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### Statistical Thermodynamics: The Big Picture

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
• Statistical Thermodynamics: The Big Picture 0:10
• Our Big Picture Goal
• Partition Function (Q)
• The Molecular Partition Function (q)
• Consider a System of N Particles
• Ensemble
• Energy Distribution Table
• Probability of Finding a System with Energy
• The Partition Function
• Microstate
• Entropy of the Ensemble
• Entropy of the System
• Expressing the Thermodynamic Functions in Terms of The Partition Function 39:21
• The Partition Function
• Pi & U
• Entropy of the System
• Helmholtz Energy
• Pressure of the System
• Enthalpy of the System
• Gibbs Free Energy
• Heat Capacity
• Expressing Q in Terms of the Molecular Partition Function (q) 59:31
• Indistinguishable Particles
• N is the Number of Particles in the System
• The Molecular Partition Function
• Quantum States & Degeneracy
• Thermo Property in Terms of ln Q
• Example: Thermo Property in Terms of ln Q

### Transcription: Statistical Thermodynamics: The Big Picture

Hello and welcome back to www.educator.com, welcome back to Physical Chemistry.0000

Today, we are going to start our discussion of statistical thermodynamics.0005

Let us jump right on in.0009

I want to go through this big picture of statistical thermodynamics is that we know why we are doing this.0013

What are all for our goal?0022

What is it that we are trying to achieve?0023

Our big picture goal is going to be the following.0025

Let me actually work in blue today.0028

Our big picture goal it is to find a way to express the thermodynamic properties of a bulk system and0036

by bulk we mean just a bunch of particles, like a block of wood as opposed to the individual molecules that make up that wood.0071

Of a bulk system, in other words the energy, the entropy, the enthalpy, the Helmholtz energy,0077

the Gibbs free energy, all of these things.0089

The constant of volume heat capacity and the pressure.0092

The basic thermodynamic functions of bulk system.0096

Our goal is to express these properties in terms of the properties and of the particles that make up the system.0099

That is what we are doing with statistical thermodynamics.0124

We start off the course with classical thermodynamics.0133

We moved on to quantum mechanics.0136

In quantum mechanics we are dealing with the individual energies and properties.0138

The individual particles out of a molecule, whatever it is.0141

Now that we have quantum mechanics, we want to go back and we want to explain0146

what we learned in classical thermodynamics via the individual particles.0151

That is it, we are just closing the circle like talked about in the overview of the course.0155

I will go back to black here, sorry about that.0161

Our primary tool in this investigation is going to be something called the partition function.0166

Our primary tool will be something called the partition function.0174

Let me actually come over here, called the partition function.0191

The symbol for the partition function is going to be a capital Q.0201

It is going to be the partition function of the system.0205

What I’m going to do is we are going to express the thermodynamic properties in terms of Q.0210

We will express the thermodynamic properties, the ones I have listed above.0218

We will be listing those and I would be expressing those properties in terms of Q.0228

I’m having a little difficulty talking today, sorry about that.0234

In terms of this Q, the partition function.0237

We then introduce q.0241

We then introduce q, this is called a molecular partition function.0248

What that means it is or should say that is the partition function for each particle in the system.0271

We can actually do that, we can write this thing called a partition function0280

for each individual particle of whatever system we happen to be dealing with.0286

It is pretty extraordinary, that it is very extraordinary.0289

For each particle of the system.0293

We will express Q, the partition function of the system in terms of q.0300

We will then have exactly what we wanted.0325

We will then have, finally, our direct relationship between the thermodynamic properties of the system0328

and the particles that make up the system.0365

That is our big picture goal, that is what we want to do.0367

We want to come up with this thing called a partition function.0376

And we want to come up with this thing called, we want to find the various partition functions0381

for whatever quantum mechanical system we would happen to be dealing with.0385

We talked about the rigid rotator, the harmonic oscillator, these are the partition functions that we are going to look at.0392

These are going to be the partition functions of the molecules.0397

We are going to express the thermodynamic properties that we learned about back when we started the course.0400

Let us see what we have got.0411

Let us start up here.0412

Consider a system of N particles and N is usually just going to be Avogadro’s number.0415

