Phase Equilibria

Phase Equilibria

Molecular Basis of Vaporization

What Temperature Implies

Raoult’s Law for Ideal Mixtures

Azeotropes

Discussion of All Phase Changes

Heating Curves

Effect of Intermolecular Forces

Phase Diagrams

Boiling Point Elevation

Freezing Point Depression

Colligative Properties

Molecular Basis of Vaporization

What Temperature Implies

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By definition, temperature is the average kinetic energy of all the molecules in a sample. The word “average” implies that not all the molecules have the same kinetic energy. The following thought experiment proves this readily. Imagine that you start with a collection of gas particles in a jar that have the same masses and speeds, moving in random directions.

They will eventually begin to collide with one another. These collisions will be elastic because the electron clouds will have the same shape before and after the collisions. If these clouds were to have been permanently deformed, that would mean that some of the pre-collision kinetic energy of the particles got used to change the shape of the electron clouds. This would mean that a chemical reaction had occurred. This is the basis of the collision theory of chemical kinetics. Our particles do not react.

By random chance, some of those particles will be oriented so they engage in head-on collisions. Because momentum and kinetic energy will be conserved, they will each leave the collisions with the opposite velocity they had before they banged into one another.

No K transfer.png

If the particles had actually collided at angle, as in the diagram below, by the very same laws of conservation of momentum and energy, they must leave the collision with unequal velocities. The values for this example collision have been provided, and take an annoying amount of work to calculate. So, don’t try to generalize a solution for the problem, but just apply the simple concepts anew to each specific problem you see on the test, where the math will have be manageable with pencil and paper. For your own understanding (and to improve your billiards skills), just recognize that the angle of impact leads to an unequal distribution of speeds after the collision. Since angled collisions are much more likely than head-on collisions in our hypothetical jar of gas particles, the kinetic energy which was initially evenly distributed will become unevenly distributed after just one angled collisions. Now imagine that two of those particles that sped up bang into one another at an angle in a second collision. The “victor” of that collision will be moving even faster. Repeat this a few hundred times, and some of the particles will be moving extremely fast.

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The Maxwell-Boltzmann distribution is a frequency histogram of the kinetic energies of particles in a sample which has a given temperature.

The Kinetic Energy cannot go below zero, and there are always some particles with REALLY high energies, which is why the distribution is right-skewed.

The “mean speed” is related to the temperature by:

T = ½ mparticlevmean2

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These same collisions that we have been mentally modelling in gasses happen in liquids, but you must factor in the cohesive intermolecular forces which act to slow the particles down.

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At any temperature, there are always some particles energetic enough to break free of the intermolecular bonds that hold them in liquid form.

The picture at left shows what will happen at  increasing temperatures. When the Temperature increases, the width of the distribution increases. More and more particles have a sufficient kinetic energy to break free and enter the gas phase.

If the liquid is in a closed container with some space at the top (bottle of water, e.g.), the liquid will keep producing gas until it reaches an equilibrium between the gas and liquid form.

This Youtube video shows how the Maxwell-Boltzmann distribution affects particles’ ability to overcome the force of gravity, but it is visually a great approximation of the particles’ energy being used to do work against a force. In the video, this force is gravity. In vaporization, the force that pulls the particles back down is the net intermolecular force between the particle of interest and the surrounding liquid particles. This net intermolecular force pulls in all directions when the particle is in the middle of the liquid, surrounded by other particles on all sides. At the surface, the net intermolecular force just comes from the particles in the liquid phase (below our energetic particle of interest), thus it points back down into the liquid.

There is no simple formula for the MCAT level to calculate exactly what the equilibrium pressure of the vapor (gas) will be at a given temperature, because it requires calculus to solve. However, there is an easy way to figure it out. The Keq of the equation:

Substance(l) → Substance(g)

will take the form of Keq = (Pressuresubstance as a gas), because the liquid reactant isn’t included in the Keq. So, like all Keq’s for endothermic reactions, this Keq (vaporization) must be directly proportional to Temperature. We can restate the Keq as follows:

Keq = (Pressuresubstance as a gas) , or

Keq = Pvap

In other words, the equilibrium pressure of the vapor above a pure liquid, Pvap ∝ T. Figuring out the exact Pvap may require using the Gibbs’ Free Energy Function and calculating for the Keq. Remember,

∆G = -RT ln Keq, so

∆G = -RT ln Pvap

Raoult’s Law for Ideal Mixtures

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For mixtures of liquids, the vapor pressure created by each liquid depends on what proportion of the liquid is one liquid (Liquid A) and what proportion is the other (Liquid B). Again, a little thought-experiment can help.

