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# LECTURE 15

last edited by 10 years, 2 months ago

Dr Cooper's Comments: Charles, please label all figures (Eg. Figure 1: Title...), otherwise both of you, good work!

Charles Price- Written in Blue

Definition of importance:

1.       Phases in a system (P):  a uniform state of matter throughout the system.

a.       Example:  in a system that has solid particles within a liquid solution: two phases exist.  The solid and liquid.

2.       Components (C):  the MINIMUM number of independents species that is able to survive on its own.

a.       Example: mixture of ethanol and water in a system is a two component system because these are two separate entities to survive on its own.

3.        Constituents: the number of chemical species that exist in a system that can independently survive on its own.

a.       Example:  A solution of sodium chloride has three constituents: the water, sodium ion, and chloride ion.

4.       Variance (F): the number of VARIABLES that can change WITHOUT ALTERING the number of phases that exist in the beginning.

a.       Examples of type of things can be altered is the pressure, temperature

With variances below is the equation which can be used to discover the exact number of variances within the system.  The variance is equal to the number of COMPONENTS subtracted by the NUMBER OF PHASES plus two.  This is described as the phase rule.

Phase Rule Equation:                        F = C - P +2 From the phase rule equation and graph placed above (FIGURE 1) it intuitively tells you that when there is a one component system present the variance will be equal to two.  While when dealing with a two component system the variance is one. Again, this number means that these are the allotted degrees that can change without altering the phases.   Now because this equation is in effect and used there is certain rules that go along with it.

Rule#1:  Four phases can never be at an equilibrium sate because when the phase rule equation applied the variance is a negative number which is not possible.

Rule#2:  When the variance is equal to zero it means two things and that is the pressure and temperature are dependent open one another so altering either will break the phase rule.  And also it means that you have reached the triple point in which all three phases coexists.

To get a phase diagram there are two methods for a one component system (one component system where F=2, P=1) you do either a thermal analysis or a more modern technique using a diamond anvil cell.  Let’s discuss the thermal analysis first:

Thermal analysis is an easy enough experiment where you simply heat something up, cool it down and record its temperature readings.  Yet when slowed down and observed it gives you the characteristics of its phase diagram via temperature because this technique pays attention to the change enthalpy that occurs. Example of the chart above (FIGURE 2): you have boiled water to purify it, it’s condensed down to a liquid.  Now the liquid is beginning to cool down.  As the liquid is being cooled the temperature drops from the room temperature to its freezing point 0°C.  Once it reaches that temperature it stays constant throughout as the liquid begins to go from the phase transition of liquid into a solid phase.  Once that liquid has completely transformed into the solid phase, the temperature begins to go below that point.  That’s what the above graph is showing as the liquid is going from a room temp.  The temperature drops steadily but when the temperature plateaus and slowing down the process to witness the phase change, it must stay at this temp until it all has transitioned from one phase to another because they both cannot exist here.

The other method using a diamond-anvil cell is like the apparatus below.  The use in this is pressure is what is observed.  Being diamond is a pretty strong material under significant amount of pressure it is used.  A sample is placed in between to diamond pieces in which the screw is turned.  Spectroscopically the shift of spectral lines observed and gives the characteristics needed.  The picture of this is below FIGURE 3. Cynthia Ayers (cayers1) Written in BLACK

Two component Systems

Looking at the general phase rule equation, F = C – P + 2, C = 2 for a two component system which it can then deduced that F = 4 – P. If the temperature is constant, the remaining variance is F’ = 2 – P; where F’ indicates that one of the degrees of freedom has been discarded.

Vapor Pressure Diagrams

In an ideal solution of two volatile liquids, the partial vapor pressure of the components can be related to the composition of the liquid mixture by Raoult’s Law: OR (6.2)

Using Raoult’s law and EQ 6.2A and EQ 6.2B the total vapor pressure, p, of the mixture is can be defined as: (6.3)

This shows that at a constant temperature the total vapor pressure changes linearly with the composition from pB to pA as xA changes from 0 to 1 this is shown in FIGURE 6-6 FIGURE 6.6 – Following Raoult’s law, this shows the variation of the total pressure of a mixture of two substances.

