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Summary of Entropy So Far

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
  • Summary of Entropy So Far 0:43
    • Defining dS
    • Fundamental Equation of Thermodynamics
    • Temperature & Volume
    • Temperature & Pressure
    • Two Important Equations for How Entropy Behaves
    • State of a System & Heat Capacity
    • Temperature-dependence of Entropy

Transcription: Summary of Entropy So Far

Hello and welcome back to and welcome back to Physical Chemistry.0000

Today, I thought we would stop and do a little summary of what we have done with entropy so far.0004

In the last couple of lessons, we have always done a lot of mathematical derivations.0009

There has been a lot of partial derivatives, a lot of substitution, a lot of math and symbolism on the page.0014

Oftentimes, when that happens it is easy to lose your way, to lose the fours from the threes.0019

If I pullback and give us a nice global perspective of what is that we did and more importantly what was that we are actually after.0024

To distill and reduce it to what is important but you want to take away from all these derivations and all these equations that we have been dealing with.0032

Let us jump right on in.0042

Basically, just like energy now we have this new state property that we call entropy.0045

Again, we have not defined what entropy is aside from saying that the best way to handle it0051

at this point is to just treat it like the general disorder of a system.0056

It is the best way to handle it.0062

I will go ahead and write that down.0066

We have this new state property of the system that was very important, it is a state property.0068

State property entropy which we been symbolizing with S and our empirical evidence with this state property we just to define it like this.0084

Our empirical experience leads us to define the differential change in entropy of a system this way as DS = DQ reversible / T0100

which just means that any process when you going from state 1 to state 2, if I actually follow a reversal path0123

when I'm making this particular change then the amount of heat has gained or lost,0131

Let us go ahead and take it from the perspective of the surroundings.0137

The heat withdrawn from the surroundings divided by the temperature at which the change takes place is some numerical value0141

that equals the change in this property in going from state 1 to state 2.0148

The definition is reasonably straightforward, it is not altogether different than when we defined DU = DQ – DW.0152

We defined energy in terms of the heat and the work.0161

Here the entropy is defined only in terms of the heat along a reversible path.0164

We wanted to investigate entropy’s behavior especially with respect to the state variables temperature, pressure, and volume.0172

We begin by combining the first and second laws, this is the second i will go ahead and write this down.0215

This is the second law and this was the first law, first law of energy and second law is entropy.0221

We combine those and we came up with something called the fundamental equation for thermodynamics.0226

The fundamental equation of thermodynamics was our beginning point.0234

We just put these two together and we moved this over here DQ, we put it here and we are rearranged it to express it like this DS = 1 / T DU + P / T DV.0249

Basically, what this does it expresses when I combine these we found that this thing expresses0267

the change in entropy of a system with respect to a change in energy or change in volume.0273

If I want to change the entropy of the system, I can do it in two ways independently.0277

I can either change the energy, change the volume, or change both.0281

This is a relationship that expresses how it changes.0284

Notice that all the variables are represented here, energy, entropy, temperature, pressure, volume,0287

that is why it is called the fundamental equation of thermodynamics.0295

This equation is one you absolutely have to memorize.0298

You have to know this.0301

As far as doing the problems are concerned, we are going to use the equations that we derived from this but this is very important.0303

You do not have to memorize it.0309

I mean you have this definition and this definition, you just substitute this in and you get this.0310

It is not a problem, it is very easy.0316

If you know these two, you already know this.0318

We start with that.0322

Basically, all we have to do is investigate for a particular system we are running our experiments,0325

we have to change the volume and the energy and we are going to see how the entropy changes.0331

Experimentally, we do not usually control the energy of the system.0339

We can control the volume but we do not normally control the energy.0343

We decided to take this equation and fiddle with it mathematically to see if we can somehow express0346

the change in entropy with respect to the other variables temperature, pressure, and volume.0353

That is what we wanted, that is the overarching goal.0359

That is why we went through all those mathematical derivations that we did.0361

When we did that under conditions of temperature and volume, here is what we came up with.0366