Put yourself mentally inside a pot containing a boiling mixture of liquids, and imagine yourself throwing out “baseballs” that are liquid particles. You will see that you have to be throwing up, towards the surface. To vaporize successfully, you must get the baseball through the surface. Thus the rate of vaporization will depend on the available surface area “lanes” for each type of liquid particle.

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In a liquid mixture, the proportion of the surface area that contacts Liquid A is the same as the proportion of the liquid is made up of Liquid A, i.e.: the molar fraction Xa.

XA = (moles of Liquid A)/(total moles of liquid in the mixture)

So, we get Raoult’s Law:

Pvapor-total above the mixture = ∑ Weighted vapor pressure for each liquid in the mixture

The “Weighted” vapor pressure is weighted by the mole fraction, so we can restate this as:

Ptot = XAPA + XBPB + XCPC + … + XNPN

where Ptot is the total vapor pressure above the mixture, XN is the molar fraction that is liquid N, and PN is what the vapor pressure of Liquid N would be at that temperature if it were by itself (pure liquid N). This is the other way the MCAT will often expect you to calculate a numerical quantity for the vapor pressure of a given liquid. Notice the similarity to Dalton’s Law of Partial Pressures, which uses the molar fraction in a gas mixture to weight the contribution of each gas species to the total pressure. Here, we use the molar fraction in a liquid mixture to weight the contribution of the vapor produced by each liquid species’ to the total vapor pressure above the mixture.

Azeotropes

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For some combinations of two liquids, there exists a certain mole fraction in the liquid phase that will generate the same mole fraction in the gaseous phase. These are called azeotropes. The graph below plots the gas-phase mole fraction vs. the liquid-phase mole fraction, and points out the azeotrope.

The reason these are important is that Azeotropes cannot be separated by fractional distillation, which relies on the disproportionate representation of the substances in the gas phases to separate the liquids.

A classic example is 95% ethanol. 2-gallon jugs of this stuff is found on lab benches everywhere. As a solvent mixture, it is really useful. Although some liquid volume may evaporate over timev, the concentration of ethanol will not change, ensuring its reliability for the standardization of reactions using it as a solvent.

440px-VLEIsoropanolWaterAzeotrope.png

Gunner Tip [don’t read if you aren’t aiming for an MCAT score of 40+]:  Compare the two images below. The image at left is how Raoult’s Law would be graphed if the liquid mixture were ideal. Ideality means that the particles of the two liquids are as attracted to each other (adhesive forces) as they are to particles of their own chemical substance (cohesive forces). If the adhesive forces and cohesive forces are not equal, the mixture deviates from Raoult’s Law.

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RaoultDeviationPressureDiagram.png

A negative deviation results when the total vapor pressure is lower than expected. Notice all the lines meet at the same endpoints, which means that the deviation happens with mixtures, not with pure samples of each liquid. For such a negative deviation to occur, it must to be harder for liquid particles to escape into the gas phase when they were mixed with another type of liquid, implying that the adhesive forces (opposite type) were greater than the cohesive forces (same type). Similarly, a positive deviation occurs when mixing the liquids reduces the net intermolecular force between the escaping particle and the liquid particles below. Notice that in each possible situation, the azeotrope is the point on the composition × axis where the slope of the Pvap vs. composition graph is zero (flat).

Discussion of All Phase Changes

Heating Curves

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The graph below plots the temperature of a sample of water vs. the Heat added. It shows two interesting things:

  1. At the phase transitions, heat is being added, but temperature doesn’t increase. That energy is going into the phase change, not into the speeds of the molecules. The energies of these phase changes are called the “latent heat of fusion” (fusion = freezing)  and the “latent heat of vaporization.”
  1. These are called Latent energies because you are adding Heat, but the heat energy appears “latent” because there is no resultant increase in temperature.
  1. Everywhere else in the graph, the slope is positive and straight. The ratio of ( Heat added/∆ Temp) is the specific heat, c. The formula for these sloped parts of the line is “Q = MCAT”: Q = mc∆T.
  1. The higher the specific heat, the more heat it takes to achieve the same temperature change.
  2. The way that this graph is plotted, the slope of each section is 1/c, so the lower the slope, the higher the specific heat
  3. Notice that the specific heat of water is higher than that of steam or ice. This is one of the reasons why liquid water is so important for water. It is a “heat buffer” that makes up 70% of your body, making sure the optimal temperature for enzyme activity is maintained over time.

