WEBVTT chemistry/physical-chemistry/hovasapian
00:00:00.000 --> 00:00:04.200
Hello and welcome back to www.educator.com and welcome back to Physical Chemistry.
00:00:04.200 --> 00:00:08.000
Today, we are going to talk about the entropy and probability.
00:00:08.000 --> 00:00:11.600
We are going to define what entropy is finally.
00:00:11.600 --> 00:00:14.000
Let us jump right on in.
00:00:14.000 --> 00:00:32.500
When we introduced the definition of entropy, when we introduced that the DS =DQ reversible/ T which was our definition of the differential element.
00:00:32.500 --> 00:00:44.200
That is the definition of entropy, we did not get the structural model.
00:00:44.200 --> 00:01:11.700
We did not require a structural model for the system in order to work with the entropy,
00:01:11.700 --> 00:01:26.100
in order to work with this state function entropy or describe its behavior .
00:01:26.100 --> 00:01:29.700
In fact, we did not even need to know what entropy was.
00:01:29.700 --> 00:01:36.500
We had these mathematical descriptions and we had these constraints of temperature, pressure, volume and we saw how entropy behaves.
00:01:36.500 --> 00:01:40.400
We are able to derive and calculate numerical values for it.
00:01:40.400 --> 00:01:47.900
The only thing that we really did was casually refer to it as a measure of the disorder or randomness of the system.
00:01:47.900 --> 00:01:55.100
I still think, in my personal opinion that disorder and randomness is actually a great way of thinking about entropy.
00:01:55.100 --> 00:02:00.500
We are going to do was to define what we mean when we say disorder and randomness.
00:02:00.500 --> 00:02:11.000
We are going to quantify, we are going to come up with some numerical way of explaining what is this disorder or randomness.
00:02:11.000 --> 00:02:24.800
When I use the words disorder and randomness, what I’m talking about is something called the distribution.
00:02:24.800 --> 00:02:28.400
When we talk about disorder or randomness, we are talking about a distribution.
00:02:28.400 --> 00:02:32.600
In this case, it is going to be the distribution of particles.
00:02:32.600 --> 00:02:36.600
We did not require a structural model, we did not care what a particular system was made up off.
00:02:36.600 --> 00:02:42.000
Whether it is particles, chairs, it could be made absolutely anything.
00:02:42.000 --> 00:02:47.100
There was this behavior that it represented, this is our empirical observation.
00:02:47.100 --> 00:02:56.100
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.
00:02:56.100 --> 00:03:00.600
Here is a structural model.
00:03:00.600 --> 00:03:09.600
What I'm going to do, I would go ahead and do this in blue, I think.
00:03:09.600 --> 00:03:16.200
Our structural model is exactly what you think it is.
00:03:16.200 --> 00:03:40.200
A system is composed of a very large number of particles.
00:03:40.200 --> 00:03:48.200
Those particles could be molecules, they could be atoms, they could be ions, whatever that you have to be discussing in that particular problem.
00:03:48.200 --> 00:03:54.500
Molecules, atoms, and ions.
00:03:54.500 --> 00:03:56.700
Let us say some things about these particles.
00:03:56.700 --> 00:04:03.400
These particles have various energies.
00:04:03.400 --> 00:04:09.500
The best way to think about this is to think about a collection of gas like the kinetic theory.
00:04:09.500 --> 00:04:19.100
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.
00:04:19.100 --> 00:04:24.800
These particles have various energies.
00:04:24.800 --> 00:04:28.400
In other words, not each particle is flying around at the same speed.
00:04:28.400 --> 00:04:31.700
There are some going faster, some that are going slower.
00:04:31.700 --> 00:04:34.700
There is a large number of them that have the same energy.
00:04:34.700 --> 00:04:41.300
There are others that also have the same energy so there might be 10,000,000 of them that are traveling at 500 kph.
00:04:41.300 --> 00:04:44.900
There might be 20,000,000 of them that are traveling at 550 kph.
00:04:44.900 --> 00:04:51.800
They are distributed if all the energy is distributed among the different numbers of particles.
00:04:51.800 --> 00:05:09.800
These particles have various energies and there is a distribution of the total energy of the system DU
00:05:09.800 --> 00:05:33.700
of the total energy of the system which is U first law / the various particles.
00:05:33.700 --> 00:05:41.800
The total energy of the system is made up of the sum of all the individual energies of the particles.
00:05:41.800 --> 00:05:50.500
If the system has 100 J of energy, those 100 J are going to be distributed among the different particles and different ways.
00:05:50.500 --> 00:05:56.400
Maybe two parts might have 1 energy, two particles might have another energy, 15 particles might have another energy.
00:05:56.400 --> 00:05:59.700
All the different energies, that is the distribution.
00:05:59.700 --> 00:06:20.900
In other words, n₁ particles have energy E₁.
00:06:20.900 --> 00:06:34.400
N₂ particles have energy E₂, and so on.
