.82	375.65	58.998	1.31
335.15	9.457	0.13	357.65	27.861	1.18	380.15	70.068	1.20
339.65	11.902	0.63	362.15	34.109	1.16	384.65	82.903	1.45
344.15	14.890	0.74	366.65	40.874	1.11	389.15	97.029	1.48
$x_1=0.3007$
324.15	5.756	0.08	346.65	18.557	1.11	369.15	49.898	1.32
328.65	7.453	0.26	351.15	22.869	1.08	373.65	59.610	1.45
333.15	9.559	0.31	355.65	28.065	1.10	378.15	70.645	1.30
337.65	11.935	0.85	360.15	34.007	1.22	382.65	83.854	1.43
342.15	14.856	0.97	364.65	41.477	1.23	387.15	98.998	1.43
$x_1=0.5011$
322.15	5.857	0.09	344.65	18.625	0.83	367.15	49.864	1.36
326.65	7.453	0.22	349.15	22.835	0.91	371.65	59.508	1.28
331.15	9.457	0.31	353.65	28.166	1.12	376.15	71.426	1.24
335.65	11.935	0.58	358.15	34.211	1.11	380.65	84.703	1.37
340.15	14.958	0.72	362.65	41.511	1.25	385.15	98.964	1.42
$x_1=0.7006$
320.15	5.688	0.11	342.65	18.014	0.92	365.15	50.204	1.23
324.65	7.317	0.27	347.15	22.733	0.92	369.65	59.983	1.23
329.15	9.253	0.31	351.65	28.031	1.12	374.15	71.528	1.31
333.65	11.460	0.67	356.15	34.244	1.14	378.65	84.941	1.35
338.15	14.788	0.81	360.65	41.409	1.15	383.15	99.779	1.40
$x_1=0.9027$
318.15	5.318	0.18	340.65	17.946	0.81	363.15	49.321	1.11
322.65	6.920	0.16	345.15	22.122	0.87	367.65	59.440	1.23
327.15	9.049	0.53	349.65	26.74	0.97	372.15	71.154	1.23
331.65	11.222	0.64	354.15	33.396	0.33	376.65	84.601	1.45
336.15	14.414	0.64	358.65	40.594	1.12	381.15	99.474	1.42
$x_1=1.0000$
305.15	2.284	0.07	327.65	9.178	0.52	350.15	28.850	1.11
309.65	3.086	0.17	332.15	11.746	0.67	354.65	35.399	1.10
314.15	4.120	0.19	336.65	14.895	0.67	359.15	43.119	1.17
318.65	5.438	0.31	341.15	18.721	0.75	363.65	52.161	1.28
323.15	7.101	0.52	345.65	23.333	0.71	368.15	62.683	1.21

* Standard uncertainties u are $u\left(T\right)$ = 0.01 K and $u\left(x\right)$ = 0.0002.

,

Table 4. Experimental VLE data for temperature T, pressure p with standard uncertainty u(p), and mole fraction for the binary system isobutanol (1) + 2-ethyl-1-hexanol (2) *.