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Effect of Intermolecular Forces

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The “vaporization region” of the graph is much wider than that for melting. This is because in a solid, the hydrogen bonds have maximal overlap (see the images below). In a liquid, the water molecules rotate, sometimes forming H-bonds, and sometimes not, which is a more unstable position, requiring an input of energy (the Heat of Fusion). In a liquid, the molecules are never more than one molecule-width apart. If they were, another molecule would fill the gap. In a gas, the space between the molecules is relatively HUGE. At those intermolecular distances, H-bonding doesn’t occur. Thus, the particles are not physically stabilized (kept from moving) in the gaseous phase, and are free to move about any which way. To reach this highly energetic (mobile) state, a GIANT Energy input is needed (the Heat of Vaporization).

Hydrogen bonds are simply the strongest level of solvent-solvent attraction. Similar, but less dramatic effects are noticed for polar compounds. The positive and negative ends of the molecular dipoles attract one another, and hold each other in relative position. A network of such interactions holds the whole liquid together. Hydrogen bonds are just an extreme form of this electrostatic attraction.

Nonpolar molecules such as alkanes still hold together because of transient dipoles. The electron pair orbiting a hydrogen is sometimes “eclipsed” from the view of adjacent molecules by the nucleus of the atom, thus creating a transient dipole. Here is a decent video explaining this effect (Dispersion  Forces aka London Forces aka Van der Waals Forces). This Youtube video discusses these weakest of intermolecular forces (Best image at 3m:11s).

To summarize: Electrostatic attraction is what holds liquids together. Hydrogen bonds are the strongest level of solvent-solvent electrostatic attraction, followed by the attraction between fixed dipoles that don’t Hydrogen-bond, and the weakest is that due to transient dipoles formed in overall non-polar molecules.

Why ionic liquids don’t matter: Yes, the ion-ion attraction is the strongest because according to Coulomb’s Law, the amount of separated charge is a whole integer value, but ionic compounds melt at a temperature of >1000K. Body temperature is < 400K, and the surface of the sun is >5000K. The MCAT doesn’t care about liquid salts, but it does care about dissolved salts.

Phase Diagrams

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The diagram below shows the typical phase changes for a pure substance (solid lines). Being able to read these phase diagrams is a critical skill for the MCAT. The vertical axis is always the ambient Pressure. The horizontal axis is the ambient Temperature. Together, the Temperature and Pressure comprise the thermodynamic conditions in which you are running the “reaction” of a phase change. The ambient conditions determine which state of matter is the most stable, and thus the most thermodynamically favored for any given combination of temperature and pressure.

Notice that as you go from lower to higher temperature on the graph (Left to Right), you pass from Solid to Liquid to Gas. Notice that in the middle temperature zone of the graph, increasing pressure squeezes gas down into a liquid, and will compress that liquid into a solid, reducing the intermolecular space at every step. Crossing a solid line means going from one phase of matter to another. The names for these specific phase transitions are listed for each direction in which you could cross the three solid lines. Notice that the Blue line represents a series of Boiling Points since the boiling point (the temperature at which Pvap = Patm increases with increasing atmospheric pressure pushing down on the surface of the liquid.

Phase Transitions.png

Image is modified from the original file on Wikimedia Commons.

Similarly, the Green Line represents a series of melting points, because as you increase the pressure, you squeeze the molecules together. For most substances, this decreased spacing makes it more thermodynamically favor to exist as a solid, since that is the densest phase.

A supremely important exception to that trend is Water. The Dashed green line has a negative slope, which implies that as you increase the pressure, water prefers to exist as a liquid (i.e.: the thermodynamically favored state is liquid water). When an ice skater glides over ice, the pressure of their skate melts the ice, leaving a thin trail of liquid water. You can do the following experiment if you haven’t witnessed this: Grab two ice cubes. Hold one loosely, and press hard on the second. The second will melt faster.

The diagram at right also highlights two important points. The first is the triple point, where all three phases of matter exist in equilibrium with one another.

Critical Point.png

Image is modified from the original file on Wikimedia Commons.