00:06:34.400 --> 00:06:44.600
The number of particles, the total number of particles so n₁ + n₂ + n₃ and so on,
00:06:44.600 --> 00:06:49.100
have to equal n which is the total number of particles in the system.
00:06:49.100 --> 00:06:55.900
If I have 100 particles in the system n₁ might be 10, n₂ might be 20, n₃ might be 70.
00:06:55.900 --> 00:06:59.500
70 + 20 + 10 =100 the total number of particles.
00:06:59.500 --> 00:07:04.900
That is one of our constraints.
00:07:04.900 --> 00:07:16.300
The number of particles with energy 1 + the number of particles + energy 2 + the number of particles with energy 3 and so on,
00:07:16.300 --> 00:07:22.900
onto the number of particles n sub i with energy sub I, that has to equal the total energy of the system.
00:07:22.900 --> 00:07:30.400
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.
00:07:30.400 --> 00:07:36.500
Nothing strange happening here, the system has a certain energy, our system is made up of a bunch of particles,
00:07:36.500 --> 00:07:43.100
we are distributing this energy over a bunch of particles.
00:07:43.100 --> 00:07:51.500
The next part is the structural model is these particles for the first one of these particles have energy.
00:07:51.500 --> 00:08:02.600
These particles occupy space, in other words, volume.
00:08:02.600 --> 00:08:07.300
I'm not saying that they themselves have volume, I'm saying that they are contained in a volume.
00:08:07.300 --> 00:08:10.600
There is some fixed volume that they are in.
00:08:10.600 --> 00:08:18.100
What ever that volume is, they are occupying that space.
00:08:18.100 --> 00:08:40.800
There is the distribution of these particles, this is the most intuitive clear one, these particles over the volume available.
00:08:40.800 --> 00:08:45.000
We have these hundred particles and we have a 1 L flask.
00:08:45.000 --> 00:08:51.600
I have this hundred particles in 1 L flask, the particles are going to arrange themselves in all kinds of different ways.
00:08:51.600 --> 00:08:56.100
Maybe these particles here, these particles there, a 100 all over the place.
00:08:56.100 --> 00:09:03.000
They are a bunch of different ways that these particles can distribute themselves in that 1 L flask, that is the volume distribution.
00:09:03.000 --> 00:09:08.700
They are going to arrange themselves in the volume up to the maximum capacity of the volume.
00:09:08.700 --> 00:09:12.600
You know this intuitively.
00:09:12.600 --> 00:09:30.900
Let us recall the fundamental equation of thermodynamics.
00:09:30.900 --> 00:09:41.300
Our fundamental equation was the following DS = 1/ T DU + P/ T DV.
00:09:41.300 --> 00:09:56.000
Notice DU DV this is energy, this is volume, there are two different independent ways of affecting the entropy of the system.
00:09:56.000 --> 00:10:05.600
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.
00:10:05.600 --> 00:10:08.300
Let us write it down.
00:10:08.300 --> 00:10:30.400
There are two independent ways of affecting the entropy of the system.
00:10:30.400 --> 00:10:41.500
One, I can change the energy that is U.
00:10:41.500 --> 00:10:48.700
Two, I can change the volume which is V.
00:10:48.700 --> 00:10:55.900
It is very important equation, the fundamental equation of thermodynamics is a relationship between all the various properties of the system,
00:10:55.900 --> 00:11:00.600
the entropy, the pressure, the temperature, the volume, the energy.
00:11:00.600 --> 00:11:02.400
It is very important.
00:11:02.400 --> 00:11:06.300
If it is expressed in terms of entropy we would write it this way DS = something.
00:11:06.300 --> 00:11:15.600
We see that there are two ways to affect the entropy, energy, and volume.
00:11:15.600 --> 00:11:25.500
Since, there are two ways affecting this entropy of the system, the energy in the volume.
00:11:25.500 --> 00:11:56.900
Therefore, it makes sense to define the entropy of the system which is S in terms of these two properties energy and volume.
00:11:56.900 --> 00:12:32.800
In terms of these two properties U and V, both one gave the following definition.
00:12:32.800 --> 00:12:36.100
Profoundly important equation, it is the fundamental equation of statistical thermodynamics.
00:12:36.100 --> 00:12:43.200
so they have the following definition, he said that the entropy of the system is equal
00:12:43.200 --> 00:12:51.200
to this constant which is Boltzmann constant × the natural log of something called O.
00:12:51.200 --> 00:12:57.300
You can use either one you want, O happens to be the classical, that is just what we use.
00:12:57.300 --> 00:13:20.100
KB is the Boltzmann constant and KB is equal to the gas constant divided by all the numbers.
00:13:20.100 --> 00:13:39.600
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.
00:13:39.600 --> 00:13:55.400
The mol and mol cancel and which you end up with is this value of KB = 1.381 × 10⁻²³ J/ °K.
