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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 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?

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