Consider a system of N articles.0421

The energies of the particles are discreet, we know that already.0431

That is what quantum was all about.0435

The energies of the particles are discreet and they are distributed over various quantum states.0437

For example, if you had some rotating molecule.0465

You know whatever it might be in the J = 1 couple of 1,000,000 of them might be in J = 2.0468

A couple of 1,000,000,000 of them might be in J = 3.0473

Different energies, the particles are the same.0476

They are distributed over the various quantum states, that is all we are saying here.0480

In any given moment, if we have add up the energies, we get the energy of the system.0483

We will add up the energies of the individual particles, you get the energy of the system.0511

We will call that E sub I.0523

Also discreet because of the individual energies are discreet, the E sub I is going to be discreet.0526

If we come back to any moment of the same system, let us say 30 seconds later, whatever,0537

1 second later, does not really matter.0548

The particles are going to be in this differ distribution of quantum states.0552

Therefore, the energy of the system is going to be different.0558

In another moment, the particles being in other quantum states, this gives rise to another,0561

I will call this not E sub I, E sub 1.0593

I take my first measurement and I get E sub 1.0595

It gives rise to another energy of the system and I will call it E sub 2.0600

What we call the thermodynamic energy of the system is an average of all of these E1, E2, E3, E4, E5.0614

If I take 100 measurements, a 1000 measurements, 500 measurements of the system at different times,0622

all the particles are going to be in different quantum states.0627

I’m going to get different energy of the system.0629

I take an average of that, that is what I call the thermodynamic energy.0632

That was what we call U.0635

What we call the thermodynamic energy of the system.0639

In other words, U is the average of many observations.0659

Make sense, I think it is particularly strange here.0670

Let us take a bunch of observations, take the average and call that number the energy of the system.0673

Instead of making a 100 or 1000, or 500 observations on the same system.0680

Instead of making a multiple, let us say multiple instead of choosing a number, multiple observations.0689

Instead of making multiple observations, let us actually choose a number.0705

I’m going to choose one of the number, it can be any number but I'm just going to choose one, a hundred.0713

Let us say we take 100 observations.0717

We average that out a 100 observations of the same system, we get the energy of the system.0719

Instead of making a 100 observations on the same system, we can also just create 100 identical systems.0724

That is it, same circumstances, same surroundings, same particles, same temperature,0743

same pressure, create 100 identical systems.0748

Each system will be in a particular quantum state.0755

It might have all hundred that are in different quantum states.0774

You might have 20 of them that are in one, 10 of them in another, 2 of them that are in another.0776

Again, this could be in various quantum states.0781

A system will be in various quantum states with the energy E sub I.0788

Let us increase that number.0803

Now, instead of 100 or 1000 or 2000 or 5000, let us create a large number of identical systems0805

and we will call that large number of identical systems N.0817

Now, let us create, in other words let us just make this 100 a really big number.0821

Because we know that the bigger number we have, the better our average.0829

Let us create a very large number of identical systems.0834

Let us call that number N.0848

We call this collection of identical systems, this large number of identical systems, we call that the ensemble.0852

That is what the ensemble means.0858

Let me go to red.0866

We call this collection an ensemble.0870

It seems to always be a bit of confusion about what ensemble is and that is it.0884

We are just taking system, we are duplicating it, a 6.02 × 10²³ × and we are calling that ensemble.0888

It is the identical system, just copied.0897

The collection is an ensemble.0899

Each system will have a particular energy.0902

Each system in the ensemble will have a particular energy E sub I.0905

The energy distribution looks like this.0937

The energy distribution, we have the number of systems in the ensemble and0944

we have a particular energy of the ensemble.0965

If there are N sub one systems that have energy 1, we might have N sub 2 of the systems, we might have energy 2.0968

We have N sub 3 of the systems are in energy 3 and so on.0977

N = the sum of all of these N sub I.0984

If I add all of the N, I'm going to get the total number of systems in the ensemble.0991

Let us say I have 6.02 × 10²³ ensemble, it does not really matter.0995

It is a really large number 500,000, whatever.1000

If 10,000 of them are energy 1, if 50,000 of them of the systems are in energy 2, that is all it says.1003