00:13:55.400 --> 00:14:00.200
Notice, it is has the same units as entropy J/ °K.
00:14:00.200 --> 00:14:11.000
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.
00:14:11.000 --> 00:14:17.000
This is the best way to think about it, the gas constant divided by other number.
00:14:17.000 --> 00:14:20.300
We have taken care of this KB, what about this O?
00:14:20.300 --> 00:14:22.700
This is the one that we are going to spend the lesson talking about.
00:14:22.700 --> 00:14:24.400
This O represents the energy and the volume distribution.
00:14:24.400 --> 00:14:51.400
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.
00:14:51.400 --> 00:14:55.700
We said that we have these particles in the system and we are going to distribute these particles.
00:14:55.700 --> 00:14:59.300
There is an energy in the system and that energy is distributed over the particles.
00:14:59.300 --> 00:15:00.900
Different particles have different energies.
00:15:00.900 --> 00:15:05.700
These particles also distribute themselves in this space available to them.
00:15:05.700 --> 00:15:12.900
O represents the number of ways that this distribution is possible.
00:15:12.900 --> 00:15:20.400
If I give you 10 particles and if I said there are 100 J of energy, distribute those 100 J among those particles and
00:15:20.400 --> 00:15:27.000
then if I gave you a 1 L flask and if I said how many different ways can you take those 10 particles and
00:15:27.000 --> 00:15:36.600
put them in that 1 L flask if I divided the 1 L flask into 20 volume elements, 20 spaces.
00:15:36.600 --> 00:15:44.400
There is some statistical probabilistic number, some combinatory number that you can come up with.
00:15:44.400 --> 00:15:48.900
The numbers actually is going to be very large, that is what O is.
00:15:48.900 --> 00:15:57.800
O is a measure of how I can distribute my energy and my volume / the number of particles that I have given to me.
00:15:57.800 --> 00:16:05.000
That number gets very huge, that is O.
00:16:05.000 --> 00:16:12.800
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.
00:16:12.800 --> 00:16:20.700
That number is the statistical entropy for that particular system in that state.
00:16:20.700 --> 00:16:26.400
Let us say some more about it.
00:16:26.400 --> 00:16:31.500
I’m going to say a few more words and I’m going to start quantifying this.
00:16:31.500 --> 00:16:35.700
A given system is in a given state, that is in microscopic state.
00:16:35.700 --> 00:16:39.300
This is the state that we experience, see, and measure.
00:16:39.300 --> 00:16:48.300
In other words, the temperature, the pressure, and the volume, that is the microscopic state of the system.
00:16:48.300 --> 00:16:57.000
O is the number of individual ways that the particles making up the system can distribute themselves over the given volume and
00:16:57.000 --> 00:17:02.100
over the given total energy in order to achieve that particular state.
00:17:02.100 --> 00:17:06.600
The temperature, pressure, volume etc, whatever it is that I happen to be measuring.
00:17:06.600 --> 00:17:11.700
These individual ways of the distributions are called microstates.
00:17:11.700 --> 00:17:18.300
When I come across a system that has a certain temperature, pressure, and volume, the particles,
00:17:18.300 --> 00:17:23.600
the energy of the system and the volume of the system are distributed among those particles.
00:17:23.600 --> 00:17:29.700
The particles spread themselves out of the volume in different ways and there is an energy distribution.
00:17:29.700 --> 00:17:31.600
There are different particles of different energies.
00:17:31.600 --> 00:17:35.800
However, that is not fixed, the particles are indistinguishable.
00:17:35.800 --> 00:17:41.500
If I have particle A here and particle B here, if I switch them and put particle B here and particle A there
00:17:41.500 --> 00:17:45.100
because they are indistinguishable they represent the same thing.
00:17:45.100 --> 00:17:51.700
Because they represent the same thing, because the particles are indistinguishable, there are millions and billions and trillions
00:17:51.700 --> 00:17:59.800
of ways of achieving the same state, the same temperature, pressure, and volume, the same microscopic state.
00:17:59.800 --> 00:18:09.100
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.
00:18:09.100 --> 00:18:12.100
O is the number of microstates.
00:18:12.100 --> 00:18:18.500
It is the different individual arrangements giving rise to a single microscopic state.
00:18:18.500 --> 00:18:35.900
Let me say that again, it is the different individual arrangements giving rise to a single microscopic state.
00:18:35.900 --> 00:18:46.300
The more individual ways, they are achieving a given state, the greater the probability of finding the system in that state.
00:18:46.300 --> 00:18:54.700
If I have 15 different ways of achieving a certain temperature, pressure, and volume, another distribution gives me 500 ways
00:18:54.700 --> 00:19:03.400
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,
00:19:03.400 --> 00:19:09.100
I’m probably did run across 1 of those 500 ways more than I’m going to run across 1 of those 15 ways.
00:19:09.100 --> 00:19:12.700
You know this intuitively, it is that simple.
