Interpretation of miscibility curves

Question

I stumbled across miscibility curves in the section Solubility of Liquids in Liquids (Organic Chemistry, by Wallwork and Grant) today. I'm not sure if my problem is with the way the content was laid out in the book or whether I'm not fully understanding the material, but I feel the book has dealt with the topic in a rather, obscure way.

I've taken snapshots of it and uploaded a portion of my text below.

What I want to know is if I interpreted the text (the bit about miscibility curves) correctly, which I seriously doubt.

Here's my interpretation:

This is the miscibility curve for phenol in water blah-blah-blah. All the points present in the area bound by the curve with the x-axis (concentration axis) are points of immiscibility. The curve itself represents the threshold of miscibility.

For example, at 20°C, phenol is soluble in water up till the concentration corresponding to point A on the curve. However, for phenol concentrations corresponding to points between A and B (lying on the line AB), the phenol-water mixture is immiscible and the phenol and water form separate layers. Although, at concentrations corresponding to point B (and beyond), the phenol is completely soluble. When for a constant temperature, the miscibility curve maps to two concentrations on the x-axis (eg; points A and B in the graph), which represent thresholds of miscibility at that temperature, the two mixtures (i.e- at A and B) are called conjugate pairs.

I labored for over two hours to come to the above conclusion, but when I saw the portion of the text which mentions that the liquids A and B exist together (at equilibrium) for the given temperature, I had to throw in the towel.

I have a very strong feeling that a large part of my 'interpretation' is horribly incorrect. If anyone's got the time, could you explain what the text actually means?

Answer 1

This is no different from what is observed in many binary systems. Here, the phenol and water exhibit a miscibility gap. As you noted, above point C, the liquid is one liquid with whatever composition it has. If one were to have a liquid at the composition corresponding to C, at a temperature above 66C. you would have one liquid. As you dropped the temperature to 66C, nothing would change.

Drop below 66C, and the liquid will phase separate in to two liquids of different compositions, given by the points on the miscibility curve. To figure out how much of each liquid you have, you do the appropriate weighted average as given by the book. That weighted average can be seen to be correct, since, say at 20C, sitting on point A you would be 100% liquid with the A composition, and sitting on point B you would be 100% liquid with the B composition. In between, you are balancing between the two liquids, which are in thermodynamic equilibrium.

If you have a book of phase diagrams handy, look at almost any phase diagram involving Pb, and you will likely find a liquid-liquid miscibility gap. For fun, look at Ag-Ni, two metals with fcc crystal structures, and you will see that the solids are immiscible, and there is even a liquid-liquid miscibility gap.

Answer 2

Your interpretation seems correct. When you have a mixture with an overall average mass fraction of X, it separates into two fluid phases in equilibrium. One of these fluid phases has mass fraction A, and the other fluid phase has mass fraction B. You can determine the fraction of the total mass that is liquid of mass fraction A and liquid of mass fraction B as follows: let f represent the fraction that is liquid of mass fraction B, and (1-f) the fraction that is liquid of mass fraction A. Then $$m(1-f)x_A+mfx_B=mX$$where m is the total mass of both phases. Cancelling out the m's yields:$$(1-f)x_A+fx_B=X$$Solving for f yields:$$f=\frac{X-x_A}{x_B-x_A}$$This equation is called the "lever rule," because it operates like as see-saw. The fraction of the total amount of material that is in phase B is proportional to the difference between the average mass fraction and the mass fraction of your substance in phase A. And, the mass fraction of material that is in phase A is proportional to the difference between the average mass fraction and the mass fraction of your substance in phase B.