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Vapor–liquid equilibria for aqueous solutions of lactic acid have been measured by Sanz et al. [30] and Vu [31]. Sanz et al. used gas chromatography to measure compositions of both phases. They reported only L₂ and L₃ oligomers of lactic acid. Both flame ionization and thermal conductivity detectors were used, which enabled quantification of the water. Details of the calculations using the peak areas were not provided. Due to the low volatility of oligomers, error in the overall lactic acid concentrations of the liquid phase is likely. Analysis of the data indicates that is nearly constant across the reported compositions, though p should change as shown in Figure 2.6, indicating that the reported mole fractions are not representative of reaction equilibria.

TABLE 2.4 Liquid Phase Mole Fractions of Lactic Acid + Water VLE at 101.33 kPa

T (K)	True x water	True	True	True	True	Apparent
378.25	0.79	0.19	0.015	0.0012	0.0001	0.22
379.25	0.8	0.18	0.014	0.0011	0.000087	0.21
380.25	0.73	0.24	0.024	0.0023	0.00023	0.30
380.75	0.72	0.25	0.025	0.0025	0.00025	0.31
381.75	0.69	0.28	0.03	0.0033	0.00037	0.35
381.85	0.71	0.26	0.027	0.0027	0.0003	0.32
383.35	0.68	0.28	0.032	0.0035	0.0004	0.36
387.35	0.58	0.36	0.054	0.008	0.0012	0.50
391.65	0.53	0.39	0.066	0.011	0.0019	0.56
399.85	0.42	0.45	0.1	0.023	0.0051	0.74
402.25	0.46	0.43	0.087	0.018	0.0036	0.67
404.05	0.46	0.43	0.088	0.018	0.0036	0.67
409.15	0.42	0.45	0.1	0.023	0.0053	0.74

The coexisting vapor compositions are given in Table 2.5 for each temperature reported by Vu [31].

Vu measured isobaric vapor–liquid equilibrium (VLE) data of lactic acid + water using a Fischer recirculating apparatus (model VLE 100D). Pure water, acetone, and ethanol were used to calibrate the pressure and temperature sensors. Heating was regulated to maintain a mean recirculation speed of 30 drops per minute. The mixtures were equilibrated for at least 12 h to ensure the equilibrium was reached, before each 0.5 mL sample was taken from condensed vapor and liquid for analyses. The equilibrium state was indicated by a constant pressure and temperature of the system. Vu further quantified the oligomers using a Hewlett‐Packard 1090 Liquid Chromatograph, equipped with an ultraviolet detector (Hitachi L400H) at a wavelength of 210 nm. The mobile phase was acetonitrile (ACN) + water in a gradient mode (0% ACN (t = 0) to 60% ACN (t = 20 min) to 90% ACN (t = 25 min) to 0% ACN (t = 28 min) at 1.0 mL/min. The Novapak C18 column (3.9 × 150 mm) was used and both ACN and water were acidified by 2 mL of 85% (w/v) phosphoric acid per 1 L of solvent, equivalent to pH = 1.3. Further details are provided in [27].

VLE data of Vu are summarized in Tables 2.4 and 2.5. The rows in the two tables at the same temperature represent the coexisting phases. The true mole fractions represent the moles of a species divided by the total moles present in the equilibrated mixture. The subscripts denote the species as defined in Equations 2.4–2.6. The apparent mole fractions in the last column are calculated by taking a basis of one true mole, recognizing that the water in the analysis includes the water released by condensation of the oligomers, thus , and the apparent moles of lactic acid are determined by .

TABLE 2.5 Vapor Phase Mole Fractions of Lactic Acid + Water VLE at 101.33 kPa.

T (K)	True y water	True	True	True	True	Apparent
378.25	1	0.00051	2.9E−07	1.7E−10	9.6E−14	5.1E−04
379.25	1	0.00072	5.8E−07	4.7E−10	3.8E−13	7.2E−04
380.25	1	0.0011	1.5E−06	1.9E−09	2.4E−12	1.1E−03
380.75	1	0.00083	7.8E−07	7.4E−10	6.9E−13	8.3E−04
381.75	1	0.00097	0.000001	1.1E−09	1.2E−12	9.7E−04
381.85	1	0.0019	0.0001	0	0	2.1E−03
383.35	1	0.0024	0.0001	0	0	2.6E−03
387.35	1	0.003	0.0001	0	0	3.2E−03
391.65	0.99	0.012	0.0005	0	0	1.3E−02
399.85	0.98	0.021	0.0009	0	0	2.3E−02
402.25	0.98	0.022	0.0009	0	0	2.4E−02
404.05	0.98	0.015	0.0006	0	0	1.6E−02
409.15	0.97	0.03	0.0013	0.0001	0	3.3E−02

The coexisting liquid compositions are given in Table 2.4 for each temperature reported by Vu [31].

Comparison of the two data sets is informative, but the Sanz et al. data should be corrected for the missing oligomers. Only approximate analysis can be performed from the published data. The oligomerization model equations can be used to convert the Sanz et al. data for comparison. The relation suggests a method to determine p, but the p values found using the Sanz et al. data with the relation range from 0.23 to 0.29 when the p values should be increasing with lactic acid concentration as in Figure 2.6. Values determined using the Sanz et al. data with the alternative relation are reasonable and range from 0.007 to 0.64 when the value K = 0.2023 is used. Thus, for the Sanz et al. data, the apparent composition can be calculated from the reported x _W and using Equation 2.10 with K = 0.2023 to determine p, and then using Equations 2.14 and 2.17 to find the apparent moles of lactic acid and water, from which the apparent mole fractions are calculated and plotted in Figure 2.7.

A similar analysis can be performed on the data of Vu. Using Equation 2.10, the p values from the data range from 0.05 to 0.22. Analysis using and comparison with shows a reasonable parity as shown in Figure 2.6, indicating that the data are much more consistent with equilibria. A comparison of the data of Vu with the manipulated data of Sanz et al. is shown in Figure 2.7. The Vu data at high concentrations are more scattered indicating some control issues with the recirculating apparatus. The data of Sanz et al. vary smoothly to higher concentrations indicating better circulation control. Though both data sets have some shortcomings, the agreement is remarkable for the complex experimental system. The data provide reasonable approximations for practitioners.

FIGURE 2.6 Parity plot of p values determined by two methods as explained in the text for data of Vu.

FIGURE 2.7 VLE data for lactic acid + water at 101.33 kPa as reported by Vu compared to the manipulated data of Sanz et al. Data of Vu are filled symbols with triangles for the vapor phase and squares for the liquid phase.