00:19:12.700 --> 00:19:20.800
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
00:19:20.800 --> 00:19:28.300
because there are more individual ways to fill up a large space than there are filling up a small space.
00:19:28.300 --> 00:19:33.100
That you know this intuitively, let us quantify this.
00:19:33.100 --> 00:19:36.700
Let us put some numbers to it.
00:19:36.700 --> 00:19:42.700
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
00:19:42.700 --> 00:19:44.800
I’m going to put them together.
00:19:44.800 --> 00:19:49.200
The energy distribution is first.
00:19:49.200 --> 00:19:56.900
We have energy distribution.
00:19:56.900 --> 00:20:22.900
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.
00:20:22.900 --> 00:20:34.900
I got energy 1 J, energy 2 make it 2 J, energy 3 that is 3 J, and so on.
00:20:34.900 --> 00:20:46.600
Just different compartments with different energies and I will just put E sub I right there.
00:20:46.600 --> 00:20:56.700
When I add up all the energies they have to equal U.
00:20:56.700 --> 00:21:00.600
Divide U compartment of various energies.
00:21:00.600 --> 00:21:08.400
It is just a bunch of different energies.
00:21:08.400 --> 00:21:14.100
Specify how many particles have which particular energy?
00:21:14.100 --> 00:21:42.700
Specify how many particles n sub i have energy e sub I.
00:21:42.700 --> 00:21:57.300
How many particles n₂ have energy e₂, and so on.
00:21:57.300 --> 00:22:09.900
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.
00:22:09.900 --> 00:22:24.600
n₁, n₂, n₃, + so on + n sub i is equal to n the total number of particles.
00:22:24.600 --> 00:22:42.200
This is the energy distribution when you specify the n's.
00:22:42.200 --> 00:22:47.600
This is the energy distribution.
00:22:47.600 --> 00:23:10.400
It is the actual specifying of how many particles n sub i have energy e sub i.
00:23:10.400 --> 00:23:53.200
The question is in how many ways can n particles be distributed according to the energy distribution n₁ n₂ n₃ and so on.
00:23:53.200 --> 00:24:01.900
I have some energy distribution n₁ n₂ n₃, these are the particles that have a particular energy e₁ e₂ e₃.
00:24:01.900 --> 00:24:07.200
If I have a total of n particles is there a way for me to count how many different ways
00:24:07.200 --> 00:24:14.400
I can actually distribute the energy over this many compartments? how can I do that?
00:24:14.400 --> 00:24:20.400
Let us go ahead and do it.
00:24:20.400 --> 00:24:24.600
Let us do this with some small number examples first.
00:24:24.600 --> 00:24:31.800
Let us suppose that the n₁=3.
00:24:31.800 --> 00:24:40.800
We have 3 particles that have an energy 1 so n₁ is 3, 3 particles energy e₁.
00:24:40.800 --> 00:24:58.100
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?
00:24:58.100 --> 00:25:02.000
How many different choices do I have for my first particle n?
00:25:02.000 --> 00:25:06.600
There are n ways to choose particle 1.
00:25:06.600 --> 00:25:14.400
If I have 10 particles I can choose any of those 10 as my first choice, to put in to be number 1.
00:25:14.400 --> 00:25:23.100
I have n -1 ways to choose particle 2.
00:25:23.100 --> 00:25:29.100
I have n -2 ways to choose particle 3.
00:25:29.100 --> 00:25:34.800
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.
00:25:34.800 --> 00:25:38.400
I have 9 ways of choosing from the other particles, I have 8.
00:25:38.400 --> 00:25:52.200
I have the following so 10 × n -1 × n -2 ways of choosing those particles.
00:25:52.200 --> 00:25:59.400
The particles are indistinguishable so it does not matter whether I choose particle 1 first or 2 first, or 3 first, they are indistinguishable.
00:25:59.400 --> 00:26:03.900
This number of ways of choosing is actually going to have redundancies.
00:26:03.900 --> 00:26:08.100
Therefore, I have to divide because it does not matter whether I choose 1, 2.
00:26:08.100 --> 00:26:12.600
Again, I'm choosing 1, 2, 3 to put them into bin number 1.
00:26:12.600 --> 00:26:18.000
I can choose 1, 2, 3 or I can choose 2, 1, 3 or 3, 2, 1.
00:26:18.000 --> 00:26:44.900
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.
00:26:44.900 --> 00:27:16.900
The particles are indistinguishable so the order of choosing is unimportant.
00:27:16.900 --> 00:27:34.300
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.
00:27:34.300 --> 00:27:39.700
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.
00:27:39.700 --> 00:27:43.000
I have repeated myself that is what we mean by the redundancies.
00:27:43.000 --> 00:27:48.100
If I choose 3, 2, 1 it is still the same particles 3, 2, 1 in that bin.
00:27:48.100 --> 00:27:52.400
It does not matter, the order, if it would all end up in that bin I cannot just keep counting those ways.
