This problem involves vapor-liquid equilibrium calculation using fugacity coefficients and
fugacity coefficients for each fluid in this binary mixture are given here in the problem
statement. So we have to use those to basically determine the temperature at which the fluid
is existing such that it has a mole fraction of 30% in the liquid phase for component 1,
70% for component 2 and that's in equilibrium with some vapor at a pressure of 2 bar pressure.
So we want to determine what that temperature is. So this is a bubble temperature calculation.
It requires an iterative solution but for this problem, we only have to show the first
iteration basically and that will provide some indication of how we'll continue to iterate
in order to find the solution. For any vapor-liquid equilibrium calculation, we can basically
say that the fugacity of each component in one phase, the fugacity of component 1 in
the vapor phase, is equal to the fugacity of component 1 in the liquid phase. And so,
if we start replacing terms and we realize we wanted to use the fugacity coefficient
model, then we have the mole fraction of component 1 times the fugacity coefficient of component
1 times the pressure, that's the fugacity of the vapor, is equal to the mole fraction
of component 1 in the liquid times the fugacity coefficient 1 in the liquid times the pressure
is equal to the fugacity of the liquid. And so we can simplify the equation right away
by canceling the pressure terms and we're furthermore told that the vapor phase is an
ideal gas. And recall that for an ideal gas, the fugacity coefficient is 1. So now we have
a somewhat simplified equation where we have a known fugacity, or no excuse me, mole fraction
in the liquid and an expression for the fugacity coefficient in terms of temperature and we're
told to make some kind of initial guess. We will begin by saying that y1=0.30. Then we
simply need the fugacity coefficient term. Now we should get an expression like the one
that just appeared on your screen where we've just taken this equation 1 plus 1.3 times
x1 which is 0.3, 30% component 1 in the liquid phase, times the exponential of T over 200
and we'll just guess for our initial guess 240 Kelvin. And in that case, y1 is equal
to about 0.688. Now we can use the same expression as we have up here, but just with subscripted
2's. So we can take y2 is equal to x2 which is equal to 0.7. Then we take the expression
for the fugacity coefficient of component 2 which is right up here. So that's equal
to 0.1 plus 0.2 times 0.7 again times the exponential of T over 280 in this case. And
this is equal to approximately 0.301. Alright and now we see that the constraint we have
to meet in order to make this make our phase equilibrium hold is that the sum of yi, so
the y1+y2, should be equal to 100%, so it should be equal to 1. And we actually made
a pretty good initial guess here and when we add 0.688 to 0.301, and we'll get 0.989
which is close to 1 so we made a pretty good initial guess as was indicated here that it
should be in the range of 230 and 280 Kelvin. And we see that the mole fraction is summing
out to be too low which indicates that the partial pressures of the two components are
being estimated as lower than they actually are. So they should actually add up to 2 bar
and they add up to slightly less than that. So that indicates that we don't have a high
enough temperature to make the components as volatile as they actually are. And so we
should increase the temperature in our next guess, something above 240 Kelvin, as we iterate
to try to find the solution for which the sum of y's is equal to 1 and when we achieve
an acceptably close value to 1, then we're done with our iteration scheme.