The Composition of Vapor

Even if the liquid and vapor are in mutual equilibrium, it does not mean the compositions are the same. If the component is more volatile, then the amount of substance in the vapor should be higher.

Dalton’s Law states that yA and yB represent the mole fractions of A and B in the gas. This can lead us to the equations of: OR (6.4)

If the mixture is ideal the total pressure, partial pressure, and mole fractions can be related into the following equation (6.5), which is used by substituting EQ 6.2 and EQ 6.3 into EQ 6.4 AND (6.5) FIGURE 6.7 – A mixture of two substances calculated using EQ 6.5

This graph plots the Composition of vapors vs. composition of the liquids for various values of pA* /pB* > 1

We see that yA is always greater than yB. This means that in the more volatile component the vapor is richer than the liquid. Since the composition of the liquid can be related to the composition of the vapor, the total vapor pressure can now be related to the composition of the vapor. This is shown in the equation EQ 6.6. Each curve is labeled with the value of pA*/pB*. (6.6)

This equation is then plotted below in FIGURE 6.8. The curves are labeled with the value of pA*/pB* FIGURE 6.8 – The dependence of vapor pressure in terms of the mole fraction of A using EQ 6.6

Both the vapor and liquid compositions are of equal interest when dealing with the distillation process this can be shown by combining FIGURE 6.8 and FIGURE 6.7 into one. FIGURE 6.9 – A combined diagram showing both the liquid and vapor phases depending on the total pressure and the mole fraction A on the system.

FIGURE 6.8 shows point a is the vapor pressure of a mixture of composition xA, and point b is the composition of the vapor of the vapor when it is in equilibrium at that specific pressure.

P = 2 when the two phases are in equilibrium, therefore F’ = 1

The horizontal axis shows the overall composition of the system, zA. Above the curves is where everything is in the liquid phase, below the curves is where everything is in the vapor phase, and between the curves has both liquid and vapor phases present. FIGURE 6.10 – A more specific pressure-composition diagram that shows the isopleth and compares the mixture at different pressures and mole fractions.

Figure 6.10 shows the effect of lowering the pressure on a liquid mixture of overall composition a. The vertical line is called an isopleth. The state of the system moves down the isopleth that passes through point a since the changes in the system do not affect the overall composition. A tie line is a line that joins two points representing phases in equilibrium.

When lowering the pressure of the system on the isopleths to point a2’’, the system must vaporize some of the liquid so the total vapor pressure falls to p2. We can then graphically conclude that the composition of the liquid is a2. On the other end of the tie line we can also see that the composition of the vapor is now a2’.

Therefore, at a specific pressure the vapor and liquid phases have a fixed composition, making the variance zero. This can be applied at any given constant pressure. This interpretation of a pressure-composition graph is generally shown in Figure 6.12. FIGURE 6.12 – a general idea of a vapor pressure diagram (also known as a pressure-composition diagram)

The Lever Rule

A point in the two-phase area of a phase diagram specifies qualitatively and quantitatively the relative amounts of both liquid and vapor present.

Measuring the distances lα and lβ and using the lever rule gives us the relative amount of the two phases (α and β) that are in equilibrium. The distance can be measured along the horizontal tie line shown in FIGURE 6.13. FIGURE 6.13 – The lever rule. The proportion of the amount of one phase to another is found by measuring the distances lα and lβ.

The lever rule can also be shown by EQUATION 6.7:

(6.7)

This is where nα and nβ are the amounts of phase α and phase β respectively. FIGURE 6.13 shows that lα is about twice as long as lβ, meaning that phase α is about twice as long as phase β.

Temperature-Composition Diagrams

Distillation is discussed with a temperature-composition diagram instead of a pressure-composition diagram. A temperature composition diagram is a different type of phase diagram where the boundaries show the composition of the phases at different temperature at a constant pressure (which is usually 1 atm). FIGURE 6.14 – A temperature composition diagram. This shows an ideal mixture where component A is more volatile than component B.

The main difference in this type of diagram is that the liquid phase is now in the lower section of the graph and the vapor phase is in the upper section of the graph. This is opposite in the pressure composition diagrams. Separating the mixture in FIGURE 6.14 would require a distillation process.