00:27:52.400 --> 00:27:56.900
There is only one way of getting that.
00:27:56.900 --> 00:28:14.900
For 3 particles, there are 3 factorial permutations.
00:28:14.900 --> 00:28:24.000
Therefore, we take this n -1 × n -2 and we divide by 3!.
00:28:24.000 --> 00:28:33.800
This gives us the number of ways of taking n particles and choosing 3 of them to actually go into bin number 1.
00:28:33.800 --> 00:28:37.700
There are these many ways of doing it, whatever n happens to be.
00:28:37.700 --> 00:28:43.700
If n is 10 it would be 10 × 9 × 8 ÷ 3!, whatever that number is.
00:28:43.700 --> 00:28:46.400
We will see some examples in just a minute.
00:28:46.400 --> 00:28:51.500
That is the first level, that is the first part, let us deal with n₂.
00:28:51.500 --> 00:29:02.300
n₂, that is 2 that means there are 2 particles in the second energy level.
00:29:02.300 --> 00:29:09.800
How many different ways now that I have chosen my 3 particles from my n I have 10 × n -1 × n -2.
00:29:09.800 --> 00:29:12.500
I have n -3 particles leftover.
00:29:12.500 --> 00:29:18.200
All those n -3 particles I'm going to pick two of them to put into the second energy level.
00:29:18.200 --> 00:29:25.100
How many different ways can I do that?
00:29:25.100 --> 00:29:37.100
If I have n -3 particles well I choose one of the particles that leaves me with n -4 for the second particle.
00:29:37.100 --> 00:29:46.100
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.
00:29:46.100 --> 00:29:55.100
I divide by 2! that takes care of the second bin.
00:29:55.100 --> 00:30:17.800
The total so far which is for 3 n energy compartment 1 and 2 in energy compartment 2, we multiply those two numbers.
00:30:17.800 --> 00:30:42.000
We have n × n -1 × n -2/ 3! × n -3 × n -4 / 2!.
00:30:42.000 --> 00:30:47.700
What if we continue?
00:30:47.700 --> 00:31:03.600
If we continue with n3, n4, n5, and so on, we get the following.
00:31:03.600 --> 00:31:20.100
We get the O is equal to n! divided by n₁!, n₂!, n₃!, and so on.
00:31:20.100 --> 00:31:46.100
This is the general expression for the number of ways, the number of individual arrangements,
00:31:46.100 --> 00:32:15.700
the number of microstates that allow n₁ particles in e1, n₂ particles in e2, etc.
00:32:15.700 --> 00:32:25.900
Given a particular distribution n₁ n₂ n₃ n₄, the total number of ways of distributing the energy of those particles
00:32:25.900 --> 00:32:31.300
over the number of particles of a number of energy compartments is this expression right here.
00:32:31.300 --> 00:32:46.800
The total number of particles factorial divided by the number of particles in each bin, each factorial and then divided.
00:32:46.800 --> 00:32:54.000
Let us do some examples and there we go.
00:32:54.000 --> 00:33:03.900
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?
00:33:03.900 --> 00:33:11.400
This is n1, this is n2, this is n3, this is n4.
00:33:11.400 --> 00:33:14.100
In this case, we have four energy compartments.
00:33:14.100 --> 00:33:20.400
We are saying we have 10 particles total so n= 10.
00:33:20.400 --> 00:33:23.400
We are saying that n1= 10.
00:33:23.400 --> 00:33:27.900
In other words, we are taking all those 10 particles and we are putting all of them into bin number 1.
00:33:27.900 --> 00:33:37.700
All those particles have an energy whatever energy e₁ happens to be, there is nothing in bin₂, nothing in bin₃, nothing in bin₄.
00:33:37.700 --> 00:33:48.200
How many different ways is it possible if I let 10 particles to arrange themselves according to this distribution?
00:33:48.200 --> 00:34:01.400
We use our equation, we have O= n !/n1! n2! n3! and n4!.
00:34:01.400 --> 00:34:09.500
N is 10 and so this is going to be 10!, n1=10, this is 10!.
00:34:09.500 --> 00:34:16.300
N2, n3, n4 are 0 so this is 0!.
00:34:16.300 --> 00:34:21.800
0! by definition is 1.
00:34:21.800 --> 00:34:28.300
Therefore, what you have is 10!/ 10! you get 1.
00:34:28.300 --> 00:34:36.700
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.
00:34:36.700 --> 00:34:42.100
There is only one way, all the 10 have to go into that one spot.
00:34:42.100 --> 00:34:46.300
You know this already.
00:34:46.300 --> 00:34:48.100
Let us change the distribution.
00:34:48.100 --> 00:34:54.400
Same thing, we are given 10 particles and we are given 4 energy states, it is the same old basic situation.
00:34:54.400 --> 00:34:56.800
I still have 4 energy states available and I have 10 particles.
