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

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

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### Entropy & Probability I

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

- Intro
- Entropy & Probability
- Structural Model
- Recall the Fundamental Equation of Thermodynamics
- Two Independent Ways of Affecting the Entropy of a System
- Boltzmann Definition
- Omega
- Energy Distribution
- The Energy Distribution
- In How Many Ways can N Particles be Distributed According to the Energy Distribution
- Example I: In How Many Ways can the Following Distribution be Achieved
- Example II: In How Many Ways can the Following Distribution be Achieved
- Example III: In How Many Ways can the Following Distribution be Achieved
- Example IV: In How Many Ways can the Following Distribution be Achieved
- Entropy & Probability, cont.

- Intro 0:00
- Entropy & Probability 0:11
- Structural Model
- Recall the Fundamental Equation of Thermodynamics
- Two Independent Ways of Affecting the Entropy of a System
- Boltzmann Definition
- Omega 16:24
- Definition of Omega
- Energy Distribution 19:43
- The Energy Distribution
- In How Many Ways can N Particles be Distributed According to the Energy Distribution
- Example I: In How Many Ways can the Following Distribution be Achieved 32:51
- Example II: In How Many Ways can the Following Distribution be Achieved 33:51
- Example III: In How Many Ways can the Following Distribution be Achieved 34:45
- Example IV: In How Many Ways can the Following Distribution be Achieved 38:50
- Entropy & Probability, cont. 40:57
- More on Distribution
- Example I Summary
- Example II Summary
- Distribution that Maximizes Omega
- If Omega is Large, then S is Large
- Two Constraints for a System to Achieve the Highest Entropy Possible
- What Happened When the Energy of a System is Increased?

### Physical Chemistry Online Course

### Transcription: Entropy & Probability I

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

*Today, we are going to talk about the entropy and probability.*0004

*We are going to define what entropy is finally.*0008

*Let us jump right on in.*0011

*When we introduced the definition of entropy, when we introduced that the DS =DQ reversible/ T which was our definition of the differential element.*0014

*That is the definition of entropy, we did not get the structural model.*0032

*We did not require a structural model for the system in order to work with the entropy,*0044

*in order to work with this state function entropy or describe its behavior .*0072

*In fact, we did not even need to know what entropy was.*0086

*We had these mathematical descriptions and we had these constraints of temperature, pressure, volume and we saw how entropy behaves.*0090

*We are able to derive and calculate numerical values for it.*0096

*The only thing that we really did was casually refer to it as a measure of the disorder or randomness of the system.*0100

*I still think, in my personal opinion that disorder and randomness is actually a great way of thinking about entropy.*0108

*We are going to do was to define what we mean when we say disorder and randomness.*0115

*We are going to quantify, we are going to come up with some numerical way of explaining what is this disorder or randomness.*0120

*When I use the words disorder and randomness, what I’m talking about is something called the distribution.*0131

*When we talk about disorder or randomness, we are talking about a distribution.*0145

*In this case, it is going to be the distribution of particles.*0148

*We did not require a structural model, we did not care what a particular system was made up off.*0152

*Whether it is particles, chairs, it could be made absolutely anything.*0156

*There was this behavior that it represented, this is our empirical observation.*0162

*We are able to use mathematics to derive other to describe different ways of how this thing behaves, we are going to give it a structural model.*0167

*Here is a structural model.*0176

*What I'm going to do, I would go ahead and do this in blue, I think.*0180

*Our structural model is exactly what you think it is.*0189

*A system is composed of a very large number of particles.*0196

*Those particles could be molecules, they could be atoms, they could be ions, whatever that you have to be discussing in that particular problem.*0220

*Molecules, atoms, and ions.*0228

*Let us say some things about these particles.*0234

*These particles have various energies.*0237

*The best way to think about this is to think about a collection of gas like the kinetic theory.*0243

*Basically, it is just a bunch of particles in a thick space that are bouncing into each other and bouncing off the walls, that is a system.*0249