There is a distribution of energies.1009

Now, the probability.1012

The probability of finding a system in a system with the energy E sub I, the basic probability,1020

you take a number of times of something can happen over the total number.1038

What is the probability of rolling a 5 when you roll a single dice?1045

There is only one way to get a 5, you roll a 5.1049

How many different possibilities are there when you can roll 1 through 6?1052

Yes, you have 6 possibilities that you can roll but only one way to roll a 5.1057

The probability of rolling a 5 is 1/ 6.1061

You remember this from algebra class.1064

The probability of finding a system with energy E sub I is symbolized by P sub I,1067

it = the number of states in that energy with that energy divided by the total number of systems.1076

That is it, very simple, very basic.1084

It just = the number of systems having E sub I energy divided by the total number systems.1087

It is the basic definition of probability.1101

Total number of systems.1103

We are going to define Q, the partition function.1108

Q is equal to the sum E to the - β E sub I, where β = 1 / K × T.1118

I’m going to go ahead and put this 1 / TT into the β.1141

Q is actually equal to the sum / I of E to the - E sub I divided by KT.1145

This division is up in the exponent.1157

This whole thing is up in the exponent.1159

In this particular case, T is the temperature in Kelvin, the absolute temperature.1162

K is something called the Boltzmann constant.1167

K is equal to 1.381 × 10⁻²³ and the unit is J/ K.1172

This is the partition function.1188

I will tell you what it is in just a minute.1190

My best advice is just deal with the mathematics.1197

This looks complicated because of the summation symbol, it is not.1200

All you are doing is adding a bunch of terms together.1202

You taking the first energy, the second energy, the third energy, you are dividing it by KT.1205

You are exponentiating it and that is just one term of the sum.1210

If we said out of the first 5 terms for the first 5 energy states, you have 5 terms of the some.1213

That is all it is and I will tell you what the partition function is in just a minute.1218

With respect to the partition function, P sub I is this.1230

It is equal to E ⁻E sub I / KT/ Q.1236

The probability of finding a system in a given energy state is equal to E raised to the energy state1246

divided by KT divided by the sum of all the possible energy states.1257

The part / the whole, the probability.1263

It is a fraction, that is all this is.1267

Let us go ahead and tell you what the partition function actually is.1270

Q, the partition function is a measure.1275

It is a numerical measure of the number of energy states that are accessible to a system or by a system1297

depending on which partition you want to use at a given temperature, at a given T.1327

There is always this sense of what is a partition function?1337

I’m still not sure what it really means.1343

This is what a partition function is.1345

Let us also talk about a system with a given set of energies.1347

At a given temperature, let us say there are 100 available energies for a given system.1355

There are 100 energies that could have at a given temperature, let us say only 5 of that energy levels are actually accessible.1366

The partition function is 5.1377

It is very important that we differentiate between accessible and available.1379

You might have, like for an example the rotational states of the diatomic molecule.1384

There is an infinite number of rotational states in a diatomic molecule.1387

Not infinite but a really large number if the molecule flies apart.1390

It will spin faster and faster and faster and faster into different quantum states.1394

J could be 50, 60, 70, 100, 200, but not all of those are accessible.1398

At a given temperature, let us say maybe only 30 of those rotational states are accessible.1405

That is what a partition function tells us.1411

A partition function is going to give you some number.1414

That number gives you roughly the number of states that are accessible to a system at that temperature.1416

As I raise the temperature more of the states become accessible, that is what is happening, that is all that is happened.1423

It is all a partition function is.1430

Again, we have P sub I is equal to E ⁻E sub I / KT all / Q which is equal to E ⁻E sub I/ KT.1433

I think all these exponentials and summations, fractions on top of fractions, it tends to look really intimidating.1455

It is not intimidating, it is just math.1464

Over the sum of the E/ KT.1467

The partition function is just adding up all these energy values and1474

then the probability of finding it in one of those energy values as you take a part / a whole, where Q is equal to this thing.1479

Let us go ahead and write it out again.1488

Q is equal to sum I / E ⁻E sub I / KT.1491

If you have 100 energy states that are accessible, you have 100 terms in that sum.1497