00:34:56.800 --> 00:35:02.100
In how many ways can the following distribution be achieved.
00:35:02.100 --> 00:35:11.700
I want 9 particles in bin 1, 0 in bin 2, 1 particle in bin 3 and 0 in bin 4.
00:35:11.700 --> 00:35:17.100
How can I choose in how many different ways can I achieve this one distribution?
00:35:17.100 --> 00:35:18.000
That is the question.
00:35:18.000 --> 00:35:26.400
In how many different ways can I achieve this one distribution, this is the microscopic state.
00:35:26.400 --> 00:35:33.900
The number of individual ways of achieving this are the different ways, they are the microstates.
00:35:33.900 --> 00:35:39.700
This is the one, the ways of achieving that there is several ways of achieving one state.
00:35:39.700 --> 00:35:46.100
There are several microstates, there are ways of achieving the on microstate.
00:35:46.100 --> 00:35:48.200
Let us do it.
00:35:48.200 --> 00:35:53.500
We have this is n1, this is n2, this is n3, and this is n4.
00:35:53.500 --> 00:35:56.800
n does not change that is equal to 10.
00:35:56.800 --> 00:36:00.500
This is what changed the distribution.
00:36:00.500 --> 00:36:12.000
O=10!/ 9!, 0!, 1!, 0!.
00:36:12.000 --> 00:36:25.700
What we end up with is 10!/ 9! which is equal to 10 × 9!/ 9! 10.
00:36:25.700 --> 00:36:32.000
This distribution I still have 10 particles, I still have 4 energy states but now the distribution is different.
00:36:32.000 --> 00:36:33.500
How they different ways can I do this?
00:36:33.500 --> 00:36:44.600
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,
00:36:44.600 --> 00:36:52.700
chances are I’m going to find in this state for this distribution instead of all 10 and packed into energy level 1.
00:36:52.700 --> 00:36:56.600
There is only one way of achieving that but there is 10 ways of achieving this.
00:36:56.600 --> 00:37:03.500
The system is going to shift and achieve all those 10 states more often.
00:37:03.500 --> 00:37:10.000
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.
00:37:10.000 --> 00:37:15.700
That is what we are saying.
00:37:15.700 --> 00:37:22.300
Change the distribution again, given 10 particles and 4 energy states and how many ways can the following distribution be achieved?
00:37:22.300 --> 00:37:35.500
With 6220 so n=6, n2=2, n3=2, and n4= 0.
00:37:35.500 --> 00:37:44.200
O = 10!/ 6! 2! 0!.
00:37:44.200 --> 00:37:52.600
When you do this under calculator you end up with 1260 ways.
00:37:52.600 --> 00:37:59.800
Clearly this is jumping, we went from 1 to 10 to 1260 just by broadening the distribution.
00:37:59.800 --> 00:38:05.200
Taking it all from 1 energy level and just letting a few more drift off into some of the other energy levels.
00:38:05.200 --> 00:38:14.100
If I come across 10 particles in 4 energy levels of the 3 distributions which 1 more likely to come across?
00:38:14.100 --> 00:38:21.000
The only one way to do, the on with 10 ways to do it, or the one with 1260 ways of achieving that distribution.
00:38:21.000 --> 00:38:26.700
If I take a snapshot of the system in any given moment , these 10 particles with this 4 energy levels
00:38:26.700 --> 00:38:31.200
chances are that I’m going to run across one of these 1260 ways.
00:38:31.200 --> 00:38:37.800
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.
00:38:37.800 --> 00:38:42.600
This is going to be my microscopic state.
00:38:42.600 --> 00:38:50.400
The particles are going to be in that distribution because there are so many ways.
00:38:50.400 --> 00:39:00.600
Example 4, given 10 particles and 4 energy states, in how many ways can the following distribution be achieved 3, 3, 2, 2?
00:39:00.600 --> 00:39:03.200
Let us see what this one gives us.
00:39:03.200 --> 00:39:12.000
We have 10!/ 3! 2! 2!.
00:39:12.000 --> 00:39:24.600
We have 25,200 ways, from 1 to 10 to 1200 to 25,000.
00:39:24.600 --> 00:39:32.600
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
00:39:32.600 --> 00:39:41.100
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?
00:39:41.100 --> 00:39:48.000
I’m going to see this one because there are 25,200 ways of achieving that distribution.
00:39:48.000 --> 00:39:58.700
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
00:39:58.700 --> 00:40:09.800
the maximum number of ways that I can actually achieve the maximum number of ways that I have in achieving a given distribution.
00:40:09.800 --> 00:40:15.800
With these constraints, this achieves the distribution that I will most likely see.
00:40:15.800 --> 00:40:23.600
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.
00:40:23.600 --> 00:40:30.800
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.
00:40:30.800 --> 00:40:38.900
Or I might find 3 3 2 2 but it is always going to be 3 and 1, 3 another 2, to another.
