(b) A vapor-liquid mixture of carbon tetrachloride(1) and $n$-heptane(2) is at $100 \mathrm{kPa}$. The activity coefficients can be derived from Wilson equation as follows:
$$
\begin{array}{l}
\ln \gamma_{1}=-\ln \left(x_{1}+x_{2} \Lambda_{12}\right)+x_{2}\left(\frac{\Lambda_{12}}{x_{1}+x_{2} \Lambda_{12}}-\frac{\Lambda_{21}}{x_{2}+x_{1} \Lambda_{21}}\right) \
\ln \gamma_{2}=-\ln \left(x_{2}+x_{1} \Lambda_{21}\right)-x_{1}\left(\frac{\Lambda_{12}}{x_{1}+x_{2} \Lambda_{12}}-\frac{\Lambda_{21}}{x_{2}+x_{1} \Lambda_{21}}\right) \
\Lambda_{12}=1.5410 \text { and } \Lambda_{21}=0.5197
\end{array}
$$
(i) Determine the temperature and its equilibrium composition at $x_{1}=0.4$.

Question

Accepted Answer

∻Solution∻ ‖ 1 ⋮ **Calculate the activity coefficients (γ1 and γ2)**: Given the mole fraction of carbon tetrachloride ($x_1$) is 0.4, we can calculate the mole fraction of $n$-heptane ($x_2$) as $x_2 = 1 - x_1 = 1 - 0.4 = 0.6$. Using the Wilson equation, we can then calculate $\ln \gamma_1$ and $\ln \gamma_2$ with the given $\Lambda_{12}$ and $\Lambda_{21}$ values. ‖ ‖ 2 ⋮ **Use the Wilson equation to find ln γ1 and ln γ2**: Plug in the values of $x_1$, $x_2$, $\Lambda_{12}$, and $\Lambda_{21}$ into the Wilson equations to get the natural logarithms of the activity coefficients. ‖ ‖ 3 ⋮ **Convert ln γ1 and ln γ2 to γ1 and γ2**: Exponentiate the results from step 2 to get the activity coefficients $\gamma_1$ and $\gamma_2$. ‖ ‖ 4 ⋮ **Determine the equilibrium composition**: The equilibrium composition can be found using Raoult's law and the given total pressure. However, additional information such as the vapor pressures of the pure components at the temperature of interest is required to proceed. Since this information is not provided, we cannot determine the equilibrium composition or the temperature. ‖ ∻Answer∻ ⚹ The activity coefficients (γ1 and γ2) can be calculated using the Wilson equation, but without the vapor pressures of the pure components, the equilibrium composition and temperature cannot be determined. ⚹ ∻Key Concept∻ ⚹ The Wilson equation is used to calculate activity coefficients in a vapor-liquid equilibrium scenario. ⚹ ∻Explanation∻ ⚹ The activity coefficients are essential for understanding the non-ideal behavior of mixtures and are used in conjunction with Raoult's law to determine the equilibrium composition. However, without the vapor pressures of the pure components, the calculation cannot be completed.◊What is the significance of the Wilson equation in calculating activity coefficients for vapor-liquid equilibrium in chemical engineering?⍭ Generate me a similar question◊⚹

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