Prediction of the Solubility of a Gas in a Liquid Using an EOS

Predict the solubility of carbon dioxide in toluene at 25°C and 1.013 bar carbon dioxide partial pressure using the Peng-Robinson equation of state.

Question

Prediction of the Solubility of a Gas in a Liquid Using an EOS

Predict the solubility of carbon dioxide in toluene at 25°C and 1.013 bar carbon dioxide partial pressure using the Peng-Robinson equation of state.

Accepted Answer

The critical properties for both carbon dioxide and toluene are given in Table 6.6-1. The binary interaction parameter for the [latex]CO _{2}[/latex]-toluene mixture is not given in Table 9.4-1. However, as the value for [latex]CO _{2}[/latex]-benzene is 0.077 and that for [latex]CO _{2}[/latex]–n-heptane is 0.10, we estimate that the [latex]CO _{2}[/latex]-toluene interaction parameter will be 0.09. Using this value and the bubble point pressure calculation in either the programs or the MATHCAD worksheet for the Peng-Robinson equation of state for mixtures (described in Appendix B and on the website this book), the following values were obtained:

[latex]x _{ CO _{2}}[/latex]	[latex]P_{ ext {tot }}( ext { bar })[/latex]	[latex]y_{ CO _{2}}[/latex]	[latex]P_{ CO _{2}}( ext { bar })=y_{ CO _{2}} P_{ ext {tot }}[/latex]
0.001	0.11	0.6579	0.072
0.002	0.20	0.7915	0.158
0.004	0.36	0.8834	0.318
0.006	0.51	0.9189	0.469
0.008	0.67	0.9378	0.628
0.010	0.83	0.9495	0.788
0.0125	1.03	0.9590	0.988
0.013	1.07	0.9605	1.028
0.015	1.23	0.9655	1.188

Therefore, using the Peng-Robinson equation of state, we estimate that at a partial pressure of 1.013 bar, carbon dioxide will be soluble in liquid toluene to the extent of 0.0128 mole fraction. This value differs from the value of 0.0077 computed in the last illustration using the Prausnitz- Shair correlation and regular solution theory. However, given the inaccuracy of both methods, this difference is not unreasonable.

Comment

Had we assumed that the [latex]CO _{2}[/latex]-toluene binary interaction parameter was zero, the predicted [latex]CO _{2}[/latex] solubility in toluene at 1.013 bar [latex]CO _{2}[/latex] partial pressure would be 0.0221 mole fraction (Problem 11.1-6).

In order to compare the results with those of the previous illustration, we can also compute the solubility of carbon dioxide in carbon disulfide. There is no binary interaction parameter reported for the [latex]CO _{2}- CS _{2}[/latex] mixture, or for any similar mixtures. If we assume the binary interaction parameter is zero, we find that the [latex]CO _{2} ext { solubility in } CS _{2} ext { at } 1.013 ext { bar } CO _{2}[/latex] partial pressure is 0.0159 mole fraction, which is greater than the measured value by a factor of 5 (Problem 11.1-7). However, if we set [latex]k_{ CO _{2}- CS _{2}}=0.2 ext {, we obtain a } CO _{2} ext { solubility of } 3.4 imes 10^{-3}[/latex], which is in excellent agreement with experiment.