We first considered entropy as a function of temperature and volume to get the total exact differential expression0376

which is this one DS = a partial with respect to the first variable holding the second variable constant × its differential + the partial of S with respect to the second variable.0388

The total differential expression is always the same, holding the other variable constant × its differential.0405

This is the basic mathematical expression.0410

This is the one of the equations if you want to memorize this, that is fine.0413

We actually identified these differential coefficients with something that easily measurable0419

which is ultimately what we want but this is the mathematical expression for this.0424

This is how we begin.0427

We first consider this to get this fundamental equation and now with some mathematical manipulation we were able to identify0429

these differential coefficients that says how does entropy change when I change temperature?0437

How does entropy change when I change volume?0442

Holding the other variable constant we were able to come up with the following.0445

In the case of DS DT, we had DS DT V = the constant volume heat capacity divided by the temperature.0449

For the change in entropy with respect to any change in volume holding the temperature constant,0462

we ended up with A / K which is the coefficient of thermal expansion divided by the coefficient of compressibility.0468

When we rewrite this equation using these things, putting these in there we get DS = CV/ T DT + A/ K DV.0477

This is one of the equations that is absolutely necessary and one of the ones that you want to begin all of your problems with0494

because it expresses the change in entropy with respect to a change in temperature and a change in volume base on things that we actually can measure or look up.0499

This is one of the things that we did.0511

Again, this is one of the equations that you must memorize.0519

This is the one that you want to bring to the table not the derivation, not all those partial derivatives, this one.0522

S as a function of temperature and volume.0529

If you hold volume constant this term is 0.0534

If you hold temperature constant this term is 0 it just becomes that.0536

This is true for all systems, liquid, solid, and gases.0539

This is the general, the most general equation.0543

We also do this for temperature and pressure.0548

For temperature and pressure we came up so let us go ahead.0552

We considered entropy as a function this time of temperature and pressure.0556

What happens when I change the temperature and what happens when I change the pressure?0565

The total differential expression mathematically is DS DT holding P constant × its differential + DS DP0570

'k holding the temperature constant × its differential that the basic mathematical expression.0584

Again, with some mathematical manipulation which constituted the previous lessons with all the derivations, we were able to identify these differential coefficients with the following.0589

In this case, DS DT under constant pressure is the same, one of the constant volume that is equal to this times0600

the constant pressure heat capacity divided by the temperature.0609

Notice the pattern, I’m holding volume constant, its constant volume divided by pressure.0613

If I’m holding pressure constant, it is going to be constant pressure heat capacity divided by the temperature.0618

This is generally true.0623

This one the DS DP, holding temperature constant that will end up being equal to the volume of the system × the coefficient of thermal expansion.0625

We got DS = CP/ T DT – V A DP this is the other equation you want to memorize.0637

Under conditions of temperature and pressure, this is the equation we want to bring to the table.0651

Under conditions of temperature and volume the other equation is what we want to bring to the table.0662

These are the two fundamental equations that you need to be memorized.0667

Let us see what came next.0674

You remember, these equations these are analogs to the equations that we did for energy.0682

Back to energy we essentially said that there are two different expressions, the total differential expressions0692

that you need to memorize in order to solve the problems for energy.0697

Those are the equations that we always started off with.0701

These are the analogs for entropy.0704

We are doing the same thing, we have already done it.0708

We resolve all those problems, all those example problems.0710

We end up doing the same thing here, we are going to be doing a ton of example problems in a couple of lessons.0713

We are just reducing it so we can make sense of what can see the four from the trees from a global perspective.0720

This is what we wanted.0727

At the previous equation that was what we wanted.0729

These equations are analogous to what we did for energy.0732

You remember those equations were as follows DU = CV DT + DU DV that was the equation for the change in energy with respect to temperature and volume.0747

And then we had DH under conditions of constant pressure, we did not consider the energy of the system but we consider the enthalpy of the system.0769

Because the enthalpy of the system accounts for any pressure, volume, work that was done so we end up with the following.0774