T/K	p/kPa	u(p)/kPa	T/K	p/kPa	u(p)/kPa	T/K	p/kPa	u(p)/kPa
$x_1=0.0000$
370.15	3.635	0.07	400.15	14.651	0.45	430.15	44.171	1.11
375.15	4.699	0.09	405.15	17.924	0.52	435.15	51.918	1.15
380.15	6.011	0.11	410.15	21.762	0.55	440.15	60.686	1.17
385.15	7.613	0.23	415.15	26.234	0.64	445.15	70.563	1.17
390.15	9.553	0.41	420.15	31.409	0.78	450.15	81.640	1.22
395.15	11.880	0.49	425.15	37.362	0.79	455.15	94.008	1.23
$x_1=0.1085$
349.15	4.126	0.15	384.15	16.745	0.42	419.15	58.659	1.12
354.15	5.212	0.18	389.15	20.594	0.55	424.15	67.385	1.12
359.15	6.299	0.10	394.15	24.771	0.58	429.15	75.942	1.31
364.15	7.623	0.10	399.15	28.947	0.79	434.15	87.046	1.29
369.15	9.205	0.21	404.15	34.958	0.81	439.15	98.320	1.29
374.15	11.306	0.23	409.15	42.326	0.92
379.15	13.833	0.30	414.15	50.204	0.97
$x_1=0.3014$
333.15	4.092	0.09	363.15	16.655	0.21	393.15	51.528	0.89
338.15	5.518	0.11	368.15	20.153	0.22	398.15	60.968	0.96
343.15	6.978	0.12	373.15	24.329	0.31	403.15	71.460	1.02
348.15	8.540	0.12	378.15	29.762	0.35	408.15	83.990	1.34
353.15	10.679	0.12	383.15	35.942	0.47
358.15	13.531	0.19	388.15	43.480	0.78
$x_1=0.5017$
328.15	4.974	0.10	353.15	16.825	0.31	378.15	48.438	0.62
333.15	6.469	0.12	358.15	21.002	0.32	383.15	58.217	0.79
338.15	8.370	0.12	363.15	26.299	0.34	388.15	70.034	0.99
343.15	10.781	0.15	368.15	32.445	0.35	393.15	83.413	1.24
348.15	13.362	0.21	373.15	39.847	0.46	398.15	97.844	1.24
$x_1=0.7041$
319.15	4.160	0.17	344.15	15.433	0.47	369.15	45.552	0.89
324.15	5.450	0.18	349.15	19.508	0.46	374.15	55.229	0.92
329.15	7.284	0.19	354.15	24.465	0.65	379.15	66.876	1.10
334.15	9.524	0.29	359.15	30.17	0.65	384.15	82.292	1.08
339.15	12.003	0.31	364.15	37.063	0.74	389.15	96.893	1.20
$x_1=0.9006$
315.15	4.126	0.15	339.15	15.365	0.57	363.15	45.246	1.19
319.15	5.246	0.15	343.15	18.693	0.87	367.15	53.396	1.11
323.15	6.570	0.29	347.15	22.598	0.85	371.15	62.564	1.21
327.15	8.098	0.29	351.15	27.012	0.84	375.15	73.090	1.21
331.15	10.102	0.32	355.15	32.488	0.96	379.15	85.008	1.24
335.15	12.445	0.42	359.15	38.387	0.97	383.15	97.776	1.24
$x_1=1.0000$
305.15	2.284	0.14	327.65	9.178	0.31	350.15	28.850	0.85
309.65	3.086	0.15	332.15	11.746	0.39	354.65	35.399	0.87
314.15	4.120	0.15	336.65	14.895	0.45	359.15	43.119	0.97
318.65	5.438	0.16	341.15	18.721	0.42	363.65	52.161	1.11
323.15	7.101	0.26	345.65	23.333	0.55	368.15	62.683	1.26

* Standard uncertainties u are $u\left(T\right)$ = 0.01 K and $u\left(x\right)$ = 0.0002.

and

Table 5. Experimental VLE data for temperature T, pressure p with standard uncertainty u(p), and mole fraction for the binary system 1-butanol (1) + 2-ethyl-1-hexanol (2) *.

