Lecture 17 Chapter 9, sections 1-3

Announce: • Remember to complete and return Dr. Krueger’s mid-term evaluations forms • Chem seminar on Friday

Outline:

Phase Transitions  ∂G  = −S  ∂   T  p

Phase Diagrams (H2O and CO2)  ∂G    = V  ∂   p T Chemical Potential  ∂G  µ =  ∂   n  p,T

Review: Some protein and nucleic acid binding events are driven by enthalpy and some by entropy. The balance between the two is governed by the solvent interactions: favorable solvent interactions Æ enthalpy controlled poor solvent interactions Æ entropy controlled

The body makes nonspontaneous reactions happen by coupling them to exergonic reactions like ATP Æ ADP

Phase Diagrams

Phase Diagrams Summarize Solid-Liquid-Gas Behavior of a Substance

Let’s start by defining a phase: form of matter that is uniform throughout in chemical composition and physical state.

phase transition: the spontaneous conversion of one phase to another that occurs at a well-defined temperature – the transition temperature

said another way, the transition temperature is the temperature at which the Gibbs free energies for both phases are? equal (remember that the system is seeking the minimum Gibbs free energy).

1 We can explain transition temperatures by understanding how the Gibbs energy changes with temperature: Recall that G is a function of p and T, so then the total derivative is?  ∂G  ∂G dG =   dp +   dT (from definition of total derivative for dG)  ∂p   ∂T  T p

But, we also know from our fundamental equation for G?: dG = Vdp - SdT

As we have done three times before, you guys can quickly fill in?  ∂G  ∂G   = Vand  =−S (equating coefficients in expressions for dG)  ∂p   ∂T  T p

Therefore, if we plot G vs. T, what is the slope? negative of the entropy

Show diagram. How do you expect the entropies of the phases to vary? Sg > Sl > Ss. Gases have the highest entropy and therefore the steepest dependence of G on temperature (then liquids and then solids). Where are the transition temperatures? the intersections of the G vs. T lines for each phase.

liquid solid gas

G

at fixed p

T TMelt TVap

Note that the above G vs. T plot gives us two points on a p-V phase diagram.

If we repeat for many pressures, then we can build up a full phase diagram – a P vs. T plot showing regions of thermodynamic stability. We have a generic one below:

2 critical point P solid liquid

gas

T

Notice the following characteristics of phase diagrams:

Phase boundaries or phase coexistence curves: shows the pressure, temperature pairs where two phases coexist in equilibrium. These boundaries are important as they indicate such important quantities as the freezing and boiling temperatures as a function of pressure.

Vapor pressure: the pressure of a vapor in equilibrium with its condensed phase at a specified temperature. For example, the vapor pressure for water at 25 C is about 25 Torr, while the vapor pressure for mercury at 25 C is ~ 0.01 Torr. The value of the vapor pressure is directly related to the intermolecular forces that govern the properties of the liquid (mercury has a greater attraction for itself than does water).

Normal boiling temperature: this is a special pressure, temperature point on the liquid-gas phase boundary because it is defined for a pressure of 1 atm. In a microscopic sense, all of the molecules now have sufficient energy to push the atmosphere out of the way and escape as a gas. Note that for equilibrium conditions, the boiling and condensation temperatures are the same.

Critical point: the temperature at which there is no longer any difference between the liquid and gas phases. As the temperature is increased in a closed system, more and more of the liquid is vaporized, leading to an increased density of the gas. Where does this lead? Eventually, a high enough temperature is reached so that ρliquid = ρgas Another way to put this is that ∆Hvap decreases as T increases until, at the critical point, ∆Hvap equals zero.

3 Thus, the two phases are indistinguishable. Of course, there is a corresponding critical pressure at the critical temperature. For water, the critical temperature is 647 K and the corresponding critical pressure is 218 atm!

Normal melting temperature: this is another special pressure, temperature point on the solid-liquid phase boundary. It is also defined for a pressure of 1 atm. Note that for equilibrium conditions, the melting and freezing temperatures are the same.

Triple point: this is a very special pressure, temperature point. It lies at the intersection of the gas-liquid and liquid-solid phase boundaries. At this point only, all three phases can coexist. For water, the triple point is 273 K and 4.58 Torr.

Let’s now look at the phase diagrams for a few representative substances:

Water phase diagram:

V

II III

critical point I 220 liquid atm P solid

gas 4.58 torr

T3 = 273 T Tc = 647

We should all be familiar with the triple point and normal boiling points of water. Remember that water is a very unusual substance and that is seen in the phase diagram. What is the key identifier of water? The unusual negative slope of the solid-liquid boundary. This is ascribed to? the unusual hydrogen bonding structure of water. This also leads to solid water floating in liquid (less dense).