*These particles have various energies.*0259

*In other words, not each particle is flying around at the same speed.*0265

*There are some going faster, some that are going slower.*0268

*There is a large number of them that have the same energy.*0272

*There are others that also have the same energy so there might be 10,000,000 of them that are traveling at 500 kph.*0275

*There might be 20,000,000 of them that are traveling at 550 kph.*0281

*They are distributed if all the energy is distributed among the different numbers of particles.*0285

*These particles have various energies and there is a distribution of the total energy of the system DU*0292

*of the total energy of the system which is U first law / the various particles.*0310

*The total energy of the system is made up of the sum of all the individual energies of the particles.*0334

*If the system has 100 J of energy, those 100 J are going to be distributed among the different particles and different ways.*0342

*Maybe two parts might have 1 energy, two particles might have another energy, 15 particles might have another energy.*0350

*All the different energies, that is the distribution.*0356

*In other words, n sub 1 particles have energy E sub 1.*0360

*N sub 2 particles have energy E sub 2, and so on.*0381

*The number of particles, the total number of particles so n sub 1 + n sub 2 + n sub 3 and so on,*0394

*have to equal n which is the total number of particles in the system.*0404

*If I have 100 particles in the system n sub 1 might be 10, n sub 2 might be 20, n sub 3 might be 70.*0409

*70 + 20 + 10 =100 the total number of particles.*0416

*That is one of our constraints.*0419

*The number of particles with energy 1 + the number of particles + energy 2 + the number of particles with energy 3 and so on,*0425

*onto the number of particles n sub i with energy sub I, that has to equal the total energy of the system.*0436

*When I had all these energies, the maximum energy that I can have is U, the total energy of the system, that is the second constraint.*0443

*Nothing strange happening here, the system has a certain energy, our system is made up of a bunch of particles,*0450

*we are distributing this energy over a bunch of particles.*0456

*The next part is the structural model is these particles for the first one of these particles have energy.*0463

*These particles occupy space, in other words, volume.*0471

*I'm not saying that they themselves have volume, I'm saying that they are contained in a volume.*0482

*There is some fixed volume that they are in.*0487

*What ever that volume is, they are occupying that space.*0490

*There is the distribution of these particles, this is the most intuitive clear one, these particles over the volume available.*0498

*We have these hundred particles and we have a 1 L flask.*0521

*I have this hundred particles in 1 L flask, the particles are going to arrange themselves in all kinds of different ways.*0525

*Maybe these particles here, these particles there, a 100 all over the place.*0531

*They are a bunch of different ways that these particles can distribute themselves in that 1 L flask, that is the volume distribution.*0536

*They are going to arrange themselves in the volume up to the maximum capacity of the volume.*0543

*You know this intuitively.*0549

*Let us recall the fundamental equation of thermodynamics.*0552

*Our fundamental equation was the following DS = 1/ T DU + P/ T DV.*0571

*Notice DU DV this is energy, this is volume, there are two different independent ways of affecting the entropy of the system.*0581

*I can change the energy or I can change the volume or I can do both, not a problem but they are independent of each other.*0596

*Let us write it down.*0605

*There are two independent ways of affecting the entropy of the system.*0608

*One, I can change the energy that is U.*0630

*Two, I can change the volume which is V.*0641

*It is very important equation, the fundamental equation of thermodynamics is a relationship between all the various properties of the system,*0649

*the entropy, the pressure, the temperature, the volume, the energy.*0656

*It is very important.*0660

*If it is expressed in terms of entropy we would write it this way DS = something.*0662

*We see that there are two ways to affect the entropy, energy, and volume.*0666

*Since, there are two ways affecting this entropy of the system, the energy in the volume.*0675

*Therefore, it makes sense to define the entropy of the system which is S in terms of these two properties energy and volume.*0685

*In terms of these two properties U and V, both one gave the following definition.*0717

*Profoundly important equation, it is the fundamental equation of statistical thermodynamics.*0753

