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# LECTURE 10

last edited by 11 years, 10 months ago

(Red: Rahil Ashraf)

Maxwell Relations

This particular function is an exact differential: eqn. 1

It is an exact differential because if we say g is some subset of this function and then dg/dy at constant x equals dh/dx at constant y and as a result you will get the same answer when evaluating those derivatives.

Another property of an exact differential is that the second derivative of the partial derivative ends up being the same. eqn. 2

Looking at equation 2, here we have a partial derivative with respect to y at constant x, if then we take the derivative of this derivative with respect to x at constant y it is the same thing reverse. Essentially it is saying if you go in the x direction and then y direction it is the same as going in the y direction and the x direction.

The variation of internal energy with volume eqn. 3

Equation 3 is an equation for internal energy which is an exact differential so it does not matter if we change entropy or volume because we end up in the same place. eqn. 3A label this as an equation

From our final equation 3, can be derived to make the partial derivatives in eqn. 3A. A property of an exact differential is that if we take a second derivative with respect to the other parameter from either figure 1 equations that has to equal the Maxwell relations. If we were substitute in T and –p we can get the Maxwell relations. The important thing about Maxwell relations is this is that we can take 4 parameters and use it in an equation that equates two parameters on one side to two completely different parameters. This allows us to calculate/transpose variables that otherwise wouldn’t be able to calculate.

Internal Pressure

We can use internal pressure as an example for Maxwell relations. Internal pressure relates the attractive forces between molecules. eqn. 4

Internal pressure is defined through equation 4 at constant temperature. eqn. 5

From equation 5, if we divide everything by dv and we ultimately get equation 6: eqn. 6

From equation 6, we can substitute these derivatives from Maxwell relations allowing us to demonstrate how the internal pressure is related to pressure and temperature by equation 7 eqn. 7

Internal Pressure of an Ideal Gas

For an ideal gas we know that p=nrt/V

If we relate equation 7 and substitute nrt in for pressure we get the follow derivative: Ultimately getting equation 8 eqn. 8, proving from chapter 2 that the internal pressure of an ideal gas is zero.

Internal Pressure of a Van Der Waals Gas

Just like an ideal gas for internal pressure, similarly we can derive an equation from the Van Der Waals equation: eqn. 9

We can derive the equation: eqn. 10

Gibbs Energy eqn. 11

From our equation of Gibbs Energy  if we take the derivative looking at the incremental small change we get : We take the derivative, keeping one constant with respect to other one and then flipping them around and doing the same. We also know that from this we can do the same thing we did for equation 11 by taking the derivative: We also know that by combining the two latter derivatives we ultimately get eqn. 12. From equation 12 we want to change pressure and temperature. If we change one variable and keep the other variable constant for pressure we get: eqn. 13. For variable pressure and constant temperature we get : eqn. 14 both derived from equation 12. Both equations 13 and 14 show how Gibbs energy changes with pressure and temperature. Figure 2r

Looking at Figure 2r if you plot Gibbs energy vs. temperature at constant pressure the slope of your graph will be –entropy (-S). If we then keep temperature constant and plot Gibbs energy vs. pressure the slope will give you the volume (S=V). This graph will be important when looking at phase transitions in Chapter 4.

To sum up Figure 2r we can see that S>0 for all substances, G always increases (This means that if you add heat to something its Gibbs energy will increase) when T increases (constant pressure and composition)

V>0 for all substances (since matter can only have positive volume), G always increases when P increases (constant T and composition).

This will help us to understand what particular phase is stable at a particular set of conditions knowing that lower the Gibbs energy a substance has the more stable. Figure 3r Figure 4r

Figures 3r and 4r demonstrates the amount of Gibbs energy something has for gas, solid and liquids. Let us look at Figure 3r, pick a temperatue of 25 °C of water and we can see that it is liquid because it has as the LOWEST Gibbs energy as a liquid (it prefers to be a liquid). Whatever point you get to first in the graph is what a particular substance will prefer to be in. Notice at high temperatures you get a gas and so Figures 3r and 4r explain why matter is liquid, solid or gas. In Figure 4r the gas Gibbs energy increases drastically with an increase in pressure.