That is a partition function, very important.1504

We sometimes leave θ and write Q = the sum / the index I of E ^- β E sub I.1507

Sometimes, we will just go ahead and leave the β and expressed in terms of that,1534

in order that they do not have to deal with the fraction in the exponent.1538

That is fine, you will see it both ways.1542

Again, where β = 1 / KT.1545

If your teacher wants to give you reason for why that is the case, they can but I would say just take it on faith at this point.1556

U, the energy of the system, we said it is the average of the energy.1568

The average of the energy is you add up all the energies and you divide by the number of systems,1576

the energy of the ensemble.1584

The sum / I N sub I E sub I / N.1587

The number of states/ a given energy × the energy itself.1598

Add up all of those and divide by the number of systems in the ensemble.1603

That would give you an average energy.1608

I pulled the N sub I/ n out, N /N sub I.1611

N/ N sub I that is equal to P sub I.1621

N sub I/ N is equal to P sub I.1628

U, which is the average energy is equal to the probability of finding it in a given energy state × the sum of the actual thing.1634

I will go ahead and put this P sub I back in here.1647

U equal to the average energy is equal to sum of the probability of finding1650

the system of the ensemble in a given energy × that energy.1657

That is one of our basic equations.1668

We found an expression for the energy in terms of the energy, in terms of the probability.1671

The probability is a function of the partition function.1676

We have expressed energy in terms of the partition function.1679

We will get better , do not worry about that.1682

In the ensemble, the systems are distributed over the various quantum states.1691

Each specific distribution is called microstate or a complexion of the ensemble.1729

What we mean by this is the following.1763

Let us say I have I have 10 systems, let us say 3 of them are in one energy, 3 of them are in another,1765

3 of them in another, and one of them is in the fourth.1773

That is one distribution, that is one microstate.1775

Let us go to another distribution.1779

What if I have 5 in one, 5 in another, and nothing in the other 3.1782

That is another distribution, that is another microstate.1789

In other words, in microstate is if I have certain number of bins, energy baskets, our certain number of systems,1792

how can I distribute the different energies among those various systems?1806

Each different one is called a microstate.1809

The number of possible microstates is denoted as capital ω.1814

We define the entropy of the ensemble.1835

When we want the entropy of the system, we just divide the entropy by the total number of systems in that ensemble.1845

In other words N, that is it.1850

We are always talking about the ensemble.1852

Anytime we talk about a system, we just take what we have and divide by the number of particles in it,1854

the number of systems in the ensemble.1859

The entropy of the ensemble is, and you have seen this before.1861

S = K × the natural log rhythm of O.1868

This is the definition of the entropy of an ensemble.1874

We have seen this equation before.1877

We have seen this definition before back when we talked about classical thermodynamics.1879

We talked about entropy first empirically but then we go ahead and gave this statistical definition of entropy.1885

I'm not going to state too much more about it now.1898

If you want, you can go back to that particular discussion and it will talk a little bit more1900

about what these individual things mean, in any case.1905

The entropy of the system is the entropy of the ensemble divided by N, the number of systems in the ensemble.1910

Therefore, S of the system is equal to S of the ensemble divided by N.1948

S of the ensemble is K × the natlog of this thing called ω divided by N.1959

In order words, to find the entropy of the system that we are dealing with, the system that we are interested in1966

which has happen to have made billions of copies of that system to create an ensemble.1971

In other words, to find the entropy of the system, we need to find this Boltzmann constant, we know.1976

How many systems we have in an ensemble, we need to find LN of ω.1982

ω is defined as N!/ N sub 1! N sub 2! N sub 3!, and so on.1987

To find S of the system, we need to find the natlog of O.2009

That is it, we are just doing some math here, that is it, nothing too crazy.2022

The natlog of ω is the natlog of N!/ what we said, N sub 1! N sub 2!, And so on.2027

That is equal to the natlog of N! – the sum because this is a product.2039

The natlog of N sub I!.2048

After some math, we also designated as math, which I'm not going to go through here.2050

What we get is the natlog of ω is equal to -N × the sum of the I P sub I LN P sub I.2058

