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Lecture Comments (2)

1 answer

Last reply by: Professor Hovasapian
Wed Jul 6, 2016 8:02 PM

Post by Qzwxec Poinm on July 6 at 12:21:09 AM

What would be an example of "other" work?

Spontaneity & Equilibrium I

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Reversible & Irreversible 0:24
    • Reversible vs. Irreversible
    • Defining Equation for Equilibrium
    • Defining Equation for Irreversibility (Spontaneity)
    • TdS ≥ dQ
  • Transformation in an Isolated System 11:22
    • Transformation in an Isolated System
  • Transformation at Constant Temperature 14:50
    • Transformation at Constant Temperature
  • Helmholtz Free Energy 17:26
    • Define: A = U - TS
    • Spontaneous Isothermal Process & Helmholtz Energy
    • Pressure-volume Work

Transcription: Spontaneity & Equilibrium I

Hello and welcome to www.educator.comn and welcome back to Physical Chemistry.0000

Today, we are going to start our discussion of spontaneity and equilibrium.0005

We are coming to the end of thermodynamics.0009

We are going to close out that circle and here is where everything absolutely starts to come together 0012

and we start to seriously apply these things to chemical reactions.0018

Let us jump right on in.0022

The following terms mean the same thing as irreversible.0026

We will often refer to a irreversible transformation as a real transformation, natural transformation, spontaneous transformation.0030

Let us go ahead and use the blue in here, this is going to be the word that we used most often.0038

When we talk about reactions you remember from General Chemistry, 0043

we talk about spontaneous reaction has nothing to do with speed, it is just a thermodynamic quality and we will talk about a little bit more.0045

Irreversible, real, natural, or spontaneous.0054

Here is what we want to do, we ask ourselves what differentiates irreversible or an ideal transformation.0059

Ideal transformations are ways of dealing with things that we have been dealing with but they are not real, 0067

they are ideal in the sense that they provide an alternate way of looking at things 0073

may help us put together some sort of a mathematical structure for it, but it is not real transformation.0077

The real world does not work that way but gives us a standard, a reference.0083

What differentiates a reversible transformation from an irreversible transformation or real one?0088

We consider the relation between the heat that flows and the entropy change that accompanies that heat flow in a given process.0093

During reversible process a system deviates only infinitesimally from equilibrium.0105

We have talked about that a lot especially when we talked about energy.0110

The system does undergo a transformation but it is essentially at equilibrium during the whole step 0113

because you are only deviating slightly from where you are so essentially you are at equilibrium.0121

Therefore, the condition of reversibility is the condition of equilibrium.0125

The defining equation for equilibrium is this one right here TDS = DQ reversible.0133

All I have done is just the rearrangement of the definition of entropy.0140

We had DS = DQ reversible/ T.0145

All I have done is multiply both sides by T so that I could write it in this form without any fractions if you will.0150

The defining condition for a system to be at equilibrium is TDS = DQ reversible.0157

Basically that means that when the system is an equilibrium that differential change in entropy that 0170

the system experiences × its temperature = the differential change in heat.0174

Not the differential change in heat.0182

The little bit of heat that flows along the reversible path.0183

This is the defining equation for equilibrium.0188

The condition for irreversibility, for a real transformation is actually the Clausius inequality.0195

The Clausius inequality, the DS is not equal to DQ/ T or DQ reversible/ T.0202

DS ≥ DQ/ T.0208

We have gotten rid of the irreversible part and now we have this greater than operator.0211

Upon rearranging we get the defining equation for irreversible and spontaneity.0218

This is important right here.0222

Once I had multiplied by that I get that this is the defining equation for a spontaneous process.0224

This is the equation that we are going to play with and manipulate.0231

It starts right here, a spontaneous process, a natural process, a real process, 0235

one that happens on its own without any external help these this is the condition for T × DS ≥ DQ.0240

We can combine these two equations by writing this.0251

Basically, all of them is greater than or equal to, where we take the quality sign to mean a reversible value of the DQ.0255

If the heat exchange happens to be done reversibly that is all the says.0263

It just include all possibilities, our concern is going to be this one, greater than.0268

It is always going to be like that because every real process is this right here so this is the important equation.0272

