Estimation of the Solubility of a Gas in a Liquid

Estimate the solubility and Henry’s law constant for carbon dioxide in a liquidmixture of toluene and carbon disulfide as a function of the $CS _{2}$ mole fraction at 25°C and a partial pressure of $CO _{2}$ of 1.013 bar.

Data: See Tables 6.6-1, 9.6-1, and 11.1-1.

Question

Estimation of the Solubility of a Gas in a Liquid

Estimate the solubility and Henry’s law constant for carbon dioxide in a liquidmixture of toluene and carbon disulfide as a function of the [latex]CS _{2}[/latex] mole fraction at 25°C and a partial pressure of [latex]CO _{2}[/latex] of 1.013 bar.

Data: See Tables 6.6-1, 9.6-1, and 11.1-1.

Accepted Answer

Equation 11.1-5 provides the starting point for the solution of this problem. Since the partial pressure of carbon dioxide and the vapor pressures of toluene and carbon disulfide are so low, the total pressure must be low, and we can assume that

[latex]x_{1}=\frac{y_{1} P(f / P)_{1}}{\gamma_{1}(T, P, \underline{x}) f_{1}^{ L }(T, P=1.013 bar ) \exp \left[\underline{V}_{1}^{ L }(P-1.013 bar ) / R T ight]}[/latex] (11.1-5a)

[latex]x_{1}^{ ID }=\frac{y_{1} P(f / P)_{1}}{f_{1}^{ L }(T, P=1.013 ext { bar }) \exp \left[\underline{V}_{1}^{ L }(P-1.013 ext { bar }) / R T ight]}[/latex] (11.1-5b)

[latex]\left(\frac{f}{P} ight)=1 \quad ext { and } \quad \exp \left[\frac{\underline{V}_{ CO _{2}}^{ L }(P-1.013 bar )}{R T} ight]=1[/latex]

Next, using the regular solution model for γ, we obtain

[latex]x_{ CO _{2}}=\frac{y_{ CO _{2}} P}{f_{ CO _{2}}^{ L }(T, P=1.013 bar ) \exp \left[\frac{\underline{V}^L_{ CO _{2}}\left(\delta_{ CO _{2}}-\bar{\delta} ight)^{2}}{R T} ight]}[/latex]

with

[latex]\bar{\delta}=\sum_{ j } \Phi_{ j } \delta_{ j }[/latex]

The reduced temperature of [latex]CO _{2} ext { is } T_{ r }=298.15 K / 304.3 K =0.98[/latex], so from the Shair-Prausnitz correlation [latex]f^{ L } / P_{c} \approx 0.60 ext { and } f^{ L } \approx 0.60 imes 73.76 bar =44.26 bar[/latex]. To calculate the activity coefficients we will assume that [latex]CO _{2}[/latex] is only slightly soluble in the solvents, so that its volume fraction is small; we will then verify this assumption. Thus, as a first guess, the contribution of [latex]CO _{2}[/latex] to δ will be neglected.

To compute the solubility of [latex]CO _{2}[/latex] in pure carbon disulfide, we note that

[latex]\bar{\delta} \approx \delta_{ CS _{2}}=10( cal / cc )^{1 / 2} \quad ext { and } \quad\left(\delta_{ CO _{2}}-\bar{\delta} ight)^{2}=16 cal / cc =66.94 J / cc[/latex]

so that

[latex]\begin{aligned}x_{ CO _{2}} &=\frac{1.013 bar }{44.26 bar imes \exp \left\{\frac{55 cc / mol imes 66.94 J / cc }{8.314 J /( mol K ) imes 298.15 K } ight\}} \&=5.18 imes 10^{-3}\end{aligned}[/latex]

(The experimental value is [latex]x_{ CO _{2}}=3.28 imes 10^{-3}[/latex].) Also,

[latex]\begin{aligned}H=P / x_{ CO _{2}} &=1.013 \operatorname{bar} /\left(5.18 imes 10^{-3} ight) \&=195.5 bar / ext { mole fraction }\end{aligned}[/latex]

The solubility of [latex]CO _{2}[/latex] in pure toluene is computed as follows:

[latex]\bar{\delta} \approx \delta_{ T }=8.9( cal / cc )^{1 / 2} \quad ext { and } \quad\left(\delta_{ CO _{2}}-\bar{\delta} ight)^{2}=8.4 cal / cc =35.15 J / cc[/latex]

so that

[latex]x_{ CO _{2}}=\frac{1.013 bar }{44.26 bar imes \exp \left\{\frac{55 imes 35.15}{8.314 imes 298.15} ight\}}=1.05 imes 10^{-2}[/latex]

and

H = 96.6 bar/mole fraction

Finally, the solubility of C[latex]CO _{2}[/latex] in a 50 mol % toluene, 50 mol % [latex]CS _{2}[/latex] mixture is found from

[latex]\begin{aligned}\underline{V}_{ ext {mix }}^{ L } &=x_{ CS _{2}} \underline{V}_{ CS _{2}}^{ L }+x_{ T } \underline{V}_{ T }^{ L } \&=0.5 imes 61 \frac{ cc }{ mol }+0.5 imes 107 \frac{ cc }{ mol } \&=84 \frac{ cc }{ mol }\end{aligned}[/latex]

[latex]\Phi_{ CS _{2}}=\frac{0.5 imes 61}{84}=0.363, \quad \Phi_{ T }=\frac{0.5 imes 107}{84}=0.637[/latex]

[latex]\begin{aligned}\bar{\delta} &=0.363 imes 10\left(\frac{ cal }{ cc } ight)^{1 / 2}+0.637 imes 8.9\left(\frac{ cal }{ cc } ight)^{1 / 2} \&=9.30( cal / cc )^{1 / 2}\end{aligned}[/latex]

and

[latex](\delta-\bar{\delta})^{2}=10.88 cal / cc =45.55 J / cc[/latex]

Thus

[latex]x_{ CO _{2}}=\frac{1.013}{44.26 imes \exp \left\{\frac{55 imes 45.52}{8.314 imes 298.15} ight\}}=8.33 imes 10^{-3}[/latex]

and

H = 121.6 bar/mole fraction

These results are plotted in Fig. 11.1-2. In all cases [latex]x_{ CO _{2}}[/latex] is small, as had initially been assumed, so that an iterative calculation is not necessary.

Table 11.1-1 “Liquid” Volumes and Solubility Parameters for Gaseous Solutes at 25°C
Gas	[latex]\underline{V}^{ L }( cc / mol )[/latex]	[latex]\delta( cal / cc )^{1 / 2}[/latex]
[latex]N _{2}[/latex]	32.4	2.58
CO	32.1	3.13
[latex]O _{2}[/latex]	33	4
Ar	57.1	5.33
[latex]CH _{4}[/latex]	52	5.68
[latex]CO _{2}[/latex]	55	6
Kr	65	6.4
[latex]C _{2} H _{4}[/latex]	65	6.6
[latex]C _{2} H _{6}[/latex]	70	6.6
[latex]Cl _{2}[/latex]	74	8.7
Source: This table originally appeared in J. M. Prausnitz and F. H. Shair, AIChE J., 7, 682 (1961). It appears here courtesy of the copyright owners, the American Institute of Chemical Engineers.

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 9.6-1 Molar Liquid Volumes and Solubility Parameters of Some Nonpolar Liquids
	[latex]\underline{V}^{ L }( cc / mol )[/latex]	[latex]\delta( cal / cc )^{1 / 2}[/latex]
Liquefied gases at 90 K
Nitrogen	38.1	5.3
Carbon monoxide	37.1	5.7
Argon	29	6.8
Oxygen	28	7.2
Methane	35.3	7.4
Carbon tetrafluoride	46	8.3
Ethane	45.7	9.5
Liquid solvents at 25°C
Perfluoro-n-heptane	226	6
Neopentane	122	6.2
Isopentane	117	6.8
n-Pentane	116	7.1
n-Hexane	132	7.3
1-Hexene	126	7.3
n-Octane	164	7.5
n-Hexadecane	294	8
Cyclohexane	109	8.2
Carbon tetrachloride	97	8.6
Ethyl benzene	123	8.8
Toluene	107	8.9
Benzene	89	9.2
Styrene	116	9.3
Tetrachloroethylene	103	9.3
Carbon disulfide	61	10
Bromine	51	11.5
Source: J. M. Prausnitz, Molecular Thermodynamics of Fluid-Phase Equilibria. 1969. Reprinted with permission from Prentice-Hall, Englewood Cliffs, N.J. Note: In regular solution theory the solubility parameter has traditionally been given in the units shown. For this reason the traditional units, rather than SI units, appear in this table.

Question 11.1.1: Estimation of the Solubility of a Gas in a Liquid Estimate t...

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