WEBVTT chemistry/physical-chemistry/hovasapian
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Hello and welcome back to www.educator.com, welcome back to Physical Chemistry.
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Today, we are going to start our discussion of statistical thermodynamics.
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Let us jump right on in.
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I want to go through this big picture of statistical thermodynamics is that we know why we are doing this.
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What are all for our goal?
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What is it that we are trying to achieve?
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Our big picture goal is going to be the following.
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Let me actually work in blue today.
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Our big picture goal it is to find a way to express the thermodynamic properties of a bulk system and
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by bulk we mean just a bunch of particles, like a block of wood as opposed to the individual molecules that make up that wood.
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Of a bulk system, in other words the energy, the entropy, the enthalpy, the Helmholtz energy,
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the Gibbs free energy, all of these things.
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The constant of volume heat capacity and the pressure.
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The basic thermodynamic functions of bulk system.
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Our goal is to express these properties in terms of the properties and of the particles that make up the system.
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That is what we are doing with statistical thermodynamics.
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We start off the course with classical thermodynamics.
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We moved on to quantum mechanics.
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In quantum mechanics we are dealing with the individual energies and properties.
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The individual particles out of a molecule, whatever it is.
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Now that we have quantum mechanics, we want to go back and we want to explain
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what we learned in classical thermodynamics via the individual particles.
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That is it, we are just closing the circle like talked about in the overview of the course.
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I will go back to black here, sorry about that.
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Our primary tool in this investigation is going to be something called the partition function.
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Our primary tool will be something called the partition function.
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Let me actually come over here, called the partition function.
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The symbol for the partition function is going to be a capital Q.
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It is going to be the partition function of the system.
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What I’m going to do is we are going to express the thermodynamic properties in terms of Q.
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We will express the thermodynamic properties, the ones I have listed above.
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We will be listing those and I would be expressing those properties in terms of Q.
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I’m having a little difficulty talking today, sorry about that.
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In terms of this Q, the partition function.
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We then introduce q.
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We then introduce q, this is called a molecular partition function.
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What that means it is or should say that is the partition function for each particle in the system.
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We can actually do that, we can write this thing called a partition function
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for each individual particle of whatever system we happen to be dealing with.
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It is pretty extraordinary, that it is very extraordinary.
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For each particle of the system.
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We will express Q, the partition function of the system in terms of q.
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We will then have exactly what we wanted.
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We will then have, finally, our direct relationship between the thermodynamic properties of the system
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and the particles that make up the system.
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That is our big picture goal, that is what we want to do.
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We want to come up with this thing called a partition function.
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And we want to come up with this thing called, we want to find the various partition functions
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for whatever quantum mechanical system we would happen to be dealing with.
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And again, we already talked about the particle in the box.
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We talked about the rigid rotator, the harmonic oscillator, these are the partition functions that we are going to look at.
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These are going to be the partition functions of the molecules.
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We are going to express the thermodynamic properties that we learned about back when we started the course.
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Let us see what we have got.
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Let us start up here.
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Consider a system of N particles and N is usually just going to be Avogadro’s number.
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Consider a system of N articles.
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The energies of the particles are discreet, we know that already.
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That is what quantum was all about.
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The energies of the particles are discreet and they are distributed over various quantum states.
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For example, if you had some rotating molecule.
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You know whatever it might be in the J = 1 couple of 1,000,000 of them might be in J = 2.
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A couple of 1,000,000,000 of them might be in J = 3.
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Different energies, the particles are the same.
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They are distributed over the various quantum states, that is all we are saying here.
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In any given moment, if we have add up the energies, we get the energy of the system.
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We will add up the energies of the individual particles, you get the energy of the system.
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We will call that E sub I.
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Also discreet because of the individual energies are discreet, the E sub I is going to be discreet.
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If we come back to any moment of the same system, let us say 30 seconds later, whatever,
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1 second later, does not really matter.
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The particles are going to be in this differ distribution of quantum states.
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Therefore, the energy of the system is going to be different.
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In another moment, the particles being in other quantum states, this gives rise to another,
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I will call this not E sub I, E₁.
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I take my first measurement and I get E₁.
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It gives rise to another energy of the system and I will call it E₂.
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What we call the thermodynamic energy of the system is an average of all of these E1, E2, E3, E4, E5.
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If I take 100 measurements, a 1000 measurements, 500 measurements of the system at different times,
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all the particles are going to be in different quantum states.
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I’m going to get different energy of the system.
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I take an average of that, that is what I call the thermodynamic energy.
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That was what we call U.
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What we call the thermodynamic energy of the system.
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In other words, U is the average of many observations.
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Make sense, I think it is particularly strange here.
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Let us take a bunch of observations, take the average and call that number the energy of the system.
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Instead of making a 100 or 1000, or 500 observations on the same system.
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Instead of making a multiple, let us say multiple instead of choosing a number, multiple observations.
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Instead of making multiple observations, let us actually choose a number.
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I’m going to choose one of the number, it can be any number but I'm just going to choose one, a hundred.
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Let us say we take 100 observations.
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We average that out a 100 observations of the same system, we get the energy of the system.
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Instead of making a 100 observations on the same system, we can also just create 100 identical systems.
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That is it, same circumstances, same surroundings, same particles, same temperature,
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same pressure, create 100 identical systems.
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Each system will be in a particular quantum state.
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It might have all hundred that are in different quantum states.
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You might have 20 of them that are in one, 10 of them in another, 2 of them that are in another.
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Again, this could be in various quantum states.
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A system will be in various quantum states with the energy E sub I.