T/K	p/kPa	u(p)/kPa	T/K	p/kPa	u(p)/kPa	T/K	p/kPa	u(p)/kPa
$x_1=0.0000$
370.15	3.635	0.08	400.15	14.651	0.32	430.15	44.171	1.09
375.15	4.700	0.17	405.15	17.924	0.42	435.15	51.918	1.09
380.15	6.011	0.11	410.15	21.762	0.41	440.15	60.686	1.18
385.15	7.613	0.19	415.15	26.234	0.71	445.15	70.563	1.21
390.15	9.553	0.21	420.15	31.409	0.74	450.15	81.640	1.21
395.15	11.880	0.21	425.15	37.362	0.95	455.15	94.008	1.21
$x_1=0.1002$
359.15	4.408	0.09	389.15	16.230	0.33	419.15	47.250	0.82
364.15	5.620	0.11	394.15	19.745	0.32	424.15	55.501	0.82
369.15	6.944	0.19	399.15	23.820	0.32	429.15	64.771	0.98
374.15	8.676	0.19	404.15	28.506	0.54	434.15	74.516	1.02
379.15	11.019	0.21	409.15	33.939	0.54	439.15	84.805	1.14
384.15	13.531	0.21	414.15	39.949	0.63	444.15	96.580	1.18
$x_1=0.3423$
340.15	4.194	0.15	370.15	17.708	0.52	400.15	55.229	0.99
345.15	5.450	0.17	375.15	21.952	0.52	405.15	64.737	1.03
350.15	6.978	0.18	380.15	27.385	0.61	410.15	76.452	1.15
355.15	8.879	0.21	385.15	33.056	0.63	415.15	87.793	1.24
360.15	11.256	0.32	390.15	38.251	0.85
365.15	14.448	0.41	395.15	46.978	0.84
$x_1=0.5029$
335.15	4.635	0.08	360.15	15.297	0.43	385.15	42.971	1.02
340.15	5.789	0.11	365.15	18.964	0.42	390.15	51.868	1.12
345.15	7.351	0.29	370.15	23.480	0.51	395.15	61.715	1.12
350.15	9.355	0.29	375.15	29.151	0.73	400.15	73.260	1.24
355.15	12.003	0.31	380.15	35.399	0.86	405.15	86.095	1.32
$x_1=0.7116$
329.15	4.635	0.09	354.15	15.942	0.53	379.15	46.876	1.03
334.15	5.959	0.15	359.15	20.170	0.61	384.15	56.010	1.17
339.15	7.657	0.27	364.15	25.144	0.75	389.15	67.385	1.24
344.15	9.864	0.28	369.15	31.460	0.73	394.15	80.798	1.12
349.15	12.614	0.31	374.15	38.523	0.86	399.15	95.331	1.40
$x_1=0.9028$
323.15	4.024	0.14	348.15	15.365	0.71	373.15	46.774	0.97
328.15	5.416	0.15	353.15	19.440	0.72	378.15	56.621	1.01
333.15	7.148	0.25	358.15	24.839	0.92	383.15	68.947	1.12
338.15	9.219	0.38	363.15	30.577	0.94	388.15	83.243	1.32
343.15	11.902	0.51	368.15	37.810	0.94	393.15	97.810	1.41
$x_1=1.0000$
320.15	3.869	0.09	345.15	14.900	0.55	370.15	45.674	1.05
325.15	5.174	0.12	350.15	18.952	0.55	375.15	55.843	1.17
330.15	6.844	0.23	355.15	23.900	0.87	380.15	67.806	1.30
335.15	8.957	0.39	360.15	29.890	0.22	385.15	81.791	1.31
340.15	11.608	0.39	365.15	37.088	0.98	390.15	98.039	1.42

* Standard uncertainties u are $u\left(T\right)$ = 0.01 K and $u\left(x\right)$ = 0.0002.

. The standard uncertainty $u\left(x_i\right)$ was calculated using the relation (1) [25]. Since the compositions of the vapor phases were not measured, the thermodynamic consistency could not be evaluated.

$$u\left(x_i\right)={\left({\displaystyle \frac{1}{n(n-1)}}{\sum _{k=1}^n(X_{i,k}-\overline{X_i})}^2\right)}^{1/2}$$

(1)

3. Results and Discussions

The p-T-x experimental vapor–liquid equilibrium data are presented in Table 3, Table 4 and Table 5. The standard uncertainties associated with vapor pressures lower than 10 kPa are at most ±0.5 kPa, while for higher vapor pressures the maximum uncertainty is ±1.5 kPa.

Table 3 shows the VLE experimental data for the isobutanol + 1-butanol system in the temperature range of (305.15–388.15) K and pressure range of (2.284–99.779) kPa.

Table 4 and Table 5 present the VLE experimental data for the two binaries containing 2-ethyl-1-hexanol. The temperature and pressure ranges for the VLE data are as follows: (305.15–455.15) K and (2.284–99.779) kPa for the isobutanol + 2-ethyl-1-hexanol system and (320.15–455.15) K and (3.635–98.039) kPa for the 1-butanol + 2-ethyl-1-hexanol system.

