Example 2. Binary Phase Equilibrium and Mass Balances

Set the composition slider at z = 0.5 and the temperature slider at T = 316 K. You are now simulating a mixture that has half blue and half red molecules.  What phase(s) is/are present?  Now heat the fluid up to 324 K.  Is what you observe in the simulation consistent with the Txy diagram?  This is a multifaceted question.

First of all, you should observe two phases.  Is there a density difference between the two phases?  The velocities of the molecules are the same in both phases (recall that T and P must be the same in the two equilibrium phases), so why is there a cluster of molecules (liquid) and relatively un-grouped molecules (vapor) if the average kinetic energy of the molecules is the same?  To answer this question, observe that it's not the same molecules in the liquid all the time; there is exchange between the two phases.  Some molecules are evaporating from the liquid and others are condensing into it.  At equilibrium the evaporation rate is the same as the condensation rate.  At any one time, the number of molecules in the vapor and liquid will vary as can be seen by the fluctuations in the magenta line on the strip chart recording (Click on the Recorder tab to observe it).  Note that the fraction of the molecules in the vapor phase, or fraction vaporized, oscillates about the value given by the lever rule from the Txy diagram (the horizontal magenta line on the strip chart recorder).  So what is happening is that there is a rough balance between the attractive forces that molecules feel in the clusters as they are attracted into the potential wells and are held close to each other and the kinetic energy of the molecules that propels them out of the attractive forces.  As they get close to several molecules that are already held together by the collective attractive forces, the net attraction is strong enough that the molecules become trapped in the potential wells of the other molecules.  However, molecules near the surface of the liquid cluster don't feel as many close pair interactions and because of this rough equality between the kinetic energy and the potential energy, they can easily leave the cluster as they are bumped, etc.  In the gas phase where the density is still low, there are not enough pair attractions to the molecules to overcome the kinetic energy and molecules will generally collide and separate.  As they collide with the liquid, there are enough pair potentials to overcome the kinetic energy and capture the molecule into the liquid phase. The high gravitational field applied in this simulation to ensure that the liquid forms at the bottom is unnatural, but tends to ensure that contact with the gas and liquid molecules happens more often so that you can see the process in real time.

Next, observe the thermodynamic equilibrium that occurs,  shown on the Txy diagram are the equilibrium compositions in the liquid (x) and vapor (y).  These are represented by the ends of the tie line on the phase diagram.  The asterisk represents the bulk or overall composition (z).  How do these values compare with the values in the simulator?  Again, instantaneous values of the phase compositions can be seen on the strip-chart recorder.  The green line represents the mole fraction of red molecules in the vapor phase; i.e., y.  This instantaneous value is found by counting the number of each color in only the vapor phase.  Then y = r/(r + b).  Similarly, x = r/(r + b) where the numbers of red and blue are counted this time only in the liquid phase.  How do the instantaneous phase compositions compare with the actual values, marked on the strip chart recorder with horizontal lines?  For any one period of time, MD simulations may somewhat under- or over-predict the compositions because these quantities fluctuate on a molecular scale.  How would you suggest to get a very accurate estimate of the phase compositions from these fluctuating simulation values?

Finally, let's observe the constraints placed on the system by the mole balances; i.e., the number of red and blue molecules remains constant.  Again look at the Txy diagram. What does the lever rule tell you about the relative quantities of the two equilibrium phases?  Is this consistent with the fraction vaporized in the simulation?  To see this, look at the strip chart recorder and compare the fluctuating fraction of vapor with the constant line for the experimental value.  Notice that at this temperature (324 K) there is only about 17% of the molecules in the vapor phase.  Now move the T slider up to 326 K.  Now what has happened to the relative amounts of the two phases?  Why is there more vapor than before?  What is the relationship between the relative position of the asterisk to the two tie-line endpoints on the Txy diagram and the fraction vaporized shown in the simulator?  By just looking at the Txy diagram, can you predict what will happen when you raise the temperature of the simulator up to 328 K?  Try it and see if you were correct. Now what do you observe about the relative amounts of the two phases?  Why?  In this case, you are barely below the dew point temperature and you have very nearly all vapor.  In fact for the small number of molecules used by the simulator, there are so few molecules in the liquid phase that it is essentially impossible to identify a liquid phase at all.  If we had enough size in our simulator so that we had 1000 molecules instead of 20, you would see a small but distinct liquid phase just as before.  However, the vapor phase would be so large that it couldn't be displayed it on your screen.

Now move the temperature up to 330 K.  What happens?  Can you again explain why in terms of the intermolecular forces and the average kinetic energy of the molecules?

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More Examples: Example 1 | Example 3 | Example 4 | Example 5