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Let us increase that number.
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Now, instead of 100 or 1000 or 2000 or 5000, let us create a large number of identical systems
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and we will call that large number of identical systems N.
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Now, let us create, in other words let us just make this 100 a really big number.
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Because we know that the bigger number we have, the better our average.
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Let us create a very large number of identical systems.
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Let us call that number N.
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We call this collection of identical systems, this large number of identical systems, we call that the ensemble.
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That is what the ensemble means.
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Let me go to red.
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We call this collection an ensemble.
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It seems to always be a bit of confusion about what ensemble is and that is it.
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We are just taking system, we are duplicating it, a 6.02 × 10²³ × and we are calling that ensemble.
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It is the identical system, just copied.
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The collection is an ensemble.
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Each system will have a particular energy.
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Each system in the ensemble will have a particular energy E sub I.
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The energy distribution looks like this.
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The energy distribution, we have the number of systems in the ensemble and
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we have a particular energy of the ensemble.
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If there are N sub one systems that have energy 1, we might have N₂ of the systems, we might have energy 2.
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We have N₃ of the systems are in energy 3 and so on.
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N = the sum of all of these N sub I.
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If I add all of the N, I'm going to get the total number of systems in the ensemble.
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Let us say I have 6.02 × 10²³ ensemble, it does not really matter.
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It is a really large number 500,000, whatever.
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If 10,000 of them are energy 1, if 50,000 of them of the systems are in energy 2, that is all it says.
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There is a distribution of energies.
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Now, the probability.
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The probability of finding a system in a system with the energy E sub I, the basic probability,
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you take a number of times of something can happen over the total number.
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What is the probability of rolling a 5 when you roll a single dice?
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There is only one way to get a 5, you roll a 5.
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How many different possibilities are there when you can roll 1 through 6?
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Yes, you have 6 possibilities that you can roll but only one way to roll a 5.
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The probability of rolling a 5 is 1/ 6.
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You remember this from algebra class.
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The probability of finding a system with energy E sub I is symbolized by P sub I,
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it = the number of states in that energy with that energy divided by the total number of systems.
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That is it, very simple, very basic.
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It just = the number of systems having E sub I energy divided by the total number systems.
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It is the basic definition of probability.
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Total number of systems.
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We are going to define Q, the partition function.
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Q is equal to the sum E to the - β E sub I, where β = 1 / K × T.
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I’m going to go ahead and put this 1 / TT into the β.
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Q is actually equal to the sum / I of E to the - E sub I divided by KT.
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This division is up in the exponent.
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This whole thing is up in the exponent.
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In this particular case, T is the temperature in Kelvin, the absolute temperature.
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K is something called the Boltzmann constant.
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K is equal to 1.381 × 10⁻²³ and the unit is J/ K.
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This is the partition function.
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I will tell you what it is in just a minute.
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My best advice is just deal with the mathematics.
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This looks complicated because of the summation symbol, it is not.
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All you are doing is adding a bunch of terms together.
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You taking the first energy, the second energy, the third energy, you are dividing it by KT.
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You are exponentiating it and that is just one term of the sum.
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If we said out of the first 5 terms for the first 5 energy states, you have 5 terms of the some.
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That is all it is and I will tell you what the partition function is in just a minute.
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Once again, we said P sub I was this thing.
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With respect to the partition function, P sub I is this.
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It is equal to E ⁻E sub I / KT/ Q.
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The probability of finding a system in a given energy state is equal to E raised to the energy state
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divided by KT divided by the sum of all the possible energy states.
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The part / the whole, the probability.
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It is a fraction, that is all this is.
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Let us go ahead and tell you what the partition function actually is.
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Q, the partition function is a measure.
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It is a numerical measure of the number of energy states that are accessible to a system or by a system
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depending on which partition you want to use at a given temperature, at a given T.
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Let us talk about this.
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There is always this sense of what is a partition function?
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I’m still not sure what it really means.
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This is what a partition function is.
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Let us also talk about a system with a given set of energies.
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At a given temperature, let us say there are 100 available energies for a given system.
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There are 100 energies that could have at a given temperature, let us say only 5 of that energy levels are actually accessible.
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The partition function is 5.
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It is very important that we differentiate between accessible and available.
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You might have, like for an example the rotational states of the diatomic molecule.
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There is an infinite number of rotational states in a diatomic molecule.
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Not infinite but a really large number if the molecule flies apart.
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It will spin faster and faster and faster and faster into different quantum states.
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J could be 50, 60, 70, 100, 200, but not all of those are accessible.
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At a given temperature, let us say maybe only 30 of those rotational states are accessible.
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That is what a partition function tells us.
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A partition function is going to give you some number.
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That number gives you roughly the number of states that are accessible to a system at that temperature.
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As I raise the temperature more of the states become accessible, that is what is happening, that is all that is happened.
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It is all a partition function is.
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Again, we have P sub I is equal to E ⁻E sub I / KT all / Q which is equal to E ⁻E sub I/ KT.
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I think all these exponentials and summations, fractions on top of fractions, it tends to look really intimidating.
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It is not intimidating, it is just math.
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Over the sum of the E/ KT.
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The partition function is just adding up all these energy values and
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then the probability of finding it in one of those energy values as you take a part / a whole, where Q is equal to this thing.
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Let us go ahead and write it out again.
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Q is equal to sum I / E ⁻E sub I / KT.
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If you have 100 energy states that are accessible, you have 100 terms in that sum.
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That is a partition function, very important.