Experimental VLE data were regressed using the PRO/II software, version 2024 (AVEVA Group plc, Cambridge, UK)., applying the default objective function (S). This function, Equation (2), minimizes the sum of squared relative deviations between the calculated vapor pressure (P_icalc) and the experimental vapor pressure (P_iexpt). Convergence is achieved when the objective function value falls below the default tolerance of 1 × 10⁻⁶.

$$S=\sum _{i=1}^N{\left(1-{\displaystyle \frac{P_{icalc}}{P_{iexpt}}}\right)}^2$$

(2)

The binary systems we studied are non-ideal in the liquid phase; therefore, we performed a regression of the VLE experimental data using models based on liquid-phase activity coefficients. The experimental VLE data for the three binary systems were regressed using the NRTL model [26,27] and the Universal Quasi Chemical (UNIQUAC) model [28]. Similar modeling approaches have also been reported in recent studies involving alcohol-based or DBU-containing systems [29,30]. Due to the large maximum percentage relative deviations for the pressure and liquid-phase composition from the regression of experimental data using the UNIQUAC model, the results obtained using the UNIQUAC model were not included.

For the isobutanol + 1-butanol system, we used only the NRTL model with three parameters in the regression of the VLE experimental data, as the PRO/II software, version 2024 (AVEVA Group plc, Cambridge, UK) database contains binary interaction parameters with only three parameters for the same thermodynamic model for this binary system. The experimental data for the other two binaries involving 2-ethyl-1-hexanol, for which there are no parameters in the database of the PRO/II software, version 2024 (AVEVA Group plc, Cambridge, UK), were regressed using the NRTL model with all possible numbers of parameters (three, five, six, and eight). Equations (3)–(7) represent the NRTL equations according to the PRO/II software, version 2024 (AVEVA Group plc, Cambridge, UK) reference manual [11].

$$\mathit{ln}{\gamma }_i={\displaystyle \frac{{\sum }_j{\tau }_{ji}{G}_{ji}{x}_{ji}}{{\sum }_k{G}_{ki}{x}_k}}+\sum _j{\displaystyle \frac{x_j{G}_{ij}}{{\sum }_k{G}_{kj}{x}_k}}\left({\tau }_{ij}-{\displaystyle \frac{{\sum }_k{x}_k{\tau }_{kj}{G}_{kj}}{{\sum }_k{G}_{kj}{x}_k}}\right)$$

(3)

$${\tau }_{ij}=a_{ij}+{\displaystyle \frac{b_{ij}}{T}}+{\displaystyle \frac{c_{ij}}{T^{2 }}}(\mathrm{unit} \mathrm{is} \mathrm{K})$$

(4)

$${\tau }_{ij}=a_{ij}+{\displaystyle \frac{b_{ij}}{RT}}+{\displaystyle \frac{c_{ij}}{{R^{2}T}^{2 }}}(\mathrm{unit} \mathrm{is} \mathrm{kcal} \mathrm{or} \mathrm{kJ})$$

(5)

$$G_{ij}=\exp \left(-{\alpha }_{ji}{\tau }_{ij}\right)$$

(6)

$${\alpha }_{ji}={\alpha }_{ji}^{\prime }+{\beta }_{ji}^{\prime }T$$

(7)

Table 6. Binary interaction parameters of the NRTL model with three parameters from the PRO/II software, version 2024 (AVEVA Group plc, Cambridge, UK) database, and those results obtained by correlating the experimental data for the system isobutanol + 1-butanol.

NRTL Binary Interaction Parameters	Isobutanol + 1-Butanol from PRO II Database	Isobutanol + 1-Butanol Results by Correlating the Experimental Data
${\mathrm{b}}_{\mathrm{i}\mathrm{j}}/\mathrm{K}$	101.7760	−2352.36720
$b_{ji}/K$	−92.3501	2143.25720
${\alpha }_{ij}^{\prime }$	0.3085	−0.01532

presents the values of the binary interaction parameters for the isobutanol + 1-butanol system from the PRO/II software, version 2024 (AVEVA Group plc, Cambridge, UK) database and the results obtained by correlating the experimental data from this study.