Table 6.6-1 The Critical and Other Constants for Selected Fluids
Substance	Symbol	Molecular Weight [latex]\left( g mol ^{-1} ight)[/latex]	[latex]T_{c}( K )[/latex]	[latex]P_{c}( MPa )[/latex]	[latex]V_{c}\left( m ^{3} / kmol ight)[/latex]	[latex]Z_{c}[/latex]	ω	[latex]T_{ ext {boil }}( K )[/latex]
Acetylene	[latex]C _{2} H _{2}[/latex]	26.038	308.3	6.14	0.113	0.271	0.184	189.2
Ammonia	[latex]NH _{3}[/latex]	17.031	405.6	11.28	0.0724	0.242	0.25	239.7
Argon	Ar	39.948	150.8	4.874	0.0749	0.291	-0.004	87.3
Benzene	[latex]C _{6} H _{6}[/latex]	78.114	562.1	4.894	0.259	0.271	0.212	353.3
n-Butane	[latex]C _{4} H _{10}[/latex]	58.124	425.2	3.8	0.255	0.274	0.193	272.7
Isobutane	[latex]C _{4} H _{10}[/latex]	58.124	408.1	3.648	0.263	0.283	0.176	261.3
1-Butene	[latex]C _{4} H _{8}[/latex]	56.108	419.6	4.023	0.24	0.277	0.187	266.9
Carbon dioxide	[latex]CO _{2}[/latex]	44.01	304.2	7.376	0.094	0.274	0.225	194.7
Carbon monoxide	CO	28.01	132.9	3.496	0.0931	0.295	0.049	81.7
Carbon tetrachloride	[latex]CCl _{4}[/latex]	153.823	556.4	4.56	0.276	0.272	0.194	349.7
n-Decane	[latex]C _{10} H _{22}[/latex]	142.286	617.6	2.108	0.603	0.247	0.49	447.3
n-Dodecane	[latex]C _{12} H _{26}[/latex]	170.34	658.3	1.824	0.713	0.24	0.562	489.5
Ethane	[latex]C _{2} H _{6}[/latex]	30.07	305.4	4.884	0.148	0.285	0.098	184.5
Ethyl ether	[latex]C _{4} H _{10} O[/latex]	74.123	466.7	3.638	0.28	0.262	0.281	307.7
Ethylene	[latex]C _{2} H _{4}[/latex]	28.054	282.4	5.036	0.129	0.276	0.085	169.4
Helium	He	4.003	5.19	0.227	0.0573	0.301	-0.387	4.21
n-Heptane	[latex]C _{7} H _{16}[/latex]	100.205	540.2	2.736	0.304	0.263	0.351	371.6
n-Hexane	[latex]C _{6} H _{14}[/latex]	86.178	507.4	2.969	0.37	0.26	0.296	341.9
Hydrogen	[latex]H _{2}[/latex]	2.016	33.2	1.297	0.065	0.305	-0.22	20.4
Hydrogen fluoride	HF	20.006	461	6.488	0.069	0.12	0.372	292.7
Hydrogen sulfide	[latex]H _{2} S[/latex]	34.08	373.2	8.942	0.0985	0.284	0.1	212.8
Methane	[latex]CH _{4}[/latex]	16.043	190.6	4.6	0.099	0.288	0.008	111.7
Naphthalene	[latex]C _{10} H _{8}[/latex]	128.174	748.4	4.05	0.41	0.267	0.302	491.1
Neon	Ne	20.183	44.4	2.756	0.0417	0.311	0	27
Nitric oxide	NO	30.006	180	6.485	0.058	0.25	0.607	121.4
Nitrogen	[latex]N _{2}[/latex]	28.013	126.2	3.394	0.0895	0.29	0.04	77.4
n-Octane	[latex]C _{8} H _{18}[/latex]	114.232	568.8	2.482	0.492	0.259	0.394	398.8
Oxygen	[latex]O _{2}[/latex]	31.999	154.6	5.046	0.0732	0.288	0.021	90.2
n-Pentane	[latex]C _{5} H _{12}[/latex]	72.151	469.6	3.374	0.304	0.262	0.251	309.2
Isopentane	[latex]C _{5} H _{12}[/latex]	72.151	460.4	3.384	0.306	0.271	0.227	301
Propane	[latex]C _{3} H _{8}[/latex]	44.097	369.8	4.246	0.203	0.281	0.152	231.1
Propylene	[latex]C _{3} H _{6}[/latex]	42.081	365	4.62	0.181	0.275	0.148	225.4
Refrigerant R12	[latex]CCl _{2} F _{2}[/latex]	120.914	385	4.124	0.217	0.28	0.176	243.4
Refrigerant HFC-134a	[latex]CH _{2} FCF _{3}[/latex]	102.03	374.23	4.06	0.198	0.258	0.332	247.1
Sulfur dioxide	[latex]SO _{2}[/latex]	64.063	430.8	7.883	0.122	0.268	0.251	263
Toluene	[latex]C _{7} H _{8}[/latex]	92.141	591.7	4.113	0.316	0.264	0.257	383.8
Water	[latex]H _{2} O[/latex]	18.015	647.3	22.048	0.056	0.229	0.344	373.2
Xenon	Xe	131.3	289.7	5.836	0.118	0.286	0.002	165
Source: Adapted from R. C. Reid, J. M. Prausnitz, and B. E. Poling, The Properties of Gases and Liquids, 4th ed.,McGraw-Hill, New York, 1986, Appendix A and other sources.