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We sometimes leave θ and write Q = the sum / the index I of E ^- β E sub I.
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Sometimes, we will just go ahead and leave the β and expressed in terms of that,
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in order that they do not have to deal with the fraction in the exponent.
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That is fine, you will see it both ways.
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Again, where β = 1 / KT.
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I'm not really going to say more about this β = 1/ KT.
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If your teacher wants to give you reason for why that is the case, they can but I would say just take it on faith at this point.
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U, the energy of the system, we said it is the average of the energy.
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The average of the energy is you add up all the energies and you divide by the number of systems,
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the energy of the ensemble.
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The sum / I N sub I E sub I / N.
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The number of states/ a given energy × the energy itself.
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Add up all of those and divide by the number of systems in the ensemble.
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That would give you an average energy.
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I pulled the N sub I/ n out, N /N sub I.
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N/ N sub I that is equal to P sub I.
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N sub I/ N is equal to P sub I.
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U, which is the average energy is equal to the probability of finding it in a given energy state × the sum of the actual thing.
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I will go ahead and put this P sub I back in here.
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U equal to the average energy is equal to sum of the probability of finding
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the system of the ensemble in a given energy × that energy.
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That is one of our basic equations.
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We found an expression for the energy in terms of the energy, in terms of the probability.
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The probability is a function of the partition function.
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We have expressed energy in terms of the partition function.
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We will get better , do not worry about that.
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In the ensemble, the systems are distributed over the various quantum states.
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Each specific distribution is called microstate or a complexion of the ensemble.
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What we mean by this is the following.
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Let us say I have I have 10 systems, let us say 3 of them are in one energy, 3 of them are in another,
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3 of them in another, and one of them is in the fourth.
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That is one distribution, that is one microstate.
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Let us go to another distribution.
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What if I have 5 in one, 5 in another, and nothing in the other 3.
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That is another distribution, that is another microstate.
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In other words, in microstate is if I have certain number of bins, energy baskets, our certain number of systems,
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how can I distribute the different energies among those various systems?
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Each different one is called a microstate.
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The number of possible microstates is denoted as capital ω.
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We define the entropy of the ensemble.
00:30:44.700 --> 00:30:50.500
When we want the entropy of the system, we just divide the entropy by the total number of systems in that ensemble.
00:30:50.500 --> 00:30:52.600
In other words N, that is it.
00:30:52.600 --> 00:30:54.200
We are always talking about the ensemble.
00:30:54.200 --> 00:30:58.700
Anytime we talk about a system, we just take what we have and divide by the number of particles in it,
00:30:58.700 --> 00:31:00.900
the number of systems in the ensemble.
00:31:00.900 --> 00:31:05.400
The entropy of the ensemble is, and you have seen this before.
00:31:05.400 --> 00:31:08.300
Except now, we are talking about ensemble instead of the system.
00:31:08.300 --> 00:31:14.100
S = K × the natural log rhythm of O.
00:31:14.100 --> 00:31:16.800
This is the definition of the entropy of an ensemble.
00:31:16.800 --> 00:31:19.400
We have seen this equation before.
00:31:19.400 --> 00:31:24.700
We have seen this definition before back when we talked about classical thermodynamics.
00:31:24.700 --> 00:31:32.100
We talked about entropy first empirically but then we go ahead and gave this statistical definition of entropy.
00:31:32.100 --> 00:31:37.700
And we talked about what it means, we talked about this idea of complexions, and a number of possible microstates.
00:31:37.700 --> 00:31:40.100
I'm not going to state too much more about it now.
00:31:40.100 --> 00:31:44.900
If you want, you can go back to that particular discussion and it will talk a little bit more
00:31:44.900 --> 00:31:50.000
about what these individual things mean, in any case.
00:31:50.000 --> 00:32:28.600
The entropy of the system is the entropy of the ensemble divided by N, the number of systems in the ensemble.
00:32:28.600 --> 00:32:38.700
Therefore, S of the system is equal to S of the ensemble divided by N.
00:32:38.700 --> 00:32:46.300
S of the ensemble is K × the natlog of this thing called ω divided by N.
00:32:46.300 --> 00:32:51.200
In order words, to find the entropy of the system that we are dealing with, the system that we are interested in
00:32:51.200 --> 00:32:56.600
which has happen to have made billions of copies of that system to create an ensemble.
00:32:56.600 --> 00:33:01.700
In other words, to find the entropy of the system, we need to find this Boltzmann constant, we know.
00:33:01.700 --> 00:33:07.600
How many systems we have in an ensemble, we need to find LN of ω.
00:33:07.600 --> 00:33:28.700
ω is defined as N!/ N₁! N₂! N₃!, and so on.
00:33:28.700 --> 00:33:42.000
To find S of the system, we need to find the natlog of O.
00:33:42.000 --> 00:33:46.800
That is it, we are just doing some math here, that is it, nothing too crazy.
00:33:46.800 --> 00:33:59.500
The natlog of ω is the natlog of N!/ what we said, N₁! N₂!, And so on.
00:33:59.500 --> 00:34:08.000
That is equal to the natlog of N! – the sum because this is a product.
00:34:08.000 --> 00:34:10.600
The natlog of N sub I!.
00:34:10.600 --> 00:34:18.600
After some math, we also designated as math, which I'm not going to go through here.
00:34:18.600 --> 00:34:39.900
What we get is the natlog of ω is equal to -N × the sum of the I P sub I LN P sub I.
00:34:39.900 --> 00:34:48.600
S of the system is equal to K LN ω / N.