The differences in the values of the binary parameters can be explained by the temperature intervals of the experimental data, which may differ from those of the VLE in the software database. The maximum percentage relative deviations for the pressure and liquid-phase composition from the regression of the experimental data for the isobutanol + 1-butanol system, using the NRTL model with three parameters, are quite small and acceptable, at −4.4378% and −6.8919%.

Figure 1 presents the vapor pressure curves as a function of temperature for the binary isobutanol + 1-butanol mixture at different concentrations, using experimental data from this study in contrast with data calculated using the binary parameters resulting from this study and those from the PRO/II software, version 2024 (AVEVA Group plc, Cambridge, UK) database. As expected, the vapor pressure increases as the concentration of isobutanol in the mixture increases, for the same temperature. Also, for the same concentration of isobutanol in the mixture, the vapor pressure values increase as the temperature increases. This graphical comparison of the vapor pressures shows a similarity between the calculated data using PRO/II software, version 2024 (AVEVA Group plc, Cambridge, UK) database parameters and the experimental data, as well as the calculated data using the parameters from this study. This statement further strengthens the validation of the experimental method that we used.

Table 7. Maximum percentage relative deviations and average percentage absolute deviations resulting from the regression of the experimental data using the NRTL model for isobutanol(1) + 2-ethyl-1-hexanol(2) and 1-butanol(1) + 2-ethyl-1-hexanol(2) binary systems.

Binary System	Number of NRTL Model Parameters	NRTL Model Parameters	Maximum Percentage Relative Deviation for the Pressure	Average Percentage Absolute Deviation for the Pressure	Maximum Percentage Relative Deviation for the Liquid-Phase Composition of the Most Volatile Component	Average Percentage Absolute Deviation for the Liquid-Phase Composition of the Most Volatile Component
isobutanol + 2-ethyl-1-hexanol	three	$b_{ij}$$, b_{ji}$$, {\alpha }_{ij}^{\prime }$	−1.8982	0.3376	15.7101	1.4094
	five	$a_{ij}$$, b_{ij}$$, a_{ji}$$, b_{ji}$$, {\alpha }_{ij}^{\prime }$	−1.8226	0.2470	8.3500	1.1548
	six	$a_{ij}$$, b_{ij}$$, a_{ji}$$, b_{ji}$$, {\alpha }_{ij}^{\prime }$$, {\beta }_{ij}^{\prime }$	−1.8516	0.2955	7.6593	1.2311
	eight	$a_{ij}$$, b_{ij}$$, c_{ij}$$, a_{ji}$$, b_{ji}$$, c_{ji}$$, {\alpha }_{ij}^{\prime }$$, {\beta }_{ij}^{\prime }$	−1.9198	0.2669	−6.6373	1.2709
1-butanol + 2-ethyl-1-hexanol	three	$b_{ij}$$, b_{ji}$$, {\alpha }_{ij}^{\prime }$	−2.8868	0.6820	−26.5523	4.2562
	five	$a_{ij}$$, b_{ij}$$, a_{ji}$$, b_{ji}$$, {\alpha }_{ij}^{\prime }$	2.0247	0.7309	19.7846	3.0232
	six	$a_{ij}$$, b_{ij}$$, a_{ji}$$, b_{ji}$$, {\alpha }_{ij}^{\prime }$$, {\beta }_{ij}^{\prime }$	1.6656	0.5800	11.6417	2.7918
	eight	$a_{ij}$$, b_{ij}$$, c_{ij}$$, a_{ji}$$, b_{ji}$$, c_{ji}$$, {\alpha }_{ij}^{\prime }$$, {\beta }_{ij}^{\prime }$	1.4753	0.3120	−6.9102	1.5897

presents the maximum percentage relative deviations and the average percentage absolute deviations for the pressure and the liquid-phase concentration of the most volatile component (1) for the binaries isobutanol + 2-ethyl-1-hexanol and 1-butanol + 2-ethyl-1-hexanol.