Change of Gibbs energy with pressure

From equation 12 of Gibbs energy if we keep temperature constant and change pressure we get: eqn. 15. If we integrate both sides we get: eqn 16. For Gibbs energy and pressure we will use molar quantities (molar Gibbs energy and molar volume) so equation 16 will become: eqn. 17. Remember that for a solid or liquid Vm (molar volume) is constant (insignificant change for liquid or solid when compressed referring back to Figure 4r) so that means we can bring the molar volume out in front of the integral for equation 17. Figure 5r

Figure 5r shows graphically what we are talking about in Gibbs energy and pressure, the volume times delta p which is the shaded area on graph.

When we look at an ideal gas we know that molar volume changes as we compress gas so we cannot take the molar volume as a constant from equation 17. From our ideal gas law we know that molar volume=RT/P. From this we can modify equation 17 and get for an ideal gas: eqn. 18. Figure 6r

Figure 6r shows an isotherm (constant temperature), volume vs. pressure, the integral in Figure 6r (V=nrt/p) calculates the area of the shaded region. eqn. 19. Equation 19 is exactly the same as equation 18 with different symbols and just says that the initial set of conditions are the standard set of conditions.

(Green: Matthew Allen-Daniels)

Stability of Phase

Phase is a characteristic of a substance where the form of matter is uniform in relation to chemical composition and physical state.  The three most common phases are solid, liquid and gas.  With concern to water, ice, liquid water, and water vapor respectively.  Phases are not static.  When one phase is spontaneously converted into another this is a phase transition.  Though they are spontaneous, phase transitions occur at specific temperatures and pressures depending on the substance.  Using water as an example again, at 1 atm, a temperature of <0C the most stable phase of water is ice, while at temperatures >0C the liquid water is the most stable phase.  This begs the question, how do phases changes effect Gibbs energy ?  <0C Gibbs decreases (liquid to solid), >0C Gibbs also deceases (solid to liquid), while the transition temperature (Ttrs) is the temperature where two phases are at equilibrium and has minimum Gibbs energy.  Remember Gibbs energy does not predict rate, so while a phase transition is spontaneous it may take eons to occur.  An example of this is a diamond.  At average temperature and pressure, graphite has a lower Gibbs energy than diamond, the process for the rearranging of each carbon atom is inexorable but takes a very long time.  Thus diamond is a metastable phase in that it is unstable but the kinetics of change are extremely slow.

Phase Diagram (Figure 3) Sample Phase Diagram

The figure above, (Figure 3.) is a phase diagram.  The diagram of a specific substance shows at what pressure and temperature the phase of matter is thermodynamically stable. All pressures and temperatures are relative in (Figure 1.) for ease of explanation.  Each phase of matter is governed by molar Gibbs energy.  At low pressure, and high temperature the substance is a vapor because a vapor has the lowest molar Gibbs energy of the three phases of matter at that specific pressure and temperature.  Conversely, if pressure is high and temperature is low the phase diagram dictates a solid has the lowest molar Gibbs energy.  The curving red lines in the figure are phase boundaries and show were at specific pressures and temperatures both phases are at equilibrium.  The Triple point will be discussed in greater detail further on but for the moment it donates a specific temperature and pressure where all three phases are at equilibrium.  The final noteworthy point on the phase diagram is the critical point.  The critical point denotes a point were at a temperature higher than the critical point the substance is a super critical fluid.  (For a review of super critical fluid refer to lecture 2 notes.)

Vapor Pressure

Vapor pressure is best understood by imagining a pure liquid encased in a sealed container.  While the vast majority of the molecules of the substance are in the liquid phase but some have the energy to escape to the gas phase.  These gas phase molecules are trapped by the container and only so many can build up and exert a pressure.  The vapor pressure is the pressure of these gas molecules at equilibrium with the liquid in the container.  A second example is the drinking bird that was shown in class.  When the bird’s head dips into water, the water is drawn into the sponge in it’s beak.  Even though the water is a liquid, some of the water escapes in to the vapor phase, cooling the gas in the head, returning the bird to it’s vertical position until the water evaporates completely.