00:34:48.600 --> 00:34:52.600
LNO is this thing.
00:34:52.600 --> 00:35:20.700
We put this thing into there, we end up getting - K × N × the sum / I P sub I LN of P sub I O divided by N.
00:35:20.700 --> 00:35:24.900
The N cancel and we get an expression for the entropy of the system.
00:35:24.900 --> 00:35:42.700
The entropy of the system is equal to - K which is Boltzmann constant × the sum / I, the probability of I × the log of the probability sub I.
00:35:42.700 --> 00:35:46.600
We found an expression for the energy, in terms of the probability.
00:35:46.600 --> 00:35:49.800
We found an expression for the entropy, in terms of the probability.
00:35:49.800 --> 00:35:53.800
The probability is expressed in terms of the partition function.
00:35:53.800 --> 00:35:56.700
We are getting to where we want to go.
00:35:56.700 --> 00:36:02.800
This is our second major equation.
00:36:02.800 --> 00:36:07.300
Let us go ahead and rewrite what we have.
00:36:07.300 --> 00:36:20.600
Our first major equation was U = which is the average energy, which is equal to the sum of the probability sub I × E sub I.
00:36:20.600 --> 00:36:30.100
And our second major equation which is entropy that is equal to -K × the sum / I, the probability of I.
00:36:30.100 --> 00:36:37.000
These are our two basic equations that we are going to start with and derive everything else.
00:36:37.000 --> 00:36:49.200
Again, where P sub I is the probability of finding a system or ensemble.
00:36:49.200 --> 00:36:53.900
Probability of finding a system in that particular energy state.
00:36:53.900 --> 00:37:01.900
Or it is also a fraction of the systems in that energy state, that is the best way to think of P sub I.
00:37:01.900 --> 00:37:09.900
It is a fraction of the systems in the ensemble that are in a given energy state E sub I.
00:37:09.900 --> 00:37:17.900
If I have a total of 1000 systems in the ensemble and if I have 100 of those systems
00:37:17.900 --> 00:37:25.300
in given energy state E₁, 100/ 1000 that means 10%, 0.10.
00:37:25.300 --> 00:37:27.700
My P sub I is 0.10.
00:37:27.700 --> 00:37:56.600
Where P sub I is the fraction of the systems in the ensemble having energy E sub I.
00:37:56.600 --> 00:37:59.300
If all of these do not make sense, do not worry.
00:37:59.300 --> 00:38:01.500
Really, do not worry, what matters here are the results.
00:38:01.500 --> 00:38:06.300
But again, I go through this as a part of your scientific literacy.
00:38:06.300 --> 00:38:12.900
If you go through this, if you see this, and you go in your book and read it, it will make your book make more sense.
00:38:12.900 --> 00:38:16.200
I think it works better that way, or perhaps you read your book and you did not quite get it,
00:38:16.200 --> 00:38:20.200
and now that you are seeing this lecture, it might make more sense.
00:38:20.200 --> 00:38:25.500
It is just another way of looking at it.
00:38:25.500 --> 00:38:44.600
We have P sub I is equal to E ⁻E sub I/ KT/ Q, that is one equation that we have.
00:38:44.600 --> 00:38:53.600
We have an expression for the partition function which is the sum / I/ E ⁻E sub I/ KT,
00:38:53.600 --> 00:38:56.000
very important partition function.
00:38:56.000 --> 00:39:02.100
We have an expression for the energy U which is the actual average energy.
00:39:02.100 --> 00:39:13.000
That is equal to the sum of the index I of the P sub I E sub I, the fraction in energy state I × the energy itself.
00:39:13.000 --> 00:39:14.300
And we have expression for the entropy.
00:39:14.300 --> 00:39:22.300
Entropy = - K × the sum/ I P sub I LN P sub I.
00:39:22.300 --> 00:39:47.000
These equations, if these 4 equations all of the thermodynamic properties, all of the thermodynamic quantities,
00:39:47.000 --> 00:39:58.100
all the thermodynamic functions can be expressed in terms of Q.
00:39:58.100 --> 00:40:03.400
In terms of Q, all we need is this, this, the energy and the entropy and
00:40:03.400 --> 00:40:09.400
we can express all the other thermodynamic functions in terms of this thing we call the partition function.
00:40:09.400 --> 00:40:15.200
Partition function, very important.
00:40:15.200 --> 00:40:18.700
Let us start first of all with Q, let us start with that equation.
00:40:18.700 --> 00:40:27.900
Q = the sum of E ⁻E sub I/ KT.
00:40:27.900 --> 00:40:42.500
We differentiate with respect to, I will go ahead and differentiate with respect to T.
00:40:42.500 --> 00:40:52.800
Therefore, DQ DT and we will hold volume constant.
00:40:52.800 --> 00:41:16.400
When you take the derivative of this, you get 1 / KT² × the sum I E sub I E ⁻E sub I/ KT.
00:41:16.400 --> 00:41:19.300
We took the derivative of Q with respect to T.
00:41:19.300 --> 00:41:23.300
Let us go ahead and go over here.
00:41:23.300 --> 00:41:42.300
P sub I is equal to E ⁻E sub I/ KT/ Q which means that if I multiply Q which means that π × Q is equal to E ⁻E sub I/ KT.
00:41:42.300 --> 00:41:44.400
It is just mathematical manipulation.
00:41:44.400 --> 00:41:57.400
If I put this back into the other equation, if I do KT² × DQ DT under constant V,
00:41:57.400 --> 00:42:27.600
that is going to equal this sum E sub I P sub I × Q = Q × the sum of the E sub I P sub I.