What does this say?0279

It says the transfer of heat during a spontaneous process is less than the entropy change 0281

for the process × the temperature at which the process takes place.0289

Once again, for a particular process, a spontaneous process the heat that transfer during the process is going to be less than 0294

the entropy change for the process × the temperature which the process takes place.0302

That is the defining equation for spontaneity and that is the one we are going to manipulate.0307

Let us go ahead and get started.0316

We have this TDS ≥ DQ.0321

Let us go ahead and recall the first law of thermodynamics, our energy.0329

We know the DU = DQ – DW.0338

Let us go ahead and rearrange that so we end up in solving for DQ.0344

We get DQ = DU + DW and we can go ahead and put this expression for DQ into here.0348

We end up with the following, we end up with TDS ≥ DU + DW.0357

Let us go ahead and rearrange this.0368

I'm going to bring both of these over to the left hand side and leave 0 out on the right hand side.0370

I have - DU - DW + TDS ≥ 0 this right here, this is our defining equation.0376

All we have done is we have taken that defining condition.0391

We have included the first law of thermodynamics so we have broadened the heat and the work and energy and now we have expressed it this way.0394

This is the defining equation for spontaneous process.0401

The negative of the differential change in energy - the work that is done in that process + the temperature × 0405

the differential change in entropy is always going to be ≥ 0.0413

This defines a spontaneous process.0418

If you have a system and this is satisfied the process is spontaneous.0421

If you know a process is spontaneous this will always hold.0426

So far we only looked at pressure, volume, work.0432

Let me actually write that.0436

We have only looked at PV work, this DW right here it includes all work not just pressure, volume, work.0447

Maybe there are some electrical works done.0461

Maybe there is some other mechanical work done.0463

This is a combination of all the work that is done not just expansion work.0469

This DW includes all work.0475

We are bringing this up because we want to make this discussion as general as possible.0486

For our purposes, I will to get to that in a minute.0491

We are not going to be concerned with other work.0495

We are only to be concerned with pressure, volume, work.0498

Sometimes when we start putting constraints on this equation, 0501

we are not even going to be concerned with pressure, volume, work because of the volume change is 0.0503

Under constant volume the pressure volume work is 0 so all of these terms are dropping out 0508

but we want to give the most general conditions for a spontaneous process and we are going to tighten up by placing some constraints on.0512

Do not worry if things start a little too complicated, it will be very simple in a minute.0521

So far we are going to look at the pressure, volume, work.0525

This DW includes all work so we have this DW = the pressure, volume, work which is the external pressure × the change in volume + 0528

DW sub other, whatever other work that is, we do not really care all that much.0546

When we put this expression in for here, we get the following.0551

We get -DW - P external DV - DW other + TDS ≥ 0.0558

This is the equation we want to concern ourselves with, this is the one we start with.0576

These two equations actually are going to be the same.0582

Sometimes we are going to talk about work cumulatively.0587

Sometimes we are going to separate work out in to pressure, volume, work, and all the other work but these equations are absolutely equivalent.0590

Sometimes we will start with this one, this derivation process for new equations.0599

Sometimes we will start with this, they are the same.0603

We have now expressed the condition of spontaneity in terms of changes of the properties of the system.0608

We have the energy and we have volume.0614

We have pressure, we have the work done, we have temperature and we have entropy.0618

All of these things we have expressed in terms of all the properties of the system.0623

This is the condition of spontaneity.0629

We do what we normally do.0637

We take this condition, we take this as the basis equation and we start putting constraints on it because in the laboratory it is not just a free flow.0639

In the laboratory, they are normally you are going to have conditions of constant temperature and pressure.0643

That was going to be the majority of it but sometimes we are going to constrain the volume.0654

Sometimes we are going to constrain the pressure or the temperature.0657

We take this and place those constraints on it the same way we did for energy, the same way we did for entropy.0661

Remember we did a constant temperature constant volume, constant temperature constant pressure.0667

We are going to do the same thing.0672

We are going to take this equation, constrain it, and derive a new set of equations under laboratory conditions, the things that we normally run across.0673

The first one I’m going to talk about is transformations in an isolated system, just for the sake of being complete.0683

Transformation in isolated system is when the system is isolated there is no exchange of energy.0690

I’m going to go back to blue, I think.0714

There is going to be no change of energy so DU = 0.0718

There is no work done in an isolated system D = 0 and there is no heat exchange DQ = 0.0726

