Physical Chemistry week 7 Monday February 18, 2013 page 1
Ideal solutions
Lever rule and tie lines
Consider a graph with pressure on the vertical axis and overall mole fraction of A (χA) on the horizontal
axis. On the left side is a point for pure B (PB*). On the right side is a point for pure A (PA*). A straight
line connects those two points. A curve connects the same two points but sags below the line. Above
both curves is pure liquid. Below both curves is pure vapor. Between the two twos there is both liquid
and vapor. Start at point C in the liquid phase. The mole fraction is χA’. A straight dotted line comes
down from z to the horizontal axis. Any point on this line has the same composition, so it’s an isopleth
line, as isopleth line means constant composition. Where this line intersects the top curve is point D.
Take any arbitrary point on this line but between the two curves and call it E. Going straight to the left
from point E until intersecting the top curve gives point H, which has composition XA’’’. Going straight to
the right from point E until intersecting the bottom curve gives point I, has composition X A’’.
nA = total number of moles of A (l + v)
nl = total number of moles of A and B in liquid
nv = total number of moles of A and B in vapor
χA =
nA
nl +nv
χA is overall
nA = χAnl + χAnv
nA = χA,lnl + χAnv
nA = nA,l + nA,v
χAnl + χAnv = χA,lnl + χA,vnv
χAnl – χA,lnl = χA,vnv – χAnv
nl(χA – χA,l) = nv(χA,v-χA)
χA = χA’
(χA’ – χA’’) = E͞H
(χA,v’’ – χA) = E͞I
Test
nlH͞E = nvE͞I
Lever rule
nαlα = nβlβ lever rule, α and β are phases, l is length, n is total moles of A
in weight fractions:
mαlα = mβlβ m is weight fraction, l is length, α and β are phases
The top curve is the bubble point line. The bottom curve is the dew point line.
Ideal solution at fixed pressure
Consider a diagram with temperature on the vertical axis and mole fraction of A on the horizontal axis.
A and B are both liquids. Pressure is constant. At the left side of the diagram is a point for TB*. On the
right side of the diagram is a point for TA*. Connecting those two points is a curve that bulges upwards.
Also connecting those same two points is a curve that sags down. Above the top curve is pure vapor.
Below the bottom curve is pure liquid. Consider an arbitrary point on the bottom curve. It has mole
fraction χA1. Go straight across from that point to the top curve. This point has mole fraction χA2. From
this point come down to the bottom curve, then go straight to the right until at the top curve. This point
has mole fraction χA3. χA2 is the composition of the vapor from χA1. Likewise, χA3 is the composition of
the vapor from χA2. Repeat to keep separating A from B. This is called distillation.
Theoretical plate
Each horizontal line as described above is a theoretical plate. More plates give better separation. If the
boiling points are close together, we need more many theoretical plates to get a good separation. For a
fractionating column at an oil refinery, the column is many feet high and can contain thousands of
theoretical plates.
Theoretical plate: maximum degree of enrichment for any one distillation
Putting beads into a fractionating column or making the fractionating column zigzag increases the
theoretical plates
b. ideal solution
Consider a wrong diagram for educational purposes. It has pressure on the vertical axis and mole
fraction of A on the horizontal axis. Temperature is constant. Consider two curves. They have the same
end points on the left side and right side and both bulge upward, but one bulges up more than the other.
At the highest point of the top curve is Pmax. This point is D. Above it is point C, which is in the pure
liquid phase. Directly below D on the bottom curve is point E. Below the bottom curve is pure vapor. At
point E there is no way to go right to determine vapor pressure, so this diagram can’t be right.
Consider instead a graph with pressure on the vertical axis and mole fraction of A on the horizontal axis.
At the left side is point PB*. At the right side is point PA*. Temperature is constant. Consider a curve that
connected PB* and PA* and bulges upward. From the top of this curve comes two more curves, one
bulging down to PA* and one bulging down to PB*. Above all the curves everything is liquid. Below all the
curves everything is vapor. Between any pair of curves there is both liquid and vapor. The point where
all three curves intersect has composition χA’. The top curve is the liquid curve. The bottom curve is the
vapor curve. This is positive deviation.
Consider a graph with temperature on the vertical axis and mole fraction of A on the horizontal axis.
Pressure is constant. On the left side is a point TB*. On the right side is point TA*. Connecting these two
points is a curve that sags down. Below this curve everything is liquid. From the lowest point on this
curve comes two more curves, one going to TA* and one going to TB* with both building upwards. Above
these curves everything is vapor. Where all three curves come together is χA’. This point is called the
azeotropic mixture. Higher vapor pressure goes along with a lower boiling point. Using distillation we
can’t purify beyond the bottom point (the azeotropic mixture). For ethanol and water we can get to
about 96% ethanol by distillation. If χA > χA’ fractional distillation gives pure A and azeotrope. If χA < χA’
fractional distillation gives pure B and azeotrope. The position of the azeotrope can be moved by
changing the pressure.
Two component liquid equilibrium
miscible: when 2 liquids are completely soluble in each other (like water and acetone), makes a single
phase
That’s why adding acetone dries out a container. The water dissolves in the acetone and the acetone
evaporates, taking the water with it.
partially miscible: n-pentanol in water. The OH end of the pentanol is hydrophilic but the chain is
hydrophobic. We get 2 phases, one for alcohol and one for water.
Consider a diagram with temperature on the vertical axis and mole fraction of A on the horizontal axis.
A curve starts at the bottom near the left side and curves up near the middle then comes back down to
the bottom near the right side. Below this curve there are two phases. Outside of this curve is one
phase. The top of this curve is the critical temperature. Where there are two phases, the lever rule
applies (two substances in equilibrium with each other). The critical temperature is where two phases
become one.