*so they have the following definition, he said that the entropy of the system is equal*0756

*to this constant which is Boltzmann constant × the natural log of something called O.*0763

*You can use either one you want, O happens to be the classical, that is just what we use.*0771

*KB is the Boltzmann constant and KB is equal to the gas constant divided by all the numbers.*0777

*If we take 8.314 if we do J/ °K mol and if we divide 6.02 × 10²³ particles / mol which is just / mol of the actual unit of it.*0800

*The mol and mol cancel and which you end up with is this value of KB = 1.381 × 10⁻²³ J/ °K.*0819

*Notice, it is has the same units as entropy J/ °K.*0835

*You can memorize 1.381 × 10⁻²³, you cannot memorize it all and just have your book in front of you and look it up or you can think of it as R÷ n.*0840

*This is the best way to think about it, the gas constant divided by other number.*0851

*We have taken care of this KB, what about this O?*0857

*This is the one that we are going to spend the lesson talking about.*0860

*This O represents the energy and the volume distribution.*0863

*Let me just say a couple of quick words about this and then I will actually get into what this is and get much more detail about this.*0864

*We said that we have these particles in the system and we are going to distribute these particles.*0891

*There is an energy in the system and that energy is distributed over the particles.*0896

*Different particles have different energies.*0899

*These particles also distribute themselves in this space available to them.*0901

*O represents the number of ways that this distribution is possible.*0906

*If I give you 10 particles and if I said there are 100 J of energy, distribute those 100 J among those particles and*0913

*then if I gave you a 1 L flask and if I said how many different ways can you take those 10 particles and*0920

*put them in that 1 L flask if I divided the 1 L flask into 20 volume elements, 20 spaces.*0927

*There is some statistical probabilistic number, some combinatory number that you can come up with.*0936

*The numbers actually is going to be very large, that is what O is.*0944

*O is a measure of how I can distribute my energy and my volume / the number of particles that I have given to me.*0949

*That number gets very huge, that is O.*0958

*When I take the log of that number and I multiply it by both constant, I'm going to get some number, that number is the entropy.*0965

*That number is the statistical entropy for that particular system in that state.*0973

*Let us say some more about it.*0981

*I’m going to say a few more words and I’m going to start quantifying this.*0986

*A given system is in a given state, that is in microscopic state.*0991

*This is the state that we experience, see, and measure.*0996

*In other words, the temperature, the pressure, and the volume, that is the microscopic state of the system.*0999

*O is the number of individual ways that the particles making up the system can distribute themselves over the given volume and*1008

*over the given total energy in order to achieve that particular state.*1017

*The temperature, pressure, volume etc, whatever it is that I happen to be measuring.*1022

*These individual ways of the distributions are called microstates.*1026

*When I come across a system that has a certain temperature, pressure, and volume, the particles,*1032

*the energy of the system and the volume of the system are distributed among those particles.*1038

*The particles spread themselves out of the volume in different ways and there is an energy distribution.*1043

*There are different particles of different energies.*1050

*However, that is not fixed, the particles are indistinguishable.*1051

*If I have particle A here and particle B here, if I switch them and put particle B here and particle A there*1056

*because they are indistinguishable they represent the same thing.*1061

*Because they represent the same thing, because the particles are indistinguishable, there are millions and billions and trillions*1065

*of ways of achieving the same state, the same temperature, pressure, and volume, the same microscopic state.*1072

*We have a bunch of microstates, a bunch of different ways of distributing it to achieve the same state, that is what we are saying.*1080

*O is the number of microstates.*1089

*It is the different individual arrangements giving rise to a single microscopic state.*1092

*Let me say that again, it is the different individual arrangements giving rise to a single microscopic state.*1098

*The more individual ways, they are achieving a given state, the greater the probability of finding the system in that state.*1116

*If I have 15 different ways of achieving a certain temperature, pressure, and volume, another distribution gives me 500 ways*1126