If we have - DU - DW + TDS ≥ 0, I used this particular one with just the generic cumulative work.0740

That is 0, that is 0, what I end up with is TDS ≥ 0 and of course I get DS ≥ 0.0756

In an isolated system, the entropy of that system will always increase.0771

By isolated means completely separated from everything so we have no interaction at all with its surroundings, 0777

no change of energy, no exchange of matter, no exchange of anything.0783

The entropy of an isolated system will always increase.0787

A little bit of copy out with respect to this one.0798

It is not just the entropy of the system that is going to increase, 0800

the truth is it is going to be the change in entropy of the universe that is actually has to be ≥ 0.0803

The entropy of the universe, the change in entropy of the universe is going to be change in entropy of the system + the change in entropy of the surroundings.0817

In this particular case, we are isolated so it is not really going to include the surroundings.0824

Let me just go ahead and bring this up here and I will talk about it later.0829

It is actually possible for a system itself to have a decrease in entropy provided that 0834

in the surroundings the entropy increase more than compensate for the decrease.0840

The total entropy change in the universe is always greater than 0.0845

This is universally true.0849

This is true for an isolated system.0851

Again, we are talking about isolated system then the DS of the system is always going to be ≥ 0 0854

but in general what we are really going to be concerned with and how often do we do isolated systems, we do not.0860

In fact of the matter is we do not.0866

There is always going to be some surrounding involved.0867

This is just for mathematical completion.0870

We just want to use this equation in isolated system, this is what happens.0872

Let me just go ahead and mention this but we will get back to it and discuss it in more detail.0876

Let us start getting into the real important aspects because we are not going to be dealing with isolated systems.0884

Let us go ahead and talk about transformations at constant temperature.0890

If we hold the temperature constant, isothermal process, here is what happens.0895

Transformation at constant temperature, let us go ahead and start with our equation - DU - DW + TDS ≥ 0.0906

We have to find a way to manipulate this.0930

Here is what we are going to do, when temperature is constant I can rewrite TDS = D of TS.0932

In another words, I can put the TS together.0948

The differential operator actually operates on both because T is constant, this T actually comes out which is why we write TDS.0950

All I did was rewrite it in its more original form.0957

The temperature × the entropy, when I take the differential of it because temperature is constant it comes out.0962

Instead of writing TDS, I’m going to write DTS so we have - DU + TDS.0968

I’m going to go ahead and move this DW over the other side.0983

You do not have to it just different ways of looking at the same equation so the derivation itself you can leave the DW on the left, you can put it on the right.0987

We are actually going to do both when I do this summary of conditions.0996

In the next lesson I'm going to bring everything to left and leave 0 on the right so ≥ DW.0999

This is just moving things around.1008

This is going to be - DU + DTS ≥ DW.1011

I'm going to go ahead and factor out the differential operator, we can do that.1023

- DU - TS ≥ DW we end up getting this thing, - the differential of this thing ≥ DW.1029

Energy, temperature, entropy, and work.1042

This combination right here, this U - TS it shows up so much.1047

We actually give it a special name so combination of U – TS.1053

Remember what we did with enthalpy, we said that enthalpy =U + PV.1060

Enthalpy is a composite function, it is a composite of the energy of the system + the pressure of the system × 1065

the volume of the system and we called it the enthalpy.1071

This U- TS it is just a composite.1074

We define this thing called A the Helmholtz energy.1078

A happens to equal U – TS, it is a composite function.1085

It is a combination of the energy, the temperature, and the entropy not altogether different than this.1091

A is the Helmholtz energy of the system.1102

If I know the energy of the system and if I know the temperature of the entropy of the system, I multiply the temperature by the entropy and 1118

subtract it from the energy of the system, I get the Helmholtz energy of the system.1124

I just put something together, it is an accounting device.1130

I do not want to deal with UT and S, so I just say U - TS is this thing called Helmholtz energy so that is what we get.1134

A is a state function, the Helmholtz energy is a state function.1144

It is a state function because it is a combination of state functions.1148

Energy, entropy, temperature, is a state function.1152

We have - D of U - TS ≥ DW.1160

Let us go ahead and we said that A = U - TS so we can write - D of A ≥ DW.1168

Or upon integration, if I integrate this I end up with - δ A ≥ W.1181

It is not δ W it is W.1191

Under conditions of constant temperature, the work in a given process for a spontaneous isothermal process, 1199

the work done during that process is going to be less than or equal to the decrease in the Helmholtz energy.1210

