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Â We learned that the governing thermodynamic potential of the closed

Â isolated system is the entropy s, given as s of u, v, and m.

Â That is the control or canonical variables are the internal energy, volume,

Â and the number of particles.

Â However, the closed isolated system is only one possible thermodynamic system.

Â In many instances it is desirable

Â to have different control variables where we replace one or

Â more of the extensive variable with their conjugate intensive variables.

Â For convenience, we start with the equation of state u of s, v, and n.

Â Remember, that from our definitions, the extensive variables and their

Â conjugate intensive variables are related through the following relationship.

Â 1:02

A thermodynamic ensemble is defined by the canonical variables of the system.

Â Now changing from one ensemble to another simply amounts to shifting from one

Â thermodynamic potential to another that depends on the new canonical variables.

Â It turns out that replacing an extensive variable for

Â its conjugate intensive variable is effectively

Â replacing the control variable with the slope of the control variable.

Â 1:30

Now this is a deep concept, and we'll illustrate this with an example so

Â that you can gain some intuition about this.

Â The mathematical method of performing this change of variables

Â is called the Legendre transform.

Â Now let's consider the following steps.

Â Let's take a function y given by f(x) which is defined as a list of x and

Â y pairs, that is the plot of y of x graphs (x1, y1),

Â (x2, y2), (x3, y3), and so on.

Â Now, equivalently, it's possible to express the same information in terms

Â of the tangent slope c of x on the corresponding y-intercepts b,

Â that is the points (c1, b1), (c2, b2), (c3, b3), and so on.

Â Now for a small change dx in the x variable,

Â the corresponding change dy is given by.

Â Now from this point on the curve, we can extend a line back to

Â x equals 0 to find the y-intercept b of x such that.

Â Now the resulting function for

Â the series of intercepts versus the slope, that is b as a function of c,

Â that is b of c, contains the same information as y equals y of x.

Â Now let's say we're interested in a change in the intercept b of c.

Â Now, this is given by.

Â 3:03

Now this is still quite abstract.

Â Let's consider a simple example.

Â Let's consider the following function, y = (x- 2) square.

Â Now, let's apply the Legendre transform to shift the variable from x to the slope c.

Â The slope of the function is simply given by twice (x- 2),

Â that is c is equal to twice (x- 2).

Â Now, this gives us a way to relate the variable x to the variable c.

Â 3:51

Now, all of the thermodynamic potentials that define specific ensembles

Â are Legendre transforms of the original potential,

Â say the internal energy, u of s, v, and m.

Â Now, let's transform the entropy, s, to the conjugate intensive

Â variable temperature, t, to gain the closed isothermal ensemble.

Â Now this gives us a new quantity that is given as the internal energy

Â minus the temperature times the entropy.

Â Now this quantity is defined as the Helmholtz free energy.

Â Now, let's take the next example of phase equilibrium.

Â Consider a closed isothermal system,

Â which contains a fluid that has a liquid and a vapor.

Â In order to describe the phase equilibria,

Â you need a description of the equation of state for the vapor and the liquid.

Â 5:04

The microscopic intuition for

Â this is to challenge two assumptions made in the ideal gas.

Â We assume that an ideal gas occupies no volume, and

Â that the molecules don't interact with each other.

Â On the other hand, the van der Waals gas accounts for

Â corrections that relaxes these two assumptions.

Â Now, there is a constant b that is added,

Â that represents the excluded volume of the molecules in the system.

Â Now, another constant, a, is added, which determines the strength

Â of the two-body attractive interactions between the molecules in the system.

Â 5:49

Now this dimensionless form reduces the number of parameters

Â to capture the temperature dependence.

Â Only a tilde now depends on the temperature.

Â Now our goal is to find the limit of stability

Â of each phase, the conditions where the phases coexist, and

Â the properties of the two phases in coexistence.

Â The simple example of phase coexistence demonstrates the essential

Â issues at work in more complex multi-component and multi-phased systems.

Â Now let's consider a plot of the pressure as a function of volume for

Â the van der Waals fluid for a given A tilda value.

Â 6:38

At large volume we get the vapor phase, and

Â at very low volume we get the liquid phase.

Â In between there emerges a region where the slope

Â of the pressure versus volume curve is positive.

Â 6:53

Now this is in violation of the second law of thermodynamics.

Â Now a way around this violation is the emergence of phase coexistence.

Â Now these set the limit of the stability of the pure single phases.

Â Now how we do find the properties of the coexisting phase?

Â Now clearly at equilibrium the chemical potential of the vapor phase

Â is equal to the chemical potential of the liquid phase.

Â 7:21

In addition, mechanical equilibrium requires the pressure of the liquid

Â phase is equal to the pressure of the vapor phase.

Â Now let's call this the coexistence pressure.

Â One intuitive way to think about this is that the energy

Â that goes into making the vapor phase is compensated

Â by the energy that is given up by the liquid phase.

Â This implies that the coexistence pressure is set such that

Â the area under the PV curve in these two regions are exactly equal.

Â 7:54

Now this technique is known as the Maxwell's construction technique.

Â And this provides a way to determine the coexistence pressure

Â of a vapor liquid equilibrium.

Â Now the phenomenon of phase coexistence is extremely crucial in general.

Â This finds applications in many important areas.

Â Now in this module, we studied classical thermodynamics and

Â the first and second laws of thermodynamics.

Â Now from the second law of thermodynamics, we derived conditions for equilibrium.

Â 8:29

Now we introduced the concept of thermodynamics ensembles.

Â And moving from one ensemble to another is a simple change of variables that is

Â equivalent to the mathematical technique of Legendre transforms.

Â Finally, we explored the idea of phase equilibrium and showed

Â a technique to identify the coexistence pressure of a vapor liquid equilibrium.

Â