Computing the Solvent Partial Pressure above a Polymer-Solvent Mixture In the processing of polymers, and also for polymer devolatilization (the removal of the solvent from the polymer), it is important to be able to calculate the equilibrium partial pressure of a solvent above solvent-polymer

Question

Computing the Solvent Partial Pressure above a Polymer-Solvent Mixture

In the processing of polymers, and also for polymer devolatilization (the removal of the solvent from the polymer), it is important to be able to calculate the equilibrium partial pressure of a solvent above solvent-polymer mixtures of different compositions. Calculate the partial pressure of benzene in benzene + polyisobutylene (PIB) mixtures at 298.15 and 312.75 K. In this calculation you can assume that polyisobutylene has a negligible vapor pressure, and that the Flory-Huggins model describes the solution behavior of this polymer + solvent mixture. Do the calculations for values of the Flory-Huggins χ parameter equal to 0.5 to 1.0.

Data: The molar volume of benzene is 88.26 [latex]cm ^{3} / mol[/latex], its molecular weight is 78, and its vapor pressures are [latex]P_{ B }^{ vap }=0.1266[/latex] bar at 298.15 K and 0.2392 bar at 312.75 K, respectively. The molecular weight of the PIB is 40,000, the monomeric unit in PIB has a molecular weight of 104, and the monomeric volume [latex]\underline{V}_{ PIB , m} ext { is } 131.9 cm ^{3} / mol[/latex] monomer.

Accepted Answer

The average number of monomer units, n, in the PIB polymer is computed as follow

[latex]\begin{aligned}n &=\frac{ ext { Molecular weight of polymer }}{ ext { Molecular weight of monomer }} \&=\frac{40000}{104}=384.6\end{aligned}[/latex]

The mole fraction [latex]x_{ B }[/latex] and the volume fraction [latex]\phi_{ B }[/latex] of benzene in terms of its weight fraction [latex]W_{ B }[/latex] are

[latex]x_{ B }=\frac{\frac{W_{ B }}{78}}{\frac{W_{ B }}{78}+\frac{W_{ PIB }}{40000}}[/latex] and [latex]\phi_{ B }=\frac{\frac{W_{ B }}{78} imes \underline{V}_{ B }}{\frac{W_{ B }}{78} imes \underline{V}_{ B }+\frac{W_{ PIB }}{40000} imes n imes \underline{V}_{ PIB , m}}[/latex]

[latex]=\frac{\frac{W_{ B }}{78} imes 88.26}{\frac{W_{ B }}{78} imes 88.26+\frac{1-W_{ B }}{40000} imes 384.6 imes 131.9}[/latex]

Also

[latex]\begin{aligned}m &=\frac{\underline{V}_{ PIB }}{\underline{V}_{ B }}=\frac{\underline{V}_{ PIB , m} imes n}{\underline{V}_{ B }} \&=\frac{131.9 imes 384.6}{88.26}=574.8\end{aligned}[/latex]

Since the PIB is (assumed to be) involatile, we only have to equate the fugacity of benzene in the vapor and liquid phases. Further, since the total pressure will be low, we use

[latex]\bar{f}_{ B }^{ L }=\bar{f}_{ B }^{ V } \quad ext { or } \quad x_{ B } \gamma_{ B } P_{ B }^{ vap }=P_{ B }=[/latex] partial pressure of benzene

where the activity coefficient of benzene is calculated from the Flory-Huggins equation, Eq. 9.5-18,

[latex]\begin{aligned}\ln \gamma_{1} &=\ln \frac{\phi_{1}}{x_{1}}+\left(1-\frac{1}{m} ight) \phi_{2}+\chi \phi_{2}^{2} \\ln \gamma_{2} &=\ln \frac{\phi_{2}}{x_{2}}+(m-1) \phi_{1}+m \chi \phi_{1}^{2}\end{aligned}[/latex] (9.5-18)

[latex]\ln \gamma_{ B }=\ln \frac{\phi_{ B }}{x_{ B }}+\left(1-\frac{1}{m} ight)\left(1-\phi_{ B } ight)+\chi\left(1-\phi_{ B } ight)^{2}[/latex]

Using this information, we obtain the following results:

These results are plotted in Figures 1 and 2. These results show that the Flory-Huggins model with a constant value of χ = 1.0 gives a reasonable representation of the experimental data of Eichinger and Flory [Trans. Farad. Soc., 64, 2053–2060 (1968)]. The results also show the significant effect of the value of the Flory χ parameter on the partial pressure predictions.

Question 10.2.7: Computing the Solvent Partial Pressure above a Polymer-Solve...

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