Question 14.8: Acetic acid is esterified in the liquid phase with ethanol a...

Data for $\Delta H_{f_{298}}^{\circ} \text { and } \Delta G_{f_{298}}^{\circ}$ are given for liquid acetic acid, ethanol, and water in Table C.4. For liquid ethyl acetate, the corresponding values are:

$\Delta H_{f_{298}^{\circ}}^{\circ}=-480,000 \mathrm{~J} \quad \text { and } \quad \Delta G_{f_{298}}^{\circ}=-332,200 \mathrm{~J}<br />$

The values of $\Delta H_{298}^{\circ} \text { and } \Delta G_{298}^{\circ}$ for the reaction are therefore:

$\Delta H_{298}^{\circ}=-480,000-285,830+484,500+277,690=-3640 \mathrm{~J}$

$\Delta G_{298}^{\circ}=-332,200-237,130+389,900+174,780=-4650 \mathrm{~J}$

By Eq. (14.11b),

$\ln K=\frac{-\Delta G^{\circ}}{R T}$    (14.11b)

For the small temperature change from 298.15 to 373.15 K, Eq. (14.15) is adequate for estimating K. Thus,

$\ln \frac{K}{K^{\prime}}=-\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T}-\frac{1}{T^{\prime}}\right)$    (14.15)

$\ln \frac{K_{373}}{K_{298}}=\frac{-\Delta H_{298}^{\circ}}{R}\left(\frac{1}{373.15}-\frac{1}{298.15}\right)$

or                      $\text { or } \quad \ln \frac{K_{373}}{6.5266}=\frac{3640}{8.314}\left(\frac{1}{373.15}-\frac{1}{298.15}\right)=-0.2951$

and            $K_{373}=(6.5266)(0.7444)=4.8586$

For the given reaction, Eq. (14.5), with x replacing y, yields:

$y_i=\frac{n_i}{n}=\frac{n_{i_0}+\nu_i \varepsilon}{n_0+\nu \varepsilon}$    (14.5)

$x_{\mathrm{AcH}}=x_{\mathrm{EtOH}}=\frac{1-\varepsilon_e}{2} \quad x_{\mathrm{EtAc}}=x_{\mathrm{H}_2 \mathrm{O}}=\frac{\varepsilon_e}{2}$

Because the pressure is low, Eq. (14.32) is applicable. In the absence of data for the activity coefficients in this complex system, we assume that the reacting species form an ideal solution. In this case Eq. (14.33) is employed, giving:

$\prod_i\left(x_i \gamma_i\right)^{\nu_i}=K$    (14.32)

$\prod_i\left(x_i\right)^{\nu_i}=K$    (14.33)

$K=\frac{x_{\mathrm{EtAc}} x_{\mathrm{H}_2 \mathrm{O}}}{x_{\mathrm{AcH}} x_{\mathrm{EtOH}}}$

Thus,              $4.8586=\left(\frac{\varepsilon_e}{1-\varepsilon_e}\right)^2$

Solution yields:

$\varepsilon_e=0.6879 \quad \text { and } \quad x_{\mathrm{EtAc}}=0.6879 / 2=0.344$

This result is in good agreement with experiment, even though the assumption of ideal solutions may be unrealistic. Carried out in the laboratory, the reaction yields a measured mole fraction of ethyl acetate at equilibrium of about 0.33.