The second important point is the critical point, a combination of pressure and temperature above which the substance can no longer be classified as liquid or gas, but as a supercritical (above-critical) fluid.

For the MCAT, be able to make statements about two different substances (in this case, a pure solvent and a solution made with that solvent) by “drawing” imaginary vertical and horizontal lines that compare the two substances when you hold the Temperature or Pressure constant, respectively.

The diagram at right shows the phase diagram for two related substances. The purple lines are the phase changes for a pure substance. The blue lines are the phase changes for a solution made with the substance as a solvent. For the MCAT, be able to make statements about two different substances by “drawing” imaginary vertical and horizontal lines that compare the two materials when you hold the Temperature or Pressure constant, respectively. For example, if we hold Pressure constant at 1 atm (Patm at sea level), and follow the horizontal line, the solution will melt (Solid → Liquid) at a lower temperature than the pure solvent. Similarly, the solution will boil at a higher temperature.

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These differences between a solution and the pure solvent are well-recognized and tested often. Their molecular basis is discussed in the next three sections.

Boiling Point Elevation

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In solids, molecules are stacked on top on one another like a produce section in a grocery store. This crystal lattice takes many shapes, depending on the shape of the substance making it. Ions tend to make a cubic lattice, like the one for NaCl shown at right.

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For liquids used as solvents in solution, solvation shells form. These are spherical arrangements of the solvent in which the solvent molecules arranged their charged parts to be near the oppositely charged ion. It is the formation of shells that underlies the process of Solvation (aka Dissolution) - the creation of solutions. The ions are strongly attracted to one another because they are full charges, but get pulled away from each other by the sheer number of partially-charged solvent molecules, whose collective charge makes the net attraction to the solvent larger. This Youtube Video shows this process of Solvation nicely.

These solvation shells are the reason for Boiling Point Elevation, because they tend to keep the solvent in the liquid form. The solvent in the shell is attracted to the solute, and gets pulled towards the center of the shell. It take extra energy input for the solvent molecules in the solvation shell to join the “regular pool” of liquid solvent particles. This energy is use to do work against the electrostatic force. The Heat of Vaporization is the energy required to turn particles of pure solvent into vapor, and the ∆ Tb (Elevation in Boiling Point) represents the additional Kinetic Energy that must be invested to turn solvent particles in the solvation shell into particles in the regular liquid pool.

Freezing Point Depression

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When solids form from liquids, the molecules “stack” into an ordered 3-d shape. Distilled water is a good example of this (pictures at right and below).

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ice.GIF

If you dissolve a solute into the water and then try to freeze it, the solute disrupts the perfect crystal lattice, which means that the molecules must be moving even less (i.e: have lower kinetic energy, thus be at a lower temperature) in order to let the hydrogen bonds force them into some sort of crystalline order which wraps around the solute in an non-regular way. This is what is responsible for the freezing-point depression.

This Youtube video on recrystallization explains this phenomenon in a parallel situation, particularly if you watch what happens starting at 5m:30s. (Feel free to watch the whole thing if you don’t understand how recrystallization purifies solid reaction products.)

Colligative Properties

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Some phenomena depend not on the nature of the dissolved solute, but rather, just on the total concentration of dissolved particles. The equations that describe colligative properties tend to use a measure of concentration called molality. Whereas molarity is (mol solute/L solution), the molality = (mol solute/kg solvent). The difference is in the denominator. Boiling point elevation is a colligative properties because the number of dissolved particles determines the number of solvation shells. Freezing point depression is colligative because the more “grit” there is within the still-forming ice crystal, the more disordered it will be, thus the less movement it can tolerate without breaking apart into the liquid form. Below are the formulas for these phenomenon.

∆ Tb = iKbm, where Kb is a constant for the solution, and m is the concentration in molality

∆ Tf = -iKfm, where the only major difference is the negative sign for “depression.”

Mnemonic: The hot one gets hotter, the cold one gets colder (Boiling Point elevation and Freezing Point Depression).

Clinical application: In Biology, a very important colligative property is the osmolality of a solution. This is paralleled by the osmolarity (1 osm = 1 mol of total solute/L solution). In the human body, the osmolarity of the plasma, which is isosmotic with the extracellular fluid (ECF) which bathes the cells, is 290 ± 5 mOsm. Osmolarity is regulated by the hormone ADH. See the Endocrine and Kidney sections for more information.

Last edit: 10-17-14