00:40:38.900 --> 00:40:42.500
That is what we are talking about here.
00:40:42.500 --> 00:40:51.400
I hope that makes sense, these are the microstates, this is O, this is the macrostate given the constraints of distribution,
00:40:51.400 --> 00:40:58.300
the number of particles and the total energy.
00:40:58.300 --> 00:41:01.000
Let us see what we got.
00:41:01.000 --> 00:41:12.100
Clearly, as the distribution broadens, the number of ways to achieve that distribution increases massively 1 to 10 to 1200 to 25,000.
00:41:12.100 --> 00:41:22.900
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.
00:41:22.900 --> 00:41:26.800
This is very important.
00:41:26.800 --> 00:41:33.700
The system will appear in the state that offers the greatest number of ways of achieving that state because
00:41:33.700 --> 00:41:44.200
the probability of finding a system in a given state depends directly on the total number of ways that that state is achievable.
00:41:44.200 --> 00:41:47.800
In example 1, there is only one way of achieving the particular distribution.
00:41:47.800 --> 00:41:55.600
The chance of finding a given system of that arrangement are 10 particles with 4 energy levels available is very slim.
00:41:55.600 --> 00:41:59.200
In any given moment you probably not going to run across that distribution.
00:41:59.200 --> 00:42:04.300
In other words, these 10 particles will not crowded to one compartment and stay that way.
00:42:04.300 --> 00:42:13.600
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.
00:42:13.600 --> 00:42:18.100
Example 4, offers a distribution as a huge number of ways of being obtained.
00:42:18.100 --> 00:42:27.600
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.
00:42:27.600 --> 00:42:35.700
Within the constraints of the sum of the total, the sum of the individual n sub i equals the total number of particles n.
00:42:35.700 --> 00:42:41.700
The sum of the number of particles in a given energy level × the energy of that level equals U the total energy.
00:42:41.700 --> 00:42:49.800
There exists a distribution, there is always one distribution that completely maximizes this O.
00:42:49.800 --> 00:42:57.900
The sheer number of microstates for this distribution is so huge that it completely dominates the landscape of probabilities.
00:42:57.900 --> 00:43:01.300
You will certainly find the system with this distribution in the state.
00:43:01.300 --> 00:43:10.300
In our example we had 1, we had 10, we had 1260, I think and we had 25,200.
00:43:10.300 --> 00:43:21.400
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.
00:43:21.400 --> 00:43:31.000
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.
00:43:31.000 --> 00:43:34.400
We are going to find 3 in 1 bin, 3 in another, 2 in another, and 2 in another.
00:43:34.400 --> 00:43:46.900
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.
00:43:46.900 --> 00:43:53.500
Sorry if I keep repeating myself, but this is profoundly important.
00:43:53.500 --> 00:44:10.500
Let us look again at this O.
00:44:10.500 --> 00:44:12.600
O we set this as n!/ n₁! n₂! n₃!, and so on.
00:44:12.600 --> 00:44:17.400
We set that S, let us go back to entropy because we are talking about entropy here.
00:44:17.400 --> 00:44:22.800
S= Boltzmann constant × nat log of this thing called O.
00:44:22.800 --> 00:44:30.300
If O is large, the log of O is large, the entropy is large.
00:44:30.300 --> 00:44:45.900
If O is large then S is large.
00:44:45.900 --> 00:45:09.500
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.
00:45:09.500 --> 00:45:18.300
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.
00:45:18.300 --> 00:45:31.200
The smaller the n sub I, the larger O becomes which means the larger the entropy.
00:45:31.200 --> 00:45:42.900
Small n sub i means distributing as many particles in as many compartments as possible to lower that number.
00:45:42.900 --> 00:45:53.400
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.
00:45:53.400 --> 00:46:04.200
That maximizes it, that lower the n₁ n₂ n₃ n₄, as low as they will go.
00:46:04.200 --> 00:46:11.100
As low as they will go that raises the O to as high as it will go which in that case was 25,200.
00:46:11.100 --> 00:46:14.200
That is an increase in entropy.
00:46:14.200 --> 00:46:22.600
This is what we mean by this order or randomness having 25,200 ways of achieving given distribution
00:46:22.600 --> 00:46:27.700
is kind of chaotic vs. only one way of achieving a distribution.
00:46:27.700 --> 00:46:37.800
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.
00:46:37.800 --> 00:46:43.200
It represents the distribution, these are the numbers.
00:46:43.200 --> 00:46:54.300
Back to this, the larger the n sub I, the larger these numbers, the smaller O is.
00:46:54.300 --> 00:47:11.100
The smaller O becomes which implies that the smaller entropy.
00:47:11.100 --> 00:48:00.700
The system will try to achieve the highest entropy possible thus the broadest distribution subject to the two constraints.
00:48:00.700 --> 00:48:07.700
Broad means we want to spread out the particles in as many bins as possible to lower the number of particles in each bin.