*of achieving that same temperature, pressure, and volume, or if I come up on that state chances are the probability says that the 500 ways,*1135

*I’m probably did run across 1 of those 500 ways more than I’m going to run across 1 of those 15 ways.*1143

*You know this intuitively, it is that simple.*1149

*This is why you never see a gas collected in one corner of the room, instead it spreads out and occupies as much of the room as possible*1153

*because there are more individual ways to fill up a large space than there are filling up a small space.*1161

*That you know this intuitively, let us quantify this.*1168

*Let us put some numbers to it.*1173

*The first thing I want to talk about is the energy distribution and then in the next lesson I’m going to talk about the volume distribution and*1177

*I’m going to put them together.*1183

*The energy distribution is first.*1185

*We have energy distribution.*1189

*We had n particles now what we are going to do is we are going to divide U the total energy into compartments of various energies.*1197

*I got energy 1 J, energy 2 make it 2 J, energy 3 that is 3 J, and so on.*1223

*Just different compartments with different energies and I will just put E sub I right there.*1235

*When I add up all the energies they have to equal U.*1246

*Divide U compartment of various energies.*1257

*It is just a bunch of different energies.*1260

*Specify how many particles have which particular energy?*1268

*Specify how many particles n sub i have energy e sub I.*1274

*How many particles n sub 2 have energy e sub 2, and so on.*1303

*5 particles have energy 1, 30 particles have energy 2, 30 particles have energy 3, 1 particle has energy 4, that is the energy distribution.*1317

*n sub 1, n sub 2, n sub 3, + so on + n sub i is equal to n the total number of particles.*1330

*This is the energy distribution when you specify the n's.*1344

*This is the energy distribution.*1362

*It is the actual specifying of how many particles n sub i have energy e sub i.*1367

*The question is in how many ways can n particles be distributed according to the energy distribution n sub 1 n sub 2 n sub 3 and so on.*1390

*I have some energy distribution n sub 1 n sub 2 n sub 3, these are the particles that have a particular energy e sub 1 e sub 2 e sub 3.*1433

*If I have a total of n particles is there a way for me to count how many different ways*1442

*I can actually distribute the energy over this many compartments? how can I do that?*1447

*Let us go ahead and do it.*1454

*Let us do this with some small number examples first.*1460

*Let us suppose that the n sub 1=3.*1464

*We have 3 particles that have an energy 1 so n sub 1 is 3, 3 particles energy e sub 1.*1472

*The n particles, the question is all those n particles how many different ways can I actually put 3 of those particles into the first energy level?*1481

*How many different choices do I have for my first particle n?*1498

*There are n ways to choose particle 1.*1502

*If I have 10 particles I can choose any of those 10 as my first choice, to put in to be number 1.*1506

*I have n -1 ways to choose particle 2.*1514

*I have n -2 ways to choose particle 3.*1523

*If I choose particle 1 that is going to be at 10 ways to choose that and I have I chosen out and 9 particles was left.*1529

*I have 9 ways of choosing from the other particles, I have 8.*1535

*I have the following so 10 × n -1 × n -2 ways of choosing those particles.*1538

*The particles are indistinguishable so it does not matter whether I choose particle 1 first or 2 first, or 3 first, they are indistinguishable.*1552

*This number of ways of choosing is actually going to have redundancies.*1559

*Therefore, I have to divide because it does not matter whether I choose 1, 2.*1564

*Again, I'm choosing 1, 2, 3 to put them into bin number 1.*1568

*I can choose 1, 2, 3 or I can choose 2, 1, 3 or 3, 2, 1.*1572

*I can choose 3, 1, 2 or 3, 2, 1 as it turns out there are three factorial ways of arranging three particles that are indistinguishable.*1578

*The particles are indistinguishable so the order of choosing is unimportant.*1605

*This n × n -1 × n -2 has redundancies and otherwise, if I choose 1 and 2 and 3 and they end up in this bin.*1637