Let us write this down, what does this say.1224

it says in a spontaneous isothermal process the work produced in the surroundings or if I just refer to the work in a process,1232

regardless of the particular point of view you are taking, the work produced in the surroundings 1265

is less than or equal to the decrease in the Helmholtz energy.1276

Just follow the math, not a problem.1295

Let me go back to blue here.1299

The DW we said is going to be equal to this pressure, volume, work, so PDV + other kinds of work. 1309

It includes all the work.1321

In general, we are not going to be concerned with other work.1324

We are only going to be concerned with pressure, volume, work.1327

In general, we will concern ourselves with the PB work only.1330

We have - δ A ≥ W, we have - δ U -TS ≥ W.1356

I can distribute the δ operator so I get – δ U + T δ S ≥ W.1373

Starting to look for a little bit familiar or if I want I can write it as, I go ahead and switch this sign and1383

move some negatives around and make the δ U positive.1391

Δ U - δ S ≤ -W It is another way of looking at this.1394

When Helmholtz energy is taught, it is taught as constant temperature and constant volume.1408

It is taught as T constant which is what we have done and V constant.1441

Normally, the chances are in your book you are going to see this introduced as if I hold temperature and volume constant what do I get.1450

That is usually how we introduce this notion of Helmholtz energy.1457

If V is constant that means that this right here, this portion of the work is 0 because if V is constant, 1463

then DV is 0 so that there is no pressure, volume, work and the only work you have is work other.1473

We just said that in general we are not going to concern ourselves with other work.1483

If we are going to drop, we have DW = P external DV + DW other.1488

If we are going to forget about the other work and if we hold volume constant and this is going to be 0 then this is going to be 0.1501

Here is what you end up getting, you end up with - DA ≥ DW that is our basic condition.1508

If DW is 0 you get – DA ≥ 0 or if you want DA ≤ 0 or the integrated version δ A≤ 0.1518

This is normally what you are going to see in your book.1535

Under conditions of constant temperature and volume, the change in Helmholtz energy for a spontaneous process is going to be less than 0.1539

It should start to look familiar.1549

You are accustomed to the δ G being less than 0 for a spontaneous process.1551

δ G happens under conditions of constant temperature and pressure which 1555

we are actually to be talking about in the next lesson but there is this other energy in the system.1558

This energy which is the energy - TS which is the Helmholtz energy and if I hold the temperature and 1563

the volume constant then this for a spontaneous process it is the change in Helmholtz energy, the δ A ≤ 0.1569

Let us go ahead and finish up with writing a couple of different versions of this.1582

δ A ≤ 0 and once again in your book you are going to be introduced to Helmholtz energy is constant temperature and volume.1587

But in general, the volume does not have to be held constant.1596

If the temperature that has to be held constant you can go ahead and leave the volume there and in this work term is not 0.1600

If we ignore all the other work, we are still to be concerned with the pressure, volume, work that is done.1607

This is the general version of this.1611

This is the specialized version that holds T and V constant.1615

V does not have to be constant here so I do not think that it has to be.1619

This is that, let us go ahead and write δ U - TS ≤ 0.1624

This is going to be δ U - T δ S ≤ 0 and of course we have δ A = δ U - T δ S.1636

This is the basic relationship that exists.1651

This is similar to the relationship that you learn in General Chemistry which we are going to learn again which is just as a little bit of a preview.1654

δ G = δ H - T δ S remember that one, this is the Helmholtz version of it.1663

Under conditions of constant temperature and volume this is what you get.1670

All of these things are equivalent.1675

This is the general equation right here, that is the general version of the equation.1677

This is the one for temperature and volume, for constant temperature volume and constant temperature and volume.1684

Please remember, you do not have to hold volume constant but you have to hold temperature constant.1691

If you hold volume constant also, you end up with 0 on the right hand side.1695

If you do not, there is some pressure, volume, work, is going to be done and there is going to be a work term.1701

Again, we are not going to be concerned with other work.1706

If there was other work, electrical or mechanical, that would also be part of the work term.1709

I hope that makes sense.1714

Thank you so much for joining us here at www.educator.com.1716

We will see you next time for a discussion of Gibbs free energy, bye. 1718