00:42:27.600 --> 00:42:50.900
This is U and U = that, the sum/ I of P sub I E sub I.
00:42:50.900 --> 00:43:02.600
KT² DQ DT is equal to Q × U.
00:43:02.600 --> 00:43:05.200
I solve for U.
00:43:05.200 --> 00:43:21.900
U is equal to KT² / Q DQ DT V, which is the same as if I take, instead of taking the derivative of Q
00:43:21.900 --> 00:43:25.800
with respect to T, if I take the derivative LN Q.
00:43:25.800 --> 00:43:31.400
The derivative of LN Q is 1 / Q DQ DT, I get the following.
00:43:31.400 --> 00:43:44.400
I get KT² D LN Q DT constant V.
00:43:44.400 --> 00:43:48.600
I have an expression for U directly in terms of Q or LN Q.
00:43:48.600 --> 00:44:14.200
In this last part because D DT or LN Q = 1 / Q DQ DT, this is energy in terms of the partition function.
00:44:14.200 --> 00:44:15.300
We have our first part.
00:44:15.300 --> 00:44:31.500
Now, the entropy of the system = -K × the sum of P sub I LN P sub I.
00:44:31.500 --> 00:44:41.300
We said that P sub I is equal to the E ⁻E sub I/ KT/ Q.
00:44:41.300 --> 00:45:01.900
LN of P sub I = -E sub I/ KT – LN Q.
00:45:01.900 --> 00:45:15.200
LN P sub I, if I take this thing and put it into here.
00:45:15.200 --> 00:45:35.600
Therefore, S is equal to -K × the sum P sub I - E sub I/ KT – LN Q.
00:45:35.600 --> 00:45:58.000
I get S = -K × -1 / KT, the sum P sub I E sub I – LN Q × the sum of the P sub I.
00:45:58.000 --> 00:46:18.700
Therefore, S is equal to 1 / T × the sum / I of the P sub I E sub I + K × LN Q.
00:46:18.700 --> 00:46:22.700
And this is because this thing is actually equal to 1.
00:46:22.700 --> 00:46:28.400
The sum of the probabilities, the sum of all of fractions always = 1.
00:46:28.400 --> 00:46:33.500
S = this is U.
00:46:33.500 --> 00:46:44.300
U / T + K LN Q.
00:46:44.300 --> 00:46:58.700
We already found U, U equal to KT² D LN Q DT constant V.
00:46:58.700 --> 00:47:04.700
Let me go ahead and put this in for that and when we do, we end up with S
00:47:04.700 --> 00:47:19.200
is equal to KT D LN Q DT under constant volume + K LN Q.
00:47:19.200 --> 00:47:26.700
We found an expression for entropy directly in terms of the partition function.
00:47:26.700 --> 00:47:31.000
Very nice.
00:47:31.000 --> 00:47:59.800
Let us see, with energy, entropy, temperature, and volume, all of the other thermodynamic properties can be derived.
00:47:59.800 --> 00:48:14.700
All of the other thermal properties can be derived.
00:48:14.700 --> 00:48:21.800
Let us begin with, let us go back to black.
00:48:21.800 --> 00:48:27.400
Let us begin with Helmholtz A = U - TS.
00:48:27.400 --> 00:48:30.100
This is the definition of the Helmholtz energy.
00:48:30.100 --> 00:48:33.600
We have an expression for U and we have an expression for S.
00:48:33.600 --> 00:49:07.000
This is equal to KT² D LN Q DT under constant V - T LN Q - KT² D LN Q DT constant V.
00:49:07.000 --> 00:49:20.500
The Helmholtz energy is equal to -KT LN Q.
00:49:20.500 --> 00:49:26.300
There you go, that is an expression for the Helmholtz energy.
00:49:26.300 --> 00:49:28.400
That one was reasonably straightforward.
00:49:28.400 --> 00:49:33.200
I just put the value of U and S in here and solve, and I end up with this.
00:49:33.200 --> 00:49:38.900
One of the fundamental questions of thermodynamics,
00:49:38.900 --> 00:49:45.600
if you remember from towards the end of the classical thermodynamics portion of the course.
00:49:45.600 --> 00:50:10.500
One of the fundamental equations of thermodynamics says DA is equal to - S DT - P DV.
00:50:10.500 --> 00:50:29.500
That means P, let me go to red, P is equal to - DA DV under constant temperature.
00:50:29.500 --> 00:50:34.600
That is what this says, this is the total differential equation.
00:50:34.600 --> 00:50:40.400
This P is just the partial derivative of this with respect to this variable.
00:50:40.400 --> 00:50:43.500
That is it, because the DA DV × the DV.
00:50:43.500 --> 00:50:47.700
The DV DV cancel, you are left with the A.
00:50:47.700 --> 00:50:48.400
That is what this means.
00:50:48.400 --> 00:51:00.300
S would be partial of A of DA DT, holding V constant.
00:51:00.300 --> 00:51:15.100
Therefore, P is equal to - DDV constant T of A.
00:51:15.100 --> 00:51:21.800
A was this, - KT LN Q.
00:51:21.800 --> 00:51:34.700
Therefore, the pressure of the system is equal to KT D LN Q DV holding temperature constant.
00:51:34.700 --> 00:51:38.700
That is quite extraordinary.
00:51:38.700 --> 00:51:46.100
You are just knocking out all these thermodynamic expressions in terms of partition function.
00:51:46.100 --> 00:51:49.300
Let us go ahead and do the enthalpy of the system.