*It is going to be the same as if I end up choosing 2, 1, and 3, it is the same particles ending up in the same bin.*1654

*I have repeated myself that is what we mean by the redundancies.*1660

*If I choose 3, 2, 1 it is still the same particles 3, 2, 1 in that bin.*1663

*It does not matter, the order, if it would all end up in that bin I cannot just keep counting those ways.*1668

*There is only one way of getting that.*1672

*For 3 particles, there are 3 factorial permutations.*1677

*Therefore, we take this n -1 × n -2 and we divide by 3!.*1695

*This gives us the number of ways of taking n particles and choosing 3 of them to actually go into bin number 1.*1704

*There are these many ways of doing it, whatever n happens to be.*1714

*If n is 10 it would be 10 × 9 × 8 ÷ 3!, whatever that number is.*1718

*We will see some examples in just a minute.*1724

*That is the first level, that is the first part, let us deal with n sub 2.*1726

*n sub 2, that is 2 that means there are 2 particles in the second energy level.*1731

*How many different ways now that I have chosen my 3 particles from my n I have 10 × n -1 × n -2.*1742

*I have n -3 particles leftover.*1750

*All those n -3 particles I'm going to pick two of them to put into the second energy level.*1752

*How many different ways can I do that?*1758

*If I have n -3 particles well I choose one of the particles that leaves me with n -4 for the second particle.*1765

*I have n -3 particles to choose from, I choose one that leaves me with n -4 particles but there are redundancies because I can choose 1 and 2 or 2 and 1.*1777

*I divide by 2! that takes care of the second bin.*1786

*The total so far which is for 3 n energy compartment 1 and 2 in energy compartment 2, we multiply those two numbers.*1795

*We have n × n -1 × n -2/ 3! × n -3 × n -4 / 2!.*1818

*What if we continue?*1842

*If we continue with n3, n4, n5, and so on, we get the following.*1848

*We get the O is equal to n! divided by n sub 1!, n sub 2!, n sub 3!, and so on.*1863

*This is the general expression for the number of ways, the number of individual arrangements,*1880

*the number of microstates that allow n sub 1 particles in e1, n sub 2 particles in e2, etc.*1906

*Given a particular distribution n sub 1 n sub 2 n sub 3 n sub 4, the total number of ways of distributing the energy of those particles*1936

*over the number of particles of a number of energy compartments is this expression right here.*1946

*The total number of particles factorial divided by the number of particles in each bin, each factorial and then divided.*1951

*Let us do some examples and there we go.*1967

*Let us do some examples, given 10 particles and four energy states e1 e2 e3 e4, in how many ways can the following distribution be achieved?*1974

*This is n1, this is n2, this is n3, this is n4.*1984

*In this case, we have four energy compartments.*1991

*We are saying we have 10 particles total so n= 10.*1994

*We are saying that n1= 10.*2000

*In other words, we are taking all those 10 particles and we are putting all of them into bin number 1.*2003

*All those particles have an energy whatever energy e sub 1 happens to be, there is nothing in bin sub 2, nothing in bin sub 3, nothing in bin sub 4.*2008

*How many different ways is it possible if I let 10 particles to arrange themselves according to this distribution?*2018

*We use our equation, we have O= n !/n1! n2! n3! and n4!.*2028

*N is 10 and so this is going to be 10!, n1=10, this is 10!.*2041

*N2, n3, n4 are 0 so this is 0!.*2049

*0! by definition is 1.*2056

*Therefore, what you have is 10!/ 10! you get 1.*2062

*If I have 10 particles and I take 4 and 4 energy states available, there is only one way that I can put those 10 particles into 1 energy state.*2068

*There is only one way, all the 10 have to go into that one spot.*2077

*You know this already.*2082

*Let us change the distribution.*2086

*Same thing, we are given 10 particles and we are given 4 energy states, it is the same old basic situation.*2088