S of the system is equal to K LN ω / N.2080

LNO is this thing.2088

We put this thing into there, we end up getting - K × N × the sum / I P sub I LN of P sub I O divided by N.2092

The N cancel and we get an expression for the entropy of the system.2121

The entropy of the system is equal to - K which is Boltzmann constant × the sum / I, the probability of I × the log of the probability sub I.2125

We found an expression for the energy, in terms of the probability.2143

We found an expression for the entropy, in terms of the probability.2146

The probability is expressed in terms of the partition function.2150

We are getting to where we want to go.2154

This is our second major equation.2157

Let us go ahead and rewrite what we have.2163

Our first major equation was U = which is the average energy, which is equal to the sum of the probability sub I × E sub I.2167

And our second major equation which is entropy that is equal to -K × the sum / I, the probability of I.2180

These are our two basic equations that we are going to start with and derive everything else.2190

Again, where P sub I is the probability of finding a system or ensemble.2197

Probability of finding a system in that particular energy state.2209

Or it is also a fraction of the systems in that energy state, that is the best way to think of P sub I.2214

It is a fraction of the systems in the ensemble that are in a given energy state E sub I.2222

If I have a total of 1000 systems in the ensemble and if I have 100 of those systems2230

in given energy state E sub 1, 100/ 1000 that means 10%, 0.10.2238

My P sub I is 0.10.2245

Where P sub I is the fraction of the systems in the ensemble having energy E sub I.2248

If all of these do not make sense, do not worry.2276

Really, do not worry, what matters here are the results.2279

But again, I go through this as a part of your scientific literacy.2281

If you go through this, if you see this, and you go in your book and read it, it will make your book make more sense.2286

I think it works better that way, or perhaps you read your book and you did not quite get it,2293

and now that you are seeing this lecture, it might make more sense.2296

It is just another way of looking at it.2300

We have P sub I is equal to E ⁻E sub I/ KT/ Q, that is one equation that we have.2305

We have an expression for the partition function which is the sum / I/ E ⁻E sub I/ KT,2324

very important partition function.2333

We have an expression for the energy U which is the actual average energy.2336

That is equal to the sum of the index I of the P sub I E sub I, the fraction in energy state I × the energy itself.2342

And we have expression for the entropy.2353

Entropy = - K × the sum/ I P sub I LN P sub I.2354

These equations, if these 4 equations all of the thermodynamic properties, all of the thermodynamic quantities,2362

all the thermodynamic functions can be expressed in terms of Q.2387

In terms of Q, all we need is this, this, the energy and the entropy and2398

we can express all the other thermodynamic functions in terms of this thing we call the partition function.2403

Partition function, very important.2409

Let us start first of all with Q, let us start with that equation.2415

Q = the sum of E ⁻E sub I/ KT.2419

We differentiate with respect to, I will go ahead and differentiate with respect to T.2428

Therefore, DQ DT and we will hold volume constant.2442

When you take the derivative of this, you get 1 / KT² × the sum I E sub I E ⁻E sub I/ KT.2453

We took the derivative of Q with respect to T.2476

Let us go ahead and go over here.2479

P sub I is equal to E ⁻E sub I/ KT/ Q which means that if I multiply Q which means that π × Q is equal to E ⁻E sub I/ KT.2483

It is just mathematical manipulation.2502

If I put this back into the other equation, if I do KT² × DQ DT under constant V,2504

that is going to equal this sum E sub I P sub I × Q = Q × the sum of the E sub I P sub I.2517

This is U and U = that, the sum/ I of P sub I E sub I.2547

KT² DQ DT is equal to Q × U.2571

I solve for U.2582

U is equal to KT² / Q DQ DT V, which is the same as if I take, instead of taking the derivative of Q2585

with respect to T, if I take the derivative LN Q.2602

The derivative of LN Q is 1 / Q DQ DT, I get the following.2606

I get KT² D LN Q DT constant V.2611

I have an expression for U directly in terms of Q or LN Q.2624

In this last part because D DT or LN Q = 1 / Q DQ DT, this is energy in terms of the partition function.2628

