This introductory physical chemistry course examines the connections between molecular properties and the behavior of macroscopic chemical systems.

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Statistical Molecular Thermodynamics

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This introductory physical chemistry course examines the connections between molecular properties and the behavior of macroscopic chemical systems.

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Module 8

This last module rounds out the course with the introduction of new state functions, namely, the Helmholtz and Gibbs free energies. The relevance of these state functions for predicting the direction of chemical processes in isothermal-isochoric and isothermal-isobaric ensembles, respectively, is derived. With the various state functions in hand, and with their respective definitions and knowledge of their so-called natural independent variables, Maxwell relations between different thermochemical properties are determined and employed to determine thermochemical quantities not readily subject to direct measurement (such as internal energy). Armed with a full thermochemical toolbox, we will explain the behavior of an elastomer (a rubber band, in this instance) as a function of temperature. Homework problems will provide you the opportunity to demonstrate mastery in the application of the above concepts. The final exam will offer you a chance to demonstrate your mastery of the entirety of the course material.

- Dr. Christopher J. CramerDistinguished McKnight and University Teaching Professor of Chemistry and Chemical Physics

Chemistry

At this point in the course, we've gone through the first, second, and third laws

of thermodynamics. This is the last week, and we're going to

wrap up with the introduction and discussion of two last state functions,

let's start with the Helmholtz free energy.

So I'll remind you that. We discover the condition for a

spontaneous process, is that the entropy change is greater than or equal to zero,

in isolated system. That is a system at constant internal

energy and constant volume. Now, if the system is not isolated, then

the entropy change that's evaluated has to consider both the system and the

surroundings, and evaluating the total entropy change is not very convenient.

That might be the total entropy of the universe we'd be interested in.

So instead, let's consider a system at constant temperature and volume, instead

of internal energy and volume. So the system's no longer isolated, and

that means that heat can flow in order to maintain constant temperature, heat from

the outside or to the outside exchanged with the system.

And as a result, the change in entropy being greater than or equal to zero for

the system alone, is no longer the criterion for a spontaneous process.

And so we can ask ourselves, what is the criterion for a spontaneous process?

A constant T and V. So before we do that, it's a new week,

let's get the intellectual juices flowing.

Ill let you take a quick self assessment that's a little bit of a review and then

we'll move on. Alright, so with that review out of the

way, you remember that heat and work are path functions, but that their sum is a

state function and it's the differential of the internal energy.

That's the first law of thermodynamics. We also have learned the second law says

that DS is greater than or equal to the del q over t or I can rearrange that and

have something I can insert in for del q, namely TdS.

Moreover, I know at constant volume that the work is going to be minus external

pressure times dV, and that's equal to 0 at constant volume.

And so, I established that I have a relationship of dU is less than or equal

to TdS at constant volume. Where the less than symbol applies for an

irreversible process, because dS will be greater than del q over T, and the equal

symbol applies for a reversible process where dS is equal to del q over T.

Of course, if the system is isolated. Then dU is equal to 0, right, that's

constant internal energy. And we would recover that dS is greater

than or equal to 0. That was actually what we determined to

be the condition for spontaneity when the system is isolated.

But when it's not, that suggests that we can rearrange this inequality, dU being

less than or equal to tDS. And say that, at constant volume, dU

minus TdS is less than or equal to 0. And again, the less than symbol holds for

irreversible processes. The equal symbol holds for reversible

processes. And this refers to constant T and V.

Given that, we can define a new state function.

Namely, U minus TS. And that defines the Helmholtz free

energy which, is abbreviated capital A. And it was first suggested by Hermann von

Helmholtz shown here on the left hand side of the slide.

And notice that the total differential, then, d of U minus TS.

That would be dU minus. And then, with the chain rule, it would

be dTS. But we're at constant temperature.

So dT is 0. And what's left is TdS.

And that's this inequality up here. So this total differential, these are

state functions. And hence, the Helmholtz Free Energy must

be a state function. Thus, at constant T and V, we have that

dA, Ud of us minus TS, is less than or equal to 0.

Less than 0 for spontaneous processes. Equal to 0 at equilibrium.

So, if we imagine a system that starts out of equilibrium, it will have some

value of the Helmholtz free energy. And as spontaneous processes occur over

time, they will lower that free energy. DA always has to be less than 0.