The change in enthalpy = CP DT + DH DP constant T DP.0783

This equation is analogous to this equation and this equation is analogous to the previous one for temperature and volume.0800

That is it, we are doing the exact same thing.0808

We want to find out how the state variable changes with respect to changes in temperature, pressure, volume.0810

We have seen two important equations for how entropy behaves.0815

We have seen DS = CV/ T DT + A / K DV this is temperature and volume dependents.0844

We have seen DS = CP / T DT – VA DP this is a temperature and pressure dependents.0856

In each case, the temperature dependence was simple.0872

In fact, it is very simple, it is just the appropriate heat capacity for the particular variable divided by the temperature.0893

It is just the appropriate heat capacity divided by T.0901

In other words, this and this.0920

When we hold V constant that goes to 0.0926

The temperature dependence of the entropy under the conditions of constant pressure, that is it, there is nothing going on here.0928

If for any reason the state of the system is described by T and some other variable, let us call it M.0936

In this case, it was T and V, in this case it was T and P.0971

If let us say, you are doing some kind of work and you are actually describing the system not with temperature volume, temperature and pressure,0975

but now by temperature and some other variable M, whatever that variable happens to be0982

then the heat capacity of the system at constant M is just the change the DM / DT.0990

It is just a change in that particular, let us do it this way.1025

DQ is the change in heat under the condition of constant M divided by the temperature change, this is just the definition of heat capacity.1030

I will write that down.1050

This is just the definition of heat capacity which is just heat per unit temperature.1051

That is the fundamental definition of heat capacity.1073

Now if we have, let us go ahead and move to next page here.1079

If I have the constant M heat capacity for the system = DQ M/ T and if we have the definition of entropy1084

which is DQ reversible / T which implies if I move this T over here I get T DS = DQ reversible.1098

I will just go ahead and put this into here and I get CM = T DS/ DT that I divide by T and I get the following.1115

I get CM/ T = basically, what this says is that the change in entropy per unit change in temperature,1130

if I'm holding variable M constant is just the constant M heat capacity divided by the temperature.1145

I already know that with respect to M being volume and N pressure.1153

When N is volume well the DS DT is CV/ T.1158

When M is pressure, it is the DS DT = CP/ T.1162

N is any other variable, whatever it happens to be, the change in entropy per unit change in temperature1167

holding that particular variable M constant is just the constant M heat capacity of the system which I can measure from this divided by temperature.1173

This is just a general expression that is all that is going on here.1186

Let us go ahead and write that down.1191

Under constraint of any state variable dependents of entropy is just we will call it the appropriate heat capacity divided by T.1192

In practice, we generally just hold either the volume constant or the pressure constant because those are the easiest thing1241

for us to do experimentally which is why the constant volume heat capacity and the constant pressure heat capacity are the most important heat capacities.1249

For any reason there are some other variable that we are dealing with, that we are holding constant it is going to be the constant M heat capacity.1261

That is all that is going on here.1269

In terms of pressure and volume, let us go ahead and do this.1275

DS DT under constant volume = CV / T and of course we have DS DT under constant pressure = CP / T.1279

I'm going to go ahead and move these T's over.1294

I’m going to rewrite this as CV = T × DS DT V and CP = T × DS/ DT constant P.1296

These last two equations, if I want to now that I have entropy I can go ahead and take this temperature × DS DT1318

under constant V or temperature × DS DT under constant P, I can take these as the definitions of heat capacity as opposed to defining heat capacity.1328

Let us say the constant volume heat capacity as the change in heat as I hold volume.1336

This was the definition of heat capacity.1344

We defined in terms of heat.1346

The heat that has lost or gained during the particular process.1348

If I want, because of these particular dependents of entropy on the heat capacity, I can actually define my heat capacities differently.1352

These two, this and this, they are alternate definitions for what a constant volume heat capacity is or what a constant pressure heat capacity is.1361

Not altogether that important but if I want to, I can use these relations.1373

There are just some other relations and definition that I can use if I need to.1377

Thank you so much for joining us here at

We will see you next time, bye.1385