The deviations are displayed for different variants of regression, using the NRTL thermodynamic model, with three, five, six, and eight binary interaction parameters. It can be observed that the maximum percentage relative deviations decrease as the number of binary interaction parameters in the model increases. The maximum percentage relative deviations for the pressure show similar values for the four variants of the NRTL model with the different number of binary interaction parameters used in the regression. However, it can be observed that the maximum percentage relative deviations for the liquid-phase composition of the most volatile component differ, and the lowest values are obtained for the NRTL model with eight parameters. This improved accuracy is attributed to the inclusion of temperature-dependent parameters, which enhance the model’s flexibility in representing non-ideal liquid-phase behavior. The binary interaction parameters for the isobutanol + 2-ethyl-1-hexanol and 1-butanol + 2-ethyl-1-hexanol binaries, obtained from the regression of the experimental data, are presented in

Table 8. The eight binary interaction parameters of the NRTL model for the binaries isobutanol + 2-ethyl-1-hexanol and 1-butanol + 2-ethyl-1-hexanol binary systems.

NRTL Binary Parameters	Isobutanol + 2-Ethyl-1-hexanol	1-Butanol + 2-Ethyl-1-hexanol
$a_{ij}$	−0.31998	−0.62742
$b_{ij}/K$	−141.92441	−9.80658
$c_{ij}/K^2$	60,605.14069	98,335.43425
$a_{ji}$	0.00301	−0.07301
$b_{ji}/K$	−75.28924	153.21176
$c_{ji}/K^2$	62,224.75031	−65,532.18635
${\alpha }_{ij}^{\prime }$	−0.87252	1.00000
${\beta }_{ij}^{\prime }/{\displaystyle \frac{1}{K}}$	0.00574	0.00900

, only for the NRTL thermodynamic model with eight parameters, because it represents the model variant that, following the regression, gave the lowest deviation values for the quantities involved.

The dispersion of the relative deviations of the calculated pressure with the NRTL thermodynamic model with eight parameters from those determined experimentally for the isobutanol + 2-ethyl-1-hexanol and 1-butanol + 2-ethyl-1-hexanol binaries is displayed in Figure 2.

It can be observed from Figure 2 that the binary system isobutanol + 2-ethyl-1-hexanol has a broader domain of relative deviations (−2% to 1.25%) compared with those for the 1-butanol + 2-ethyl-1-hexanol binary system (−1% to −1.5%).

Figure 3 displays the relative deviations between the calculated concentration using the NRTL model with eight parameters of the most volatile component in the liquid phase of the mixture and the experimentally determined concentrations of the most volatile component in the liquid phase of the mixture for the binary systems isobutanol + 2-ethyl-1-hexanol and 1-butanol + 2-ethyl-1-hexanol. For the deviations in the concentrations of the liquid phase, we can observe a greater dispersion of the values of the deviations for both binary systems, ranging from −7% to 7%.

The vapor pressure curves as a function of temperature for the binary isobutanol + 2-ethyl-1-hexanol mixture at different concentrations are displayed in Figure 4. Figure 5 shows the vapor pressure curves for the binary 1-butanol + 2-ethyl-1-hexanol.

From both figures, it can be observed that the values of the vapor pressure calculated using the PRO/II software, version 2024 (AVEVA Group plc, Cambridge, UK)and the NRTL thermodynamic model with eight binary interaction parameters overlap with the experimental values of the vapor pressure for the entire concentration domain. This indicates a good correlation between the experimental and the calculated values of the vapor pressure.

The VLE data for two binary systems that include 2-ethyl-1-hexanol, obtained from this study, were compared with the data calculated using the IDEAL [31] and UNIFAC (UNIquac Functional-group Activity Coefficients) [32] models, as the VLE data in the literature are insufficient and incomplete for comparison. The improved UNIFAC Dortmund was used to calculate the VLE data, as it includes adjustable parameters with temperature dependence implied in the calculation of activity coefficients [33]. Similar methodology was employed by Caqueret et al. (2023), who compared NRTL and UNIFAC Dortmund models for binary mixtures of ethyl acetate and branched alkanes at 101.3 kPa [34]. The IDEAL model was included as a comparison to evaluate the deviation from ideal behavior of the studied binary systems. The comparison of the VLE data for the isobutanol + 2-ethyl-1-hexanol (Figure 6) and 1-butanol + 2-ethyl-1-hexanol (Figure 7) systems was realized by calculus using the models: first NRTL, with the binary interaction parameters obtained in this study, then IDEAL and UNIFAC, and finally, plotting the data in the T-x-y diagrams at constant pressures of 20 and 80 kPa.