The above diagram shows higher p and T than one usually sees. Consequently there are some other interesting things: critical point: T=647 K and p=218 atm

4 many phases of solid water: they are all solid forms of water, but with very different properties. All are fairly exotic. Interestingly Ice VII melts at 373 K. As with all phase diagrams remember that the data only apply to equilibrium conditions

Now, let’s take a look at a more typical phase diagram – CO2

critical 73 atm point

liquid P solid

gas

5 atm

T3 = 216 K T Tc = 304 K

CO2 has a more characteristic liquid-solid boundary, where the melting temperature increases as a function of pressure. Think of it as increasing pressure holding the molecules together so Tmelt increases. Also shows more ‘normal’ behavior of solid sinking in liquid.

Also confirm our everyday experience that solid CO2 sublimes directly to vapor – 1 atm is below the triple point pressure of 5 atm. (Normal sublimation temp is –78 °C or 195 K.) From another point of view, liquid CO2 at 1 atm is never the most stable phase, no matter the temperature.

Do Will's CO2 Demo 1) Draw in 1 atm phase change 2) Draw in pressurization and melting 3) Draw in depressurization, evaporation and cooling

Supercritical CO2 (compressed CO2 above the critical temperature, 304 K) is useful as a low temperature solvent because many organics are soluble in CO2. For example, this is how coffee is decaffeinated. If a traditional higher temperature separation were attempted, you wouldn’t be left with

5 any flavor! Supercritical CO2 is also useful in many industrial cleaning processes –it dissolves things like a liquid but flows more freely like a gas.

Dependence on phase stability on the conditions

It is generally true that at low temperatures that the most stable phase is the solid. However as we saw earlier, because the rate of change of the free energy with temperature is different for each phase, eventually the liquid and gas phases become the most thermodynamically stable as the temperature is raised. How does G vary with T?

 ∂G  = −S  ∂   T  p

This shift in stability comes about because of the difference in the way the free energy of each phase depends on pressure. Let’s look more closely at this pressure dependence. Recall how G varies with p?  ∂G    = V  ∂   p T Thus, the slope of a G vs. p plot is the volume (usually molar V and molar G). For most things how does molar V depend on phase? Vg > Vl > Vs

Look at G vs. P plots at various temps (like Fig 9.12 from text)

liquid liquid solid solid G G

gas gas T < Ttriple T = Ttriple

P P

solid solid

liquid fluid G G T > Ttriple gas T < Tcritical T > Tcritical

P P

6 Note that for water Vs > Vl so the (third) diagram looks different, essentially with the liquid and solid curves interchanged. This is because of the unusual water phase diagram that allows P increase at const T to go from gas to solid to water. Usually P increase takes things from liquid to solid.

Chemical Potential

Consider a system consisting of two phases of a pure substance in equilibrium with each other (i.e., liquid water and water vapor). What is the total free energy of the system? And should we use A or G?

l g G = G + G

If dn moles are transferred from the liquid to the gas phases (at constant T and P),

 ∂G g   ∂G l  dG =   dn g +   dn l  ∂ g   ∂ l   n  P,T  n  P,T

Since dnl = -dng (conservation of matter!)

 ∂G g   ∂G l   dG =   −   dn g  ∂n g   ∂n l    P,T   P,T 

The partial derivatives in this equation are called chemical potentials:

 ∂G g   ∂G l  µ g =   and µ l =    ∂ g   ∂ l   n  p,T  n  p,T

dG can be rewritten simply:

dG = (µg - µl)dng

If the two phases are in equilibrium, then what does dG have to be? dG = 0 (there is no thermodynamic driving force in either the condensation or vaporization direction), this requires that

µg = µl

If the two phases are not in equilibrium, then a spontaneous transfer of matter will occur. For example if µg > µl, to make the process spontaneous (dG < 0), then dng would have to be negative. In other words, molecules from the gas phase will be spontaneously transferred to the liquid phase.

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This is where the term chemical potential comes from. Matter “flows” from situations of higher chemical potential to situations of lower chemical potential.

Note that if we start from our definition of chemical potential:  ∂G g  µ g =   how do we rearrange to solve for G?  ∂ g   n  p,T

dG = µdn ∫∫ G(n,T, p) = nµ(T, p)

So, this sure makes it look like µ is just the molar Gibbs energy. For a pure, single substance this is exactly what µ is. The Gibbs energy and chemical potential are the same thing.

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