*I still have 4 energy states available and I have 10 particles.*2094

*In how many ways can the following distribution be achieved.*2097

*I want 9 particles in bin 1, 0 in bin 2, 1 particle in bin 3 and 0 in bin 4.*2102

*How can I choose in how many different ways can I achieve this one distribution?*2112

*That is the question.*2117

*In how many different ways can I achieve this one distribution, this is the microscopic state.*2118

*The number of individual ways of achieving this are the different ways, they are the microstates.*2126

*This is the one, the ways of achieving that there is several ways of achieving one state.*2134

*There are several microstates, there are ways of achieving the on microstate.*2140

*Let us do it.*2146

*We have this is n1, this is n2, this is n3, and this is n4.*2148

*n does not change that is equal to 10.*2153

*This is what changed the distribution.*2157

*O=10!/ 9!, 0!, 1!, 0!.*2160

*What we end up with is 10!/ 9! which is equal to 10 × 9!/ 9! 10.*2172

*This distribution I still have 10 particles, I still have 4 energy states but now the distribution is different.*2186

*How they different ways can I do this?*2192

*10 different ways because there are 10 different ways of achieving this one distribution, if I come across this system 10 particles 4 energy levels,*2193

*chances are I’m going to find in this state for this distribution instead of all 10 and packed into energy level 1.*2204

*There is only one way of achieving that but there is 10 ways of achieving this.*2213

*The system is going to shift and achieve all those 10 states more often.*2216

*There is 10 different ways so chances are the probability is that I'm going to run across 1 of these 10 instead of that 1.*2223

*That is what we are saying.*2230

*Change the distribution again, given 10 particles and 4 energy states and how many ways can the following distribution be achieved?*2236

*With 6220 so n=6, n2=2, n3=2, and n4= 0.*2242

*O = 10!/ 6! 2! 0!.*2255

*When you do this under calculator you end up with 1260 ways.*2264

*Clearly this is jumping, we went from 1 to 10 to 1260 just by broadening the distribution.*2272

*Taking it all from 1 energy level and just letting a few more drift off into some of the other energy levels.*2280

*If I come across 10 particles in 4 energy levels of the 3 distributions which 1 more likely to come across?*2285

*The only one way to do, the on with 10 ways to do it, or the one with 1260 ways of achieving that distribution.*2294

*If I take a snapshot of the system in any given moment , these 10 particles with this 4 energy levels*2301

*chances are that I’m going to run across one of these 1260 ways.*2307

*Chances are that when I look at that, I'm going to have 6, 2, 2, 0 that this is the distribution I’m going to see.*2311

*This is going to be my microscopic state.*2318

*The particles are going to be in that distribution because there are so many ways.*2322

*Example 4, given 10 particles and 4 energy states, in how many ways can the following distribution be achieved 3, 3, 2, 2?*2330

*Let us see what this one gives us.*2340

*We have 10!/ 3! 2! 2!.*2343

*We have 25,200 ways, from 1 to 10 to 1200 to 25,000.*2352

*If I have 10 particles and 4 energy states in how many ways if, I take a snapshot, if I just sort of come across the system*2364

*that has 10 particles and 4 energy states available, which distribution I’m most likely going to see when I take a picture of it?*2372

*I’m going to see this one because there are 25,200 ways of achieving that distribution.*2381

*In fact, given the constraints of 10 particles and the constraint of 4 energy levels, this number right here the 3 3 2 2 represents*2388

*the maximum number of ways that I can actually achieve the maximum number of ways that I have in achieving a given distribution.*2399

*With these constraints, this achieves the distribution that I will most likely see.*2410

*The probability is that because there are 25,200 ways of achieving this one distribution, that is the one that I'm going to see.*2416

*If I take a snapshot, I’m going to find 3 in bin 1, 3 in bin 2, 2 in bin 3, 2 in bin 4.*2423

*Or I might find 3 3 2 2 but it is always going to be 3 and 1, 3 another 2, to another.*2431