Figure 6 and Figure 7 show that for the two studied binaries, the T-x-y diagrams calculated using the NRTL model, which contains parameters derived from experimental data, are quite different from those calculated using the UNIFAC and IDEAL models. The bubble-point curve traced with the NRTL model shows significant differences compared to the other two models used at 20 kPa in the concentration range of 0–0.5 for the isobutanol + 2-ethyl-1-hexanol system and the concentration range of 0–0.9 for the 1-butanol + 2-ethyl-1-hexanol system. For the isobutanol + 2-ethyl-1-hexanol binary, it can be observed that the dew-point curves calculated with all mentioned thermodynamic models at 20 kPa are almost identical. In comparison, at 80 kPa, some differences can be observed. The dew-point curves exhibit considerable differences at 20 kPa and 80 kPa across the entire concentration range for the binary 1-butanol + 2-ethyl-1-hexanol system, as predicted by the NRTL model, compared to the UNIFAC and IDEAL models. This way, we can conclude that the differences between the bubble- and dew-point curves calculated using the completed NRTL thermodynamic model and those predicted with the UNIFAC or calculated with the IDEAL thermodynamic models increase with pressure for both binaries. The two studied binaries display significant non-ideal behavior in the liquid phase, primarily due to molecular interactions such as hydrogen bonding and polarity differences between the components. These effects become more relevant at elevated temperatures, where deviations from ideality are typically more pronounced. This behavior justifies the use of the NRTL model, which can account for such non-idealities through fitted interaction parameters.

4. Conclusions

VLE experimental data are reported within the temperature range of 305.15 to 455.15 K and pressures up to 99.779 kPa for the following binary systems: isobutanol + 1-butanol, isobutanol + 2-ethyl-1-hexanol, and 1-butanol + 2-ethyl-1-hexanol.

Vapor–liquid equilibrium data for these binaries were determined using a static equilibrium apparatus built in our laboratory, which has been used and validated in many of our studies. The p-T-x experimental data obtained were regressed using the NRTL model. The low relative deviations indicate a strong correlation between experimental and calculated data. Overall, the results demonstrate a satisfactory agreement between the model and the experimental data, with maximum relative deviations of up to 7% in the liquid-phase composition of the more volatile component and up to 4.5% in pressure.

The VLE data in the literature are insufficient and incomplete to make the comparison with the VLE data obtained in this study for the isobutanol + 2-ethyl-1-hexanol and 1-butanol + 2-ethyl-1-hexanol systems; therefore, the data were compared with calculated data using a model that predicts data, namely UNIFAC, and with calculated data using the IDEAL model to evaluate the deviation from ideal behavior.

The T-x-y diagrams that we calculated and plotted at two different pressures using the NRTL model, which contains parameters from this study of the involved binary systems, show a different behavior than the IDEAL and UNIFAC models. The bubble-point curve traced with the NRTL model shows significant differences compared to the other two models. The differences between the bubble- and dew-point curves calculated using the completed NRTL thermodynamic model and those predicted with UNIFAC or calculated with the IDEAL thermodynamic model increase with pressure for the isobutanol + 2-ethyl-1-hexanol and 1-butanol + 2-ethyl-1-hexanol binaries.

The binary systems investigated display non-ideal behavior in the liquid phase; nevertheless, none of them form azeotrope mixtures under the studied conditions. To accurately describe this non-ideality—particularly relevant at elevated temperatures—the NRTL model can be used, as it incorporates binary interaction parameters fitted to experimental VLE data, ensuring a reliable representation of the phase behavior of such binary systems.

The VLE experimental data, together with the resulting binary parameters of the NRTL model for the isobutanol + 1-butanol, isobutanol + 2-ethyl-1-hexanol, and 1-butanol + 2-ethyl-1-hexanol binaries, are sufficiently accurate and can be used in various studies of chemical process simulation involving the studied binary systems.