For more information, please see full course syllabus of Physical Chemistry

For more information, please see full course syllabus of Physical Chemistry

### 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
- Statistical Thermodynamics: The Big Picture
- 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
- 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)

- 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

### Physical Chemistry Online Course

### 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 and*0036

*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 explain*0146

*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 function*0280

*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 system*0328

*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 functions*0381

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

*And again, we already talked about the particle in the box.*0389

*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 systems*0805

*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 and*0944

*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

*Once again, we said P sub I was this thing.*1223

*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 state*1246

*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 system*1297

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

*Let us talk about this.*1335

*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 and*1474

*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

*I'm not really going to say more about this β = 1/ KT.*1552

*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 finding*1650

*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

*Except now, we are talking about ensemble instead of the system.*1865

*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

*And we talked about what it means, we talked about this idea of complexions, and a number of possible microstates.*1892

*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 more*1900

*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 in*1966

*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 systems*2230

*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 and*2398

*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 Q*2585

*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 S*2819

*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

*A was this, - KT LN Q.*3075

*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 temperature*3198

*× 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 functions*4287

*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

*Let us go ahead and talk about energy.*4423

*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

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