Table 9.4-1 Binary Interaction Parameters [latex]k_{12}[/latex] for the Peng-Robinson Equation of State*
	[latex]C _{2} H _{4}[/latex]	[latex]C _{2} H _{6}[/latex]	[latex]C _{3} H _{6}[/latex]	[latex]C _{3} H _{8}[/latex]	[latex]i- C _{4} H _{10}[/latex]	[latex]n- C _{4} H _{10}[/latex]	[latex]i- C _{5} H _{12}[/latex]	[latex]n- C _{6} H _{14}[/latex]	[latex]C _{6} H _{6}[/latex]	[latex]c- C _{6} H _{12}[/latex]	[latex]n- C _{7} H _{16}[/latex]	[latex]n- C _{8} H _{18}[/latex]	[latex]n- C _{10} H _{22}[/latex]	[latex]N _{2}[/latex]	[latex]CO[/latex]	[latex]CO _{2}[/latex]	[latex]SO _{2}[/latex]	[latex]H _{2} S[/latex]
[latex]CH _{4}[/latex]	0.022	-0.003	0.033	0.016	0.026	0.019	0.026	0.04	0.055	0.039	0.035	0.05	0.049	0.03	0.03	0.09	0.136	0.08
[latex]C _{2} H _{4}[/latex]		0.01				0.092			0.031		0.014		0.025	0.086	-0.022	0.056
[latex]C _{2} H _{6}[/latex]			0.089	0.001	-0.007	0.01	0.008	-0.04	0.042	0.018	0.007	0.019	0.014	0.044	0.026	0.13		0.086
[latex]C _{3} H _{6}[/latex]				0.007	-0.014									0.09	0.026	0.093		0.08
[latex]C _{3} H _{8}[/latex]					-0.007	0.003	0.027	0.001	0.023		0.006	0	0	0.078	0.03	0.12		0.08
[latex]i- C _{4} H _{10}[/latex]						0								0.1	0.04	0.13		0.047
[latex]n- C _{4} H _{10}[/latex]							0.017	-0.006			0.003	0.007	0.008	0.087	0.04	0.135		0.07
[latex]i- C _{5} H _{12}[/latex]							0.06		0.018	0.004				0.092	0.04	0.121		0.06
[latex]n- C _{5} H _{12}[/latex]									0.01	-0.004	0.007	0		0.1	0.04	0.125		0.063
[latex]n- C _{6} H _{14}[/latex]										0.013	-0.008			0.15	0.04	0.11		0.06
[latex]C _{6} H _{6}[/latex]											0.001	0.003	0.1	0.164	0.11	0.077	0.015
[latex]c- C _{6} H _{12}[/latex]														0.14	0.1	0.105
[latex]n- C _{7} H _{16}[/latex]												0		0.1	0.04	0.1		0.06
[latex]n- C _{8} H _{18}[/latex]														0.1	0.04	0.12		0.06
[latex]n- C _{10} H _{22}[/latex]														0.11	0.04	0.114		0.033
[latex]N _{2}[/latex]															0.012	-0.02	0.08	0.17
CO																0.03		0.054
[latex]CO _{2}[/latex]																	0.136	0.097
[latex]SO _{2}[/latex]
[latex]H _{2} S[/latex]
*Obtained from data in “Vapor-Liquid Equilibria for Mixtures of Low-Boiling Substances,” by H. Knapp, R. D¨oring, L. Oellrich, U. Pl¨ocker, and J. M. Prausnitz, DECHEMA Chemistry Data Series, Vol. VI, Frankfurt/Main, 1982, and other sources. Blanks indicate no data are available from which the k12 could be evaluated. In such case use estimates from mixtures of similar compounds.

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