We have our first part.2654

Now, the entropy of the system = -K × the sum of P sub I LN P sub I.2655

We said that P sub I is equal to the E ⁻E sub I/ KT/ Q.2671

LN of P sub I = -E sub I/ KT – LN Q.2681

LN P sub I, if I take this thing and put it into here.2702

Therefore, S is equal to -K × the sum P sub I - E sub I/ KT – LN Q.2715

I get S = -K × -1 / KT, the sum P sub I E sub I – LN Q × the sum of the P sub I.2735

Therefore, S is equal to 1 / T × the sum / I of the P sub I E sub I + K × LN Q.2758

And this is because this thing is actually equal to 1.2779

The sum of the probabilities, the sum of all of fractions always = 1.2783

S = this is U.2788

U / T + K LN Q.2793

We already found U, U equal to KT² D LN Q DT constant V.2804

Let me go ahead and put this in for that and when we do, we end up with S2819

is equal to KT D LN Q DT under constant volume + K LN Q.2825

We found an expression for entropy directly in terms of the partition function.2839

Very nice.2847

Let us see, with energy, entropy, temperature, and volume, all of the other thermodynamic properties can be derived.2851

All of the other thermal properties can be derived.2880

Let us begin with, let us go back to black.2895

Let us begin with Helmholtz A = U - TS.2902

This is the definition of the Helmholtz energy.2907

We have an expression for U and we have an expression for S.2910

This is equal to KT² D LN Q DT under constant V - T LN Q - KT² D LN Q DT constant V.2913

The Helmholtz energy is equal to -KT LN Q.2947

There you go, that is an expression for the Helmholtz energy.2960

That one was reasonably straightforward.2966

I just put the value of U and S in here and solve, and I end up with this.2968

One of the fundamental questions of thermodynamics,2973

if you remember from towards the end of the classical thermodynamics portion of the course.2979

One of the fundamental equations of thermodynamics says DA is equal to - S DT - P DV.2985

That means P, let me go to red, P is equal to - DA DV under constant temperature.3010

That is what this says, this is the total differential equation.3029

This P is just the partial derivative of this with respect to this variable.3034

That is it, because the DA DV × the DV.3040

The DV DV cancel, you are left with the A.3043

That is what this means.3048

S would be partial of A of DA DT, holding V constant.3048

Therefore, P is equal to - DDV constant T of A.3060

Therefore, the pressure of the system is equal to KT D LN Q DV holding temperature constant.3082

That is quite extraordinary.3095

You are just knocking out all these thermodynamic expressions in terms of partition function.3099

Let us go ahead and do the enthalpy of the system.3106

The enthalpy of the system is defined as the energy + the pressure × the volume.3109

We are using these or sometimes look exactly alike.3119

I will just put them in, we have expressions for these.3122

We have KT² DLN Q DT constant V + the P which we said was KT D LN Q DV constant T × V.3124

Therefore, this is equal to KT × T D LN Q DT under constant V + V × D LN Q DT constant T.3146

Very beautiful, absolutely stunningly beautiful.3167

That is enthalpy.3172

Let us go ahead and go to Gibb’s free energy which is the most important for chemists.3176

K = U + PV - TS.3184

If we put all of these all in, I have it here, I will just write it all out.3192

U was KT² D LN Q DT under constant volume + KT × D LN Q DV at constant temperature3198

× V - T × K LN Q + KT × D LN Q DT under constant V.3214

This is equal to, when I multiply this, when I multiply that and add some terms,3235

I end up with G = KT D LN Q DV under constant temperature × V - LN Q.3240

This gives me an expression for the Gibb’s free energy.3263

Heat capacity is very important.3270

The heat capacity is the partial derivative of the energy with respect to temperature under constant volume.3275

We have an expression for the energy of the system.3284

We have got DDT constant volume of this expression which is KT² D LN Q DT constant volume.3288

This is going to end up equaling K × T² D² LN Q DT² under constant volume + D LN Q DT under constant volume × QT.3306

Therefore, the direct expression for the constant volume heat capacity is KT × T D² LN Q DT² under constant volume + 2 × D LN Q DT constant volume.3328

This is a direct expression for the constant volume heat capacity.3352

Normally, what we would be doing is we are going to be finding an expression for the energy.3357