00:51:49.300 --> 00:51:58.900
The enthalpy of the system is defined as the energy + the pressure × the volume.
00:51:58.900 --> 00:52:01.800
We are using these or sometimes look exactly alike.
00:52:01.800 --> 00:52:04.400
I will just put them in, we have expressions for these.
00:52:04.400 --> 00:52:25.900
We have KT² DLN Q DT constant V + the P which we said was KT D LN Q DV constant T × V.
00:52:25.900 --> 00:52:47.100
Therefore, this is equal to KT × T D LN Q DT under constant V + V × D LN Q DT constant T.
00:52:47.100 --> 00:52:52.400
Very beautiful, absolutely stunningly beautiful.
00:52:52.400 --> 00:52:56.600
That is enthalpy.
00:52:56.600 --> 00:53:03.800
Let us go ahead and go to Gibb’s free energy which is the most important for chemists.
00:53:03.800 --> 00:53:11.700
K = U + PV - TS.
00:53:11.700 --> 00:53:18.600
If we put all of these all in, I have it here, I will just write it all out.
00:53:18.600 --> 00:53:34.200
U was KT² D LN Q DT under constant volume + KT × D LN Q DV at constant temperature
00:53:34.200 --> 00:53:54.800
× V - T × K LN Q + KT × D LN Q DT under constant V.
00:53:54.800 --> 00:54:00.400
This is equal to, when I multiply this, when I multiply that and add some terms,
00:54:00.400 --> 00:54:22.700
I end up with G = KT D LN Q DV under constant temperature × V - LN Q.
00:54:22.700 --> 00:54:30.100
This gives me an expression for the Gibb’s free energy.
00:54:30.100 --> 00:54:34.800
Heat capacity is very important.
00:54:34.800 --> 00:54:44.400
The heat capacity is the partial derivative of the energy with respect to temperature under constant volume.
00:54:44.400 --> 00:54:48.600
We have an expression for the energy of the system.
00:54:48.600 --> 00:55:06.100
We have got DDT constant volume of this expression which is KT² D LN Q DT constant volume.
00:55:06.100 --> 00:55:28.300
This is going to end up equaling K × T² D² LN Q DT² under constant volume + D LN Q DT under constant volume × QT.
00:55:28.300 --> 00:55:51.700
Therefore, the direct expression for the constant volume heat capacity is KT × T D² LN Q DT² under constant volume + 2 × D LN Q DT constant volume.
00:55:51.700 --> 00:55:56.700
This is a direct expression for the constant volume heat capacity.
00:55:56.700 --> 00:56:00.400
Normally, what we would be doing is we are going to be finding an expression for the energy.
00:56:00.400 --> 00:56:04.400
I'm just taking the partial derivative of that with respect to temperature directly.
00:56:04.400 --> 00:56:07.100
We are not going to be using this expression.
00:56:07.100 --> 00:56:10.300
And again, the almost important of the thermodynamic functions,
00:56:10.300 --> 00:56:13.700
we are mostly to be concerned with the energy and the constant volume heat capacity.
00:56:13.700 --> 00:56:16.300
Occasionally, we will deal with pressure.
00:56:16.300 --> 00:56:19.700
If for any reason you need to go to the other thermodynamic functions, that is fine.
00:56:19.700 --> 00:56:21.200
But again, this is an overview.
00:56:21.200 --> 00:56:24.400
I wanted to show you what our big picture goal is.
00:56:24.400 --> 00:56:30.500
When we get the expression for the energy in the individual cases, we will just differentiate with respect to T.
00:56:30.500 --> 00:56:32.900
We will write that down.
00:56:32.900 --> 00:56:58.900
In general, we will be concerned with U and CV.
00:56:58.900 --> 00:57:26.200
We will normally find an expression for U then differentiate directly, and differentiate with respect to T directly.
00:57:26.200 --> 00:57:31.200
We have actually done it, we have done what we are set out to do.
00:57:31.200 --> 00:57:34.600
Let me go ahead and go to blue here.
00:57:34.600 --> 00:57:50.800
We have done it, thermodynamic properties expressed in terms of Q or LN Q.
00:57:50.800 --> 00:58:07.900
Thermodynamic properties expressed in terms of Q.
00:58:07.900 --> 00:58:36.700
Q, by definition is related to the energy states of the system E sub I.
00:58:36.700 --> 00:58:40.800
These are the E sub I.
00:58:40.800 --> 00:58:53.000
The E sub I are related to the energies of the individual particles making up the system, the small E sub I.
00:58:53.000 --> 00:59:10.100
E sub I are related to the energies of the particles making up the system.
00:59:10.100 --> 00:59:23.900
Let us say that, of the particles making up the system, the small E sub I.
00:59:23.900 --> 00:59:31.500
We expressed these thermodynamic functions in terms of the partition function of the system.
00:59:31.500 --> 00:59:42.800
Now, we are going to express it in terms of the partition function of the individual particles, the molecular partition function.
00:59:42.800 --> 00:59:55.000
We will now express Q in terms of Q, in terms of small q, the molecular partition function.
00:59:55.000 --> 00:59:59.800
Because we have expressions in terms of Q, if we have an expression for Q in terms of small q,
00:59:59.800 --> 01:00:08.500
we put that in wherever we see a Q and we have an expression for thermodynamic properties in terms of the small q.
01:00:08.500 --> 01:00:23.400
The molecular partition function.
01:00:23.400 --> 01:00:31.400
The E sub I that we talked about above that is made up of the energies of the individual particles,
01:00:31.400 --> 01:00:40.400
E₁ + E₂ + E₃, and so on.