Sublimation vapor pressure is vapor in equilibrium with a solid. (Figure 4) Illustration of vapor pressure in sealed container

Boiling Point

A common method for preparing pasta or rice is to boil it.  Heat is applied to food submerged in water until the water bubbles.  But is the formation of bubbles an accurate indication of boiling?   Thermodynamically, boiling is when a liquid is allowed to freely vaporize.  For this to occur, the vapor pressure of an open container must equal external pressure (in the case of boiling pasta this is atmospheric pressure).  The bubbles rising from the pot are vaporized water rising to the top of the water.  The actual temperature were the external pressure is equal to vapor pressure is referred to as boiling temperature.  This is the minimum temperature needed to be achieve true boiling.

•         At 1 atm. boiling temperature is referred to as normal boiling point (Tb)
• ·       At 1 bar. boiling temperature is referred to as standard boiling point

Because 1 bar is a bit smaller than 1 atm. Standard boiling point is marginally smaller that normal boiling point.

Boiling only occurs in an open container.  If the container is heated but closed, as the temperature goes up, the vapor pressure and the density of the vapor increases.  At the same time, the density of the liquid slowly decrease.  If the heating continues, eventually the density of the vapor equals the density of the remaining liquid and the two distinct phases disappear.  This temperature is called the critical temperature (Tc).  The vapor pressure at the critical temperature is referred to as critical pressure (Pc).  Above this critical temperature, instead of a definite surface between two phases, only one phase exists called a supercritical fluid. (Figure 5)   Illustration of critical point in sealed container.    a.) Liquid (dark blue) being heated, vapor pressure (light blue) is low.     b.) Heating continues, vapor pressure increases, density of liquid decreases.    c.) TC, and PC achieved, vapor and liquid have become supercritical fluid

Melting Points and Triple Points

Melting temperature is the temperature where the solid and liquid phases of a substance at a set pressure at equilibrium coexist. The freezing temperature is the same as the melting temperature because substances freeze at the same temperature that they melt.  This can be visualized by an ice cube melting when placed in an environment that is higher than 0 C, but liquid water freezes into ice in an environment that is at least 0C.  As is similar to boiling point, normal freezing point (Tp) is the freezing temperature where the pressure is 1 atm.  Likewise, standard freezing point is the freezing point when the pressure is 1 bar.  The normal freezing point can also be called the normal melting point.  The point where all three phases exist simultaneously in a perfect equilibrium is the triple point and can be seen on the phase diagram. (Figure 1)  For all three phases to exist, both pressure and temperature must be exact.  A slight deviation will cause at least one of the phases to no longer exist.  For pure water, the triple point is 273.16 K and 611 Pa.

Three Example Phase Diagrams (Figure 6) Phase Diagram of carbon dioxide.  From the above figure it possible to explain why dry ice (solid carbon dioxide) sublimates at atmospheric pressure.  To see liquid carbon dioxide, pressure must be increased to at least 5.12 atmospheres. (Figure 7)   Phase Diagram of Water.  Note the numerous solid phases of water (I-XI).  These are different conformations caused by rearranging of hydrogen bonds in water’s unique open structure.  They are only seen at high pressures (Figure 8) Phase Diagram of Helium.  Note the similarities to carbon dioxide.  Many diagrams appears some what similar

Thermodynamics of Phase Transitions

Molar Gibbs energy (Gm) is important for determining phase transitions and can be called chemical potential (μ).  For a phase transition to occur the phases must be a equilibrium. Thermodynamic definition of equilibrium: when equilibrium exists, the chemical potential (μ) must be uniform throughout the sample no matter how many phases are present. (Figure 9) Illustration of Thermodynamic Equilibrium.  Even though there are two phases in the sample, the chemical potentials are the same so the system is at equilibrium

If the temperature is low, as long as the pressure is not too low, solid phase has the lowest chemical potential and is the most stable.  Chemical potential changes with temperature. This change explains why there are phase changes in a substance.

This can be shown by the flowing equations (Equation 1) (Equation 2)

The first equation denotes that as temperature rises chemical potential decrees.  Graphing μ vs. T, the slope becomes steeper the higher the temperature, and thus the higher the entropy.  The second equation plots μ against pressure, showing the slope is equal to molar volume.  Thus, raising the pressure raises μ. (Figure 10) Chemical Potential (μ) vs. Temperatuere.  Note the solid blue line shows lowest chemical potential, and this highest stability for a particular phase at a specific temperature.