We are interested in finding volatility and relative
volatility at any given temperature for the binary mixture. We would also like to develop an equilibrium
curve (y vs. x) for this system at one atmosphere. Vapor pressure data is provided:
log of P is equal to A minus B over t plus C. Note that P is the vapor pressure in millimeter of mercury and t
is in degrees Centigrade.
Solution:
Volatility is defined as the ratio of vapor pressure of the
pure component and the system pressure. For example, at 90 C, vapor pressure of benzene can be calculated
as 1020 millimeter of mercury, thus giving a volatility of 1020 over 760 as
1.343. Similarly, vapor pressure of
toluene is found to be 225.21 millimeter and a volatility of 0.296. Ratio of these two volatilities is called
relative volatility, alpha, and its value is equal to 1.343 over 0.296 or
4.534.
When vapor and liquid are in equilibrium, their fugacity
values are the same. Dew point is the
temperature at which the first drop of dew is formed upon condensation. During calculation of dew point, it is
assumed that the liquid phase is ideal, so for a binary mixture, y could be
given as K1 time (K2 minus 1) over (K2 minus K1).
Mole fraction of benzene in the vapor phase, y, is 0.903 at 90
degrees C.
Bubble point is the temperature at which the first bubble is
formed upon boiling. During calculation
of bubble point, it is assumed that the vapor phase is ideal, so for a binary
mixture, x could be given as (K2 minus K1) over (K2
minus K1). Mole fraction of
benzene in the liquid phase, x, is 0.672 at 90 degrees C.
This point with x equal to 0.672 and y equal to 0.903 could
be plotted as an equilibrium point.
When x is zero, the mixture is made of 100 percent less volatile
component, toluene. Using the system
pressure as the vapor pressure, we could calculate the boiling point of toluene
as 130.13 degrees Centigrade. On the
other hand, if x is 1.0, the mixture is made of 100 percent more volatile
component, benzene. The boiling point
of benzene is calculated as 80.1 degrees Centigrade.
The mixture boiling range is 80.1 degrees to 130.13 degrees
Centigrade. In order to plot or develop
an equilibrium curve, we could chose some convenient points, say 90, 100, 110,
and 120 degrees and repeat the calculations to get the values of P1,
P2, K1, K2, alpha, y, and x. Then we can plot these points to draw a
smooth equilibrium curve.
We do notice that volatilities of benzene and toluene vary
significantly over the mixture range, however, relative volatility does not
vary that much. One can take an average
of the relative volatility values and thus simplify calculations. It is customary to use geometric average for
relative volatility values.