I'm just taking the partial derivative of that with respect to temperature directly.3360

We are not going to be using this expression.3364

And again, the almost important of the thermodynamic functions,3367

we are mostly to be concerned with the energy and the constant volume heat capacity.3370

Occasionally, we will deal with pressure.3374

If for any reason you need to go to the other thermodynamic functions, that is fine.3376

But again, this is an overview.3380

I wanted to show you what our big picture goal is.3381

When we get the expression for the energy in the individual cases, we will just differentiate with respect to T.3384

We will write that down.3390

In general, we will be concerned with U and CV.3393

We will normally find an expression for U then differentiate directly, and differentiate with respect to T directly.3419

We have actually done it, we have done what we are set out to do.3446

Let me go ahead and go to blue here.3451

We have done it, thermodynamic properties expressed in terms of Q or LN Q.3454

Thermodynamic properties expressed in terms of Q.3471

Q, by definition is related to the energy states of the system E sub I.3488

These are the E sub I.3517

The E sub I are related to the energies of the individual particles making up the system, the small E sub I.3521

E sub I are related to the energies of the particles making up the system.3533

Let us say that, of the particles making up the system, the small E sub I.3550

We expressed these thermodynamic functions in terms of the partition function of the system.3564

Now, we are going to express it in terms of the partition function of the individual particles, the molecular partition function.3571

We will now express Q in terms of Q, in terms of small q, the molecular partition function.3583

Because we have expressions in terms of Q, if we have an expression for Q in terms of small q,3595

we put that in wherever we see a Q and we have an expression for thermodynamic properties in terms of the small q.3600

The molecular partition function.3608

The E sub I that we talked about above that is made up of the energies of the individual particles,3623

E sub 1 + E sub 2 + E sub 3, and so on.3631

The energy of the system is the sum of the energies of the individual particles.3640

The partition function of the system is therefore going to be a product of the partition functions of the individual particles.3648

That is how this works, some product.3658

That is the whole idea behind the log, the exponential.3661

That is why the log shows up in these problems.3665

Let us say this again.3671

The partition function of the system Q can be written as the product of the partition functions for each particle in the system.3675

The molecular partition function is Q.3719

For indistinguishable particles which is going to be pretty much all that we talk about when we have a liter of nitrogen gas.3724

You can tell one molecule of nitrogen gas from another molecule of nitrogen gas.3732

For indistinguishable particles, Q is equal to Q ⁺N!, this is the expression.3736

If I have N particles, 6.82 × 10²³, I find a partition function for each particle.3762

I raise that partition function to the nth power, I divide by N!, that would be give me the partition function of the system.3769

That is what this is, very very important equation right here, for indistinguishable particles.3778

Is there another expression for the distinguishable particles?3784

Yes, it is just that without the denominator.3787

And if your teacher feels like discussing that, if it comes in a problem, we will deal with it then, not a problem.3789

But for the most part, it is going to be indistinguishable particles.3795

N is the number of particles in the system.3801

Let us see what we have got here.3805

N is the number of particles in the system.3808

Q, the small partition function, it is the same definition as Q except now we use the individual energies not of the system.3823

The individual energies of the particles, the atoms, and molecules.3832

It is going to be the sum / the index I of E ⁻I E sub I/ KT.3837

We are taking the sum / quantum states.3848

You will see a minute in the quantum levels.3855

If I have a diatomic nitrogen molecule and I want to find the partition function of its vibrational partition function.3857

The vibrational partition function as you know, first quantum state R = 0, R =1, R =2, R =3, those are the different quantum states.3865

Each one has an energy, I put those in here for the E sub I and add it all up.3877

That is how get my vibrational partition function for that molecule, for vibration.3883

If I want the partition function for rotation, there is a difference of energies.3890

If I want one for translation, it is a different set of energies.3894

We will get to that in subsequent lessons.3897

The molecular partition function is exactly the same as what is qualitatively is this.3900

The molecular partition function is a measure, it is a numerical measure.3909

It actually gives you the number of states that are accessible.3922

A partition function is a measure of the number of quantum states,3926

the number of energy states that are accessible to the particle at a given temperature.3940