*That is what we are talking about here.*2439

*I hope that makes sense, these are the microstates, this is O, this is the macrostate given the constraints of distribution,*2442

*the number of particles and the total energy.*2451

*Let us see what we got.*2458

*Clearly, as the distribution broadens, the number of ways to achieve that distribution increases massively 1 to 10 to 1200 to 25,000.*2461

*With just a small change in the distribution the system will appear in the state distribution that offers the greatest number of ways of achieving that state.*2472

*This is very important.*2483

*The system will appear in the state that offers the greatest number of ways of achieving that state because*2487

*the probability of finding a system in a given state depends directly on the total number of ways that that state is achievable.*2494

*In example 1, there is only one way of achieving the particular distribution.*2504

*The chance of finding a given system of that arrangement are 10 particles with 4 energy levels available is very slim.*2508

*In any given moment you probably not going to run across that distribution.*2515

*In other words, these 10 particles will not crowded to one compartment and stay that way.*2519

*If they have other compartments available to them as far as energy is concern, within the constraints of total energy and total number of particles.*2524

*Example 4, offers a distribution as a huge number of ways of being obtained.*2533

*Therefore, chances are very high that if we come up on the system it will be in a state that is consistent with this distribution.*2538

*Within the constraints of the sum of the total, the sum of the individual n sub i equals the total number of particles n.*2547

*The sum of the number of particles in a given energy level × the energy of that level equals U the total energy.*2556

*There exists a distribution, there is always one distribution that completely maximizes this O.*2562

*The sheer number of microstates for this distribution is so huge that it completely dominates the landscape of probabilities.*2570

*You will certainly find the system with this distribution in the state.*2578

*In our example we had 1, we had 10, we had 1260, I think and we had 25,200.*2581

*Within the constraints of the particular problem, the 10 and 4, the distribution is 25,200 is so massive, it is so much bigger than this and this.*2590

*If we ever come across the system with 10 particles and 4 energy levels, we are going to find the distribution 3 3 2 2 or 3 2 3 2 or 2 2 3 3.*2601

*We are going to find 3 in 1 bin, 3 in another, 2 in another, and 2 in another.*2611

*That is the distribution the particles will arrange themselves in the way that offers the greatest number of ways of achieving that distribution, that microscopic state.*2614

*Sorry if I keep repeating myself, but this is profoundly important.*2627

*Let us look again at this O.*2633

*O we set this as n!/ n sub 1! n sub 2! n sub 3!, and so on.*2650

*We set that S, let us go back to entropy because we are talking about entropy here.*2652

*S= Boltzmann constant × nat log of this thing called O.*2657

*If O is large, the log of O is large, the entropy is large.*2663

*If O is large then S is large.*2670

*The smaller the n sub I, these individual n sub I, n sub I, the smaller these numbers are the larger O becomes because this is the ratio for the numerator/ denominator.*2686

*If I have a certain number of particles that is fixed, the smaller the denominator is the bigger my ratio is, the bigger O is going to be.*2709

*The smaller the n sub I, the larger O becomes which means the larger the entropy.*2718

*Small n sub i means distributing as many particles in as many compartments as possible to lower that number.*2731

*Instead of 10 particles in 1 bin, it is a lot better to have a fewer number in this the 3 3 2 2 for the particular arrangement.*2743

*That maximizes it, that lower the n sub 1 n sub 2 n sub 3 n sub 4, as low as they will go.*2753

*As low as they will go that raises the O to as high as it will go which in that case was 25,200.*2764

*That is an increase in entropy.*2771

*This is what we mean by this order or randomness having 25,200 ways of achieving given distribution*2774

*is kind of chaotic vs. only one way of achieving a distribution.*2782

*It is a highly ordered system having 25,200 ways of doing the same thing that is random, that is disordered, that is chaotic, that is what we mean.*2788

*It represents the distribution, these are the numbers.*2798

*Back to this, the larger the n sub I, the larger these numbers, the smaller O is.*2803

