Chapter 13, Problem 13.68P

Use Eq. (13.13) to reduce one of the following isothermal data sets, and compare the result with that obtained by application of Eq. (13.19). Recall that reduction means developing a numerical expression for G^E/RT as a function of composition.
(a) Methylethylketone(1)/toluene(2) at 50°C: Table 13.1.

1. Acetone(1)/methanol( 2) at 55°C: Prob. 13.34.
2. Methyl tert-butyl ether( l)/dichloromethane( 2) at 35°C: Prob. 13.37.
3. Acetonitrile(1)/benzene(2) at 45°C: Prob. 13.40.

Second-virial-coefficient data are as follows:

	Part (a)	Part (b)	Part (c)	Part (d)
B11/cm3?mol-1	-1840	-1440	-2060	-4500
B12/cm3?mol-1	-1800	-1150	-860	-1300
B22/cm3?mol-1	-1150	-1040	-790	-1000

Table 13.1: VLE Data for Methyl Ethyl Ketone(l)/Toluene(2) at 50°C

P/kPa	x1y1	${\hat{f}}_1^t=y_{1}P$	${\hat{f}}_2^t=y_{2}P$	${\gamma }_1$	${\gamma }_2$
12.30 ( $P_2^{sat}$ )	0.0000	0.0000	0.000	12.300 ( $P_2^{sat}$ )		1.000
15.51	0.0895	0.2716	4.212	11.298	1.304	1.009
18.61	0.1981	0.4565	8.496	10.114	1.188	1.026
21.63	0.3193	0.5934	12.835	8.795	1.114	1.050
24.01	0.4232	0.6815	16.363	7.697	1.071	1.078
25.92	0.5119	0.7440	19.284	6.636	1.044	1.105
27.96	0.6096	0.8050	22.508	5.542	1.023	1.135
30.12	0.7135	0.8639	26.021	4.099	1.010	1.163
31.75	0.7934	0.9048	28.727	3.023	1.003	1.189
34.15	0.9102	0.9590	32.750	1.400	0.997	1.268
36.09 ( $P_1^{sat}$ )	1.0000	1.0000	36.090 ( $P_1^{sat}$ )	0.000	1.000

13.34. The following is a set of VLE data for the system acetone(1)/methanol(2) at 55°C:

P/kPa	x1	y1	P/kPa	x1	y1
68.728	0.0000	0.0000	97.646	0.5052	0.5844
72.278	0.0287	0.0647	98.462	0.5432	0.6174
75.279	0.0570	0.1295	99.811	0.6332	0.6772
77.524	0.0858	0.1848	99.950	0.6605	0.6926
78.951	0.1046	0.2190	100.278	0.6945	0.7124
82.528	0.1452	0.2694	100.467	0.7327	0.7383
86.762	0.2173	0.3633	100.999	0.7752	0.7729
90.088	0.2787	0.4184	101.059	0.7922	0.7876
93.206	0.3579	0.4779	99.877	0.9080	0.8959
95.017	0.4050	0.5135	99.799	0.9448	0.9336
96.365	0.4480	0.5512	96.885	1.0000	1.0000

Extracted from D. C. Freshwater and K. A. Pike, J. Chem. Eng. Data. vol. 12, pp. 179-183. 1967.

1. Basing calculations on Eq. (13.24), find parameter values for the Margules equation that provide the best fit of G^E/RT to the data, and prepare a Pxy diagram that compares the experimental points with curves determined from the correlation.
2. Repeat (a) for the van Laar equation.
3. Repeat (a) for the Wilson equation.
4. Using Barker’s method, find parameter values for the Margules equation that provide the best fit of the P- x₁data. Prepare a diagram showing the residuals $\delta P$ and $\delta y_1$ plotted vs. x₁.
5. Repeat (d) for the van Laar equation.
6. Repeat (d) for the Wilson equation.

13.37. VLE data for methyl tert-butyl ether(1)/dichloromethane(2) at 308.15 K are as follows:

P/kPa	x1y1	P/kPa	x1	y1
85265	0.0000	0.0000	59.651	0.5036	0.3686
83.402	0.0330	0.0141	56.833	0.5749	0.4564
82.202	0.0579	0.0253	53.689	0.6736	0.5882
80.481	0.0924	0.0416	51.620	0.7676	0.7176
76.719	0.1665	0.0804	50.455	0.8476	0.8238
72.422	0.2482	0.1314	49.926	0.9093	0.9002
68.005	0.3322	0.1975	49.720	0.9529	0.9502
65.096	0.3880	0.2457	49.624	1.0000	1.0000

Extracted from F. A. Mato, C. Bcrru. and A. Pcncloux, J. Cheni. Eng. Data. Vol. 36, pp. 259-262. 1991.