01:00:40.400 --> 01:00:47.900
The energy of the system is the sum of the energies of the individual particles.
01:00:47.900 --> 01:00:58.500
The partition function of the system is therefore going to be a product of the partition functions of the individual particles.
01:00:58.500 --> 01:01:00.900
That is how this works, some product.
01:01:00.900 --> 01:01:04.700
That is the whole idea behind the log, the exponential.
01:01:04.700 --> 01:01:11.100
That is why the log shows up in these problems.
01:01:11.100 --> 01:01:14.700
Let us say this again.
01:01:14.700 --> 01:01:59.600
The partition function of the system Q can be written as the product of the partition functions for each particle in the system.
01:01:59.600 --> 01:02:04.200
The molecular partition function is Q.
01:02:04.200 --> 01:02:12.500
For indistinguishable particles which is going to be pretty much all that we talk about when we have a liter of nitrogen gas.
01:02:12.500 --> 01:02:16.300
You can tell one molecule of nitrogen gas from another molecule of nitrogen gas.
01:02:16.300 --> 01:02:42.500
For indistinguishable particles, Q is equal to Q ⁺N!, this is the expression.
01:02:42.500 --> 01:02:49.600
If I have N particles, 6.82 × 10²³, I find a partition function for each particle.
01:02:49.600 --> 01:02:58.200
I raise that partition function to the nth power, I divide by N!, that would be give me the partition function of the system.
01:02:58.200 --> 01:03:04.400
That is what this is, very very important equation right here, for indistinguishable particles.
01:03:04.400 --> 01:03:07.500
Is there another expression for the distinguishable particles?
01:03:07.500 --> 01:03:09.600
Yes, it is just that without the denominator.
01:03:09.600 --> 01:03:15.300
And if your teacher feels like discussing that, if it comes in a problem, we will deal with it then, not a problem.
01:03:15.300 --> 01:03:21.500
But for the most part, it is going to be indistinguishable particles.
01:03:21.500 --> 01:03:25.100
N is the number of particles in the system.
01:03:25.100 --> 01:03:28.300
Let us see what we have got here.
01:03:28.300 --> 01:03:43.500
N is the number of particles in the system.
01:03:43.500 --> 01:03:52.400
Q, the small partition function, it is the same definition as Q except now we use the individual energies not of the system.
01:03:52.400 --> 01:03:56.700
The individual energies of the particles, the atoms, and molecules.
01:03:56.700 --> 01:04:07.800
It is going to be the sum / the index I of E ⁻I E sub I/ KT.
01:04:07.800 --> 01:04:14.800
We are taking the sum / quantum states.
01:04:14.800 --> 01:04:17.600
You will see a minute in the quantum levels.
01:04:17.600 --> 01:04:25.100
If I have a diatomic nitrogen molecule and I want to find the partition function of its vibrational partition function.
01:04:25.100 --> 01:04:37.600
The vibrational partition function as you know, first quantum state R = 0, R =1, R =2, R =3, those are the different quantum states.
01:04:37.600 --> 01:04:43.300
Each one has an energy, I put those in here for the E sub I and add it all up.
01:04:43.300 --> 01:04:50.100
That is how get my vibrational partition function for that molecule, for vibration.
01:04:50.100 --> 01:04:54.400
If I want the partition function for rotation, there is a difference of energies.
01:04:54.400 --> 01:04:57.100
If I want one for translation, it is a different set of energies.
01:04:57.100 --> 01:05:00.100
We will get to that in subsequent lessons.
01:05:00.100 --> 01:05:09.300
The molecular partition function is exactly the same as what is qualitatively is this.
01:05:09.300 --> 01:05:22.200
The molecular partition function is a measure, it is a numerical measure.
01:05:22.200 --> 01:05:25.800
It actually gives you the number of states that are accessible.
01:05:25.800 --> 01:05:40.100
A partition function is a measure of the number of quantum states,
01:05:40.100 --> 01:06:01.700
the number of energy states that are accessible to the particle at a given temperature.
01:06:01.700 --> 01:06:10.400
Let us say there are 250 vibrational states available for carbon monoxide that are available.
01:06:10.400 --> 01:06:16.900
At a given temperature, let us say 300 K, or I just say 298 K, room temperature.
01:06:16.900 --> 01:06:20.900
Let us say that only 5 of those states are accessible.
01:06:20.900 --> 01:06:28.700
In other words, the molecule does not have enough energy to get to the 50th state of the 49th state, or the 10th state.
01:06:28.700 --> 01:06:34.900
It only has enough energy to occupy state 1, 2, 3, 4, 5, two different degrees.
01:06:34.900 --> 01:06:39.000
Most the molecules might be in state 1 and 2, and maybe a couple in 3, 4, 5.
01:06:39.000 --> 01:06:43.000
But at a given temperature, it just cannot vibrate anymore than that.
01:06:43.000 --> 01:06:47.900
They are the states that are available at a given temperature, this is what is accessible.
01:06:47.900 --> 01:06:55.500
That is what the molecular partition function does when you calculate this, when you actually get a number like 3.4.
01:06:55.500 --> 01:07:04.100
That is telling you that at that temperature, there is really mostly about 3.4 states that are accessible.
01:07:04.100 --> 01:07:12.000
If in a 3.4 that means that the 4th state is, there is a couple of particles in that state and maybe even in the 5th.
01:07:12.000 --> 01:07:15.000
But in general, it is going to be the 1st, 2nd, 3rd.