*The smaller O becomes which implies that the smaller entropy.*2814

*The system will try to achieve the highest entropy possible thus the broadest distribution subject to the two constraints.*2831

*Broad means we want to spread out the particles in as many bins as possible to lower the number of particles in each bin.*2881

*The lower these numbers, the higher O, the higher the entropy.*2888

*A system is going to try to achieve the highest entropy possible thus the broadest distribution subject to the two constraints.*2891

*And I will go ahead and put those constraints again.*2900

*The sum of the individual n sub i =n, this is the sigma notation and the sum of n sub i × the energy of that particular bin is equal to the total energy.*2902

*Within these two constraints, the system will try to achieve the broadest distribution possible that it can.*2918

*In the case of the 10 particles and 4 bins it was 3 3 2 2.*2925

*That was its broadest distribution possible.*2930

*When the energy of a system is increased, the energy distribution broadens and the particles occupied this broader distribution.*2945

*In other words, O rises which means the entropy arises.*3000

*If I have a certain energy of the system and all of a sudden I pump some more energy in the system, if I increase the energy of the system,*3008

*by increasing the energy of the system now I have allowed more energy, a lot more bins.*3013

*Therefore, if I have more bins, the particles that are in these other bins are going to filter off and move into those other bins.*3026

*They are going to be fewer particles in each individual bin.*3034

*Again, if there are fewer particles in each individual bin that means the number of ways that*3038

*the denominator of the O is to get smaller which means O goes up.*3043

*Numerically, if O goes up the entropy goes up.*3048

*If I have the 123 and 4, all of a sudden if I pump some more energy into that, now all of a sudden I introduced.*3053

*We are going back to the example where I have 10 particles and I have the 4 bins.*3063

*We said we had 3 3 2 2 under the constraints of 10 and 4, this was the broadest distribution I could have which is going to be 25,200 microstates.*3068

*Let us say if I increase the energy of the system now introduce let us say 3 more energy levels in here, these particles can now move to those.*3079

*It is going to move as much as possible.*3088

*Maybe one of these particles end up going over here so that goes to 1.*3091

*Maybe one of these particles that goes here and maybe about one of these particles ends up coming over here because I have added energy.*3095

*I have broaden the energy distribution .*3104

*I have allowed more bin, now they are going to distribute themselves in such a way that its broader.*3106

*It is going to achieve a broader distribution.*3110

*If I do 10! / 3! 2! 1 1, now it is going to be a lot higher than 25,200.*3113

*The entropy is going to go up but you know this already.*3125

*Here is how you know this.*3127

*You know this already from the work that we did in the previous lessons.*3132

*You know the DS =1/ T DU + P/ T DV.*3139

*This 1/ T is positive, it is always positive because the Kelvin temperature is always going to be about 0.*3149

*1/ T is always positive which means that if you increase the energy of the system, you increase the entropy.*3153

*This is a simple math.*3160

*You know this already from your experience, you have dealt with this.*3163

*What we have given is a statistical, we have given a probabilistic explanation for these increase in entropy.*3167

*We know why when we increase the energy we are increasing the energy distribution because we increase the energy distribution O goes up.*3175

*When O goes up, the entropy goes up.*3185

*That is what is happening here.*3188

*What we have done here, we just given.*3190

* I have to slow down.*3199

*The statistical and I will go ahead and say probabilistic.*3202

*In other words, the microscopic reason for our classical observation which is our macroscopic observation.*3215

*I hope this has made sense, increasing the energy of the system broadens the energy distribution.*3237

*As you broaden the energy distribution, the number of particles that can achieve the distribution become the spread out, they themselves broaden out,*3243

*when that happens the numerator of our O ends up getting smaller so O gets higher.*3252

*When O gets higher, because S is equal to KB LN O S gets higher.*3259

*I hope that makes sense.*3268

*Thank you so much for joining us here at www.educator.com.*3270

*We will see you next time for a continuation of this discussion, bye.*3272

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