Let us say there are 250 vibrational states available for carbon monoxide that are available.3962

At a given temperature, let us say 300 K, or I just say 298 K, room temperature.3970

Let us say that only 5 of those states are accessible.3977

In other words, the molecule does not have enough energy to get to the 50th state of the 49th state, or the 10th state.3981

It only has enough energy to occupy state 1, 2, 3, 4, 5, two different degrees.3989

Most the molecules might be in state 1 and 2, and maybe a couple in 3, 4, 5.3995

But at a given temperature, it just cannot vibrate anymore than that.3999

They are the states that are available at a given temperature, this is what is accessible.4003

That is what the molecular partition function does when you calculate this, when you actually get a number like 3.4.4008

That is telling you that at that temperature, there is really mostly about 3.4 states that are accessible.4015

If in a 3.4 that means that the 4th state is, there is a couple of particles in that state and maybe even in the 5th.4024

But in general, it is going to be the 1st, 2nd, 3rd.4032

That said, that is all the partition function is.4035

It is a numerical measure of the number of quantum states that are accessible to a particle at a given temperature.4037

Accessible not available.4043

Quantum states can be degenerate, as we now.4047

For example, the rotational degeneracy is 2J +1.4050

The degeneracy is the number of quantum states that have that particular energy.4056

The degeneracy of 5 for a given level means that 5 different quantum states have that same energy.4062

Quantum states can be degenerate.4069

In other words, have the same energy E sub I.4083

If we include degeneracy in our definition of the molecular partition function, we get the following.4097

This is the one that we are going to be using.4102

Including degeneracy, our molecular partition functions as follows.4105

Q is equal to the sum, the index I G sub I, the degeneracy × E ⁻E sub I/ KT.4115

Our sum is over the energy levels.4128

Our sum is over energy levels not states.4136

This degeneracy takes care of all the states.4150

Q, we said is equal to Q ⁺nth/ N!.4157

In the expressions for the thermodynamic functions, we used LN Q not Q.4169

Therefore, let us take the log of this and see what we get.4198

LN Q is equal to LN of Q ⁺N/ N!.4213

That is equal to N LN Q - LN N!.4222

By sterling's formula, we have LN of N! is actually equal to N LN N – N.4231

We have an expression for this so we can substitute back into that.4250

LN³ is equal to N LN³ - this thing N LN of N – N.4256

Therefore, we have LN Q = N LN – N LN + N.4269

LN Q, we can express Q in terms of q.4281

Using this expression for LN Q, we put it back into the expressions for the thermodynamic functions4287

and we have our thermodynamic functions now in terms of q.4295

Using this expression for LN Q, we can now express all of the thermodynamic functions in terms of LN q.4301

Our connection is complete.4338

We had the thermodynamic properties, a classical thermodynamic properties.4350

The bulk properties of a system that we developed empirically back in the 19th century.4358

We related that to the partition function of the system Q and related that to the partition function, the particles.4366

The circle is closed.4377

We began with classical thermodynamics, we went on to quantum mechanics.4378

Quantum mechanics deals with particles.4383

We have this thing called partition function.4385

We can use properties of the particles to express the thermodynamic properties of the bulk system.4388

The circle for physical chemistry is closed.4395

Let us go ahead and do an example here so that we see.4399

An example of a thermodynamic property in terms of q, in terms of LN q.4404

Energy is equal to KT² D LN Q DT V.4425

We said that LN Q is equal to N LN q - N LN N + N.4434

We put this expression into here.4447

We get U is equal to K × T² × D DT under constant V of N LN Q – N LN N.4450

Everything is basically in dropout, when you take the derivative, these are constants.4469

We take the derivative of them with respect to temperature, they just going to go to 0.4472

What you end up with here is N KT² D DT of LN Q.4476

We will just leave it as D LN Q DT constant V + 0 + 0.4484

Therefore, the energy of the system is equal to the number of particles in the system × K × T² ×4494

the temperature derivative of the natlog of the molecular partition function, holding volume constant.4505

There you go, that is it.4512

Thank you so much for joining us here at www.educator.com.4516

We will see you next time for a continuation of statistical thermodynamics.4518