The data are well correlated by the three-parameter Margules equation [an extension of Eq. (13.39)]:

$\frac{G^E}{RT}=\left(A_{21}{x}_1+A_{12}{x}_2-C{x}_1{x}_2\right)x_1{x}_2$

Implied by this equation are the expressions:

$In{\gamma }_1=x_2^2\left[A_{12}+2\left(A_{21}-A_{12}-C\right)x_1+3C{x}_1^2\right]$

$In{\gamma }_2=x_1^2\left[A_{12}+2\left(A_{12}-A_{21}-C\right)x_2+3C{x}_2^2\right]$

1. Basing calculations on Eq. (13.24), find the values of parameters Aj₂, A₂|. and C that provide the best fit of G^E/RT to the data.
2. Prepare a plot of In ${\gamma }_1$ In ${\gamma }_2$ , and $\begin{array}{l}\\ G^E/\left(x_1{x}_{2}RT\right)\end{array}$ vs. $x_1$ showing both the correlation and experimental values.
3. Prepare a Pxy diagram [see Fig. 13.8(a)] that compares the experimental data with the correlation determined in (a).
4. Prepare a consistency-test diagram like Fig. 13.9.
5. Using Barker’s method, find the values of parameters A₁₂. A₂₁, and C that provide the best fit of the P-x₁data. Prepare a diagram showing the residuals $\delta P$ and $\delta y_1$ plotted vs. x₁

Figure 13.8: The diethyl ketone(l)/n-hexane(2) system at 65°C. (a) Pxy data and their correlations.

13.40. Following are VLE data for the system acetonitrile(1)/benzene(2) at 45°C

P/kPa	x1	y1	P/kPa	x1y1
29.819	0.0000	0.0000	36.978	0.5458	0.5098
31.957	0.0455	0.1056	36.778	0.5946	0.5375
33.553	0.0940	0.1818	35.792	0.7206	0.6157
35.285	0.1829	0.2783	34.372	0.8145	0.6913
36.457	0.2909	03607	32.331	0.8972	0.7869
36.996	0.3980	0.4274	30.038	0.9573	0.8916
37.068	0.5069	0.4885	27.778	1.0000	1.0000

The data are well correlated by the three-parameter Margules equation (see Prob. 13.37).
(?)
Basing calculations on Eq. (13.24), Find the values of parameters A₁₂, A₂₁, and C that provide the best fit of G^E/RT to the data.
(b)
Prepare a plot of In ${\gamma }_1$ , In ${\gamma }_2$ , and $\begin{array}{l}\\ G^E/\left(x_1{x}_{2}RT\right)\end{array}$vs. $x_1$ showing both the correlation and experimental values.

1. Prepare a Pxy diagram [see Fig. 13.8(a)] that compares the experimental data with the correlation determined in (a).
2. Prepare a consistency-test diagram like Fig. 13.9.
3. Using Barker’s method, find the values of parameters A₁₂, A₂₁, and C that provide the best Fit of the P- x₁data. Prepare a diagram showing the residuals $\delta P$ and $\delta y_1$ plotted vs. x₁.

13.37. VLE data for methyl tert-butyl ether(1)/dichloromethane(2) at 308.15 K are as follows:

P/kPa	x1y1	P/kPa	x1	y1
85265	0.0000	0.0000	59.651	0.5036	0.3686
83.402	0.0330	0.0141	56.833	0.5749	0.4564
82.202	0.0579	0.0253	53.689	0.6736	0.5882
80.481	0.0924	0.0416	51.620	0.7676	0.7176
76.719	0.1665	0.0804	50.455	0.8476	0.8238
72.422	0.2482	0.1314	49.926	0.9093	0.9002
68.005	0.3322	0.1975	49.720	0.9529	0.9502
65.096	0.3880	0.2457	49.624	1.0000	1.0000

Extracted from F. A. Mato, C. Bcrru. and A. Pcncloux, J. Cheni. Eng. Data. Vol. 36, pp. 259-262. 1991.

The data are well correlated by the three-parameter Margules equation [an extension of Eq. (13.39)]:

$\frac{G^E}{RT}=\left(A_{21}{x}_1+A_{12}{x}_2-C{x}_1{x}_2\right)x_1{x}_2$

Implied by this equation are the expressions:

$In{\gamma }_1=x_2^2\left[A_{12}+2\left(A_{21}-A_{12}-C\right)x_1+3C{x}_1^2\right]$

$In{\gamma }_2=x_1^2\left[A_{12}+2\left(A_{12}-A_{21}-C\right)x_2+3C{x}_2^2\right]$

1. Basing calculations on Eq. (13.24), find the values of parameters Aj₂, A₂|. and C that provide the best fit of G^E/RT to the data.
2. Prepare a plot of In ${\gamma }_1$ In ${\gamma }_2$ , and $\begin{array}{l}\\ G^E/\left(x_1{x}_{2}RT\right)\end{array}$ vs. $x_1$ showing both the correlation and experimental values.
3. Prepare a Pxy diagram [see Fig. 13.8(a)] that compares the experimental data with the correlation determined in (a).
4. Prepare a consistency-test diagram like Fig. 13.9.
5. Using Barker’s method, find the values of parameters A₁₂. A₂₁, and C that provide the best fit of the P-x₁data. Prepare a diagram showing the residuals $\delta P$ and $\delta y_1$ plotted vs. x₁

Figure 13.8: The diethyl ketone(l)/n-hexane(2) system at 65°C. (a) Pxy data and their correlations.