01:07:15.000 --> 01:07:16.800
That said, that is all the partition function is.
01:07:16.800 --> 01:07:22.700
It is a numerical measure of the number of quantum states that are accessible to a particle at a given temperature.
01:07:22.700 --> 01:07:27.100
Accessible not available.
01:07:27.100 --> 01:07:29.800
Quantum states can be degenerate, as we now.
01:07:29.800 --> 01:07:35.800
For example, the rotational degeneracy is 2J +1.
01:07:35.800 --> 01:07:41.700
The degeneracy is the number of quantum states that have that particular energy.
01:07:41.700 --> 01:07:48.700
The degeneracy of 5 for a given level means that 5 different quantum states have that same energy.
01:07:48.700 --> 01:08:03.600
Quantum states can be degenerate.
01:08:03.600 --> 01:08:17.100
In other words, have the same energy E sub I.
01:08:17.100 --> 01:08:22.500
If we include degeneracy in our definition of the molecular partition function, we get the following.
01:08:22.500 --> 01:08:25.000
This is the one that we are going to be using.
01:08:25.000 --> 01:08:35.000
Including degeneracy, our molecular partition functions as follows.
01:08:35.000 --> 01:08:47.700
Q is equal to the sum, the index I G sub I, the degeneracy × E ⁻E sub I/ KT.
01:08:47.700 --> 01:08:55.700
Our sum is over the energy levels.
01:08:55.700 --> 01:09:10.600
Our sum is over energy levels not states.
01:09:10.600 --> 01:09:17.100
This degeneracy takes care of all the states.
01:09:17.100 --> 01:09:29.000
Q, we said is equal to Q ⁺nth/ N!.
01:09:29.000 --> 01:09:58.600
In the expressions for the thermodynamic functions, we used LN Q not Q.
01:09:58.600 --> 01:10:13.200
Therefore, let us take the log of this and see what we get.
01:10:13.200 --> 01:10:22.400
LN Q is equal to LN of Q ⁺N/ N!.
01:10:22.400 --> 01:10:31.300
That is equal to N LN Q - LN N!.
01:10:31.300 --> 01:10:50.200
By sterling's formula, we have LN of N! is actually equal to N LN N – N.
01:10:50.200 --> 01:10:56.400
We have an expression for this so we can substitute back into that.
01:10:56.400 --> 01:11:09.400
LN³ is equal to N LN³ - this thing N LN of N – N.
01:11:09.400 --> 01:11:21.600
Therefore, we have LN Q = N LN – N LN + N.
01:11:21.600 --> 01:11:27.300
LN Q, we can express Q in terms of q.
01:11:27.300 --> 01:11:35.400
Using this expression for LN Q, we put it back into the expressions for the thermodynamic functions
01:11:35.400 --> 01:11:40.800
and we have our thermodynamic functions now in terms of q.
01:11:40.800 --> 01:12:18.100
Using this expression for LN Q, we can now express all of the thermodynamic functions in terms of LN q.
01:12:18.100 --> 01:12:30.600
Our connection is complete.
01:12:30.600 --> 01:12:38.100
We had the thermodynamic properties, a classical thermodynamic properties.
01:12:38.100 --> 01:12:45.700
The bulk properties of a system that we developed empirically back in the 19th century.
01:12:45.700 --> 01:12:57.600
We related that to the partition function of the system Q and related that to the partition function, the particles.
01:12:57.600 --> 01:12:58.500
The circle is closed.
01:12:58.500 --> 01:13:02.800
We began with classical thermodynamics, we went on to quantum mechanics.
01:13:02.800 --> 01:13:05.000
Quantum mechanics deals with particles.
01:13:05.000 --> 01:13:08.000
We have this thing called partition function.
01:13:08.000 --> 01:13:15.000
We can use properties of the particles to express the thermodynamic properties of the bulk system.
01:13:15.000 --> 01:13:19.300
The circle for physical chemistry is closed.
01:13:19.300 --> 01:13:23.700
Let us go ahead and do an example here so that we see.
01:13:23.700 --> 01:13:42.900
An example of a thermodynamic property in terms of q, in terms of LN q.
01:13:42.900 --> 01:13:45.300
Let us go ahead and talk about energy.
01:13:45.300 --> 01:13:54.500
Energy is equal to KT² D LN Q DT V.
01:13:54.500 --> 01:14:07.500
We said that LN Q is equal to N LN q - N LN N + N.
01:14:07.500 --> 01:14:09.900
We put this expression into here.
01:14:09.900 --> 01:14:29.100
We get U is equal to K × T² × D DT under constant V of N LN Q – N LN N.
01:14:29.100 --> 01:14:31.900
Everything is basically in dropout, when you take the derivative, these are constants.
01:14:31.900 --> 01:14:35.900
We take the derivative of them with respect to temperature, they just going to go to 0.
01:14:35.900 --> 01:14:43.900
What you end up with here is N KT² D DT of LN Q.
01:14:43.900 --> 01:14:53.900
We will just leave it as D LN Q DT constant V + 0 + 0.
01:14:53.900 --> 01:15:05.600
Therefore, the energy of the system is equal to the number of particles in the system × K × T² ×
01:15:05.600 --> 01:15:12.600
the temperature derivative of the natlog of the molecular partition function, holding volume constant.
01:15:12.600 --> 01:15:16.400
There you go, that is it.
01:15:16.400 --> 01:15:18.500
Thank you so much for joining us here at www.educator.com.
01:15:18.500 --> 01:01:15.000
We will see you next time for a continuation of statistical thermodynamics.