Determination of Dew Point and Bubble Point Pressures
a. Estimate the bubble point and dew point pressures of a 20 mol % ethyl acetate, 80 mol % benzene mixture at 75°C.
b. Estimate the bubble point and dewpoint temperatures of a 20mol%ethyl acetate, 80 mol% benzene mixture at 1.013 bar.
Data: See previous illustration.
a. Bubble point and dew point pressures at T = 75°C. If a P-x-y diagram is available, such as the one constructed in the previous illustration, the bubble point and dew point pressures can immediately be read from such a diagram. From Fig. 1 of the previous illustration, the bubble point pressure is found to be 1.0954 bar (where a bubble of vapor with an ethyl acetate mole fraction of 0.3367 is obtained). Similarly, from the same figure, the dew point pressure is found to be 0.9971 bar (where a drop of liquid is formed with an ethyl acetate mole fraction of 0.0834).
However, if such a diagram is not available, the bubble point and dew point pressures must be found by direct calculation. As the liquid composition is known \left(x_{ EA }=0.2\right),
so that the activity coefficients can be computed, the bubble point is easily found, by solving the equation
P_{\text {bubble }}( bar )=0.2 \cdot \exp \left[\frac{\alpha}{\left[1+\frac{\alpha \cdot 0.2}{\beta \cdot 0.8}\right]^{2}}\right] \cdot 0.946+0.8 \cdot \exp \left[\frac{\beta}{\left[1+\frac{\beta \cdot 0.8}{\alpha \cdot 0.2}\right]^{2}}\right] \cdot 0.862
and then the vapor mole fraction is obtained from
y_{ EA }=\frac{0.2 \cdot \exp \left[\frac{\alpha}{\left[1+\frac{\alpha \cdot 0.2}{\beta \cdot 0.8}\right]^{2}}\right] \cdot 0.946}{P_{ bubble }( bar )}
Note that this is a direct (that is, noniterative) calculation and gives the same bubble point results as were obtained above from the graphical solution.
Finding the dewpoint temperature ismore complicated, as the liquid composition (which also appears in the expressions for the activity coefficients) is unknown. The equation to be solved is
\begin{aligned}y_{ EA } &=0.2 \\=& \frac{x_{ EA } \cdot \exp \left[\frac{\alpha}{\left[1+\frac{\alpha \cdot x_{ EA }}{\beta \cdot\left(1-x_{ EA }\right)}\right]^{2}}\right] \cdot 0.946}{P_{\text {bubble }}( bar )}\end{aligned}
=\frac{x_{ EA } \cdot \exp \left[\frac{\alpha}{\left[\left[1+\frac{\alpha \cdot x_{ EA }}{\beta \cdot\left(1-x_{ EA }\right)}\right]^{2}\right.}\right] \cdot 0.946}{x_{ EA } \cdot \exp \left[\frac{\alpha}{\left[1+\frac{\alpha \cdot x_{ EA }}{\beta \cdot\left(1-x_{ EA }\right)}\right]^{2}}\right] \cdot 0.946+\left(1-x_{ EA }\right) \cdot \exp \left[\frac{\beta}{\left[1+\frac{\beta \cdot\left(1-x_{ EA }\right)}{\alpha \cdot x_{ EA }}\right]^{2}}\right]\cdot0.862}
which is solved iteratively (or using a program such asMATHCAD) for the dew point mole fraction of ethyl acetate, x_{ EA }. The mole fraction found is then used in the equation
P_{\text {bubble }}( bar )=x_{ EA } \cdot \exp \left[\frac{\alpha}{\left[1+\frac{\alpha \cdot x_{ EA }}{\beta \cdot\left(1-x_{ EA }\right)}\right]^{2}}\right] \cdot 0.946
+\left(1-x_{ EA }\right) \cdot \exp \left[\frac{\beta}{\left[1+\frac{\beta \cdot\left(1-x_{ EA }\right)}{\alpha \cdot x_{ EA }}\right]^{2}}\right] \cdot 0.862
to calculate the bubble point pressure. This leads to the same solution as was found graphically (after the complete P-x-y diagram had been constructed.)
b. Bubble point and dew point temperatures at P= 1.013 bar. If a T-x-y diagram is available, such as the one constructed in the previous illustration, the bubble point and dew point temperatures can immediately be read from such a diagram. Using Fig. 2 of the previous illustration, the bubble point temperature is found to be 345.78 K (when a bubble of vapor with an ethyl acetate mole fraction of 0.3360 is formed). Similarly, from the same figure, the dew point pressure is found to be 348.67 K (where a drop of liquid is formed with an ethyl acetate mole fraction of 0.0833).
However, if such a diagram is not available, the bubble point and dew point temperatures must be found by direct calculation. The bubble point temperature is more easily found, since the liquid composition is known, by solving the following equation
P=1.013=0.2 \cdot \exp \left[\frac{\alpha}{\left[1+\frac{\alpha \cdot 0.2}{\beta \cdot 0.8}\right]^{2}}\right] \cdot P_{ EA }^{ vap }(T)+0.8 \cdot \exp \left[\frac{\beta}{\left[1+\frac{\beta \cdot 0.8}{\alpha \cdot 0.2}\right]^{2}}\right] \cdot P_{ BZ }^{ vap }(T)
and using the solution in the equation below to obtain the vapor composition:
y_{ EA }=\frac{0.2 \cdot \exp \left[\frac{\alpha}{\left[1+\frac{\alpha \cdot 0.2}{\beta \cdot 0.8}\right]^{2}}\right] \cdot P_{ EA }^{ vap }(T)}{P=1.013 bar }
Finding the dew point temperature is more complicated, as both the liquid composition and the temperature are unknown. The two equations below must be solved simultaneously for the bubble point temperature and the liquid composition (which is best done with a computer program):
P=1.013=x_{ EA } \cdot \exp \left[\frac{\alpha}{\left[1+\frac{\alpha \cdot x_{ EA }}{\beta \cdot\left(1-x_{ EA }\right)}\right]^{2}}\right] \cdot P_{ EA }^{ vap }(T)
+\left(1-x_{ EA }\right) \cdot \exp \left[\frac{\beta}{\left[1+\frac{\beta \cdot\left(1-x_{ EA }\right)}{\alpha \cdot x_{ EA }}\right]^{2}}\right] \cdot P_{ BZ }^{ vap }(T)
and
y_{ EA }=0.2=\frac{x_{ EA } \cdot \exp \left[\frac{\alpha}{\left[1+\frac{\alpha \cdot x_{ EA }}{\beta \cdot\left(1-x_{ EA }\right)}\right]^{2}}\right] \cdot P_{ EA }^{ vap }(T)}{P=1.013 bar }
y_{ EA }=0.2=\frac{x_{ EA } \cdot \exp \left[\frac{\alpha}{\left[1+\frac{\alpha \cdot x_{ EA }}{\beta \cdot\left(1-x_{ EA }\right)}\right]^{2}}\right] \cdot P_{ EA }^{ vap }(T)}{P=1.013 bar }
This leads to the same solution as was found graphically (after the complete T-x-y diagram had been constructed.)