So the Helmholtz free energy is going down, down, down, down, down until

finally, the system achieves equilibrium. At which point, the Helmholtz free energy

is minimized and it stays at that value until the system is perturbed from

equilibrium again, indefinitely. So spontaneous processes will be

reversible equilibrium processes. So before we explore that state function

a little more, I can't resist doing a little bit of history of chemistry and

physics. So von Helmholtz was a fascinating

individual in the history of thermodynamics.

He has a wonderful quote that most physical scientists keep close to heart.

It says, whoever in the pursuit of science, seeks after immediate practical

utility, may generally rest assured that he will seek in vain.

And this quote is often trotted out as a basis for a fundamental scientific

research, curiosity driven research, you just never know what might come of it.

What's ironic to some extent is that Helmholtz himself was remarkably

successful in pulling out practical utility.

So for instance, Helmholtz invented the opthalmoscope.

So that device that an optometrist or a doctor uses to look at the inside of your

eye. That was invented by von Helmholtz in the

nineteenth century and he figured out the way to structure the optics and the light

beams so that people could see the inside of the eye, and it revolutionized the

medicine of the eye. In fact he was a physician, he was also a

professor of physiology and psychology, he did fundamental work on the

transmission of nerve signals. Optimology, auditory perception, oh and

thermodynamics just on the side. He ended up with academic appointments

beginning in Konigsberg, then Bonn, then Heidelberg, and ultimately he moved to

Berlin where he was a professor of physics.

And a senior colleague of Max Planck, who appeared early in this course, as one of

the founders of quantum mechanics. And so, one of the wonderful things about

Germany, is that it tends to honor its scientists a great deal.

And in this case, on the 100th anniversary of his death, the German

Bundespost issued a stamp honoring von Helmholtz.

You see here a picture of an eye to recognize what he did in, in the area of

physiology. As well as other features that honored

his scientific career. So a real towering figure in early

thermodynamics and in early German science in the nineteenth century.

All right, well enough digression on history.

Let's return to our state function, Helmholtz free energy equals U minus TS,

that's the definition. And so if I were to consider an

isothermal change, that is one where temperature is going to be held constant.

I have delta A is equal to delta U minus T delta S.

Recalling that our condition for spontaneity is dA less than or equal to

0. I could write that for a non

infinitesimal change as delta U minus T delta S less than or equal to 0.

And just emphasizing, again, this is under constant temperature.

And volume conditions. And so what it says is that if a process

is going to take place spontaneously, there needs to be a compromise.

So there's an energy term, and an entropy term.

So if I increase the entropy because it's preceded by a negative sign, that will

make delta A more negative. Delta U, on the other hand.

Rather than becoming more positive, would need to become more negative to make

delta A more negative. So these two can balance one another.

And entropy becomes more important at high temperature, because it's multiplied

by temperature. A process where delta A is positive will

not take place spontaneously at constant T and V.

Instead, you'll have to add something to the system, perhaps work, in order to

drive that process. And so why was Helmholtz so interested in

this quantity, the Helmholtz free energy? And it's because it gives you insight

into the work you can extract from a system.

So given that we have delta A is equal to delta U minus T delta S.

And given that A is a state function, we can evaluate a change in the Helmholtz

free energy. Simply by knowing the beginning point and

the end point, and indeed we can follow a reversible path which tends to allow us

to compute more readily what the change is.

And so, on a reversible path I would have that delta S is equal to the reversible

heat divided by temperature. So T delta S is just the reversible heat.

And so if I plug that in, I will get that delta A is equal to delta U minus the

reversible heat. But we know from the first law of

thermodynamics, that delta U is equal to the reversible heat plus the reversible

work. So if I take away the reversible heat I

am left with. Delta A is equal to the reversible work.

So isothermal reversible work that is. Given a process for which delta A is less

than 0, that is a spontaneous process, the reversible work will be the maximum

work. Arbeit is work in German, capital A, and

that's why Helmholtz used A as the symbol for the Helmholtz free energy.

That can be extracted from the system if you do anything irreversibly.

Then the entropy inequality will come into play.

You'll have wasted some of the heat energy, and you won't be able to do as

much work. If delta A is greater than 0, well then

the process is not spontaneous. And W reversible, the reversible work is

the minimum work that must be done on the system.

In order to make the process occur. And again if there are irreversible

processes, you'll need to increase the amount of work in order to accomplish

those. All right.

Well, that is the Helmholtz free energy in a nutshell.

Next, I want to take a look at another kind of free energy, and in particular

the Gibbs free energy.

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