00:48:07.700 --> 00:48:11.600
The lower these numbers, the higher O, the higher the entropy.
00:48:11.600 --> 00:48:19.700
A system is going to try to achieve the highest entropy possible thus the broadest distribution subject to the two constraints.
00:48:19.700 --> 00:48:22.100
And I will go ahead and put those constraints again.
00:48:22.100 --> 00:48:38.200
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.
00:48:38.200 --> 00:48:45.100
Within these two constraints, the system will try to achieve the broadest distribution possible that it can.
00:48:45.100 --> 00:48:50.200
In the case of the 10 particles and 4 bins it was 3 3 2 2.
00:48:50.200 --> 00:49:05.200
That was its broadest distribution possible.
00:49:05.200 --> 00:49:59.700
When the energy of a system is increased, the energy distribution broadens and the particles occupied this broader distribution.
00:49:59.700 --> 00:50:08.600
In other words, O rises which means the entropy arises.
00:50:08.600 --> 00:50:13.400
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,
00:50:13.400 --> 00:50:26.000
by increasing the energy of the system now I have allowed more energy, a lot more bins.
00:50:26.000 --> 00:50:34.400
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.
00:50:34.400 --> 00:50:38.300
They are going to be fewer particles in each individual bin.
00:50:38.300 --> 00:50:43.600
Again, if there are fewer particles in each individual bin that means the number of ways that
00:50:43.600 --> 00:50:48.400
the denominator of the O is to get smaller which means O goes up.
00:50:48.400 --> 00:50:53.500
Numerically, if O goes up the entropy goes up.
00:50:53.500 --> 00:51:03.100
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.
00:51:03.100 --> 00:51:08.200
We are going back to the example where I have 10 particles and I have the 4 bins.
00:51:08.200 --> 00:51:19.300
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.
00:51:19.300 --> 00:51:28.300
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.
00:51:28.300 --> 00:51:31.600
It is going to move as much as possible.
00:51:31.600 --> 00:51:34.900
Maybe one of these particles end up going over here so that goes to 1.
00:51:34.900 --> 00:51:44.000
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.
00:51:44.000 --> 00:51:45.800
I have broaden the energy distribution .
00:51:45.800 --> 00:51:50.600
I have allowed more bin, now they are going to distribute themselves in such a way that its broader.
00:51:50.600 --> 00:51:53.000
It is going to achieve a broader distribution.
00:51:53.000 --> 00:52:04.700
If I do 10! / 3! 2! 1 1, now it is going to be a lot higher than 25,200.
00:52:04.700 --> 00:52:07.400
The entropy is going to go up but you know this already.
00:52:07.400 --> 00:52:12.500
Here is how you know this.
00:52:12.500 --> 00:52:19.300
You know this already from the work that we did in the previous lessons.
00:52:19.300 --> 00:52:28.700
You know the DS =1/ T DU + P/ T DV.
00:52:28.700 --> 00:52:33.400
This 1/ T is positive, it is always positive because the Kelvin temperature is always going to be about 0.
00:52:33.400 --> 00:52:40.600
1/ T is always positive which means that if you increase the energy of the system, you increase the entropy.
00:52:40.600 --> 00:52:43.300
This is a simple math.
00:52:43.300 --> 00:52:47.200
You know this already from your experience, you have dealt with this.
00:52:47.200 --> 00:52:55.400
What we have given is a statistical, we have given a probabilistic explanation for these increase in entropy.
00:52:55.400 --> 00:53:04.700
We know why when we increase the energy we are increasing the energy distribution because we increase the energy distribution O goes up.
00:53:04.700 --> 00:53:08.000
When O goes up, the entropy goes up.
00:53:08.000 --> 00:53:10.400
That is what is happening here.
00:53:10.400 --> 00:53:19.400
What we have done here, we just given.
00:53:19.400 --> 00:53:22.400
I have to slow down.
00:53:22.400 --> 00:53:35.600
The statistical and I will go ahead and say probabilistic.
00:53:35.600 --> 00:53:57.100
In other words, the microscopic reason for our classical observation which is our macroscopic observation.
00:53:57.100 --> 00:54:03.400
I hope this has made sense, increasing the energy of the system broadens the energy distribution.
00:54:03.400 --> 00:54:12.100
As you broaden the energy distribution, the number of particles that can achieve the distribution become the spread out, they themselves broaden out,
00:54:12.100 --> 00:54:19.000
when that happens the numerator of our O ends up getting smaller so O gets higher.
00:54:19.000 --> 00:54:28.600
When O gets higher, because S is equal to KB LN O S gets higher.
00:54:28.600 --> 00:54:30.400
I hope that makes sense.
00:54:30.400 --> 00:54:32.200
Thank you so much for joining us here at www.educator.com.
00:54:32.200 --> 00:54:35.000
We will see you next time for a continuation of this discussion, bye.